few_shots: 
Example 1:
Concept/skill: 
Recognize and represent proportional relationships between quantities.

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?


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:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


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:
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.

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?


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
problem_activity: Optional activity
In this activity, students explore percentages that describe parts of a whole. They find both
\(B\)
and
\(C\)
, where
\(A\%\)
of
\(B\)
is
\(C\)
, in the context of available and consumed water on a hike.
Students who use a double number line may notice that the value of
\(B\)
is the same in both questions, so the same double number line can be used to solve both parts of the problem. To solve the second question, however, the diagram needs to be partitioned with more tick marks.
As in the previous task, students may solve using other strategies, including by simply multiplying or dividing, i.e.,
\((1.5) \boldcdot 2 = 3.0\)
and
\(\frac{80}{100} \boldcdot (3.0) = 2.4\)
. Encourage them to also explain their reasoning with a double number line, table, or tape diagram. Monitor for at least one student using each of these representations.
Launch
Give students quiet think time to complete the activity and then time to share their explanation with a partner. Follow with a whole-class discussion.
Student Facing
During the first part of a hike, Andre drank 1.5 liters of the water he brought.
If this is 50% of the water he brought, how much water did he bring?
If he drank 80% of his water on his entire hike, how much did he drink?
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?
Decide if each scenario is possible.
Andre plans to bring his dog on his next hike, along with 150% as much water as he brought on this hike.
Andre plans to drink 150% of the water he brought on his hike.
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
After students create a double number line diagram with tick marks at 50% and 100%, some may struggle to know how to fit 80% in between. Encourage them to draw and label tick marks at 10% increments or work with a table instead. Some students may think that the second question is asking for the amount of water Andre drank on the second part of the hike. Clarify that it is asking for his total water consumption on the entire hike.
Activity Synthesis
Select 1–2 students who used a double number line, a table of equivalent ratios, and a tape diagram to share their strategies. As students explain, illustrate and display those representations for all to see.
Ask students how they knew what 100% means in the context. At this point it is not necessary for students to formally conceptualize the two ways percentages are used (to describe parts of whole, and to describe comparative relationships). Drawing their attention to concrete and contextualized examples of both, however, serves to build this understanding intuitively.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their answers for the second question, present an incorrect answer and explanation. For example, “If Andre drank 80% of his water on his entire hike, then he drank 1.2 liters of water because 0.8 times 1.5 liters is 1.2 liters.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss in partners, listen for students who identify and clarify the assumption in the statement. For example, the author assumed that Andre brought 1.5 liters for his entire hike; however, the problem states that Andre drank 1.5 liters of the water he brought. This will remind students to carefully identify the quantity they are finding a percentage of.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


aligns: no