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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings | 1: Scale Drawings Last updated Save as PDF Page ID 34976 Illustrative Mathematics OpenUp Resources 1.1: Scaled Copies 1.1.2: Corresponding Parts and Scale Factors 1.1.3: Making Scaled Copies 1.1.4: Scaled Relationships 1.1.5: The Size of the Scale Factor 1.1.6: Scaling and Area 1.2: Scale Drawings 1.2.1: Scale Drawings 1.2.2: Scale Drawings and Maps 1.2.3: Creating Scale Drawings 1.2.4: Changing Scales in Scale Drawings 1.2.5: Scales without Units 1.2.6: Units in Scale Drawings 1.3: Let's Put It to Work 1.3.1: Draw It to Scale | libretexts | 2025-03-17T19:52:06.516389 | 2020-01-25T01:41:07 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1: Scale Drawings",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page | 1.1: Scaled Copies Last updated Save as PDF Page ID 34977 Illustrative Mathematics OpenUp Resources 1.1.2: Corresponding Parts and Scale Factors 1.1.3: Making Scaled Copies 1.1.4: Scaled Relationships 1.1.5: The Size of the Scale Factor 1.1.6: Scaling and Area | libretexts | 2025-03-17T19:52:06.602898 | 2020-01-25T01:41:08 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1: Scaled Copies",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.2%3A_Corresponding_Parts_and_Scale_Factors | 1.1.2: Corresponding Parts and Scale Factors
Lesson
Let's describe features of scaled copies.
Exercise \(\PageIndex{1}\): Number Talk: Multiplying by a Unit Fraction
Find each product mentally.
\(\frac{1}{4}\cdot 32\)
\((7.2)\cdot\frac{1}{9}\)
\(\frac{1}{4}\cdot (5.6)\)
Exercise \(\PageIndex{2}\): Corresponding Parts
One road sign for railroad crossings is a circle with a large X in the middle and two R’s—with one on each side. Here is a picture with some points labeled and two copies of the picture. Drag and turn the moveable angle tool to compare the angles in the copies with the angles in the original.
-
Complete this table to show
corresponding parts
in the three pictures.
original Copy 1 Copy 2 point \(L\) segment \(LM\) segment \(ED\) point \(X\) angle \(KLM\) angle \(XYZ\) Table \(\PageIndex{1}\) - Is either copy a scaled copy of the original road sign? Explain your reasoning.
- Use the moveable angle tool to compare angle \(KLM\) with its corresponding angles in Copy 1 and Copy 2. What do you notice?
- Use the moveable angle tool to compare angle \(NOP\) with its corresponding angles in Copy 1 and Copy 2. What do you notice?
Exercise \(\PageIndex{3}\): Scaled Triangles
Here is Triangle O, followed by a number of other triangles.
Your teacher will assign you two of the triangles to look at.
- For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
- As a group, identify all the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
-
List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
Triangle O \(3\) \(4\) \(5\) Table \(\PageIndex{2}\) - Explain or show how each copy has been scaled from the original (Triangle O).
Are you ready for more?
Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.
Summary
A figure and its scaled copy have corresponding parts , or parts that are in the same position in relation to the rest of each figure. These parts could be points, segments, or angles. For example, Polygon 2 is a scaled copy of Polygon 1.
-
Each point in Polygon 1 has a
corresponding point
in Polygon 2.
For example, point \(B\) corresponds to point \(H\) and point \(C\) corresponds to point \(I\). -
Each segment in Polygon 1 has a
corresponding segment
in Polygon 2.
For example, segment \(AF\) corresponds to segment \(GL\). -
Each angle in Polygon 1 also has a
corresponding angle
in Polygon 2.
For example, angle \(DEF\) corresponds to angle \(JKL\).
The scale factor between Polygon 1 and Polygon 2 is 2, because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1. The angle measures in Polygon 2 are the same as the corresponding angle measures in Polygon 1. For example, the measure of angle \(JKL\) is the same as the measure of angle \(DEF\).
Glossary Entries
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{4}\)
The second H-shaped polygon is a scaled copy of the first.
- Show one pair of corresponding points and two pairs of corresponding sides in the original polygon and its copy. Consider using colored pencils to highlight corresponding parts or labeling some of the vertices.
- What scale factor takes the original polygon to its smaller copy? Explain or show your reasoning.
Exercise \(\PageIndex{5}\)
Figure B is a scaled copy of Figure A. Select all of the statements that must be true:
- Figure B is larger than Figure A.
- Figure B has the same number of edges as Figure A.
- Figure B has the same perimeter as Figure A.
- Figure B has the same number of angles as Figure A.
- Figure B has angles with the same measures as Figure A.
Exercise \(\PageIndex{6}\)
Polygon B is a scaled copy of Polygon A.
- What is the scale factor from Polygon A to Polygon B? Explain your reasoning.
- Find the missing length of each side marked with ? in Polygon B.
- Determine the measure of each angle marked with ? in Polygon A.
Exercise \(\PageIndex{7}\)
Compare each equation with a number that makes it true.
- \(8\cdot\underline{\qquad}=40\)
- \(8+\underline{\qquad}=40\)
- \(21\div\underline{\qquad}=7\)
- \(21-\underline{\qquad}=7\)
- \(21\cdot\underline{\qquad}=7\) | libretexts | 2025-03-17T19:52:06.686630 | 2020-03-29T00:12:19 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.2%3A_Corresponding_Parts_and_Scale_Factors",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.2: Corresponding Parts and Scale Factors",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.3%3A_Making_Scaled_Copies | 1.1.3: Making Scaled Copies
Lesson
Let's draw scaled copies.
Exercise \(\PageIndex{1}\): More or Less?
For each problem, select the answer from the two choices.
-
The value of \(25\cdot (8.5)\) is:
- More than 205
- Less than 205
-
The value of \((9.93)\cdot (0.984)\) is:
- More than 10
- Less than 10
-
The value of \((0.24)\cdot (0.67)\) is:
- More than 0.2
- Less than 0.2
Exercise \(\PageIndex{2}\): Drawing Scaled Copies
- Draw a scaled copy of either Figure A or B using a scale factor of \(3\).
- Draw a scaled copy of either Figure C or D using a scale factor of \(\frac{1}{2}\).
Exercise \(\PageIndex{3}\): Which Operations? (Part 1)
Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy.
Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings.
- What operation do you think Diego used to calculate the lengths for his drawing?
- What operation do you think Jada used to calculate the lengths for her drawing?
- Did each method produce a scaled copy of the polygon? Explain your reasoning.
Exercise \(\PageIndex{4}\): Which Operations? (Part 2)
Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.
- Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.
- Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.
Are you ready for more?
The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.
Summary
Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.
For example, to make a scaled copy of triangle \(ABC\) where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle \(DEF\), each side is 4 times as long as the corresponding side in triangle \(ABC\).
Glossary Entries
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{5}\)
Here are 3 polygons.
Draw a scaled copy of Polygon A using a scale factor of 2.
Draw a scaled copy of Polygon B using a scale factor of \(\frac{1}{2}\).
Draw a scaled copy of Polygon C using a scale factor of \(\frac{3}{2}\).
Exercise \(\PageIndex{6}\)
Quadrilateral A has side lengths 6, 9, 9, and 12. Quadrilateral B is a scaled copy of Quadrilateral A, with its shortest side of length 2. What is the perimeter of Quadrilateral B?
Exercise \(\PageIndex{7}\)
Here is a polygon on a grid.
Draw a scaled copy of this polygon that has a perimeter of 30 units. What is the scale factor? Explain how you know.
Exercise \(\PageIndex{8}\)
Priya and Tyler are discussing the figures shown below. Priya thinks that B, C, and D are scaled copies of A. Tyler says B and D are scaled copies of A. Do you agree with Priya, or do you agree with Tyler? Explain your reasoning.
(From Unit 1.1.1) | libretexts | 2025-03-17T19:52:06.853249 | 2020-03-29T00:11:44 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.3%3A_Making_Scaled_Copies",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.3: Making Scaled Copies",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.4%3A_Scaled_Relationships | 1.1.4: Scaled Relationships
Lesson
Let's find relationships between scaled copies.
Exercise \(\PageIndex{1}\): Three Quadrilaterals (Part 1)
Each of these polygons is a scaled copy of the others.
- Name two pairs of corresponding angles. What can you say about the sizes of these angles?
- Check your prediction by measuring at least one pair of corresponding angles using a protractor. Record your measurements to the nearest \(5^{\circ}\).
Exercise \(\PageIndex{2}\): Three Quadrilaterals (Part 2)
Each of these polygons is a scaled copy of the others. You already checked their corresponding angles.
1. The side lengths of the polygons are hard to tell from the grid, but there are other corresponding distances that are easier to compare. Identify the distances in the other two polygons that correspond to \(DB\) and \(AC\), and record them in the table.
| quadrilateral | distance that corresponds to \(DB\) | distance that corresponds to \(AC\) |
|---|---|---|
| \(ABCD\) | \(DB=4\) | \(AC=6\) |
| \(EFGH\) | ||
| \(IJKL\) |
2. Look at the values in the table. What do you notice?
Pause here so your teacher can review your work.
3. The larger figure is a scaled copy of the smaller figure.
- If \(AE=4\), how long is the corresponding distance in the second figure? Explain or show your reasoning.
- If \(IK=5\), how long is the corresponding distance in the first figure? Explain or show your reasoning.
Exercise \(\PageIndex{3}\): Scaled or Not Scaled?
Here are two quadrilaterals.
- Mai says that Polygon \(ZSCH\) is a scaled copy of Polygon \(XJYN\), but Noah disagrees. Do you agree with either of them? Explain or show your reasoning.
-
Record the corresponding distances in the table. What do you notice?
quadrilateral horizontal distance vertical distance \(XJYN\) \(XY=\) \(JN=\) \(ZSCH\) \(ZC=\) \(SH=\) Table \(\PageIndex{2}\) - Measure at least three pairs of corresponding angles in \(XJYN\) and \(ZSCH\) using a protractor. Record your measurements to the nearest \(5^{\circ}\). What do you notice?
- Do these results change your answer to the first question? Explain.
- Here are two more quadrilaterals.
Kiran says that Polygon \(EFGH\) is a scaled copy of \(ABCD\), but Lin disagrees. Do you agree with either of them? Explain or show your reasoning.
Are you ready for more?
All side lengths of quadrilateral \(MNOP\) are 2, and all side lengths of quadrilateral \(QRST\) are 3. Does \(MNOP\) have to be a scaled copy of \(QRST\)? Explain your reasoning.
Exercise \(\PageIndex{4}\): Comparing Pictures of Birds
Here are two pictures of a bird. Find evidence that one picture is not a scaled copy of the other. Be prepared to explain your reasoning.
Summary
When a figure is a scaled copy of another figure, we know that:
- All distances in the copy can be found by multiplying the corresponding distances in the original figure by the same scale factor, whether or not the endpoints are connected by a segment.
For example, Polygon \(STUVWX\) is a scaled copy of Polygon \(ABCDEF\). The scale factor is 3. The distance from \(T\) to \(X\) is 6, which is three times the distance from \(B\) to \(F\).
- All angles in the copy have the same measure as the corresponding angles in the original figure, as in these triangles.
These observations can help explain why one figure is not a scaled copy of another.
For example, even though their corresponding angles have the same measure, the second rectangle is not a scaled copy of the first rectangle, because different pairs of corresponding lengths have different scale factors, \(2\cdot\frac{1}{2}=1\) but \(3\cdot\frac{2}{3}=2\).
Glossary Entries
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{5}\)
Select all the statements that must be true for any scaled copy Q of Polygon P.
- The side lengths are all whole numbers.
- The angle measures are all whole numbers.
- Q has exactly 1 right angle.
- If the scale factor between P and Q is \(\frac{1}{5}\), then each side length of P is multiplied by \(\frac{1}{5}\) to get the corresponding side length of Q.
- If the scale factor is 2, each angle in P is multiplied by 2 to get the corresponding angle in Q.
- Q has 2 acute angles and 3 obtuse angles.
Exercise \(\PageIndex{6}\)
Here is Quadrilateral \(ABCD\).
Quadrilateral \(PQRS\) is a scaled copy of Quadrilateral \(ABCD\). Point \(P\) corresponds to \(A\), \(Q\) to \(B\), \(R\) to \(C\), and \(S\) to \(D\).
If the distance from \(P\) to \(R\) is 3 units, what is the distance from \(Q\) to \(S\)? Explain your reasoning.
Exercise \(\PageIndex{7}\)
Figure 2 is a scaled copy of Figure 1.
- Identify the points in Figure 2 that correspond to the points \(A\) and \(C\) in Figure 1. Label them \(P\) and \(R\). What is the distance between \(P\) and \(R\)?
- Identify the points in Figure 1 that correspond to the points \(Q\) and \(S\) in Figure 2. Label them \(B\) and \(D\). What is the distance between \(B\) and \(D\)?
- What is the scale factor that takes Figure 1 to Figure 2?
- \(G\) and \(H\) are two points on Figure 1, but they are not shown. The distance between \(G\) and \(H\) is 1. What is the distance between the corresponding points on Figure 2?
Exercise \(\PageIndex{8}\)
To make 1 batch of lavender paint, the ratio of cups of pink paint to cups of blue paint is 6 to 5. Find two more ratios of cups of pink paint to cups of blue paint that are equivalent to this ratio. | libretexts | 2025-03-17T19:52:06.931135 | 2020-03-29T00:11:05 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.4%3A_Scaled_Relationships",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.4: Scaled Relationships",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.5%3A_The_Size_of_the_Scale_Factor | 1.1.5: The Size of the Scale Factor
Lesson
Let's look at the effects of different scale factors.
Exercise \(\PageIndex{1}\): Number Talk: Missing Factor
Solve each equation mentally.
\(16x=176\)
\(16x=8\)
\(16x=1\)
\(\frac{1}{5}x=1\)
\(\frac{2}{5}x=1\)
Exercise \(\PageIndex{2}\): Card Sort: Scaled Copies
Your teacher will give you a set of cards. On each card, Figure A is the original and Figure B is a scaled copy.
- Sort the cards based on their scale factors. Be prepared to explain your reasoning.
- Examine cards 10 and 13 more closely. What do you notice about the shapes and sizes of the figures? What do you notice about the scale factors?
- Examine cards 8 and 12 more closely. What do you notice about the figures? What do you notice about the scale factors?
Are you ready for more?
Triangle B is a scaled copy of Triangle A with scale factor \(\frac{1}{2}\).
- How many times bigger are the side lengths of Triangle B when compared with Triangle A?
- Imagine you scale Triangle B by a scale factor of \(\frac{1}{2}\) to get Triangle C. How many times bigger will the side lengths of Triangle C be when compared with Triangle A?
- Triangle B has been scaled once. Triangle C has been scaled twice. Imagine you scale triangle A \(n\) times to get Triangle N, always using a scale factor of \(\frac{1}{2}\). How many times bigger will the side lengths of Triangle N be when compared with Triangle A?
Exercise \(\PageIndex{3}\): Scaling A Puzzle
Your teacher will give you 2 pieces of a 6-piece puzzle.
1. If you drew scaled copies of your puzzle pieces using a scale factor of \(\frac{1}{2}\), would they be larger or smaller than the original pieces? How do you know?
2. Create a scaled copy of each puzzle piece on a blank square, with a scale factor of \(\frac{1}{2}\).
3. When everyone in your group is finished, put all 6 of the original puzzle pieces together like this:
Next, put all 6 of your scaled copies together. Compare your scaled puzzle with the original puzzle. Which parts seem to be scaled correctly and which seem off? What might have caused those parts to be off?
4. Revise any of the scaled copies that may have been drawn incorrectly.
5. If you were to lose one of the pieces of the original puzzle, but still had the scaled copy, how could you recreate the lost piece?
Exercise \(\PageIndex{4}\): Missing Figure, Factor, or Copy
1. What is the scale factor from the original triangle to its copy? Explain or show your reasoning.
2. The scale factor from the original trapezoid to its copy is 2. Draw the scaled copy.
3. The scale factor from the original figure to its copy is \(\frac{3}{2}\). Draw the original figure.
4. What is the scale factor from the original figure to the copy? Explain how you know.
5. The scale factor from the original figure to its scaled copy is 3. Draw the scaled copy.
Summary
The size of the scale factor affects the size of the copy. When a figure is scaled by a scale factor greater than 1, the copy is larger than the original. When the scale factor is less than 1, the copy is smaller. When the scale factor is exactly 1, the copy is the same size as the original.
Triangle \(DEF\) is a larger scaled copy of triangle \(ABC\), because the scale factor from \(ABC\) to \(DEF\) is \(\frac{3}{2}\). Triangle \(ABC\) is a smaller scaled copy of triangle \(DEF\), because the scale factor from \(DEF\) to \(ABC\) is \(\frac{2}{3}\).
This means that triangles \(ABC\) and \(DEF\) are scaled copies of each other. It also shows that scaling can be reversed using reciprocal scale factors, such as \(\frac{2}{3}\) and \(\frac{3}{2}\).
In other words, if we scale Figure A using a scale factor of 4 to create Figure B, we can scale Figure B using the reciprocal scale factor, \(\frac{1}{4}\), to create Figure A.
Glossary Entries
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Reciprocal
Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{5}\)
Rectangles P, Q, R, and S are scaled copies of one another. For each pair, decide if the scale factor from one to the other is greater than 1, equal to 1, or less than 1.
- from P to Q
- from P to R
- from Q to S
- from Q to R
- from S to P
- from R to P
- from P to S
Exercise \(\PageIndex{6}\)
Triangle S and Triangle L are scaled copies of one another.
- What is the scale factor from S to L?
- What is the scale factor from L to S?
- Triangle M is also a scaled copy of S. The scale factor from S to M is \(\frac{3}{2}\). What is the scale factor from M to S?
Exercise \(\PageIndex{7}\)
Are two squares with the same side lengths scaled copies of one another? Explain your reasoning.
Exercise \(\PageIndex{8}\)
Quadrilateral A has side lengths 2, 3, 5, and 6. Quadrilateral B has side lengths 4, 5, 8, and 10. Could one of the quadrilaterals be a scaled copy of the other? Explain.
(From Unit 1.1.2)
Exercise \(\PageIndex{9}\)
Select all the ratios that are equivalent to the ratio \(12:3\)
- \(6:1\)
- \(1:4\)
- \(4:1\)
- \(24:6\)
- \(15:6\)
- \(1,200:300\)
- \(112:13\) | libretexts | 2025-03-17T19:52:07.006025 | 2020-03-29T00:10:19 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.5%3A_The_Size_of_the_Scale_Factor",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.5: The Size of the Scale Factor",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.6%3A_Scaling_and_Area | 1.1.6: Scaling and Area
Lesson
Let's build scaled shapes and investigate their areas.
Exercise \(\PageIndex{1}\): Scaling a Pattern Block
Use the applets to explore the pattern blocks. Work with your group to build the scaled copies described in each question.
-
How many blue rhombus blocks does it take to build a scaled copy of Figure A:
- Where each side is twice as long?
- Where each side is 3 times as long?
- Where each side is 4 times as long?
-
How many green triangle blocks does it take to build a scaled copy of Figure B:
- Where each side is twice as long?
- Where each side is 3 times as long?
- Using a scale factor of 4?
-
How many red trapezoid blocks does it take to build a scaled copy of Figure C:
- Using a scale factor of 2?
- Using a scale factor of 3?
- Using a scale factor of 4?
- Make a prediction: How many blocks would it take to build scaled copies of these shapes using a scale factor of 5? Using a scale factor of 6? Be prepared to explain your reasoning.
Exercise \(\PageIndex{2}\): Scaling More Pattern Blocks
Your teacher will assign your group one of these figures, each made with original-size blocks.
- In the applet, move the slider to see a scaled copy of your assigned shape, using a scale factor of 2. Use the original-size blocks to build a figure to match it. How many blocks did it take?
- Your classmate thinks that the scaled copies in the previous problem will each take 4 blocks to build. Do you agree or disagree? Explain you reasoning.
- Move the slider to see a scaled copy of your assigned shape using a scale factor of 3. Start building a figure with the original-size blocks to match it. Stop when you can tell for sure how many blocks it would take. Record your answer.
- Predict: How many blocks would it take to build scaled copies using scale factors 4, 5, and 6? Explain or show your reasoning.
- How is the pattern in this activity the same as the pattern you saw in the previous activity? How is it different?
Are you ready for more?
- How many blocks do you think it would take to build a scaled copy of one yellow hexagon where each side is twice as long? Three times as long?
- Figure out a way to build these scaled copies.
- Do you see a pattern for the number of blocks used to build these scaled copies? Explain your reasoning.
Exercise \(\PageIndex{3}\): Area of Scaled Parallelograms and Triangles
- Your teacher will give you a figure with measurements in centimeters. What is the area of your figure? How do you know?
-
Work with your partner to draw scaled copies of your figure, using each scale factor in the table. Complete the table with the measurements of your scaled copies.
scale factor base (cm) height (cm) area (cm 2 ) \(1\) \(2\) \(3\) \(\frac{1}{2}\) \(\frac{1}{3}\) Table \(\PageIndex{1}\) - Compare your results with a group that worked with a different figure. What is the same about your answers? What is different?
- If you drew scaled copies of your figure with the following scale factors, what would their areas be? Discuss your thinking. If you disagree, work to reach an agreement. Be prepared to explain your reasoning.
| scale factor | area (cm 2 ) |
|---|---|
| \(5\) | |
| \(\frac{3}{5}\) |
Summary
Scaling affects lengths and areas differently. When we make a scaled copy, all original lengths are multiplied by the scale factor. If we make a copy of a rectangle with side lengths 2 units and 4 units using a scale factor of 3, the side lengths of the copy will be 6 units and 12 units, because \(2\cdot 3=6\) and \(4\cdot 3=12\).
The area of the copy, however, changes by a factor of (scale factor) 2 . If each side length of the copy is 3 times longer than the original side length, then the area of the copy will be 9 times the area of the original, because \(3\cdot 3\), or \(3^{2}\), equals 9.
In this example, the area of the original rectangle is 8 units 2 and the area of the scaled copy is 72 units 2 , because \(9\cdot 8=72\). We can see that the large rectangle is covered by 9 copies of the small rectangle, without gaps or overlaps. We can also verify this by multiplying the side lengths of the large rectangle: \(6\cdot 12=72\).
Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two-dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length \(l\) and width \(w\). If we scale the rectangle by a scale factor of \(s\), we get a rectangle with length \(s\cdot l\) and width \(s\cdot w\). The area of the scaled rectangle is \(A=(s\cdot l)\cdot (s\cdot w)\), so \(A=(s^{2})\cdot (l\cdot w)\). The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.
Glossary Entries
Definition: Area
Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.
For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac{1}{2}\) square unit.
Definition: Corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
Definition: Reciprocal
Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\).
Definition: Scale Factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4\cdot (1.5)=6\), \(5\cdot (1.5)=7.5\), and \(6\cdot (1.5)=9\).
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{4}\)
On the grid, draw a scaled copy of Polygon Q using a scale factor of 2. Compare the perimeter and area of the new polygon to those of Q.
Exercise \(\PageIndex{5}\)
A right triangle has an area of 36 square units.
If you draw scaled copies of this triangle using the scale factors in the table, what will the areas of these scaled copies be? Explain or show your reasoning.
| scale factor | area (units 2 ) |
|---|---|
| \(1\) | \(36\) |
| \(2\) | |
| \(3\) | |
| \(5\) | |
| \(\frac{1}{2}\) | |
| \(\frac{2}{3}\) |
Exercise \(\PageIndex{6}\)
Diego drew a scaled version of a Polygon P and labeled it Q.
If the area of Polygon P is 72 square units, what scale factor did Diego use to go from P to Q? Explain your reasoning.
Exercise \(\PageIndex{7}\)
Here is an unlabeled polygon, along with its scaled copies Polygons A–D. For each copy, determine the scale factor. Explain how you know.
(From Unit 1.1.2)
Exercise \(\PageIndex{8}\)
Solve each equation mentally.
- \(\frac{1}{7}\cdot x=1\)
- \(x\cdot\frac{1}{11}=1\)
- \(1\div\frac{1}{5}=x\)
(From Unit 1.1.5) | libretexts | 2025-03-17T19:52:07.089832 | 2020-03-29T00:09:25 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.01%3A_New_Page/1.1.6%3A_Scaling_and_Area",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.6: Scaling and Area",
"author": "Illustrative Mathematics"
} |
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1: Scale Drawings
Pre-Algebra I (Illustrative Mathematics - Grade 7)
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Sat, 25 Jan 2020 01:41:09 GMT
1.2: Scale Drawings
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OpenUp Resources | libretexts | 2025-03-17T19:52:07.165657 | 2020-01-25T01:41:09 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2: Scale Drawings",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.1%3A_Scale_Drawings | 1.2.1: Scale Drawings
Lesson
Let's explore scale drawings.
Exercise \(\PageIndex{1}\): What is a Scale Drawing?
Here are some drawings of a school bus, a quarter, and the subway lines around Boston, Massachusetts. The first three drawings are scale drawings of these objects.
The next three drawings are not scale drawings of these objects.
Discuss with your partner what a scale drawing is.
Exercise \(\PageIndex{2}\): Sizing Up a Basketball Court
Your teacher will give you a scale drawing of a basketball court. The drawing does not have any measurements labeled, but it says that 1 centimeter represents 2 meters.
- Measure the distances on the scale drawing that are labeled a–d to the nearest tenth of a centimeter. Record your results in the first row of the table.
- The statement “1 cm represents 2 m” is the scale of the drawing. It can also be expressed as “1 cm to 2 m,” or “1 cm for every 2 m.” What do you think the scale tells us?
-
How long would each measurement from the first question be on an actual basketball court? Explain or show your reasoning.
measurement (a) length of court (b) width of court (c) hoop to hoop (d) 3 point line to sideline scale drawing actual court Table \(\PageIndex{1}\) -
On an actual basketball court, the bench area is typically 9 meters long.
- Without measuring, determine how long the bench area should be on the scale drawing.
- Check your answer by measuring the bench area on the scale drawing. Did your prediction match your measurement?
Exercise \(\PageIndex{3}\): Tall Structures
Here is a scale drawing of some of the world’s tallest structures.
- About how tall is the actual Willis Tower? About how tall is the actual Great Pyramid? Be prepared to explain your reasoning.
- About how much taller is the Burj Khalifa than the Eiffel Tower? Explain or show your reasoning.
- Measure the line segment that shows the scale to the nearest tenth of a centimeter. Express the scale of the drawing using numbers and words.
Are you ready for more?
The tallest mountain in the United States, Mount Denali in Alaska, is about 6,190 m tall. If this mountain were shown on the scale drawing, how would its height compare to the heights of the structures? Explain or show your reasoning.
Summary
Scale drawings are two-dimensional representations of actual objects or places. Floor plans and maps are some examples of scale drawings. On a scale drawing:
- Every part corresponds to something in the actual object.
- Lengths on the drawing are enlarged or reduced by the same scale factor.
- A scale tells us how actual measurements are represented on the drawing. For example, if a map has a scale of “1 inch to 5 miles” then a \(\frac{1}{2}\)-inch line segment on that map would represent an actual distance of 2.5 miles
Sometimes the scale is shown as a segment on the drawing itself. For example, here is a scale drawing of a stop sign with a line segment that represents 25 cm of actual length.
The width of the octagon in the drawing is about three times the length of this segment, so the actual width of the sign is about \(3\cdot 25\), or 75 cm.
Because a scale drawing is two-dimensional, some aspects of the three-dimensional object are not represented. For example, this scale drawing does not show the thickness of the stop sign.
A scale drawing may not show every detail of the actual object; however, the features that are shown correspond to the actual object and follow the specified scale.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
The Westland Lysander was an aircraft used by the Royal Air Force in the 1930s. Here are some scale drawings that show the top, side, and front views of the Lysander.
Use the scales and scale drawings to approximate the actual lengths of:
- the wingspan of the plane, to the nearest foot
- the height of the plane, to the nearest foot
- the length of the Lysander Mk. I, to the nearest meter
Exercise \(\PageIndex{5}\)
A blueprint for a building includes a rectangular room that measures 3 inches long and 5.5 inches wide. The scale for the blueprint says that 1 inch on the blueprint is equivalent to 10 feet in the actual building. What are the dimensions of this rectangular room in the actual building?
Exercise \(\PageIndex{6}\)
Here is a scale map of Lafayette Square, a rectangular garden north of the White House.
- The scale is shown in the lower right corner. Find the actual side lengths of Lafayette Square in feet.
- Use an inch ruler to measure the line segment of the graphic scale. About how many feet does one inch represent on this map?
Exercise \(\PageIndex{7}\)
Here is Triangle A. Lin created a scaled copy of Triangle A with an area of 72 square units.
- How many times larger is the area of of the scaled copy compared to that of Triangle A?
- What scale factor did Lin apply to the Triangle A to create the copy?
- What is the length of bottom side of the scaled copy?
(From Unit 1.1.6) | libretexts | 2025-03-17T19:52:07.239507 | 2020-03-29T00:16:39 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.1: Scale Drawings",
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.2%3A_Scale_Drawings_and_Maps | 1.2.2: Scale Drawings and Maps
Lesson
Let's use scale drawings to solve problems.
Exercise \(\PageIndex{1}\): A Train And a Car
Two cities are 243 miles apart.
- It takes a train 4 hours to travel between the two cities at a constant speed.
- A car travels between the two cities at a constant speed of 65 miles per hour.
Which is traveling faster, the car or the train? Be prepared to explain your reasoning.
Exercise \(\PageIndex{2}\): Driving on I-90
- A driver is traveling at a constant speed on Interstate 90 outside of Chicago. If she traveled from Point A to Point B in 8 minutes, did she obey the speed limit of 55 miles per hour? Explain your reasoning.
- A traffic helicopter flew directly from Point A to Point B in 8 minutes. Did the helicopter travel faster or slower than the driver? Explain or show your reasoning.
Use the Distance tool to measure the shortest distance between two points.
Exercise \(\PageIndex{3}\): Biking through Kansas
A cyclist rides at a constant speed of 15 miles per hour. At this speed, about how long would it take the cyclist to ride from Garden City to Dodge City, Kansas?
Are you ready for more?
Jada finds a map that says, “Note: This map is not to scale.” What do you think this means? Why is this information important?
Summary
Maps with scales are useful for making calculations involving speed, time, and distance. Here is a map of part of Alabama.
Suppose it takes a car 1 hour and 30 minutes to travel at constant speed from Birmingham to Montgomery. How fast is the car traveling?
To make an estimate, we need to know about how far it is from Birmingham to Montgomery. The scale of the map represents 20 miles, so we can estimate the distance between these cities is about 90 miles.
Since 90 miles in 1.5 hours is the same speed as 180 miles in 3 hours, the car is traveling about 60 miles per hour.
Suppose a car is traveling at a constant speed of 60 miles per hour from Montgomery to Centreville. How long will it take the car to make the trip? Using the scale, we can estimate that it is about 70 miles. Since 60 miles per hour is the same as 1 mile per minute, it will take the car about 70 minutes (or 1 hour and 10 minutes) to make this trip.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
Here is a map that shows parts of Texas and Oklahoma.
- About how far is it from Amarillo to Oklahoma City? Explain your reasoning.
- Driving at a constant speed of 70 miles per hour, will it be possible to make this trip in 3 hours? Explain how you know.
Exercise \(\PageIndex{5}\)
A local park is in the shape of a square. A map of the local park is made with the scale 1 inch to 200 feet.
- If the park is shown as a square on the map, each side of which is one foot long, how long is each side of the square park?
- If a straight path in the park is 900 feet long, how long would the path be when represented on the map? | libretexts | 2025-03-17T19:52:07.305243 | 2020-03-29T00:16:15 | {
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"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.2%3A_Scale_Drawings_and_Maps",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.2: Scale Drawings and Maps",
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.3%3A_Creating_Scale_Drawings | 1.2.3: Creating Scale Drawings
Lesson
Let's create our own scale drawings.
Exercise \(\PageIndex{1}\): Number Talk: Which is Greater?
Without calculating, decide which quotient is larger.
\(11\div 23\) or \(7\div 13\)
\(0.63\div 2\) or \(0.55\div 3\)
\(15\div\frac{1}{3}\) or \(15\div\frac{1}{4}\)
Exercise \(\PageIndex{2}\): Bedroom Floor Plan
Here is a rough sketch of Noah’s bedroom (not a scale drawing).
Noah wants to create a floor plan that is a scale drawing.
- The actual length of Wall C is 4 m. To represent Wall C, Noah draws a segment 16 cm long. What scale is he using? Explain or show your reasoning.
- Find another way to express the scale.
- Discuss your thinking with your partner. How do your scales compare?
- The actual lengths of Wall A, Wall B, and Wall D are 2.5 m, 2.75 m, and 3.75 m. Determine how long these walls will be on Noah’s scale floor plan.
- Use the Point tool
and the Segment tool
to draw the walls of Noah's scale floor plan in the applet.
Are you ready for more?
If Noah wanted to draw another floor plan on which Wall C was 20 cm, would 1 cm to 5 m be the right scale to use? Explain your reasoning.
Exercise \(\PageIndex{3}\): Two Maps of Utah
A rectangle around Utah is about 270 miles wide and about 350 miles tall. The upper right corner that is missing is about 110 miles wide and about 70 miles tall.
-
Make a scale drawing of Utah where 1 centimeter represents 50 miles.
Make a scale drawing of Utah where 1 centimeter represents 75 miles. - How do the two drawings compare? How does the choice of scale influence the drawing?
Summary
If we want to create a scale drawing of a room's floor plan that has the scale “1 inch to 4 feet,” we can divide the actual lengths in the room (in feet) by 4 to find the corresponding lengths (in inches) for our drawing.
Suppose the longest wall is 15 feet long. We should draw a line 3.75 inches long to represent this wall, because \(15\div 4=3.75\).
There is more than one way to express this scale. These three scales are all equivalent, since they represent the same relationship between lengths on a drawing and actual lengths:
- \(1\) inch to \(4\) feet
- \(\frac{1}{2}\) inch to \(2\) feet
- \(\frac{1}{4}\) inch to \(1\) foot
Any of these scales can be used to find actual lengths and scaled lengths (lengths on a drawing). For instance, we can tell that, at this scale, an 8-foot long wall should be 2 inches long on the drawing because \(\frac{1}{4}\cdot 8=2\).
The size of a scale drawing is influenced by the choice of scale. For example, here is another scale drawing of the same room using the scale 1 inch to 8 feet.
Notice this drawing is smaller than the previous one. Since one inch on this drawing represents twice as much actual distance, each side length only needs to be half as long as it was in the first scale drawing.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
An image of a book shown on a website is 1.5 inches wide and 3 inches tall on a computer monitor. The actual book is 9 inches wide.
- What scale is being used for the image?
- How tall is the actual book?
Exercise \(\PageIndex{5}\)
The flag of Colombia is a rectangle that is 6 ft long with three horizontal strips.
- The top stripe is 2 ft tall and is yellow.
- The middle stripe is 1 ft tall and is blue.
- The bottom stripe is also 1 ft tall and is red.
- Create a scale drawing of the Colombian flag with a scale of 1 cm to 2 ft.
- Create a scale drawing of the Colombian flag with a scale of 2 cm to 1 ft.
Exercise \(\PageIndex{6}\)
These triangles are scaled copies of each other.
For each pair of triangles listed, the area of the second triangle is how many times larger than the area of the first?
- Triangle G and Triangle F
- Triangle G and Triangle B
- Triangle B and Triangle F
- Triangle F and Triangle H
- Triangle G and Triangle H
- Triangle H and Triangle B
(From Unit 1.1.6)
Exercise \(\PageIndex{7}\)
Here is an unlabeled rectangle, followed by other quadrilaterals that are labeled.
- Select all quadrilaterals that are scaled copies of the unlabeled rectangle. Explain how you know.
- On graph paper, draw a different scaled version of the original rectangle.
(From Unit 1.1.3) | libretexts | 2025-03-17T19:52:07.377150 | 2020-03-29T00:15:24 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.3: Creating Scale Drawings",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.4%3A_Changing_Scales_in_Scale_Drawings | 1.2.4: Changing Scales in Scale Drawings
Lesson
Let's explore different scale drawings of the same actual thing.
Exercise \(\PageIndex{1}\): Appropriate Measurements
1. If a student uses a ruler like this to measure the length of their foot, which choices would be appropriate measurements? Select all that apply. Be prepared to explain your reasoning.
- \(9\frac{1}{4}\) inches
- \(9\frac{5}{64}\) inches
- \(23.47659\) centimeters
- \(23.5\) centimeters
- \(23.48\) centimeters
2. Here is a scale drawing of an average seventh-grade student's foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. Estimate the length of the larger foot.
Exercise \(\PageIndex{2}\): Same Plot, Different Drawings
Here is a map showing a plot of land in the shape of a right triangle.
- Your teacher will assign you a scale to use. On centimeter graph paper, make a scale drawing of the plot of land. Make sure to write your scale on your drawing.
- What is the area of the triangle you drew? Explain or show your reasoning.
- How many square meters are represented by 1 square centimeter in your drawing?
- After everyone in your group is finished, order the scale drawings from largest to smallest. What do you notice about the scales when your drawings are placed in this order?
Are you ready for more?
Noah and Elena each make a scale drawing of the same triangular plot of land, using the following scales. Make a prediction about the size of each drawing. How would they compare to the scale drawings made by your group?
- Noah uses the scale 1 cm to 200 m.
- Elena uses the scale 2 cm to 25 m.
Exercise \(\PageIndex{3}\): A New Drawing of the Playground
Here is a scale drawing of a playground.
The scale is 1 centimeter to 30 meters.
- Make another scale drawing of the same playground at a scale of 1 centimeter to 20 meters.
- How do the two scale drawings compare?
Summary
Sometimes we have a scale drawing of something, and we want to create another scale drawing of it that uses a different scale. We can use the original scale drawing to find the size of the actual object. Then we can use the size of the actual object to figure out the size of our new scale drawing.
For example, here is a scale drawing of a park where the scale is 1 cm to 90 m.
The rectangle is 10 cm by 4 cm, so the actual dimensions of the park are 900 m by 360 m, because \(10\cdot 90=900\) and \(4\cdot 90=360\).
Suppose we want to make another scale drawing of the park where the scale is 1 cm to 30 meters. This new scale drawing should be 30 cm by 12 cm, because \(900\div 30=30\) and \(360\div 30=12\).
Another way to find this answer is to think about how the two different scales are related to each other. In the first scale drawing, 1 cm represented 90 m. In the new drawing, we would need 3 cm to represent 90 m. That means each length in the new scale drawing should be 3 times as long as it was in the original drawing. The new scale drawing should be 30 cm by 12 cm, because \(3\cdot 10=30\) and \(3\cdot 4=12\).
Since the length and width are 3 times as long, the area of the new scale drawing will be 9 times as large as the area of the original scale drawing, because \(3^{2}=9\).
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
Here is a scale drawing of a swimming pool where 1 cm represents 1 m.
- How long and how wide is the actual swimming pool?
- Will a scale drawing where 1 cm represents 2 m be larger or smaller than this drawing?
- Make a scale drawing of the swimming pool where 1 cm represents 2 m.
Exercise \(\PageIndex{5}\)
A map of a park has a scale of 1 inch to 1,000 feet. Another map of the same park has a scale of 1 inch to 500 feet. Which map is larger? Explain or show your reasoning.
Exercise \(\PageIndex{6}\)
On a map with a scale of 1 inch to 12 feet, the area of a restaurant is 60 in 2 . Han says that the actual area of the restaurant is 720 ft 2 . Do you agree or disagree? Explain your reasoning.
Exercise \(\PageIndex{7}\)
If Quadrilateral Q is a scaled copy of Quadrilateral P created with a scale factor of 3, what is the perimeter of Q?
(From Unit 1.1.3)
Exercise \(\PageIndex{8}\)
Triangle \(DEF\) is a scaled copy of triangle \(ABC\). For each of the following parts of triangle \(ABC\), identify the corresponding part of triangle \(DEF\).
- angle \(ABC\)
- angle \(BCA\)
- segment \(AC\)
- segment \(BA\)
(From Unit 1.1.2) | libretexts | 2025-03-17T19:52:07.447485 | 2020-03-29T00:14:58 | {
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"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.4%3A_Changing_Scales_in_Scale_Drawings",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.4: Changing Scales in Scale Drawings",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.5%3A_Scales_without_Units | 1.2.5: Scales without Units
Lesson
Let's explore a different way to express scales.
Exercise \(\PageIndex{1}\): One to One Hundred
A map of a park says its scale is 1 to 100.
- What do you think that means?
- Give an example of how this scale could tell us about measurements in the park.
Exercise \(\PageIndex{2}\): Apollo Lunar Module
Your teacher will give you a drawing of the Apollo Lunar Module. It is drawn at a scale of 1 to 50.
- The “legs” of the spacecraft are its landing gear. Use the drawing to estimate the actual length of each leg on the sides. Write your answer to the nearest 10 centimeters. Explain or show your reasoning.
- Use the drawing to estimate the actual height of the Apollo Lunar Module to the nearest 10 centimeters. Explain or show your reasoning.
- Neil Armstrong was 71 inches tall when he went to the surface of the Moon in the Apollo Lunar Module. How tall would he be in the drawing if he were drawn with his height to scale? Show your reasoning.
- Sketch a stick figure to represent yourself standing next to the Apollo Lunar Module. Make sure the height of your stick figure is to scale. Show how you determined your height on the drawing.
Are you ready for more?
The table shows the distance between the Sun and 8 planets in our solar system.
- If you wanted to create a scale model of the solar system that could fit somewhere in your school, what scale would you use?
- The diameter of Earth is approximately 8,000 miles. What would the diameter of Earth be in your scale model?
| planet | average distance (millions of miles) |
|---|---|
| Mercury | \(35\) |
| Venus | \(67\) |
| Earth | \(93\) |
| Mars | \(142\) |
| Jupiter | \(484\) |
| Saturn | \(887\) |
| Uranus | \(1,784\) |
| Neptune | \(2,795\) |
Exercise \(\PageIndex{3}\): Same Drawing, Different Scales
A rectangular parking lot is 120 feet long and 75 feet wide.
- Lin made a scale drawing of the parking lot at a scale of 1 inch to 15 feet. The drawing she produced is 8 inches by 5 inches.
- Diego made another scale drawing of the parking lot at a scale of 1 to 180. The drawing he produced is also 8 inches by 5 inches.
- Explain or show how each scale would produce an 8 inch by 5 inch drawing.
- Make another scale drawing of the same parking lot at a scale of 1 inch to 20 feet. Be prepared to explain your reasoning.
- Express the scale of 1 inch to 20 feet as a scale without units. Explain your reasoning.
Summary
In some scale drawings, the scale specifies one unit for the distances on the drawing and a different unit for the actual distances represented. For example, a drawing could have a scale of 1 cm to 10 km.
In other scale drawings, the scale does not specify any units at all. For example, a map may simply say the scale is 1 to 1,000. In this case, the units for the scaled measurements and actual measurements can be any unit, so long as the same unit is being used for both. So if a map of a park has a scale 1 to 1,000, then 1 inch on the map represents 1,000 inches in the park, and 12 centimeters on the map represent 12,000 centimeters in the park. In other words, 1,000 is the scale factor that relates distances on the drawing to actual distances, and \(\frac{1}{1000}\) is the scale factor that relates an actual distance to its corresponding distance on the drawing.
A scale with units can be expressed as a scale without units by converting one measurement in the scale into the same unit as the other (usually the unit used in the drawing). For example, these scales are equivalent:
- 1 inch to 200 feet
- 1 inch to 2,400 inches (because there are 12 inches in 1 foot, and \(200\cdot 12=2,400\))
- 1 to 2,400
This scale tells us that all actual distances are 2,400 times their corresponding distances on the drawing, and distances on the drawing are \(\frac{1}{2400}\) times the actual distances they represent.
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{4}\)
A scale drawing of a car is presented in the following three scales. Order the scale drawings from smallest to largest. Explain your reasoning. (There are about 1.1 yards in a meter, and 2.54 cm in an inch.)
- 1 in to 1 ft
- 1 in to 1 m
- 1 in to 1 yd
Exercise \(\PageIndex{5}\)
Which scales are equivalent to 1 inch to 1 foot? Select all that apply.
- 1 to 12
- \(\frac{1}{12}\) to 1
- 100 to 0.12
- 5 to 60
- 36 to 3
- 9 to 108
Exercise \(\PageIndex{6}\)
A model airplane is built at a scale of 1 to 72. If the model plane is 8 inches long, how many feet long is the actual airplane?
Exercise \(\PageIndex{7}\)
Quadrilateral A has side lengths 3, 6, 6, and 9. Quadrilateral B is a scaled copy of A with a shortest side length equal to 2. Jada says, “Since the side lengths go down by 1 in this scaling, the perimeter goes down by 4 in total.” Do you agree with Jada? Explain your reasoning.
(From Unit 1.1.3)
Exercise \(\PageIndex{8}\)
Polygon B is a scaled copy of Polygon A using a scale factor of 5. Polygon A’s area is what fraction of Polygon B’s area?
(From Unit 1.1.6)
Exercise \(\PageIndex{9}\)
Figures R, S, and T are all scaled copies of one another. Figure S is a scaled copy of R using a scale factor of 3. Figure T is a scaled copy of S using a scale factor of 2. Find the scale factors for each of the following:
- From T to S
- From S to R
- From R to T
- From T to R
(From Unit 1.1.5) | libretexts | 2025-03-17T19:52:07.521709 | 2020-03-29T00:14:27 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.5: Scales without Units",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.02%3A_New_Page/1.2.6%3A_Units_in_Scale_Drawings | 1.2.6: Units in Scale Drawings
Lesson
Let's use different scales to describe the same drawing.
Exercise \(\PageIndex{1}\): Centimeters in a Mile
There are 2.54 cm in an inch, 12 inches in a foot, and 5,280 feet in a mile. Which expression gives the number of centimeters in a mile? Explain your reasoning.
- \(\frac{2.54}{12\cdot 5,280}\)
- \(5,280\cdot 12\cdot (2.54)\)
- \(\frac{1}{5,280\cdot 12\cdot (2.54)}\)
- \(5,280+12+2.54\)
- \(\frac{5,280\cdot 12}{2.54}\)
Exercise \(\PageIndex{2}\): Card Sort: Scales
Your teacher will give you some cards with a scale on each card.
- Sort the cards into sets of equivalent scales. Be prepared to explain how you know that the scales in each set are equivalent. Each set should have at least two cards.
-
Trade places with another group and check each other’s work. If you disagree about how the scales should be sorted, work to reach an agreement.
Pause here so your teacher can review your work. - Next, record one of the sets with three equivalent scales and explain why they are equivalent.
Exercise \(\PageIndex{3}\): The World's Largest Flag
As of 2016, Tunisia holds the world record for the largest version of a national flag. It was almost as long as four soccer fields. The flag has a circle in the center, a crescent moon inside the circle, and a star inside the crescent moon.
-
Complete the table. Explain or show your reasoning.
flag length flag height height of crescent moon actual \(396\) m \(99\) m at \(1\) to \(2,000\) scale \(13.2\) cm Table \(\PageIndex{1}\) -
Complete each scale with the value that makes it equivalent to the scale of 1 to 2,000. Explain or show your reasoning.
- 1 cm to ____________ cm
- 1 cm to ____________ m
- 1 cm to ____________ km
- 2 m to _____________ m
- 5 cm to ___________ m
- ____________ cm to 1,000 m
- ____________ mm to 20 m
-
- What is the area of the large flag?
- What is the area of the smaller flag?
- The area of the large flag is how many times the area of the smaller flag?
Exercise \(\PageIndex{4}\): Pondering Pools
Your teacher will give you a floor plan of a recreation center.
- What is the scale of the floor plan if the actual side length of the square pool is 15 m? Express your answer both as a scale with units and without units.
- Find the actual area of the large rectangular pool. Show your reasoning.
- The kidney-shaped pool has an area of 3.2 cm 2 on the drawing. What is its actual area? Explain or show your reasoning.
Are you ready for more?
- Square A is a scaled copy of Square B with scale factor 2. If the area of Square A is 10 units 2 , what is the area of Square B?
- Cube A is a scaled copy of Cube B with scale factor 2. If the volume of Cube A is 10 units 3 , what is the volume of Cube B?
- The four-dimensional Hypercube A is a scaled copy of Hypercube B with scale factor 2. If the “volume” of Hypercube A is 10 units 4 , what do you think the “volume” of Hypercube B is?
Summary
Sometimes scales come with units, and sometimes they don’t. For example, a map of Nebraska may have a scale of 1 mm to 1 km. This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska. Notice that there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are \(1,000\cdot 1,000\) or 1,000,000 millimeters in 1 kilometer. So, the same scale without units is 1 to 1,000,000, which means that each unit of distance on the map represents 1,000,000 units of distance in Nebraska. This is true for any choice of unit to express the scale of this map.
Sometimes when a scale comes with units, it is useful to rewrite it without units. For example, let's say we have a different map of Rhode Island, and we want to use the two maps to compare the size of Nebraska and Rhode Island. It is important to know if the maps are at the same scale. The scale of the map of Rhode Island is 1 inch to 10 miles. There are 5,280 feet in 1 mile, and 12 inches in 1 foot, so there are 63,360 inches in 1 mile (because \(5,280\cdot 12=63,360\)). Therefore, there are 633,600 inches in 10 miles. The scale of the map of Rhode Island without units is 1 to 633,600. The two maps are not at the same scale, so we should not use these maps to compare the size of Nebraska to the size of Rhode Island.
Here is some information about equal lengths that you may find useful.
Customary Units
1 foot (ft) = 12 inches (in)
1 yard (yd) = 36 inches
1 yard = 3 feet
1 mile = 5,280 feet
Metric Units
1 meter (m) = 1,000 millimeters (mm)
1 meter = 100 centimeters
1 kilometer (km) = 1,000 meters
Equal Lengths in Different Systems
1 inch = 2.54 centimeters
1 foot \(\approx\) 0.30 meter
1 mile \(\approx\) 1.61 kilometers
1 centimeter \(\approx\) 0.39 inch
1 meter \(\approx\) 39.37 inches
1 kilometer \(\approx\) 0.62 mile
Glossary Entries
Definition: Scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.
Definition: Scale Drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.
Practice
Exercise \(\PageIndex{5}\)
The Empire State Building in New York City is about 1,450 feet high (including the antenna at the top) and 400 feet wide. Andre wants to make a scale drawing of the front view of the Empire State Building on an \(8\frac{1}{2}\)-inch-by-\(11\)-inch piece of paper. Select a scale that you think is the most appropriate for the scale drawing. Explain your reasoning.
- 1 inch to 1 foot
- 1 inch to 100 feet
- 1 inch to 1 mile
- 1 centimeter to 1 meter
- 1 centimeter to 50 meters
- 1 centimeter to 1 kilometer
Exercise \(\PageIndex{6}\)
Elena finds that the area of a house on a scale drawing is 25 square inches. The actual area of the house is 2,025 square feet. What is the scale of the drawing?
Exercise \(\PageIndex{7}\)
Which of these scales are equivalent to 3 cm to 4 km? Select all that apply. Recall that 1 inch is 2.54 centimeters.
- 0.75 cm to 1 km
- 1 cm to 12 km
- 6 mm to 2 km
- 0.3 mm to 40 m
- 1 inch to 7.62 km
Exercise \(\PageIndex{8}\)
These two triangles are scaled copies of one another. The area of the smaller triangle is 9 square units. What is the area of the larger triangle? Explain or show how you know.
Exercise \(\PageIndex{9}\)
Water costs $1.25 per bottle. At this rate, what is the cost of:
- 10 bottles?
- 20 bottles?
- 50 bottles?
Exercise \(\PageIndex{10}\)
The first row of the table shows the amount of dish detergent and water needed to make a soap solution.
- Complete the table for 2, 3, and 4 batches.
- How much water and detergent is needed for 8 batches? Explain your reasoning.
| number of batches | cups of water | cups of detergent |
|---|---|---|
| 1 | 6 | 1 |
| 2 | ||
| 3 | ||
| 4 | | libretexts | 2025-03-17T19:52:07.603062 | 2020-03-29T00:13:56 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.2.6: Units in Scale Drawings",
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.03%3A_New_Page | 1.3: Let's Put It to Work Last updated Save as PDF Page ID 34979 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:07.675360 | 2020-01-25T01:41:10 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.3: Let's Put It to Work",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/1.03%3A_New_Page/1.3.1%3A_Draw_It_to_Scale | 1.3.1: Draw It to Scale
Lesson
Let's draw a floor plan.
Exercise \(\PageIndex{1}\): Which Measurements Matter?
Which measurements would you need in order to draw a scale floor plan of your classroom? List which parts of the classroom you would measure and include in the drawing. Be as specific as possible.
Exercise \(\PageIndex{2}\): Creating a Floor Plan (Part 1)
- On a blank sheet of paper, make a rough sketch of a floor plan of the classroom. Include parts of the room that the class has decided to include or that you would like to include. Accuracy is not important for this rough sketch, but be careful not to omit important features like a door.
- Trade sketches with a partner and check each other’s work. Specifically, check if any parts are missing or incorrectly placed. Return their work and revise your sketch as needed.
-
Discuss with your group a plan for measuring. Work to reach an agreement on:
- Which classroom features must be measured and which are optional.
- The units to be used.
- How to record and organize the measurements (on the sketch, in a list, in a table, etc.).
- How to share the measuring and recording work (or the role each group member will play).
- Gather your tools, take your measurements, and record them as planned. Be sure to double-check your measurements.
- Make your own copy of all the measurements that your group has gathered, if you haven’t already done so. You will need them for the next activity.
Exercise \(\PageIndex{3}\): Creating a Floor Plan (Part 2)
Your teacher will give you several paper options for your scale floor plan.
- Determine an appropriate scale for your drawing based on your measurements and your paper choice. Your floor plan should fit on the paper and not end up too small.
-
Use the scale and the measurements your group has taken to draw a scale floor plan of the classroom. Make sure to:
- Show the scale of your drawing.
- Label the key parts of your drawing (the walls, main openings, etc.) with their actual measurements.
- Show your thinking and organize it so it can be followed by others.
Are you ready for more?
- If the flooring material in your classroom is to be replaced with 10-inch by 10-inch tiles, how many tiles would it take to cover the entire room? Use your scale drawing to approximate the number of tiles needed.
- How would using 20-inch by 20-inch tiles (instead of 10-inch by 10-inch tiles) change the number of tiles needed? Explain your reasoning.
Exercise \(\PageIndex{4}\): Creating a Floor Plan (part 3)
- Trade floor plans with another student who used the same paper size as you. Discuss your observations and thinking.
- Trade floor plans with another student who used a different paper size than you. Discuss your observations and thinking.
- Based on your discussions, record ideas for how your floor plan could be improved. | libretexts | 2025-03-17T19:52:07.735280 | 2020-03-29T00:18:23 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.3.1: Draw It to Scale",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/01%3A_Scale_Drawings/101%3A_New_Page/111%3A_What_are_Scaled_Copies | 1.1.1: What are Scaled Copies?
Lesson
Let's explore scaled copies.
Exercise \(\PageIndex{1}\): Printing Portraits
Here is a portrait of a student. Move the slider under each image, A–E, to see it change.
- How is each one the same as or different from the original portrait of the student?
- Some of the sliders make scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning.
- What do you think “scaled copy” means?
Exercise \(\PageIndex{2}\): Scaling F
Here is an original drawing of the letter F and some other drawings.
- Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.
- Examine all the scaled copies more closely, specifically, the lengths of each part of the letter F. How do they compare to the original? What do you notice?
- On the grid, draw a different scaled copy of the original letter F.
Exercise \(\PageIndex{3}\): Pairs of Scaled Polygons
Your teacher will give you a set of cards that have polygons drawn on a grid. Mix up the cards and place them all face up.
-
Take turns with your partner to match a pair of polygons that are scaled copies of one another.
- For each match you find, explain to your partner how you know it’s a match.
- For each match your partner finds, listen carefully to their explanation, and if you disagree, explain your thinking.
- When you agree on all of the matches, check your answers with the answer key. If there are any errors, discuss why and revise your matches.
- Select one pair of polygons to examine further. Use the grid below to produce both polygons. Explain or show how you know that one polygon is a scaled copy of the other.
Are you ready for more?
Is it possible to draw a polygon that is a scaled copy of both Polygon A and Polygon B? Either draw such a polygon, or explain how you know this is impossible.
Summary
What is a scaled copy of a figure? Let’s look at some examples.
The second and third drawings are both scaled copies of the original Y.
However, here, the second and third drawings are not scaled copies of the original W.
The second drawing is spread out (wider and shorter). The third drawing is squished in (narrower, but the same height).
We will learn more about what it means for one figure to be a scaled copy of another in upcoming lessons.
Glossary Entries
Definition: Scaled Copy
A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).
Practice
Exercise \(\PageIndex{4}\)
Here is a figure that looks like the letter A, along with several other figures. Which figures are scaled copies of the original A? Explain how you know.
Exercise \(\PageIndex{5}\)
Tyler says that Figure B is a scaled copy of Figure A because all of the peaks are half as tall.
Do you agree with Tyler? Explain your reasoning.
Exercise \(\PageIndex{6}\)
Here is a picture of the Rose Bowl Stadium in Pasadena, CA.
Here are some copies of the picture. Select all the pictures that are scaled copies of the original picture.
Exercise \(\PageIndex{7}\)
Complete each equation with a number that makes it true.
- \(5\cdot \underline{\qquad}=15\)
- \(4\cdot\underline{\qquad}=32\)
- \(6\cdot\underline{\qquad}=9\)
- \(12\cdot\underline{\qquad}=3\) | libretexts | 2025-03-17T19:52:07.830525 | 2020-03-29T00:12:54 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "1.1.1: What are Scaled Copies?",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships | 2: Introducing Proportional Relationships Last updated Save as PDF Page ID 34983 Illustrative Mathematics OpenUp Resources 2.1: Representing Proportional Relationships with Tables 2.1.1: One of These Things is Not Like the Others 2.1.2: Introducing Proportional Relationships with Tables 2.1.3: More about Constant of Proportionality 2.2: Representing Proportional Relationships with Equations 2.2.1: Proportional Relationships and Equations 2.2.2: Two Equations for Each Relationship 2.2.3: Using Equations to Solve Problems 2.3: Comparing Proportional and Nonproportional Relationships 2.3.1: Comparing Relationships with Tables 2.3.2: Comparing Relationships with Equations 2.3.3: Solving Problems about Proportional Relationships 2.4: Representing Proportional Relationships with Graphs 2.4.1: Introducing Graphs of Proportional Relationships 2.4.2: Interpreting Graphs of Proportional Relationships 2.4.3: Using Graphs to Compare Relationships 2.4.4: Two Graphs for Each Relationship 2.5: Let's Put it to Work 2.5.1: Four Representations 2.5.2: Using Water Efficiently | libretexts | 2025-03-17T19:52:07.918730 | 2020-01-25T01:41:13 | {
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"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2: Introducing Proportional Relationships",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page | 2.1: Representing Proportional Relationships with Tables Last updated Save as PDF Page ID 34984 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:08.065367 | 2020-01-25T01:41:14 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.1: Representing Proportional Relationships with Tables",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.1%3A_One_of_These_Things_is_Not_Like_the_Others | 2.1.1: One of These Things is Not Like the Others
Lesson
Let's remember what equivalent ratios are.
Exercise \(\PageIndex{1}\): Remembering Double Number Lines
1. Complete the double number line diagram with the missing numbers.
2. What could each of the number lines represent? Invent a situation and label the diagram.
3. Make sure your labels include appropriate units of measure.
Exercise \(\PageIndex{2}\): Mystery Mixtures
Your teacher will show you three mixtures. Two taste the same, and one is different.
- Which mixture tastes different? Describe how it is different.
-
Here are the recipes that were used to make the three mixtures:
- 1 cup of water with teaspoons of powdered drink mix
- 2 cups of water with teaspoon of powdered drink mix
- 1 cup of water with teaspoon of powdered drink mix
Which of these recipes is for the stronger tasting mixture? Explain how you know.
Are you ready for more?
Salt and sugar give two distinctly different tastes, one salty and the other sweet. In a mixture of salt and sugar, it is possible for the mixture to be salty, sweet or both. Will any of these mixtures taste exactly the same?
- Mixture A: 2 cups water, 4 teaspoons salt, 0.25 cup sugar
- Mixture B: 1.5 cups water, 3 teaspoons salt, 0.2 cup sugar
- Mixture C: 1 cup water, 2 teaspoons salt, 0.125 cup sugar
Exercise \(\PageIndex{3}\): Crescent Moons
Here are four different crescent moon shapes.
- What do Moons A, B, and C all have in common that Moon D doesn't?
- Use numbers to describe how Moons A, B, and C are differnt from Moon D.
- Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Are you ready for more?
Can you make one moon cover another by changing its size? What does that tell you about its dimensions?
GeoGebra Applet tbmsMsJZ
Summary
When two different situations can be described by equivalent ratios , that means they are alike in some important way.
An example is a recipe. If two people make something to eat or drink, the taste will only be the same as long as the ratios of the ingredients are equivalent. For example, all of the mixtures of water and drink mix in this table taste the same, because the ratios of cups of water to scoops of drink mix are all equivalent ratios.
| water (cups) | drink mix (scoops) |
|---|---|
| 3 | 1 |
| 12 | 4 |
| 1.5 | 0.5 |
If a mixture were not equivalent to these, for example, if the ratio of cups of water to scoops of drink mix were \(6:4\), then the mixture would taste different.
Notice that the ratios of pairs of corresponding side lengths are equivalent in figures A, B, and C. For example, the ratios of the length of the top side to the length of the left side for figures A, B, and C are equivalent ratios. Figures A, B, and C are scaled copies of each other; this is the important way in which they are alike.
If a figure has corresponding sides that are not in a ratio equivalent to these, like figure D, then it’s not a scaled copy. In this unit, you will study relationships like these that can be described by a set of equivalent ratios.
Glossary Entries
Definition: Equivalent Ratios
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
Practice
Exercise \(\PageIndex{4}\)
Which one of these shapes is not like the others? Explain what makes it different by representing each width and height pair with a ratio.
Exercise \(\PageIndex{5}\)
In one version of a trail mix, there are 3 cups of peanuts mixed with 2 cups of raisins. In another version of trail mix, there are 4.5 cups of peanuts mixed with 3 cups of raisins. Are the ratios equivalent for the two mixes? Explain your reasoning.
Exercise \(\PageIndex{6}\)
For each object, choose an appropriate scale for a drawing that fits on a regular sheet of paper. Not all of the scales on the list will be used.
Objects
- A person
- A football field (120 yards by 53\(\frac{1}{3}\) yards)
- The state of Washington (about 240 miles by 360 miles)
- The floor plan of a house
- A rectangular farm (6 miles by 2 mile)
Scales
- 1 in : 1 ft
- 1 cm : 1 m
- 1: 1000
- 1 ft: 1 mile
- 1: 100,000
- 1 mm: 1 km
- 1: 10,000,000
(From Unit 1.2.6)
Exercise \(\PageIndex{7}\)
Which scale is equivalent to 1 cm to 1 km?
- 1 to 1000
- 10,000 to 1
- 1 to 100,000
- 100,000 to 1
- 1 to 1,000,000
(From Unit 1.2.5)
Exercise \(\PageIndex{8}\)
- Find 3 different ratios that are equivalent to \(7:3\).
- Explain why these ratios are equivalent.
(From Unit 1.2.4) | libretexts | 2025-03-17T19:52:08.139659 | 2020-04-02T19:44:05 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.1%3A_One_of_These_Things_is_Not_Like_the_Others",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.1.1: One of These Things is Not Like the Others",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.2%3A_Introducing_Proportional_Relationships_with_Tables | 2.1.2: Introducing Proportional Relationships with Tables
Lesson
Let's solve problems involving proportional relationships using tables.
Exercise \(\PageIndex{1}\): NOtice and Wonder: Paper Towels by The Case
Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases.
What do you notice about the table? What do you wonder?
Exercise \(\PageIndex{2}\): Feeding a Crowd
-
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many people will 10 cups of rice serve?
-
How many cups of rice are needed to serve 45 people?
cups of rice number of people 2 6 3 9 10 45 Table \(\PageIndex{1}\)
-
A recipe says that 6 spring rolls will serve 3 people. Complete the table.
number of spring rolls number of people 6 3 30 40 28 Table \(\PageIndex{2}\)
Exercise \(\PageIndex{3}\): Making Bread Dough
A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many cups of flour do they use with 20 tablespoons of honey?
- How many cups of flour do they use with 13 tablespoons of honey?
- How many tablespoons of honey do they use with 20 cups of flour?
- What is the proportional relationship represented by this table?
| honey (tbsp) | flour (c) |
|---|---|
| 8 | 10 |
| 20 | |
| 13 | |
| 20 |
Exercise \(\PageIndex{4}\): Quarters and Dimes
4 quarters are equal in value to 10 dimes.
- How many dimes equal the value of 6 quarters?
- How many dimes equal the value of 14 quarters?
- What value belongs next to the 1 in the table? What does it mean in this context?
| number of quarters | number of dimes |
|---|---|
| 1 | |
| 4 | 20 |
| 6 | |
| 14 |
Are you ready for more?
Pennies made before 1982 are 95% copper and weigh about 3.11 grams each. (Pennies made after that date are primarily made of zinc). Some people claim that the value of the copper in one of these pennies is greater than the face value of the penny. Find out how much copper is worth right now, and decide if this claim is true.
Summary
If the ratios between two corresponding quantities are always equivalent, the relationship between the quantities is called a proportional relationship .
This table shows different amounts of milk and chocolate syrup. The ingredients in each row, when mixed together, would make a different total amount of chocolate milk, but these mixtures would all taste the same.
| tablespoons of chocolate syrup | cups of milk |
|---|---|
| \(4\) | \(1\) |
| \(6\) | \(1\frac{1}{2}\) |
| \(8\) | \(2\) |
| \(\frac{1}{2}\) | \(\frac{1}{8}\) |
| \(12\) | \(3\) |
| \(1\) | \(\frac{1}{4}\) |
Notice that each row in the table shows a ratio of tablespoons of chocolate syrup to cups of milk that is equivalent to \(4:1\).
About the relationship between these quantities, we could say:
- The relationship between amount of chocolate syrup and amount of milk is proportional.
- The relationship between the amount of chocolate syrup and the amount of milk is a proportional relationship.
- The table represents a proportional relationship between the amount of chocolate syrup and amount of milk.
- The amount of milk is proportional to the amount of chocolate syrup.
We could multiply any value in the chocolate syrup column by \(\frac{1}{4}\) to get the value in the milk column. We might call \(\frac{1}{4}\) a unit rate , because \(\frac{1}{4}\) cups of milk are needed for 1 tablespoon of chocolate syrup. We also say that \(\frac{1}{4}\) is the constant of proportionality for this relationship. It tells us how many cups of milk we would need to mix with 1 tablespoon of chocolate syrup.
Glossary Entries
Definition: Constant of Proportionality
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.
In this example, the constant of proportionality is 3, because \(2\cdot 3=6\), \(3\cdot 3=9\), and \(5\cdot 3=15\). This means that there are 3 apples for every 1 orange in the fruit salad.
| number of oranges | number of apples |
|---|---|
| 2 | 6 |
| 3 | 9 |
| 5 | 15 |
Definition: Equivalent Ratios
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
Definition: Proportional Relationship
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.
For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.
We can write this relationship as \(p=4s\). This equation shows that \(s\) is proportional to \(p\).
| \(s\) | \(p\) |
|---|---|
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
| 10 | 40 |
Practice
Exercise \(\PageIndex{5}\)
When Han makes chocolate milk, he mixes 2 cups of milk with 3 tablespoons of chocolate syrup. Here is a table that shows how to make batches of different sizes. Use the information in the table to complete the statements. Some terms are used more than once.
- The table shows a proportional relationship between ______________ and ______________.
- The scale factor shown is ______________.
- The constant of proportionality for this relationship is______________.
- The units for the constant of proportionality are ______________ per ______________.
Bank of Terms: tablespoons of chocolate syrup, 4, cups of milk, cup of milk, \(\frac{3}{2}\)
Exercise \(\PageIndex{6}\)
A certain shade of pink is created by adding 3 cups of red paint to 7 cups of white paint.
-
How many cups of red paint should be added to 1 cup of white paint?
cups of white paint cups of red paint 1 7 3 Table \(\PageIndex{9}\) - What is the constant of proportionality?
Exercise \(\PageIndex{7}\)
A map of a rectangular park has a length of 4 inches and a width of 6 inches. It uses a scale of 1 inch for every 30 miles.
- What is the actual area of the park? Show how you know.
- The map needs to be reproduced at a different scale so that it has an area of 6 square inches and can fit in a brochure. At what scale should the map be reproduced so that it fits on the brochure? Show your reasoning.
(From Unit 1.2.6)
Exercise \(\PageIndex{8}\)
Noah drew a scaled copy of Polygon P and labeled it Polygon Q.
If the area of Polygon P is 5 square units, what scale factor did Noah apply to Polygon P to create Polygon Q? Explain or show how you know.
(From Unit 1.1.6)
Exercise \(\PageIndex{9}\)
Select all the ratios that are equivalent to each other.
- \(4:7\)
- \(8:15\)
- \(16:28\)
- \(2:3\)
- \(20:35\) | libretexts | 2025-03-17T19:52:08.233119 | 2020-04-02T19:42:56 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.2%3A_Introducing_Proportional_Relationships_with_Tables",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.1.2: Introducing Proportional Relationships with Tables",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.3%3A_More_about_Constant_of_Proportionality | 2.1.3: More about Constant of Proportionality
Lesson
Let's solve more problems involving proportional relationships using tables.
Exercise \(\PageIndex{1}\): Equal Measures
Use the numbers and units from the list to find as many equivalent measurements as you can. For example, you might write “30 minutes is \(\frac{1}{2}\) hour.”
You can use the numbers and units more than once.
\(\begin{array}{llll}{1}&{\frac{1}{2}}&{0.3}&{\text{centimeter}}\\{12}&{40}&{24}&{\text{meter}}\\{0.4}&{0.01}&{\frac{1}{10}}&{\text{hour}}\\{60}&{3\frac{1}{3}}&{6}&{\text{feet}}\\{50}&{30}&{2}&{\text{minute}}\\{}&{}&{\frac{2}{5}}&{\text{inch}}\end{array}\)
Exercise \(\PageIndex{2}\): Centimeters and Millimeters
There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters.
There are two ways of thinking about this proportional relationship.
-
If you know the length of something in centimeters, you can calculate its length in millimeters.
- Complete the table.
-
What is the constant of proportionality?
length (cm) length (mm) 9 12.5 50 88.49 Table \(\PageIndex{1}\)
-
If you know the length of something in millimeters, you can calculate its length in centimeters.
- Complete the table.
-
What is the constant of proportionality?
length (mm) length (cm) 70 245 4 699.1 Table \(\PageIndex{2}\)
- How are these two constants of proportionality related to each other?
-
Complete each sentence:
- To convert from centimeters to millimeters, you can multiply by ________.
- To convert from millimeters to centimeters, you can divide by ________ or multiply by ________.
Are you ready for more?
- How many square millimeters are there in a square centimeter?
- How do you convert square centimeters to square millimeters? How do you convert the other way?
Exercise \(\PageIndex{3}\): Pittsburgh to Phoenix
On its way from New York to San Diego, a plane flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix traveling at a constant speed.
Complete the table as you answer the questions. Be prepared to explain your reasoning.
| segment | time | distance | speed |
|---|---|---|---|
| Pittsburgh to Saint Louis | 1 hour | 550 miles | |
| Saint Louis to Albuquerque | 1 hour 42 minutes | ||
| Albuquerque to Phoenix | 330 miles |
- What is the distance between Saint Louis and Albuquerque?
- How many minutes did it take to fly between Albuquerque and Phoenix?
- What is the proportional relationship represented by this table?
- Diego says the constant of proportionality is 550. Andre says the constant of proportionality is \(9\frac{1}{6}\). Do you agree with either of them? Explain your reasoning.
Summary
When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled. The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.
We can multiply any number in the first column by \(\frac{2}{3}\) to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is \(\frac{2}{3}\). This means that the bug’s pace is \(\frac{2}{3}\) seconds per centimeter.
This table represents the same situation, except the columns are switched.
We can multiply any number in the first column by \(\frac{3}{2}\) to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is \(\frac{3}{2}\). This means that the bug’s speed is \(\frac{3}{2}\) centimeters per second.
Notice that \(\frac{3}{2}\) is the reciprocal of \(\frac{2}{3}\). When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column to get the values in the second.
Glossary Entries
Definition: Constant of Proportionality
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.
In this example, the constant of proportionality is 3, because \(2\cdot 3=6\), \(3\cdot 3=9\), and \(5\cdot 3=15\). This means that there are 3 apples for every 1 orange in the fruit salad.
| number of oranges | number of apples |
|---|---|
| 2 | 6 |
| 3 | 9 |
| 5 | 15 |
Definition: Equivalent Ratios
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\cdot\frac{1}{2}=4\) and \(6\cdot\frac{1}{2}=3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
Definition: Proportional Relationship
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.
For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.
We can write this relationship as \(p=4s\). This equation shows that \(s\) is proportional to \(p\).
| \(s\) | \(p\) |
|---|---|
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
| 10 | 40 |
Practice
Exercise \(\PageIndex{4}\)
Noah is running a portion of a marathon at a constant speed of 6 miles per hour.
Complete the table to predict how long it would take him to run different distances at that speed, and how far he would run in different time intervals.
| time in hours | miles traveled at 6 miles per hour |
|---|---|
| \(1\) | |
| \(\frac{1}{2}\) | |
| \(1\frac{1}{3}\) | |
| \(1\frac{1}{2}\) | |
| \(9\) | |
| \(4\frac{1}{2}\) |
Exercise \(\PageIndex{5}\)
One kilometer is 1000 meters.
1. Complete the tables. What is the interpretation of the constant of proportionality in each case?
| meters | kilometers |
|---|---|
| 1,000 | 1 |
| 250 | |
| 12 | |
| 1 |
The constant of proportionality tells us that:
| kilometers | meters |
|---|---|
| 1 | 1,000 |
| 5 | |
| 20 | |
| 0.3 |
The constant of proportionality tells us that:
2. What is the relationship between two constants of proportionality?
Exercise \(\PageIndex{6}\)
Jada and Lin are comparing inches and feet. Jada says that the constant of proportionality is 12. Lin says it is \(\frac{1}{12}\). Do you agree with either of them? Explain your reasoning.
Exercise \(\PageIndex{7}\)
The area of the Mojave desert is 25,000 square miles. A scale drawing of the Mojave desert has an area of 10 square inches. What is the scale of the map?
(From Unit 1.2.6)
Exercise \(\PageIndex{8}\)
Which of these scales is equivalent to the scale 1 cm to 5 km? Select all that apply.
- 3 cm to 15 km
- 1 mm to 150 km
- 5 cm to 1 km
- 5 mm to 2.5 km
- 1 mm to 500 m
(From Unit 1.2.5)
Exercise \(\PageIndex{9}\)
Which one of these pictures is not like the others? Explain what makes it different using ratios.
(From Unit 2.1.1) | libretexts | 2025-03-17T19:52:08.328475 | 2020-04-02T19:42:09 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.01%3A_New_Page/2.1.3%3A_More_about_Constant_of_Proportionality",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.1.3: More about Constant of Proportionality",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page | 2.2: Representing Proportional Relationships with Equations Last updated Save as PDF Page ID 34985 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:08.399780 | 2020-01-25T01:41:15 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.2: Representing Proportional Relationships with Equations",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.1%3A_Proportional_Relationships_and_Equations | 2.2.1: Proportional Relationships and Equations
Lesson
Let's write equations describing proportional relationships.
Exercise \(\PageIndex{1}\): NUmber Talk: Division
Find each quotient mentally.
\(645\div 100\)
\(645\div 50\)
\(48.6\div 30\)
\(48.6\div x\)
Exercise \(\PageIndex{2}\): Feeding a Crowd, Revisited
1. A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many people will 1 cup of rice serve?
- How many people will 3 cups of rice serve? 12 cups? 43 cups?
- How many people will \(x\) cups of rice serve?
| cups of dry rice | number of people |
|---|---|
| 1 | |
| 2 | 6 |
| 3 | |
| 12 | |
| 43 | |
| \(x\) |
2. A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
- How many people will 1 spring roll serve?
- How many people will 10 spring rolls serve? 16 spring rolls? 25 spring rolls?
- How many people will \(n\) spring rolls serve?
| number of spring rolls | number of people |
|---|---|
| 1 | |
| 6 | 3 |
| 10 | |
| 16 | |
| 25 | |
| \(n\) |
3. How was completing this table different from the previous table? How as it the same?
Exercise \(\PageIndex{3}\): Denver to Chicago
A plane flew at a constant speed between Denver and Chicago. It took the plane 1.5 hours to fly 915 miles.
-
Complete the table.
time (hours) distance (miles) speed (miles per hour) 1 1.5 915 2 2.5 \(t\) Table \(\PageIndex{3}\) - How far does the plane fly in one hour?
- How far would the plane fly in \(t\) hours at this speed?
- If \(d\) represents the distance that the plane flies at this speed for \(t\) hours, write an equation that relates \(t\) and \(d\).
- How far would the plane fly in 3 hours at this speed? in 3.5 hours? Explain or show your reasoning.
Are you ready for more?
A rocky planet orbits Proxima Centauri, a star that is about 1.3 parsecs from Earth. This planet is the closest planet outside of our solar system.
- How long does it take light from Proxima Centauri to reach Earth? (A parsec is about 3.26 light years. A light year is the distance light travels in one year.)
- There are two twins. One twin leaves on a spaceship to explore the planet near Proxima Centauri traveling at 90% of the speed of light, while the other twin stays home on Earth. How much does the twin on Earth age while the other twin travels to Proxima Centauri? (Do you think the answer would be the same for the other twin? Consider researching “The Twin Paradox” to learn more.)
Exercise \(\PageIndex{4}\): Revisiting a Bread Dough
A bakery uses 8 tablespoons of honey for every 10 cups of flour to make bread dough. Some days they bake bigger batches and some days they bake smaller batches, but they always use the same ratio of honey to flour.
- Complete the table.
- If \(f\) is the cups of flour needed for \(h\) tablespoons of honey, write an equation that relates \(f\) and \(h\).
- How much flour is needed for 15 tablespoons of honey? 17 tablespoons? Explain or show your reasoning.
| honey (tbsp) | flour (c) |
|---|---|
| 1 | |
| 8 | 10 |
| 16 | |
| 20 | |
| \(h\) |
Summary
The table shows the amount of red paint and blue paint needed to make a certain shade of purple paint, called Venusian Sunset.
Note that “parts” can be any unit for volume. If we mix 3 cups of red with 12 cups of blue, you will get the same shade as if we mix 3 teaspoons of red with 12 teaspoons of blue.
| red paint (parts) | blue paint (parts) |
|---|---|
| \(3\) | \(12\) |
| \(1\) | \(4\) |
| \(7\) | \(28\) |
| \(\frac{1}{4}\) | \(1\) |
| \(r\) | \(4r\) |
The last row in the table says that if we know the amount of red paint needed, \(r\), we can always multiply it by 4 to find the amount of blue paint needed, \(b\), to mix with it to make Venusian Sunset. We can say this more succinctly with the equation \(b=4r\). So the amount of blue paint is proportional to the amount of red paint and the constant of proportionality is 4.
We can also look at this relationship the other way around.
If we know the amount of blue paint needed, \(b\), we can always multiply it by \(\frac{1}{4}\) to find the amount of red paint needed, \(r\), to mix with it to make Venusian Sunset. So \(r=\frac{1}{4}b\). The amount of blue paint is proportional to the amount of red paint and the constant of proportionality \(\frac{1}{4}\).
| blue paint (parts) | red paint (parts) |
|---|---|
| \(12\) | \(3\) |
| \(4\) | \(1\) |
| \(28\) | \(7\) |
| \(1\) | \(\frac{1}{4}\) |
| \(b\) | \(\frac{1}{4}b\) |
In general, when \(y\) is proportional to \(x\), we can always multiply \(x\) by the same number \(k\)—the constant of proportionality—to get \(y\). We can write this much more succinctly with the equation \(y=kx\).
Practice
Exercise \(\PageIndex{5}\)
A certain ceiling is made up of tiles. Every square meter of ceiling requires 10.75 tiles. Fill in the table with the missing values.
| square meters of ceiling | number of tiles |
|---|---|
| \(1\) | |
| \(10\) | |
| \(100\) | |
| \(a\) |
Exercise \(\PageIndex{6}\)
On a flight from New York to London, an airplane travels at a constant speed. An equation relating the distance traveled in miles, \(d\), to the number of hours flying, \(t\), is \(t=\frac{1}{500}d\). How long will it take the airplane to travel 800 miles?
Exercise \(\PageIndex{7}\)
Each table represents a proportional relationship. For each, find the constant of proportionality, and write an equation that represents the relationship.
| \(s\) | \(P\) |
|---|---|
| \(2\) | \(8\) |
| \(3\) | \(12\) |
| \(5\) | \(20\) |
| \(10\) | \(40\) |
Constant of proportionality:
Equation: \(P=\)
| \(d\) | \(C\) |
|---|---|
| \(2\) | \(6.28\) |
| \(3\) | \(9.42\) |
| \(5\) | \(15.7\) |
| \(10\) | \(31.4\) |
Constant of proportionality:
Equation: \(C=\)
Exercise \(\PageIndex{8}\)
A map of Colorado says that the scale is 1 inch to 20 miles or 1 to 1,267,200. Are these two ways of reporting the scale the same? Explain your reasoning.
(From Unit 1.2.5)
Exercise \(\PageIndex{9}\)
Here is a polygon on a grid.
- Draw a scaled copy of the polygon using a scale factor 3. Label the copy A.
- Draw a scaled copy of the polygon with a scale factor \(\frac{1}{2}\). Label it B.
- Is Polygon A a scaled copy of Polygon B? If so, what is the scale factor that takes B to A?
(From Unit 1.1.3) | libretexts | 2025-03-17T19:52:08.499000 | 2020-04-02T19:46:03 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.1%3A_Proportional_Relationships_and_Equations",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.2.1: Proportional Relationships and Equations",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.2%3A_Two_Equations_for_Each_Relationship | 2.2.2: Two Equations for Each Relationship
Lesson
Let's investigate the equations that represent proportional relationships.
Exercise \(\PageIndex{1}\): Missing Figures
Here are the second and fourth figures in a pattern.
- What do you think the first and third figures in the pattern look like?
- Describe the 10th figure in the pattern.
Exercise \(\PageIndex{2}\): Meters and Centimeters
There are 100 centimeters (cm) in every meter (m).
| length (m) | length (cm) |
|---|---|
| \(1\) | \(100\) |
| \(0.94\) | |
| \(1.67\) | |
| \(57.24\) | |
| \(x\) |
| length (cm) | length (m) |
|---|---|
| \(100\) | \(1\) |
| \(250\) | |
| \(78.2\) | |
| \(123.9\) | |
| \(y\) |
- Complete each of the tables.
- For each table, find the constant of proportionality.
- What is the relationship between these constants of proportionality?
- For each table, write an equation for the proportional relationship. Let \(x\) represent a length measured in meters and \(y\) represent the same length measured in centimeters.
Are you ready for more?
- How many cubic centimeters are there in a cubic meter?
- How do you convert cubic centimeters to cubic meters?
- How do you convert the other way?
Exercise \(\PageIndex{3}\): Filling a Water Cooler
It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let \(w\) be the number of gallons of water in the cooler after \(t\) minutes.
-
Which of the following equations represent the relationship between \(w\) and \(t\)? Select
all
that apply.
- \(w=1.6t\)
- \(w=0.625t\)
- \(t=1.6w\)
- \(t=0.625w\)
- What does 1.6 tell you about the situation?
- What does 0.625 tell you about the situation?
- Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of \(w\) and \(t\) when it takes 3 minutes to fill the cooler with 1 gallon of water.
- Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.
Exercise \(\PageIndex{4}\): Feeding Shrimp
At an aquarium, a shrimp is fed \(\frac{1}{5}\) gram of food each feeding and is fed 3 times each day.
- How much food does a shrimp get fed in one day?
-
Complete the table to show how many grams of food the shrimp is fed over different numbers of days.
number of days food in grams \(1\) \(7\) \(30\) Table \(\PageIndex{3}\)
Figure \(\PageIndex{1}\) - What is the constant of proportionality? What does it tell us about the situation?
- If we switched the columns in the table, what would be the constant of proportionality? Explain your reasoning.
- Use \(d\) for number of days and \(f\) for amount of food in grams that a shrimp eats to write two equations that represent the relationship between \(d\) and \(f\).
- If a tank has 10 shrimp in it, how much food is added to the tank each day?
- If the aquarium manager has 300 grams of shrimp food for this tank of 10 shrimp, how many days will it last? Explain or show your reasoning.
Summary
If Kiran rode his bike at a constant 10 miles per hour, his distance in miles, \(d\), is proportional to the number of hours, \(t\), that he rode. We can write the equation \(d=10t\). With this equation, it is easy to find the distance Kiran rode when we know how long it took because we can just multiply the time by 10.
We can rewrite the equation:
\[\begin{aligned} d&=10t \\ \left(\frac{1}{10}\right) d&=t\\t&=\left(\frac{1}{10}\right)d \end{aligned}\nonumber\]
This version of the equation tells us that the amount of time he rode is proportional to the distance he traveled, and the constant of proportionality is \(\frac{1}{10}\). That form is easier to use when we know his distance and want to find how long it took because we can just multiply the distance by \(\frac{1}{10}\).
When two quantities \(x\) and \(y\) are in a proportional relationship, we can write the equation \(y=kx\) and say, “\(y\) is proportional to \(x\).” In this case, the number \(k\) is the corresponding constant of proportionality. We can also write the equation \(x=\frac{1}{k}y\) and say, “\(x\) is proportional to \(y\).” In this case, the number \(\frac{1}{k}\) is the corresponding constant of proportionality. Each one can be useful depending on the information we have and the quantity we are trying to figure out.
Practice
Exercise \(\PageIndex{5}\)
The table represents the relationship between a length measured in meters and the same length measured in kilometers.
| meters | kilometers |
|---|---|
| \(1,000\) | \(1\) |
| \(3,500\) | |
| \(500\) | |
| \(75\) | |
| \(1\) | |
| \(x\) |
- Complete the table.
- Write an equation for converting the number of meters to kilometers. Use \(x\) for number of meters and \(y\) for number of kilometers.
Exercise \(\PageIndex{6}\)
Concrete building blocks weigh 28 pounds each. Using \(b\) for the number of concrete blocks and \(w\) for the weight, write two equations that relate the two variables. One equation should begin with \(w=\) and the other should begin with \(b=\).
Exercise \(\PageIndex{7}\)
A store sells rope by the meter. The equation \(p=0.8L\) represents the price \(p\) (in dollars) of a piece of nylon rope that is \(L\) meters long.
- How much does the nylon rope cost per meter?
- How long is a piece of nylon rope that costs $1.00?
Exercise \(\PageIndex{8}\)
The table represents a proportional relationship. Find the constant of proportionality and write an equation to represent the relationship.
| \(a\) | \(y\) |
|---|---|
| \(2\) | \(\frac{2}{3}\) |
| \(3\) | \(1\) |
| \(10\) | \(\frac{10}{3}\) |
| \(12\) | \(4\) |
Constant of proportionality: __________
Equation: \(y=\)
(From Unit 2.2.1)
Exercise \(\PageIndex{9}\)
On a map of Chicago, 1 cm represents 100 m. Select all statements that express the same scale.
- 5 cm on the map represents 50 m in Chicago.
- 1 mm on the map represents 10 m in Chicago.
- 1 km in Chicago is represented by 10 cm the map.
- 100 cm in Chicago is represented by 1 m on the map.
(From Unit 1.2.2) | libretexts | 2025-03-17T19:52:08.582223 | 2020-04-02T19:45:31 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.2%3A_Two_Equations_for_Each_Relationship",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.2.2: Two Equations for Each Relationship",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.3%3A_Using_Equations_to_Solve_Problems | 2.2.3: Using Equations to Solve Problems
Lesson
Let's use equations to solve problems involving proportional relationships.
Exercise \(\PageIndex{1}\): Number Talk: Quotients with Decimal Points
Without calculating, order the quotients of these expressions from least to greatest.
\(42.6\div 0.07\)
\(42.6\div 70\)
\(42.6\div 0.7\)
\(426\div 70\)
Place the decimal point in the appropriate location in the quotient: \(42.6\div 7=608571\)
Use this answer to find the quotient of one of the previous expressions.
Exercise \(\PageIndex{2}\): Concert Ticket Sales
A performer expects to sell 5,000 tickets for an upcoming concert. They want to make a total of $311,000 in sales from these tickets.
- Assuming that all tickets have the same price, what is the price for one ticket?
- How much will they make if they sell 7,000 tickets?
- How much will they make if they sell 10,000 tickets? 50,000? 120,000? a million? \(x\) tickets?
- If they make $404,300, how many tickets have they sold?
- How many tickets will they have to sell to make $5,000,000?
Exercise \(\PageIndex{3}\): Recycling
Aluminum cans can be recycled instead of being thrown in the garbage. The weight of 10 aluminum cans is 0.16 kilograms. The aluminum in 10 cans that are recycled has a value of $0.14.
- If a family threw away 2.4 kg of aluminum in a month, how many cans did they throw away? Explain or show your reasoning.
- What would be the recycled value of those same cans? Explain or show your reasoning.
- Write an equation to represent the number of cans \(c\) given their weight \(w\).
- Write an equation to represent the recycled value \(r\) of \(c\) cans.
- Write an equation to represent the recycled value \(r\) of \(w\) kilograms of aluminum.
Are you ready for more?
The EPA estimated that in 2013, the average amount of garbage produced in the United States was 4.4 pounds per person per day. At that rate, how long would it take your family to produce a ton of garbage? (A ton is 2,000 pounds.)
Summary
Remember that if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form \(y=kx\). Sometimes writing an equation is the easiest way to solve a problem.
For example, we know that Denali, the highest mountain peak in North America, is 20,300 feet above sea level. How many miles is that? There are 5,280 feet in 1 mile. This relationship can be represented by the equation
\(f=5,280m\)
where \(f\) represents a distance measured in feet and \(m\) represents the same distance measured miles. Since we know Denali is 20,310 feet above sea level, we can write
\(20,310=5,280m\)
So \(m=\frac{20,310}{5,280}\), which is approximately 3.85 miles.
Practice
Exercise \(\PageIndex{4}\)
A car is traveling down a highway at a constant speed, described by the equation \(d=65t\), where \(d\) represents the distance, in miles, that the car travels at this speed in \(t\) hours.
- What does the 65 tell us in this situation?
- How many miles does the car travel in 1.5 hours?
- How long does it take the car to travel 26 miles at this speed?
Exercise \(\PageIndex{5}\)
Elena has some bottles of water that each holds 17 fluid ounces.
- Write an equation that relates the number of bottles of water (\(b\)) to the total volume of water (\(w\)) in fluid ounces.
- How much water is in 51 bottles?
- How many bottles does it take to hold 51 fluid ounces of water?
Exercise \(\PageIndex{6}\)
There are about 1.61 kilometers in 1 mile. Let \(x\) represent a distance measured in kilometers and \(y\) represent the same distance measured in miles. Write two equations that relate a distance measured in kilometers and the same distance measured in miles.
(From Unit 2.2.2)
Exercise \(\PageIndex{7}\)
In Canadian coins, 16 quarters is equal in value to 2 toonies.
| number of quarters | number of toonies |
|---|---|
| \(1\) | |
| \(16\) | \(2\) |
| \(20\) | |
| \(24\) |
- Complete the table.
- What does the value next to 1 mean in this situation?
(From Unit 2.1.2)
Exercise \(\PageIndex{8}\)
Each table represents a proportional relationship. For each table:
- Fill in the missing parts of the table.
- Draw a circle around the constant of proportionality.
| \(x\) | \(y\) |
|---|---|
| \(2\) | \(10\) |
| \(15\) | |
| \(7\) | |
| \(1\) |
| \(a\) | \(b\) |
|---|---|
| \(12\) | \(3\) |
| \(20\) | |
| \(10\) | |
| \(1\) |
| \(m\) | \(n\) |
|---|---|
| \(5\) | \(3\) |
| \(10\) | |
| \(18\) | |
| \(1\) |
(From Unit 2.1.2)
Exercise \(\PageIndex{9}\)
Describe some things you could notice in two polygons that would help you decide that they were not scaled copies.
(From Unit 1.1.4) | libretexts | 2025-03-17T19:52:08.658891 | 2020-04-02T19:44:47 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.02%3A_New_Page/2.2.3%3A_Using_Equations_to_Solve_Problems",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.2.3: Using Equations to Solve Problems",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page | 2.3: Comparing Proportional and Nonproportional Relationships Last updated Save as PDF Page ID 34986 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:08.734755 | 2020-01-25T01:41:15 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.3: Comparing Proportional and Nonproportional Relationships",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.1%3A_Comparing_Relationships_with_Tables | 2.3.1: Comparing Relationships with Tables
Lesson
Let's explore how proportional relationships are different from other relationships.
Exercise \(\PageIndex{1}\): Adjusting a Recipe
A lemonade recipe calls for the juice of 5 lemons, 2 cups of water, and 2 tablespoons of honey.
Invent four new versions of this lemonade recipe:
- One that would make more lemonade but taste the same as the original recipe.
- One that would make less lemonade but taste the same as the original recipe.
- One that would have a stronger lemon taste than the original recipe.
- One that would have a weaker lemon taste than the original recipe.
Exercise \(\PageIndex{2}\): Visiting the State Park
Entrance to a state park costs $6 per vehicle, plus $2 per person in the vehicle.
-
How much would it cost for a car with 2 people to enter the park? 4 people? 10 people? Record your answers in the table.
number of people in vehicle total entrance cost in dollars \(2\) \(4\) \(10\) Table \(\PageIndex{1}\) - For each row in the table, if each person in the vehicle splits the entrance cost equally, how much will each person pay?
- How might you determine the entrance cost for a bus with 50 people?
- Is the relationship between the number of people and the total entrance cost a proportional relationship? Explain how you know.
Are you ready for more?
What equation could you use to find the total entrance cost for a vehicle with any number of people?
Exercise \(\PageIndex{3}\): Running Laps
Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.
Han's run:
| distance (laps) | time (minutes) | minutes per lap |
|---|---|---|
| \(2\) | \(4\) | |
| \(4\) | \(9\) | |
| \(6\) | \(15\) | |
| \(8\) | \(23\) |
Clare's run:
| distance (laps) | time (minutes) | minutes per lap |
|---|---|---|
| \(2\) | \(5\) | |
| \(4\) | \(10\) | |
| \(6\) | \(15\) | |
| \(8\) | \(20\) |
- Is Han running at a constant pace? Is Clare? How do you know?
- Write an equation for the relationship between distance and time for anyone who is running at a constant pace.
Summary
Here are the prices for some smoothies at two different smoothie shops:
Smoothie Shop A
| smoothie size (oz) | price ($) | dollars per ounce |
|---|---|---|
| \(8\) | \(6\) | \(0.75\) |
| \(12\) | \(9\) | \(0.75\) |
| \(16\) | \(12\) | \(0.75\) |
| \(s\) | \(0.75s\) | \(0.75\) |
Smoothie Shop B
| smoothie size (oz) | prize ($) | dollars per ounce |
|---|---|---|
| \(8\) | \(6\) | \(0.75\) |
| \(12\) | \(8\) | \(0.67\) |
| \(16\) | \(10\) | \(0.625\) |
| \(s\) | \(???\) | \(???\) |
For Smoothie Shop A, smoothies cost $0.75 per ounce no matter which size we buy. There could be a proportional relationship between smoothie size and the price of the smoothie. An equation representing this relationship is \(p=0.75s\) where \(s\) represents size in ounces and \(p\) represents price in dollars. (The relationship could still not be proportional, if there were a different size on the menu that did not have the same price per ounce.)
For Smoothie Shop B, the cost per ounce is different for each size. Here the relationship between smoothie size and price is definitely not proportional.
In general, two quantities in a proportional relationship will always have the same quotient. When we see some values for two related quantities in a table and we get the same quotient when we divide them, that means they might be in a proportional relationship—but if we can't see all of the possible pairs, we can't be completely sure. However, if we know the relationship can be represented by an equation is of the form \(y=kx\), then we are sure it is proportional.
Practice
Exercise \(\PageIndex{4}\)
Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would the constant of proportionality be?
-
How loud a sound is depending on how far away you are.
distance to listener (ft) sound level (dB) \(5\) \(85\) \(10\) \(79\) \(20\) \(73\) \(40\) \(67\) Table \(\PageIndex{6}\) -
The cost of fountain drinks at Hot Dog Hut.
volume (fluid ounces) cost ($) \(16\) \($1.49\) \(20\) \($1.59\) \(30\) \($1.89\) Table \(\PageIndex{7}\)
Exercise \(\PageIndex{5}\)
A taxi service charges $1.00 for the first \(\frac{1}{10}\) mile then $0.10 for each additional \(\frac{1}{10}\) mile after that.
Fill in the table with the missing information then determine if this relationship between distance traveled and price of the trip is a proportional relationship.
| distance traveled (mi) | price (dollars) |
|---|---|
| \(\frac{9}{10}\) | |
| \(2\) | |
| \(3\frac{1}{10}\) | |
| \(10\) |
Exercise \(\PageIndex{6}\)
A rabbit and turtle are in a race. Is the relationship between distance traveled and time proportional for either one? If so, write an equation that represents the relationship.
Turtle’s run:
| distance (meters) | time (minutes) |
|---|---|
| \(108\) | \(2\) |
| \(405\) | \(7.5\) |
| \(540\) | \(10\) |
| \(1,768.5\) | \(32.75\) |
Rabbit's run:
| distance (meters) | time (minutes) |
|---|---|
| \(800\) | \(1\) |
| \(900\) | \(5\) |
| \(1,107.5\) | \(20\) |
| \(1,524\) | \(32.5\) |
Exercise \(\PageIndex{7}\)
For each table, answer: What is the constant of proportionality?
| \(a\) | \(b\) |
|---|---|
| \(2\) | \(14\) |
| \(5\) | \(35\) |
| \(9\) | \(63\) |
| \(\frac{1}{3}\) | \(\frac{7}{3}\) |
| \(a\) | \(b\) |
|---|---|
| \(3\) | \(60\) |
| \(5\) | \(600\) |
| \(8\) | \(960\) |
| \(12\) | \(1440\) |
| \(a\) | \(b\) |
|---|---|
| \(75\) | \(3\) |
| \(200\) | \(8\) |
| \(1525\) | \(61\) |
| \(10\) | \(0.4\) |
| \(a\) | \(b\) |
|---|---|
| \(4\) | \(10\) |
| \(6\) | \(15\) |
| \(22\) | \(55\) |
| \(3\) | \(7\frac{1}{2}\) |
(From Unit 2.1.2)
Exercise \(\PageIndex{8}\)
Kiran and Mai are standing at one corner of a rectangular field of grass looking at the diagonally opposite corner. Kiran says that if the the field were twice as long and twice as wide, then it would be twice the distance to the far corner. Mai says that it would be more than twice as far, since the diagonal is even longer than the side lengths. Do you agree with either of them?
(From Unit 1.1.4) | libretexts | 2025-03-17T19:52:08.836316 | 2020-04-02T19:47:52 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.1%3A_Comparing_Relationships_with_Tables",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.3.1: Comparing Relationships with Tables",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.2%3A_Comparing_Relationships_with_Equations | 2.3.2: Comparing Relationships with Equations
Lesson
Let's develop methods for deciding if a relationship is proportional.
Exercise \(\PageIndex{1}\): Notice and Wonder: Patterns with Rectangles
Do you see a pattern? What predictions can you make about future rectangles in the set if your pattern continues?
Exercise \(\PageIndex{2}\): More Conversions
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
-
Use the equation \(F=\frac{9}{5}C+32\), where \(F\) represents degrees Fahrenheit and \(C\) represents degrees Celsius, to complete the table.
temperature (\(^{\circ}\)C) temperature (\(^{\circ}\)F) \(20\) \(4\) \(175\) Table \(\PageIndex{1}\) -
Use the equation \(c=2.54n\), where \(c\) represents the length in centimeters and \(n\) represents the length in inches, to complete the table.
length (in) length (cm) \(10\) \(8\) \(3\frac{1}{2}\) Table \(\PageIndex{2}\) - Are these proportional relationships? Explain why or why not.
Exercise \(\PageIndex{3}\): Total Edge Length, Surface Area, and Volume
Here are some cubes with different side lengths. Complete each table. Be prepared to explain your reasoning.
-
How long is the total edge length of each cube?
side length total edge length \(3\) \(5\) \(9\frac{1}{2}\) \(s\) Table \(\PageIndex{3}\) -
What is the surface area of each cube?
side length surface area \(3\) \(5\) \(9\frac{1}{2}\) \(s\) Table \(\PageIndex{4}\) -
What is the volume of each cube?
side length volume \(3\) \(5\) \(9\frac{1}{2}\) \(s\) Table \(\PageIndex{5}\) - Which of these relationships is proportional? Explain how you know.
- Write equations for the total edge length \(E\), total surface area \(A\), and volume \(V\) of a cube with side length \(s\).
Are you ready for more?
- A rectangular solid has a square base with side length \(l\), height 8, and volume \(V\). Is the relationship between \(l\) and \(V\) a proportional relationship?
- A different rectangular solid has length \(l\), width 10, height 5, and volume \(V\). Is the relationship between \(l\) and \(V\) a proportional relationship?
- Why is the relationship between the side length and the volume proportional in one situation and not the other?
Exercise \(\PageIndex{4}\): All Kinds of Equations
Here are six different equations.
\[\begin{array}{lllll}{y=4+x}&{\qquad}&{y=4x}&{\qquad}&{y=\frac{4}{x}}\\{y=\frac{x}{4}}&{\qquad}&{y=4^{x}}&{\qquad}&{y=x^{4}}\end{array}\nonumber\]
- Predict which of these equations represent a proportional relationship.
- Complete each table using the equation that represents the relationship.
- Do these results change your answer to the first question? Explain your reasoning.
- What do the equations of the proportional relationships have in common?
Summary
If two quantities are in a proportional relationship, then their quotient is always the same. This table represents different values of \(a\) and \(b\), two quantities that are in a proportional relationship.
| \(a\) | \(b\) | \(\frac{b}{a}\) |
|---|---|---|
| \(20\) | \(100\) | \(5\) |
| \(3\) | \(15\) | \(5\) |
| \(11\) | \(55\) | \(5\) |
| \(1\) | \(5\) | \(5\) |
Notice that the quotient of \(b\) and \(a\) is always 5. To write this as an equation, we could say \(\frac{b}{a}=5\). If this is true, then \(b=5a\). (This doesn’t work if \(a=0\), but it works otherwise.)
If quantity \(y\) is proportional to quantity \(x\), we will always see this pattern: \(\frac{y}{x}\) will always have the same value. This value is the constant of proportionality, which we often refer to as \(k\). We can represent this relationship with the equation \(\frac{y}{x}=k\) (as long as \(x\) is not 0) or \(y=kx\).
Note that if an equation cannot be written in this form, then it does not represent a proportional relationship.
Practice
Exercise \(\PageIndex{5}\)
The relationship between a distance in yards (\(y\)) and the same distance in miles (\(m\)) is described by the equation \(y=1760m\).
-
Find measurements in yards and miles for distances by completing the table.
distance measured in miles distance measured in yards \(1\) \(5\) \(3,520\) \(17,600\) Table \(\PageIndex{7}\) - Is there a proportional relationship between a measurement in yards and a measurement in miles for the same distance? Explain why or why not.
Exercise \(\PageIndex{6}\)
Decide whether or not each equation represents a proportional relationship.
- The remaining length (\(L\)) of 120-inch rope after \(x\) inches have been cut off: \(120-x=L\)
- The total cost (\(t\)) after 8% sales tax is added to an item's price (\(p\)): \(1.08p=t\)
- The number of marbles each sister gets (\(x\)) when \(m\) marbles are shared equally among four sisters: \(x=\frac{m}{4}\)
- The volume (\(V\)) of a rectangular prism whose height is 12 cm and base is a square with side lengths \(s\) cm: \(V=12s^{2}\)
Exercise \(\PageIndex{7}\)
1. Use the equation \(y=\frac{5}{2}x\) to complete the table.
Is \(y\) proportional to \(x\) and \(y\)? Explain why or why not.
| \(x\) | \(y\) |
|---|---|
| \(2\) | |
| \(3\) | |
| \(6\) |
2. Use the equation \(y=3.2x+5\) to complete the table.
Is \(y\) proportional to \(x\) and \(y\)? Explain why or why not.
| \(x\) | \(y\) |
|---|---|
| \(1\) | |
| \(2\) | |
| \(4\) |
Exercise \(\PageIndex{8}\)
To transmit information on the internet, large files are broken into packets of smaller sizes. Each packet has 1,500 bytes of information. An equation relating packets to bytes of information is given by \(b=1,500p\) where \(p\) represents the number of packets and \(b\) represents the number of bytes of information.
- How many packets would be needed to transmit 30,000 bytes of information?
- How much information could be transmitted in 30,000 packets?
- Each byte contains 8 bits of information. Write an equation to represent the relationship between the number of packets and the number of bits.
(From Unit 2.2.3) | libretexts | 2025-03-17T19:52:08.925037 | 2020-04-02T19:47:21 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.2%3A_Comparing_Relationships_with_Equations",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.3.2: Comparing Relationships with Equations",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.3%3A_Solving_Problems_about_Proportional_Relationships | 2.3.3: Solving Problems about Proportional Relationships
Lesson
Let's solve problems about proportional relationships.
Exercise \(\PageIndex{1}\): What Do You Want to Know?
Consider the problem: A person is running a distance race at a constant rate. What time will they finish the race?
What information would you need to be able to solve the problem?
Exercise \(\PageIndex{2}\): Info Gap: Biking and Rain
Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.
If your teacher gives you the problem card :
- Silently read your card and think about what information you need to be able to answer the question.
- Ask your partner for the specific information that you need.
-
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently.
- Read the data card and discuss your reasoning.
If your teacher gives you the data card :
- Silently read your card.
-
Ask your partner
“What specific information do you need?”
and wait for them to
ask
for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
- Read the problem card and solve the problem independently.
- Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Exercise \(\PageIndex{3}\): Moderating Comments
A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.
- Person 1 worked for 210 minutes and checked a total of 50,000 comments.
- Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
-
Person 3 worked for 120 minutes, at a rate represented by \(c=331t\),
where is the number of comments checked and is the time in minutes. - Person 4 worked for 150 minutes, at a rate represented by \(t=\left(\frac{3}{800}\right)c\).
- Order the people from greatest to least in terms of total number of comments checked.
- Order the people from greatest to least in terms of how fast they checked the comments.
Are you ready for more?
- Write equations for each job applicant that allow you to easily decide who is working the fastest.
- Make a table that allows you to easily compare how many comments the four job applicants can check.
Summary
Whenever we have a situation involving constant rates, we are likely to have a proportional relationship between quantities of interest.
- When a bird is flying at a constant speed, then there is a proportional relationship between the flying time and distance flown.
- If water is filling a tub at a constant rate, then there is a proportional relationship between the amount of water in the tub and the time the tub has been filling up.
- If an aardvark is eating termites at a constant rate, then there is proportional relationship between the number of termites the aardvark has eaten and the time since it started eating.
Sometimes we are presented with a situation, and it is not so clear whether a proportional relationship is a good model. How can we decide if a proportional relationship is a good representation of a particular situation?
- If you aren’t sure where to start, look at the quotients of corresponding values. If they are not always the same, then the relationship is definitely not a proportional relationship.
- If you can see that there is a single value that we always multiply one quantity by to get the other quantity, it is definitely a proportional relationship.
After establishing that it is a proportional relationship, setting up an equation is often the most efficient way to solve problems related to the situation.
Practice
Exercise \(\PageIndex{4}\)
For each situation, explain whether you think the relationship is proportional or not. Explain your reasoning.
- The weight of a stack of standard 8.5x11 copier paper vs. number of sheets of paper.
- The weight of a stack of different-sized books vs. the number of books in the stack.
Exercise \(\PageIndex{5}\)
Every package of a certain toy also includes 2 batteries.
- Are the number of toys and number of batteries in a proportional relationship? If so, what are the two constants of proportionality? If not, explain your reasoning.
- Use \(t\) for the number of toys and \(b\) for the number of batteries to write two equations relating the two variables.
\(b=\qquad\qquad t=\)
Exercise \(\PageIndex{6}\)
Lin and her brother were born on the same date in different years. Lin was 5 years old when her brother was 2.
-
Find their ages in different years by filling in the table.
Lin's age Her brother's age \(5\) \(2\) \(6\) \(15\) \(25\) Table \(\PageIndex{1}\) - Is there a proportional relationship between Lin's age and her brother's age? Explain your reasoning.
Exercise \(\PageIndex{7}\)
A student argues that \(y=\frac{x}{9}\) does not represent a proportional relationship between \(x\) and \(y\) because we need to multiply one variable by the same constant to get the other one and not divide it by a constant. Do you agree or disagree with this student?
(From Unit 2.3.2)
Exercise \(\PageIndex{8}\)
Quadrilateral A has side lengths 3, 4, 5, and 6. Quadrilateral B is a scaled copy of Quadrilateral A with a scale factor of 2. Select all of the following that are side lengths of Quadrilateral B.
- \(5\)
- \(6\)
- \(7\)
- \(8\)
- \(9\)
(From Unit 1.1.3) | libretexts | 2025-03-17T19:52:08.998613 | 2020-04-02T19:46:49 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.03%3A_New_Page/2.3.3%3A_Solving_Problems_about_Proportional_Relationships",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.3.3: Solving Problems about Proportional Relationships",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page | 2.4: Representing Proportional Relationships with Graphs Last updated Save as PDF Page ID 34987 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:09.071520 | 2020-01-25T01:41:16 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.4: Representing Proportional Relationships with Graphs",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.1%3A_Introducing_Graphs_of_Proportional_Relationships | 2.4.1: Introducing Graphs of Proportional Relationships
Lesson
Let's see how graphs of proportional relationships differ from graphs of other relationships.
Exercise \(\PageIndex{1}\): Notice These Points
- Plot the points \((0,10), (1,8), (2,6), (3,4), (4,2)\).
- What do you notice about the graph?
Exercise \(\PageIndex{2}\): T-shirts for Sale
Some T-shirts cost $8 each.
| \(x\) | \(y\) |
|---|---|
| \(1\) | \(8\) |
| \(2\) | \(16\) |
| \(3\) | \(24\) |
| \(4\) | \(32\) |
| \(5\) | \(40\) |
| \(6\) | \(48\) |
-
Use the table to answer these questions.
- What does \(x\) represent?
- What does \(y\) represent?
- Is there a proportional relationship between \(x\) and \(y\)?
- Plot the pairs in the table on the coordinate plane.
- What do you notice about the graph?
Exercise \(\PageIndex{3}\): Matching Tables and Graphs
Your teacher will give you papers showing tables and graphs.
- Examine the graphs closely. What is the same and what is different about the graphs?
- Sort the graphs into categories of your choosing. Label each category. Be prepared to explain why you sorted the graphs the way you did.
-
Take turns with a partner to match a table with a graph.
- For each match you find, explain to your partner how you know it is a match.
-
For each match your partner finds, listen carefully to their explanation. If you disagree, work to reach an agreement.
Pause here so your teacher can review your work.
- Trade places with another group. How are their categories the same as your group's categories? How are they different?
- Return to your original place. Discuss any changes you may wish to make to your categories based on what the other group did.
- Which of the relationships are proportional?
- What have you noticed about the graphs of proportional relationships? Do you think this will hold true for all graphs of proportional relationships?
Are you ready for more?
- All the graphs in this activity show points where both coordinates are positive. Would it make sense for any of them to have one or more coordinates that are negative?
- The equation of a proportional relationship is of the form \(y=kx\), where \(k\) is a positive number, and the graph is a line through \((0,0)\). What would the graph look like if \(k\) were a negative number?
Summary
One way to represent a proportional relationship is with a graph. Here is a graph that represents different amounts that fit the situation, “Blueberries cost $6 per pound.”
Different points on the graph tell us, for example, that 2 pounds of blueberries cost $12, and 4.5 pounds of blueberries cost $27.
Sometimes it makes sense to connect the points with a line, and sometimes it doesn’t. We could buy, for example, 4.5 pounds of blueberries or 1.875 pounds of blueberries, so all the points in between the whole numbers make sense in the situation, so any point on the line is meaningful.
If the graph represented the cost for different numbers of sandwiches (instead of pounds of blueberries), it might not make sense to connect the points with a line, because it is often not possible to buy 4.5 sandwiches or 1.875 sandwiches. Even if only points make sense in the situation, though, sometimes we connect them with a line anyway to make the relationship easier to see.
Graphs that represent proportional relationships all have a few things in common:
- Points that satisfy the relationship lie on a straight line.
- The line that they lie on passes through the origin , \((0,0)\).
Here are some graphs that do not represent proportional relationships:
These points do not lie on a line.
This is a line, but it doesn't go through the origin.
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
Which graphs could represent a proportional relationship?
- A
- B
- C
- D
Exercise \(\PageIndex{5}\)
A lemonade recipe calls for \(\frac{1}{4}\) cup of lemon juice for every cup of water.
-
Use the table to answer these questions.
- What does \(x\) represent?
- What does \(y\) represent?
- Is there a proportional relationship between \(x\) and \(y\)?
- Plot the pairs in the table in a coordinate plane.
| \(x\) | \(y\) |
|---|---|
| \(1\) | \(\frac{1}{4}\) |
| \(2\) | \(\frac{1}{2}\) |
| \(3\) | \(\frac{3}{4}\) |
| \(4\) | \(1\) |
Exercise \(\PageIndex{6}\)
Select all the pieces of information that would tell you \(x\) and \(y\) have a proportional relationship. Let \(y\) represent the distance in meters between a rock and a turtle's current position and \(x\) represent the time in minutes the turtle has been moving.
- \(y=3x\)
- After 4 minutes, the turtle has walked 12 feet away from the rock.
- The turtle walks for a bit, then stops for a minute before walking again.
- The turtle walks away from the rock at a constant rate.
(From Unit 2.3.3)
Exercise \(\PageIndex{7}\)
Decide whether each table could represent a proportional relationship. If the relationship could be proportional, what would be the constant of proportionality?
-
The sizes you can print a photo.
width of photo (inches) height of photo (inches) \(2\) \(3\) \(4\) \(6\) \(5\) \(7\) \(8\) \(10\) Table \(\PageIndex{3}\) -
The distance from which a lighthouse is visible.
height of a lighthouse (feet) distance it can be seen (miles) \(20\) \(6\) \(45\) \(9\) \(70\) \(11\) \(95\) \(13\) Table \(\PageIndex{4}\)
(From Unit 2.3.1) | libretexts | 2025-03-17T19:52:09.153171 | 2020-04-02T19:50:21 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.1%3A_Introducing_Graphs_of_Proportional_Relationships",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.4.1: Introducing Graphs of Proportional Relationships",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.2%3A_Interpreting_Graphs_of_Proportional_Relationships | 2.4.2: Interpreting Graphs of Proportional Relationships
Lesson
Let's read stories from the graphs of proportional relationships.
Exercise \(\PageIndex{1}\): What Could the Graph Represent?
Here is a graph that represents a proportional relationship.
- Invent a situation that could be represented by this graph.
- Label the axes with the quantities in your situation.
- Give the graph a title.
- There is a point on the graph. What are its coordinates? What does it represent in your situation?
Exercise \(\PageIndex{2}\): Tyler's Walk
Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.
- The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
- The table representing Tyler's walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
- What does the point \((0,0)\) mean in this situation?
- How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
- What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler's walk? Where do you see it in the graph?
| time (seconds) | distance (meters) |
|---|---|
| \(0\) | \(0\) |
| \(20\) | \(25\) |
| \(30\) | \(37.5\) |
| \(40\) | \(50\) |
| \(1\) |
Are you ready for more?
If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?
Exercise \(\PageIndex{3}\): Seagulls Eat What?
4 seagulls ate 10 pounds of garbage. Assume this information describes a proportional relationship.
- Plot a point that shows the number of seagulls and the amount of garbage they ate.
- Use a straight edge to draw a line through this point and \((0,0)\).
- Plot the point \((1,k)\) on the line. What is the value of \(k\)? What does the value of \(k\) tell you about this context?
Summary
For the relationship represented in this table, \(y\) is proportional to \(x\). We can see in the table that \(\frac{5}{4}\) is the constant of proportionality because it’s the \(y\) value when \(x\) is 1.
The equation \(y=\frac{5}{4}x\) also represents this relationship.
| \(x\) | \(y\) |
|---|---|
| \(4\) | \(5\) |
| \(5\) | \(\frac{25}{4}\) |
| \(8\) | \(10\) |
| \(1\) | \(\frac{5}{4}\) |
Here is the graph of this relationship.
If \(y\) represents the distance in feet that a snail crawls in \(x\) minutes, then the point \((4,5)\) tells us that the snail can crawl 5 feet in 4 minutes.
If \(y\) represents the cups of yogurt and \(x\) represents the teaspoons of cinnamon in a recipe for fruit dip, then the point \((4,5)\) tells us that you can mix 4 teaspoons of cinnamon with 5 cups of yogurt to make this fruit dip.
We can find the constant of proportionality by looking at the graph, because \(\frac{5}{4}\) is the \(y\)-coordinate of the point on the graph where the \(x\)-coordinate is 1. This could mean the snail is traveling \(\frac{5}{4}\) feet per minute or that the recipe calls for \(1\frac{1}{4}\) cups of yogurt for every teaspoon of cinnamon.
In general, when \(y\) is proportional to \(x\), the corresponding constant of proportionality is the \(y\)-value when \(x=1\).
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
There is a proportional relationship between the number of months a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is $47.94. The point \((6,47.94)\) is shown on the graph below.
- What is the constant of proportionality in this relationship?
- What does the constant of proportionality tell us about the situation?
- Add at least three more points to the graph and label them with their coordinates.
- Write an equation that represents the relationship between \(C\), the total cost of the subscription, and \(m\), the number of months.
Exercise \(\PageIndex{5}\)
The graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix. Label the point \((1,k)\) on the graph, find the value of \(k\), and explain its meaning.
Exercise \(\PageIndex{6}\)
To make a friendship bracelet, some long strings are lined up then taking one string and tying it in a knot with each of the other strings to create a row of knots. A new string is chosen and knotted with the all the other strings to create a second row. This process is repeated until there are enough rows to make a bracelet to fit around your friend's wrist.
Are the number of knots proportional to the number of rows? Explain your reasoning.
(From Unit 2.3.3)
Exercise \(\PageIndex{7}\)
What information do you need to know to write an equation relating two quantities that have a proportional relationship?
(From Unit 2.3.3) | libretexts | 2025-03-17T19:52:09.312624 | 2020-04-02T19:49:44 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.4.2: Interpreting Graphs of Proportional Relationships",
"author": "Illustrative Mathematics"
} |
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.3%3A_Using_Graphs_to_Compare_Relationships | 2.4.3: Using Graphs to Compare Relationships
Lesson
Let's graph more than one relationship on the same grid.
Exercise \(\PageIndex{1}\): Number Talk: Fraction Multiplication and Division
Find each product or quotient mentally.
\(\frac{2}{3}\cdot\frac{1}{2}\)
\(\frac{4}{3}\cdot\frac{1}{4}\)
\(4\div\frac{1}{5}\)
\(\frac{9}{6}\div\frac{1}{2}\)
Exercise \(\PageIndex{2}\): Race to the Bumper Cars
Diego, Lin, and Mai went from the ticket booth to the bumper cars.
1. Use each description to complete the table representing that person’s journey.
- Diego left the ticket booth at the same time as Tyler. Diego jogged ahead at a steady pace and reached the bumper cars in 30 seconds.
- Lin left the ticket booth at the same time as Tyler. She ran at a steady pace and arrived at the bumper cars in 20 seconds.
- Mai left the booth 10 seconds later than Tyler. Her steady jog enabled her to catch up with Tyler just as he arrived at the bumper cars.
| Diego's time (seconds) | Diego's distance (meters) |
|---|---|
| \(0\) | |
| \(15\) | |
| \(30\) | \(50\) |
| \(1\) |
| Lim's time (seconds) | Lin's distance (meters) |
|---|---|
| \(0\) | |
| \(25\) | |
| \(20\) | \(50\) |
| \(1\) |
| Mai's time (seconds) | Mai's distance (meters) |
|---|---|
| \(0\) | |
| \(25\) | |
| \(40\) | \(50\) |
| \(1\) |
2. Using a different color for each person, draw a graph of all four people’s journeys (including Tyler's from the other day).
- Drag the names to the correct lines to label them.
- If you choose to, you can use the Paint Brush tool to change the color of each line. Select the tool, click on a color in the palette below the graph, and then click on a line. Click on the Move tool (the arrow) before changing to a new paint brush color.
- You can hide any points you create with the checkbox below the graph.
3. Which person is moving the most quickly? How is that reflected in the graph?
Are you ready for more?
Write equations to represent each person’s relationship between time and distance.
Exercise \(\PageIndex{3}\): Space Rocks and the Price of Rope
1. Meteoroid Perseid 245 and Asteroid x travel through the solar system. Explore the applet to learn about the distance they had each traveled after a given time.
Is Asteroid x traveling faster or slower than Perseid 245? Explain how you know.
2. The graph shows the price of different lengths of two types of rope.
If you buy $1.00 of each kind of rope, which one will be longer? Explain how you know.
Summary
Here is a graph that shows the price of blueberries at two different stores. Which store has a better price?
We can compare points that have the same \(x\) value or the same \(y\) value. For example, the points \((2,12)\) and \((3,12)\) tell us that at store B you can get more pounds of blueberries for the same price.
The points \((3,12)\) and \((3,18)\) tell us that at store A you have to pay more for the same quantity of blueberries. This means store B has the better price.
We can also use the graphs to compare the constants of proportionality. The line representing store B goes through the point \((1,4)\), so the constant of proportionality is 4. This tells us that at store B the blueberries cost $4 per pound. This is cheaper than the $6 per pound unit price at store A.
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
The graphs below show some data from a coffee shop menu. One of the graphs shows cost (in dollars) vs. drink volume (in ounces), and one of the graphs shows calories vs. drink volume (in ounces).
__________________ vs volume
_____________________ vs volume
- Which graph is which? Give them the correct titles.
- Which quantities appear to be in a proportional relationship? Explain how you know.
- For the proportional relationship, find the constant of proportionality. What does that number mean?
Exercise \(\PageIndex{5}\)
Lin and Andre biked home from school at a steady pace. Lin biked 1.5 km and it took her 5 minutes. Andre biked 2 km and it took him 8 minutes.
- Draw a graph with two lines that represent the bike rides of Lin and Andre.
- For each line, highlight the point with coordinates \((1,k)\) and find \(k\).
- Who was biking faster?
Exercise \(\PageIndex{6}\)
Match each equation to its graph.
- \(y=2x\)
- \(y=\frac{4}{5}x\)
- \(y=\frac{1}{4}x\)
- \(y=\frac{2}{3}x\)
- \(y=\frac{4}{3}x\)
- \(y=\frac{3}{2}x\) | libretexts | 2025-03-17T19:52:09.389276 | 2020-04-02T19:49:12 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.3%3A_Using_Graphs_to_Compare_Relationships",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.4.3: Using Graphs to Compare Relationships",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.04%3A_New_Page/2.4.4%3A_Two_Graphs_for_Each_Relationship | 2.4.4: Two Graphs for Each Relationship
Lesson
Let's use tables, equations, and graphs to answer questions about proportional relationships.
Exercise \(\PageIndex{1}\): True or False: Fractions and Decimals
Decice whether each equation is true or false. Be prepared to explain your reasoning.
- \(\frac{3}{2}\cdot 16=3\cdot 8\)
- \(\frac{3}{4}\div\frac{1}{2}=\frac{6}{4}\div\frac{1}{4}\)
- \((2.8)\cdot (13)=(0.7)\cdot (52)\)
Exercise \(\PageIndex{2}\): Tables, Graphs, and Equations
Explore the graph. Start by dragging the gray bar on the left across the screen until you can see both the table and the graph. Notice the values in the table and the coordinates of the labeled point. Grab the point and move it around.
- What stays the same and what changes in the table? in the equation? on the graph?
- Choose one row in the table and write it here. To what does this row correspond on the graph?
| \(x\) | \(y\) |
|---|---|
Grab and drag the point until you see the equation \(y=\frac{3}{2}x\).
- Do not move the point. Choose three rows from the table, other than the origin. Record \(x\) and \(y\), and compute \(\frac{y}{x}\).
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
- What do you notice? What does this have to do with the equation of the line?
- Do not move the point. Check the box to view the coordinates \((1,?)\). What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?
- Drag the point to a different location. Record the equation of the line, the coordinates of three points, and the value of \(\frac{y}{x}\).
Equation of the line: _______________________________
| \(x\) | \(y\) | \(\frac{y}{x}\) |
|---|---|---|
- Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.
Are you ready for more?
The graph of an equation of the form \(y=kx\), where \(k\) is a positive number, is a line through \((0,0)\) and the point \((1,k)\).
- Name at least one line through \((0,0)\) that cannot be represented by an equation like this.
- If you could draw the graphs of all of the equations of this form in the same coordinate plane, what would it look like?
Exercise \(\PageIndex{3}\): Hot Dog Eating Contest
Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.
- The points shown on the first set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.
- Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point. Write an equation for each line. Use to represent time in minutes, and to represent number of hot dogs.
- For each equation, what does the constant of proportionality tell you?
- The points shown on the second set of axes display information about Andre’s and Jada’s consumption. Which point indicates Andre’s consumption? Which indicates Jada’s consumption? Label them.
- Draw lines from the origin through each of the two points. Write an equation for each line. What does the constant of proportionality tell you in each case?
Summary
Imagine that a faucet is leaking at a constant rate and that every 2 minutes, 10 milliliters of water leaks from the faucet. There is a proportional relationship between the volume of water and elapsed time.
- We could say that the elapsed time is proportional to the volume of water. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of \(\frac{1}{5}\) of a minute per milliliter.
- We could say that the volume of water is proportional to the elapsed time. The corresponding constant of proportionality tells us that the faucet is leaking at a rate of 5 milliliters per minute.
Let’s use \(v\) to represent volume in milliliters and \(t\) to represent time in minutes. Here are graphs and equations that represent both ways of thinking about this relationship:
Even though the relationship between time and volume is the same, we are making a different choice in each case about which variable to view as the independent variable. The graph on the left has \(v\) as the independent variable, and the graph on the right has \(t\) as the independent variable.
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
At the supermarket you can fill your own honey bear container. A customer buys 12 oz of honey for $5.40.
- How much does honey cost per ounce?
- How much honey can you buy per dollar?
- Write two different equations that represent this situation. Use \(h\) for ounces of honey and \(c\) for cost in dollars.
- Choose one of your equations, and sketch its graph. Be sure to label the axes.
Exercise \(\PageIndex{5}\)
The point \((3,\frac{6}{5})\) lies on the graph representing a proportional relationship. Which of the following points also lie on the same graph? Select all that apply.
- \((1,0.4)\)
- \(1.5, \frac{6}{10})\)
- \(\frac{6}{5},3)\)
- \((4,\frac{11}{5})\)
- \((15,6)\)
Exercise \(\PageIndex{6}\)
A trail mix recipe asks for 4 cups of raisins for every 6 cups of peanuts. There is proportional relationship between the amount of raisins, \(r\) (cups), and the amount of peanuts, \(p\) (cups), in this recipe.
- Write the equation for the relationship that has constant of proportionality greater than 1. Graph the relationship.
- Write the equation for the relationship that has constant of proportionality less than 1. Graph the relationship.
Exercise \(\PageIndex{7}\)
Here is a graph that represents a proportional relationship.
- Come up with a situation that could be represented by this graph.
- Label the axes with the quantities in your situation.
- Give the graph a title.
- Choose a point on the graph. What do the coordinates represent in your situation?
(From Unit 2.4.2) | libretexts | 2025-03-17T19:52:09.467455 | 2020-04-02T19:48:40 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.4.4: Two Graphs for Each Relationship",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.5%3A_Let's_Put_it_to_Work | 2.5: Let's Put it to Work Last updated Save as PDF Page ID 38102 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:09.538728 | 2020-04-02T19:39:09 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.5%3A_Let's_Put_it_to_Work",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.5: Let's Put it to Work",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.5%3A_Let's_Put_it_to_Work/2.5.1%3A_Four_Representations | 2.5.1: Four Representations
Lesson
Let's contrast relationships that are and are not proportional in four different ways.
Exercise \(\PageIndex{1}\): Which is the Bluest?
1. Which group of blocks is the bluest?
2. Order the groups of blocks from least blue to bluest.
Exercise \(\PageIndex{2}\): One Scenario, Four Representations
-
Select two things from different lists. Make up a situation where there is a
proportional relationship
between quantities that involve these things.
-
creatures
- starfish
- centipedes
- earthworms
- dinosaurs
-
length
- centimeters
- cubits
- kilometers
- parsecs
-
time
- nanoseconds
- minutes
- years
- millennia
-
volume
- milliliters
- gallons
- bushels
- cubic miles
-
body parts
- legs
- eyes
- neurons
- digits
-
area
- square microns
- acres
- hides
- square light-years
-
weight
- nanograms
- ounces
- deben
- metric tonnes
-
substance
- helium
- oobleck
- pitch
- glue
-
creatures
- Select two other things from the lists, and make up a situation where there is a relationship between quantities that involve these things, but the relationship is not proportional.
-
Your teacher will give you two copies of the “One Scenario, Four Representations” sheet. For each of your situations, describe the relationships in detail. If you get stuck, consider asking your teacher for a copy of the sample response.
- Write one or more sentences describing the relationship between the things you chose.
- Make a table with titles in each column and at least 6 pairs of numbers relating the two things.
- Graph the situation and label the axes.
- Write an equation showing the relationship and explain in your own words what each number and letter in your equation means.
- Explain how you know whether each relationship is proportional or not proportional. Give as many reasons as you can.
Exercise \(\PageIndex{3}\): Make a Poster
Create a visual display of your two situations that includes all the information from the previous activity.
Summary
The constant of proportionality for a proportional relationship can often be easily identified in a graph, a table, and an equation that represents it. Here is an example of all three representations for the same relationship. The constant of proportionality is circled:
On the other hand, some relationships are not proportional. If the graph of a relationship is not a straight line through the origin, if the equation cannot be expressed in the form \(y=kx\), or if the table does not have a constant of proportionality that you can multiply by any number in the first column to get the associated number in the second column, then the relationship between the quantities is not a proportional relationship.
Practice
Exercise \(\PageIndex{4}\)
The equation \(c=2.95g\) shows how much it costs to buy gas at a gas station on a certain day. In the equation, \(c\) represents the cost in dollars, and \(g\) represents how many gallons of gas were purchased.
- Write down at least four (gallons of gas, cost) pairs that fit this relationship.
- Create a graph of the relationship.
- What does 2.95 represent in this situation?
- Jada’s mom remarks, “You can get about a third of a gallon of gas for a dollar.” Is she correct? How did she come up with that?
Exercise \(\PageIndex{5}\)
There is a proportional relationship between a volume measured in cups and the same volume measured in tablespoons. 3 cups is equivalent to 48 tablespoons, as shown in the graph.
- Plot and label at least two more points that represent the relationship.
- Use a straightedge to draw a line that represents this proportional relationship.
- For which value y is \((1,y)\) on the line you just drew?
- What is the constant of proportionality for this relationship?
- Write an equation representing this relationship. Use \(c\) for cups and \(t\) for tablespoons. | libretexts | 2025-03-17T19:52:09.608321 | 2020-04-02T19:51:36 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "2.5.1: Four Representations",
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/02%3A_Introducing_Proportional_Relationships/2.5%3A_Let's_Put_it_to_Work/2.5.2%3A_Using_Water_Efficiently | 2.5.2: Using Water Efficiently
Lesson
Let's investigate saving water.
Exercise \(\PageIndex{1}\): Comparing Baths and Showers
Some people say that it uses more water to take a bath than a shower. Others disagree.
- What information would you collect in order to answer the question?
- Estimate some reasonable values for the things you suggest.
Exercise \(\PageIndex{2}\): Saving Water: Bath or Shower?
- Describe a method for comparing the water usage for a bath and a shower.
- Find out values for the measurements needed to use the method you described. You may ask your teacher or research them yourself.
- Under what conditions does a bath use more water? Under what conditions does a shower use more water?
Exercise \(\PageIndex{3}\): Representing Water Usage
- Continue considering the problem from the previous activity. Name two quantities that are in a proportional relationship. Explain how you know they are in a proportional relationship.
- What are two constants of proportionality for the proportional relationship? What do they tell us about the situation?
- On graph paper, create a graph that shows how the two quantities are related. Make sure to label the axes.
- Write two equations that relate the quantities in your graph. Make sure to record what each variable represents. | libretexts | 2025-03-17T19:52:09.665504 | 2020-04-02T19:51:06 | {
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"title": "2.5.2: Using Water Efficiently",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/03%3A_Untitled_Chapter_3 | 3: Measuring Circles Last updated Save as PDF Page ID 34990 Illustrative Mathematics OpenUp Resources 3.1: Circumference of a Circle 3.1.1: How Well Can You Measure? 3.1.2: Exploring Circles 3.1.3: Exploring Circumference 3.1.4: Applying Circumference 3.1.5: Circumference and Wheels 3.2: Area of a Circle 3.2.1: Estimating Areas 3.2.2: Exploring the Area of a Circle 3.2.3: Relating Area to Circumference 3.2.4: Applying Area of Circles 3.3: Let's Put it to Work 3.3.1: Distinguishing Circumference and Area 3.3.2: Stained-Glass Windows | libretexts | 2025-03-17T19:52:09.748094 | 2020-01-25T01:41:19 | {
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"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "3: Measuring Circles",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/03%3A_Untitled_Chapter_3/3.01%3A_New_Page | 3.1: Circumference of a Circle Last updated Save as PDF Page ID 34991 Illustrative Mathematics OpenUp Resources | libretexts | 2025-03-17T19:52:09.818128 | 2020-01-25T01:41:20 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/03%3A_Untitled_Chapter_3/3.01%3A_New_Page",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "3.1: Circumference of a Circle",
"author": "Illustrative Mathematics"
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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/03%3A_Untitled_Chapter_3/3.01%3A_New_Page/3.1.1%3A_How_Well_Can_You_Measure | 3.1.1: How Well Can You Measure?
Lesson
Let's see how accurately we can measure.
Exercise \(\PageIndex{1}\): Estimating a Percentage
A student got 16 out of 21 questions correct on a quiz. Use mental estimation to answer these questions.
- Did the student answer less than or more than 80% of the questions correctly?
- Did the student answer less than or more than 75% of the questions correctly?
Exercise \(\PageIndex{2}\): Perimeter of a Square
Your teacher will give you a picture of 9 different squares and will assign your group 3 of these squares to examine more closely.
-
For each of your assigned squares, measure the length of the diagonal and the perimeter of the square in centimeters.
Check your measurements with your group. After you come to an agreement, record your measurements in the table.diagonal (cm) perimeter (cm) square A square B square C square D square E square F square G square H square I Table \(\PageIndex{1}\) - Plot the diagonal and perimeter values from the table on the coordinate plane.
-
What do you notice about the points on the graph?
Pause here so your teacher can review your work. - Record measurements of the other squares to complete your data.
Exercise \(\PageIndex{3}\): Area of a Square
1. In the table, record the length of the diagonal for each of your assigned squares from the previous activity. Next, calculate the area of each of your squares.
| diagonal (cm) | area (cm 2 ) | |
|---|---|---|
| square A | ||
| square B | ||
| square C | ||
| square D | ||
| square E | ||
| square F | ||
| square G | ||
| square H | ||
| square I |
Pause here so your teacher can review your work. Be prepared to share your values with the class.
2. Examine the class graph of these values. What do you notice?
3. How is the relationship between the diagonal and area of a square the same as the relationship between the diagonal and perimeter of a square from the previous activity? How is it different?
Are you ready for more?
Here is a rough map of a neighborhood.
There are 4 mail routes during the week.
- On Monday, the mail truck follows the route A-B-E-F-G-H-A, which is 14 miles long.
- On Tuesday, the mail truck follows the route B-C-D-E-F-G-B, which is 22 miles long.
- On Wednesday, the truck follows the route A-B-C-D-E-F-G-H-A, which is 24 miles long.
- On Thursday, the mail truck follows the route B-E-F-G-B.
How long is the route on Thursdays?
Summary
When we measure the values for two related quantities, plotting the measurements in the coordinate plane can help us decide if it makes sense to model them with a proportional relationship. If the points are close to a line through \((0,0)\), then a proportional relationship is a good model. For example, here is a graph of the values for the height, measured in millimeters, of different numbers of pennies placed in a stack.
Because the points are close to a line through \((0,0)\), the height of the stack of pennies appears to be proportional to the number of pennies in a stack. This makes sense because we can see that the heights of the pennies only vary a little bit.
An additional way to investigate whether or not a relationship is proportional is by making a table. Here is some data for the weight of different numbers of pennies in grams, along with the corresponding number of grams per penny.
| number of pennies | grams | grams per penny |
|---|---|---|
| \(1\) | \(3.1\) | \(3.1\) |
| \(2\) | \(5.6\) | \(2.8\) |
| \(5\) | \(13.1\) | \(2.6\) |
| \(10\) | \(25.6\) | \(2.6\) |
Though we might expect this relationship to be proportional, the quotients are not very close to one another. In fact, the metal in pennies changed in 1982, and older pennies are heavier. This explains why the weight per penny for different numbers of pennies are so different!
Practice
Exercise \(\PageIndex{4}\)
Estimate the side length of a square that has a 9 cm long diagonal.
Exercise \(\PageIndex{5}\)
Select all quantities that are proportional to the diagonal length of a square.
- Area of the square
- Perimeter of the square
- Side length of the square
Exercise \(\PageIndex{6}\)
Diego made a graph of two quantities that he measured and said, “The points all lie on a line except one, which is a little bit above the line. This means that the quantities can’t be proportional.” Do you agree with Diego? Explain.
Exercise \(\PageIndex{7}\)
The graph shows that while it was being filled, the amount of water in gallons in a swimming pool was approximately proportional to the time that has passed in minutes.
- About how much water was in the pool after 25 minutes?
- Approximately when were there 500 gallons of water in the pool?
- Estimate the constant of proportionality for the gallons of water per minute going into the pool. | libretexts | 2025-03-17T19:52:09.900282 | 2020-04-07T20:02:45 | {
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/03%3A_Untitled_Chapter_3/3.01%3A_New_Page/3.1.1%3A_How_Well_Can_You_Measure",
"book_url": "https://commons.libretexts.org/book/math-34975",
"title": "3.1.1: How Well Can You Measure?",
"author": "Illustrative Mathematics"
} |
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