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 to multiply a fraction or whole number by a fraction.

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

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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
problem_activity: Narrative
This activity develops students’ understanding of the vertical method of recording partial quotients and their ability to use it to perform division. One of the quotients here involves a remainder, prompting students to interpret it. Students are reminded that the term “remainder” is used to describe “leftovers” when dividing.
MLR2 Collect and Display.
Synthesis: Direct attention to words collected and displayed from the previous lessons. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Launch
Groups of 2
Activity
3 minutes: independent work time on the first 2 problems.
Pause after problem 2 to discuss students’ responses.
Display the different ways that students decompose 389 to divide it by 7.
“Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?” (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.)
“When we divide and end up with leftovers we call them remainders, because they represent what is remaining after we divide into equal groups.”
Display:
\(389 = 7 \times 55 + 4\)
“How does this equation show that
\(389 \div 7\)
has a remainder?” (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.)
3 minutes: independent work time on the last 2 problems.
As students work on the last two problems monitor for students who:
start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600).
use the fewest steps to find the quotient.
Student Facing
Jada used partial quotients to find out how many groups of 7 are in 389.
Analyze Jada’s steps in the algorithm.
[IMG]
Look at the three numbers above 389. What do they represent?
Look at the three subtractions below 389. What do they represent?
What is another way you can decompose 389 to divide by 7?
Is 389 a multiple of 7? Explain your reasoning.
Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702.
Is 702 a multiple of 3? Explain your reasoning.
Student Response
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for free access to Student Response.
Activity Synthesis
Display responses that show variation in how students decomposed 702 in problem 3.
Highlight that there are countless ways of using partial quotients to divide a number, but it may be more efficient to divide larger groups than smaller groups (for example, removing 350 once, as opposed to removing 70 five times. Removing smaller groups is just as valid, however.)
To help students see the relationship between corresponding partial products and partial quotients, consider illustrating them visually. Some examples:
[IMG]
[IMG]
[IMG]


aligns: yes