few_shots: Example 1:
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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
Answer: B

Example 2:
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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

Example 3:
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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
problem_activity: Warm-up
The goal of this opening activity is to activate, through a familiar context, what students know about functions from middle school.
Students first encounter a relationship in which two quantities—the number of bagels bought and price—do not form a function. They see that for some numbers of bagels bought, there are multiple possible prices. The relationship between the number of bagels bought and the
best
price, however, do form a function, because there is only one possible best price for each number of bagels.
Students contrast the two relationships by reasoning about possible prices, completing a table of values, and observing the graphs of the relationships.
Launch
Arrange students in groups of 4. Give students a few minutes of quiet time to think about the first question, and then a couple of minutes to share their thinking with their group. Pause for a class discussion.
Invite students to share an explanation as to how each person in the situation could be right. If possible, record and display students’ reasoning for all to see. After the reasoning behind each price is shared, direct students to the table in the second question. Ask students to write down “best price” in the header of the second column and then complete the table.
Student Facing
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[IMG]
A customer at a bagel shop is buying 13 bagels. The shopkeeper says, “That would be $16.25.”
Jada, Priya, and Han, who are in the shop, all think it is a mistake.
Jada says to her friends, “Shouldn’t the total be \$13.25?”
Priya says, “I think it should be \$13.00.”
Han says, “No, I think it should be \$11.25.”
Explain how the shopkeeper, Jada, Priya, and Han could all be right.
Your teacher will give you instructions for completing the table.
| number                         of bagels |   |
|------------------------------------------|---|
|                    1                     |   |
|                    2                     |   |
|                    3                     |   |
|                    4                     |   |
|                    5                     |   |
|                    6                     |   |
|                    7                     |   |
|                    8                     |   |
|                    9                     |   |
|                    10                    |   |
|                    11                    |   |
|                    12                    |   |
|                    13                    |   |
Student Response
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Activity Synthesis
Consider displaying a table to summarize the different possibilities for calculating the price of 13 bagels (or the prices for 6 or more bagels), as well as the table showing the best price for each number of bagels bought.
| number                         of bagels | shopkeeper’s                         price | Jada’s                         price | Priya’s                         price | Han’s                         price |
|------------------------------------------|--------------------------------------------|--------------------------------------|---------------------------------------|-------------------------------------|
|                    1                     |                    1.25                    |                                      |                                       |                                     |
|                    2                     |                    2.50                    |                                      |                                       |                                     |
|                    3                     |                    3.75                    |                                      |                                       |                                     |
|                    4                     |                    5.00                    |                                      |                                       |                                     |
|                    5                     |                    6.25                    |                                      |                                       |                                     |
|                    6                     |                    7.50                    |                 6.00                 |                                       |                                     |
|                    7                     |                    8.75                    |                 7.25                 |                                       |                                     |
|                    8                     |                   10.00                    |                 8.50                 |                                       |                                     |
|                    9                     |                   11.25                    |                 9.75                 |                 8.00                  |                                     |
|                    10                    |                   12.50                    |                11.00                 |                 9.25                  |                                     |
|                    11                    |                   13.75                    |                12.25                 |                 10.50                 |                                     |
|                    12                    |                   15.00                    |                12.00                 |                 11.75                 |                10.00                |
|                    13                    |                   16.25                    |                13.25                 |                 13.00                 |                11.25                |
| number                         of bagels | best price |
|------------------------------------------|------------|
|                    1                     |    1.25    |
|                    2                     |    2.50    |
|                    3                     |    3.75    |
|                    4                     |    5.00    |
|                    5                     |    6.25    |
|                    6                     |    6.00    |
|                    7                     |    7.25    |
|                    8                     |    8.50    |
|                    9                     |    8.00    |
|                    10                    |    9.25    |
|                    11                    |   10.50    |
|                    12                    |   10.00    |
|                    13                    |   11.25    |
Ask students,
“’Number of bagels’ and ’price’ do not form a
function
, but ’number of bagels’ and ’best price’ do form a function. Why is this? What do you recall about functions?”
“If we graph the relationship between ’number of bagels’ and ’price,’ what do you think the graph would look like?”
“If we graph the relationship between ’number of bagels’ and ’best price,’ what would the graph look like?”
After students make some predictions, display the two graphs for all to see. In the first graph, solid blue dots represent the shopkeeper’s price, open green circles Jada’s price, red squares Priya’s price, and yellow triangles Han’s price. In the second graph, each X represents the best price for each number of bagel.
[IMG]
Because there are multiple possible prices when the number of bagels is 6 or greater, price is not a function of the number of bagels bought.​​​​​​
[IMG]
Because there is only one best price for a particular number of bagel, best price is a function of the number of bagels bought.
Emphasize that a function assigns one output to each input. Clarify that the word “function” in mathematics has a very specific meaning that does not necessarily agree with how “function” is used in everyday life (for instance, as in the sentence “The function of a bridge is to connect two sides of a river.”).


correct_option_index: 1