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.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
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: Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
problem_activity: Activity
In this activity, students practice mentally dissecting a prism with a non-rectangular base into simpler prisms. The dissection corresponds to a dissection of the base into simpler figures. This expands on students’ ability to calculate the area of a base of a figure that has a rectangular base because here the base is not a rectangle. This prepares students to calculate the volume of this figure and other figures in future lessons.
As students work in their groups, monitor for the different ways students are decomposing or constructing the base of the figure into more familiar shapes.
Launch
Arrange students in groups of 3. Display the image for all to see throughout the activity. Tell students this is what the heart-shaped box mentioned in the task statement looks like.
[IMG]
Give students 2–3 minutes of quiet work time followed by a whole-class discussion.
Engagement: Internalize Self Regulation.
Check for understanding by inviting students to rephrase directions in their own words. Provide a project checklist that chunks the various steps of the activity into a set of manageable tasks.
Supports accessibility for: Organization; Attention
Conversing: MLR5 Co-Craft Questions.
Display the image of the heart-shaped box and initial task statement without revealing the questions that follow. Ask students to write mathematical questions about the situation. Invite students to share their questions with their group and choose 1 question from their group to share with the class. Listen for questions that connect to finding the area of the base of the heart-shaped prism (e.g., “What’s the most 1 inch by 1 inch by 1 inch chocolates that can fit along the bottom of the box?”). This helps students produce the language of mathematical questions and begin reasoning about different approaches for finding the area of a non-rectangular shape.
Design Principle(s): Cultivate conversation
Student Facing
A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Here are the measurements of the base.
[IMG]
To calculate the volume of the box, three different students have each drawn line segments showing how they plan on finding the area of the heart-shaped base.
[IMG]
For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area.
Although all three methods could work, one of them requires measurements that are not provided. Which one is it?
Between you and your partner, decide which of you will use which of the remaining two methods.
Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base.
Trade with a partner and check each other’s work. If you disagree, work to reach an agreement.
Return their work. Calculate the volume of the box of chocolates.
Student Response
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Student Facing
Are you ready for more?
The box has 30 pieces of chocolate in it, each with a volume of 1 in
3
. If all the chocolates melt into a solid layer across the bottom of the box, what will be the height of the layer?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share whose method they decided to use and why. Ask students:
“Whose method could not be used? Why not?” (Lin’s, because we don’t know the base and  height of the trapezoids.)
“How did you find the areas of the base?”
“What was different about the base of this figure in comparison to other bases we have worked with?” (This base needs to be decomposed to calculate its area.)
“What was the first thing you did to find the volume?” (Use the area of the base and multiply it by the height of the figure.)
“Why would a chocolatier want to know the volume of a heart shaped box like this?” (He may want to know how many candies can fit inside of a box.)
Explain to students that they might encounter figures that have non-rectangular bases in future activities or lessons. It will be important for them to think about different strategies to calculate the area of the base.


aligns: no