few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
In grade 3, students recognized that area is additive. They learned to find the area of a rectilinear figure by decomposing it into non-overlapping rectangles and adding their areas. Here students extend that understanding to non-rectangular shapes. They compose tangram pieces—consisting of triangles and a square—into shapes with certain areas. The square serves as a unit square. Because students have only one square, they need to use these principles in their reasoning:
If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
If a figure is decomposed and rearranged to compose another figure, then its area is the same as the area of the original figure.
Each question in the task aims to elicit discussions about these two principles. Though they may seem obvious, these principles still need to be stated explicitly (at the end of the lesson), as more-advanced understanding of the area of complex figures depends on them.
The terms
compose
,
decompose
, and
rearrange
will be formalized in an upcoming lesson, but throughout this lesson, look for opportunities to demonstrate their use as students describe their work with the tangram pieces. When students use “make” or “build,” “break,” and “move around,” recast their everyday terms using the more formal ones.
As students work, notice how they compose the pieces to create shapes with certain areas. Look for students whose reasoning illustrates the ideas outlined in the Activity Synthesis.
Demonstrate the use of the word “compose” by repeating students' everyday language use and then recast using the formal terms here.
Digital
Launch
Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.
Square: 1
Small triangles: 4
Medium triangle: 1
Large triangles: 2
It is important not to give them more than these pieces.
Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.
Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of
Harry Drew
in GeoGebra.
Speaking, Conversing: MLR2 Collect and Display.
Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit.
Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it.
Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.
Use your shapes to create a new shape with an area of 1 square unit that is not a square. Draw your shape on paper and label it with its area.
Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.
Use your shapes to create a
different
shape with an area of 2 square units. Draw your shape and label it with its area.
Use your shapes to create a new shape with an area of 4 square units. Draw your shape and label it with its area.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find a way to use all of your pieces to compose a single large square. What is the area of this large square?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.
Square: 1
Small triangles: 4
Medium triangle: 1
Large triangles: 2
It is important not to give them more than these pieces.
Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.
Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of
Harry Drew
in GeoGebra.
Speaking, Conversing: MLR2 Collect and Display.
Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Your teacher will give you one square and some small, medium, and large right triangles. The area of the square is 1 square unit.
Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.
Use your shapes to create a new shape with an area of 1 square unit that is not a square. Trace your shape.
Use your shapes to create a new shape with an area of 2 square units. Trace your shape.
Use your shapes to create a
different
shape with an area of 2 square units. Trace your shape.
Use your shapes to create a new shape with an area of 4 square units. Trace your shape.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find a way to use all of your pieces to compose a single large square. What is the area of this large square?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may consider the area to be the number of pieces in the compositions, instead of the number of square units. Remind them of the meaning of area, or prompt them to review the definition of area discussed in the warm-up activity.
Because the 2 large triangles in the tangram set can be arranged to form a square, students may consider that square to be the square unit rather than the smaller square composed of 2 small triangles. Ask students to review the task statement and verify the size of the unit square.
Activity Synthesis
Invite previously identified students (whose work illustrates the ideas shown here) to share. Name these moves explicitly as they come up:
compose
,
decompose
, and
rearrange
.
First question: Two small triangles can be composed into a square that matches up exactly with the given square piece. This means that the two squares—the composite and the unit square—have the same area.
Tell students, “We say that if a region can be placed on top of another region so that they match up exactly, then they have the same
area
.”
Second question: Two small triangles can be rearranged to compose a different figure but the area of that composite is still 1 square unit. These three shapes—each composed of two triangles—have the same area. If we rotate the first figure, it can be placed on top of the second so that they match up exactly. The third one has a different shape than the other two, but because it is made up of the same two triangles, it has the same area.
[IMG]
Emphasize: “If a figure is decomposed and rearranged as a new figure, the
area
of the new figure is the same as the area of the original figure.”
Third and fourth questions: The composite figures could be formed in several ways: with only small triangles, with two triangles and a medium triangle, or with two small triangles and a square.
Last question: A large triangle is needed here.  To find its area, we need to either compose 4 smaller triangles into a large triangle, or to see that the large triangle could be decomposed into 4 smaller triangles, which can then be composed into 2 unit squares.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: area, compose, decompose, and rearrange.
Supports accessibility for: Conceptual processing; Language


aligns: no