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: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
problem_activity: Narrative
The purpose of this activity is for students to practice generating two different patterns from two different rules and observe and quantify relationships between corresponding terms. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of Priya’s terms is 3 times greater than each of Jada’s corresponding terms and each of Jada’s terms is
\(\frac{1}{3}\)
of Priya’s corresponding terms.
When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).
Launch
Groups of 2
“You and your partner will each complete some problems about patterns independently. After you’re done, discuss your work with your partner.”
Activity
5 minutes: independent work time
3 minutes: partner discussion
“Look back at your work and make any revisions based on what you learned from discussing with your partner.”
1–2 minutes: independent work time
Monitor for students who:
revise their thinking based on partner discussion
use multiplication expressions to represent the relationships between the patterns
Student Facing
Partner A
Use each rule to create a pattern.
Jada’s rule: Start at 0. Keep adding 2.
[IMG]
Priya’s rule: Start at 0. Keep adding 6.
[IMG]
If the patterns continue:
What number will be in Priya’s pattern when Jada’s pattern has 34? Explain or show your reasoning.
What number will be in Jada’s pattern when Priya’s pattern has 120? Explain or show your reasoning.
What relationships do you notice between the numbers in Priya's pattern and the numbers in Jada's pattern?
Partner B
Use each rule to create a pattern.
Jada’s rule: Start at 0. Keep adding 3.
[IMG]
Priya’s rule: Start at 0. Keep adding 9.
[IMG]
If the patterns continue,
What number will be in Priya’s pattern when Jada’s pattern has 54? Explain or show your reasoning.
What number will be in Jada’s pattern when Priya’s pattern has 198? Explain or show your reasoning.
What relationships do you notice between the numbers in Priya’s pattern and the numbers in Jada’s pattern?
Student Response
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Activity Synthesis
Ask previously identified students to share their thinking.
Display completed patterns from both sets of rules.
“How are the patterns from the pairs of rules the same? How are they different?” (Some of the numbers are in all four patterns. Some numbers aren’t in any pattern like 5 and 11. In both sets of rules, Priya’s numbers are 3 times greater than Jada’s numbers and Jada’s numbers are
\(\frac{1}{3}\)
Priya’s numbers.)
“How would you describe the relationship between Priya's pattern and Jada's pattern?” (Priya's numbers are all 3 times as much as Jada's corresponding numbers.)
“What can I multiply the numbers in Priya’s pattern by to get the corresponding numbers in Jada’s pattern?” (
\(\frac{1}{3}\)
, each number in Jada's pattern is
\(\frac{1}{3}\)
the corresponding number in Priya's pattern.)


aligns: yes