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 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
- 100 can be thought of as a bundle of ten tens — called a “hundred.”
- The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
problem_activity: Narrative
The purpose of this activity is for students to learn stage 2 of the Greatest of Them All center. Students use digit cards to create the greatest possible number. As each student draws a card, they choose where to write it on the recording sheet. Once a digit is placed, it can’t be moved. Students compare their numbers using
\(<\)
,
\(>\)
, or
\(=\)
. The player with the greater number in each round gets a point.
Students should remove cards that show 10 from their deck.
MLR2 Collect and Display.
Synthesis: Direct attention to the words collected and displayed on the anchor chart from the previous lessons. Invite students to borrow place value language from the display as needed.
Advances: Conversing, Reading
Required Materials
Materials to Gather
Number cards 0–10
Materials to Copy
Greatest of Them All Stage 2 Recording Sheet
Launch
Groups of 2
Give each group a set of number cards and each student a recording sheet.
“Now you will be playing the Greatest of Them All center with your partner.”
“You will try to make the greatest three-digit number you can.”
Display number cards and recording sheet.
Demonstrate picking a card.
“If I pick a (2), I need to decide whether I want to put it in the hundreds, tens, or ones place to make the largest three-digit number.”
“Where do you think I should put it?” (I think it should go in the ones place because it is a low number. In the hundreds place, it would only be 200.)
30 seconds: quiet think time
Share responses.
“At the same time, my partner is picking cards and building a number, too.”
“Take turns picking a card and writing each digit in a space.”
“Read your comparison aloud to your partner.”
Activity
“Now play a few rounds with your partner.”
15 minutes: partner work time
Activity Synthesis
Select a group to share a comparison statement. For example:
\(654 > 349 \)
and
\(349 < 654\)
.
“I noticed that partners had different comparison statements for the same numbers. How can they both be true?”
“If I draw an 8, where should I choose to place it and why?” (I would put it in the hundreds place since it’s almost the highest number I could
draw
. I might not get 9. With 800, I have a good chance for my number to be larger than my partner’s.)


aligns: yes