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: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
problem_activity: Narrative
The purpose of this activity is for students to analyze a common error when using the standard algorithm to subtract decimals. This activity is recommended if students need additional practice with the standard algorithm when the two numbers do not have the same number of decimal places. The standard addition algorithm requires students to subtract digits with the same place value. In the given example the two numbers are “right aligned” as when subtracting whole numbers and this leads to the error of subtracting hundredths from tenths, tenths from ones, and so on as if these digits had the same place value.
When students identify and correct Elena's error they construct viable arguments and critique the reasoning of others (MP3).
This activity uses
MLR3 Clarify, Critique, and Correct
. Advances: Reading, Writing, Representing.
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
6–8 minutes: partner work time
MLR3 Clarify, Critique, Correct
Display Elena’s solution and explanation.
Read the explanation aloud.
“What do you think Elena means? Is anything unclear?” (She says the answer will be less than 622.3, but she doesn’t say how much less. The answer will be a lot less than 615.21.)
“With your partner, work together to write a revised solution and explanation.”
(Optional) Display and review the following criteria:
Explanation of mistakes
Specific words and phrases
Decimal point
Place value
Labeled diagram
Correct solution
3–5 minutes: partner work time
Select 1–2 groups to share their revised explanation with the class.
“What is the same and different about the revised solutions and explanations?”
Student Facing
Find the value of
\(622.35 - 71.4\)
. Explain or show your reasoning.
Elena and Andre found the value of
\(622.35 - 71.4\)
. Who do you agree with? Explain or show your reasoning.
[IMG]
Student Response
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Advancing Student Thinking
If a student does not have a strategy to solve problem 1, ask them to explain how they would find the value of
\(622 - 51\)
.
Activity Synthesis
Display Andre’s solution and explanation.
“Why do you think Andre wrote 71.40 instead of 72.4?” (It helps him keep the place values and decimals in line so he doesn’t mix up the places when he is subtracting. 71.40 is equivalent to 71.4.)
Display Elena’s solution.
“How can you see without finding the difference that Elena’s answer is not correct?” (Taking 71 away from 622 is a lot less than 615.
\(620-70=550\)
so the answer should be around 550.)


aligns: no