few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
problem_activity: Narrative
The purpose of this activity is for students to learn stage 2 of the Target Numbers center. Students start with 25, choose a number card and decide whether to add that number of tens or ones. Students play six rounds and the student who gets as close to 95 as possible without going over is the winner.
MLR8 Discussion Supports.
Prior to playing the game, invite students to make sense of the directions and take turns sharing their understanding with their partner. Listen for and clarify any questions about the directions.
Advances: Reading, Representing
Required Materials
Materials to Gather
Connecting cubes in towers of 10 and singles
Number cards 0–10
Materials to Copy
Target Numbers Stage 2 Recording Sheet
Launch
Groups of 2
Give each group a set of number cards, two recording sheets, and access to connecting cubes in towers of 10 and singles.
“Take out all of the cards that show 0 or 10.”
“We are going to learn a new way to play Target Numbers. This time, your goal is to get as close to 95 as you can, without going over. You only get 6 turns. On each turn, you get to decide whether you want to add tens or ones to your number. Let’s play a round together. Each partner starts at 25.”
Demonstrate drawing a number card.
“I drew a [4]. Should I add 4 ones or 4 tens? Why?” (You can add 4 tens because that will give you 65. You still have 5 rounds to get to 95. You can add 4 ones to make 29. That gives you more time to add tens.)
30 seconds: quiet think time
30 seconds: partner discussion
Share responses.
“I’m going to add 4 tens, 2 tens and 4 tens is 6 tens. So my sum is 65.”
“Once my partner and I agree on the sum, the partner who drew the number card records an equation. Since 25 is already filled in for me, I will record the number as tens (or ones) and the sum.”
Demonstrate completing the equation.
“After I write my equation, the sum becomes my starting number for the next round, so I write it in as the first number in the next equation.”
If needed, play another round with the class.
“Take turns drawing the number card, finding the sum and writing an equation. Each player plays 6 rounds. Whoever gets closest to 95 without going over is the winner.”
Activity
10 minutes: partner work time
Activity Synthesis
“When you draw the number card, what is your method for deciding whether to add that many tens or ones?” (I think about how close I am to 95. That helps me think about whether to add ones or tens.)


aligns: no