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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

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: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
problem_activity: Activity
This activity introduces students to graphic scales. Students interpret them, use them to find actual measurements (heights of tall buildings), and express them non-graphically. In earlier lessons, students used markings on an index card or sheet of paper to measure a drawing and create scaled copies. Here, they use non-standard measuring tools again to solve problems (MP5). Students make markings on an unmarked straightedge to measure scaled lengths on a drawing, and then use the given scale to determine actual lengths. This measuring strategy builds on students’ work with measurement in grades 1 and 2 (i.e., placing multiple copies of a shorter object end to end and expressing lengths in terms of the number of objects).
As students work, encourage them to be as precise as possible in making their marks and in estimating lengths that are less than 1 scale-segment long. Most distances students are measuring here are not line segments but rather distances between a point (the tip of a building) and a line (the ground).
Monitor the different ways students reason about scaled and actual distances. Here are two likely approaches for finding the difference between the Burj Khalifa and the Eiffel Tower (the second question):
Find the actual height of each tower and then find their difference.
Find the difference in scaled heights, and then use the scale to find the difference in actual heights.
Select students who use these approaches to share during the whole-class discussion in this sequence.
Launch
Display the scale drawing of the structures. Before beginning the work of the task, students may be interested in or eager to share the locations of the structures. Consider taking a few minutes to elicit what they know, or display a world map showing the locations of the structures.
Ask students what the segment labeled with “0 m” and “100 m” might mean. Some students are likely to say that it also conveys a scale. Verify that a scale can indeed be communicated graphically; an actual distance is not represented by a numerical measurement, but rather, by the length of the segment.
Provide access to index cards or sheets of paper students can use to measure. Tell students to check their answers to the first question with a partner. Tell them to discuss as necessary until they reach an agreement before proceeding to work on the rest of the problems. Give students 4–5 minutes of quiet work time and partner discussion.
Student Facing
Here is a scale drawing of some of the world’s tallest structures.
[IMG]
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.
Student Response
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for free access to Student Response.
Student Facing
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.
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 not measure heights of the buildings at a right angle from the ground line. Remind students that heights are to be measured perpendicular to the ground or base line.
If needed, demonstrate how to use the edge of a sheet of paper or an index card to measure lengths on a scale drawing with a graphic scale.
Activity Synthesis
Poll the class for their answers to the first question. Ask what might be some sources of discrepancies. The main issue here is measurement error, but there are also different methods to make the measurements and estimate the heights of the buildings. Two methods to estimate the heights of the buildings (and their limitations) are:
Estimate how many times “taller” each building is compared to the line segment giving the scale. Then multiply this number by 100 m. The accuracy here is not very good unless, for example, the height of the building is very close to being a whole number times the length of the scale.
Measure the segment giving the scale (in centimeters, for example), and then express the scale using centimeters (for example, 0.7 cm represents 100 m). Then measure each building and use the scale to find the actual height. Estimating and rounding will be necessary when measuring the scale and when measuring the buildings.
Ask previously selected and sequenced students to highlight the different approaches for comparing the heights of the Burj Khalifa and the Eiffel Tower. The main difference between the two strategies is the order of arithmetic. Taking the difference of the actual building heights means multiplying scaled heights first and then subtracting. Taking the difference of the scaled heights and then applying the scale factor means subtracting scaled heights first and then multiplying by the scale factor.
If time permits, discuss the scale drawing of the towers more broadly. Consider asking:
Besides height information, what other information about the towers does the drawing show? (Widths of the buildings and their overall shapes.)
What information does it
not
show? (The depth of each building, any projections or protrusions, shapes of different parts of the buildings, etc.)
How is this scale drawing the same as that of the basketball court? (They both show information on a single plane and are drawn using a scale.) How are they different? (The basketball court is a flat surface, like the drawing of the court. The drawing of the towers is a side view or front view; the actual objects represented are not actually flat objects.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no