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 a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
problem_activity: Activity
The goal of this activity is for students to be introduced to the idea that variability is important when analyzing and comparing data sets. Students will compare data sets, and make a decision as to which data set represents a better scenario. This activity prepares students to calculate variability in order to compare data sets in a later lesson. Monitor for students who calculate any
statistics
to inform their decisions. Make note of the students who only calculate a measure of center vs. students who calculate both a measure of center and a measure of variability.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Students will work individually. Tell them to use statistics to inform their decision.
Student Facing
In order from least to greatest, here are the prices per gallon of gas at two different gas stations over the past 7 days.
Station A:
2.38, 2.68, 2.82, 2.86, 2.99, 3.26, 3.59
Station B:
2.84, 2.85, 2.88, 2.95, 2.98, 3.03, 3.05
Suppose that these gas stations were the closest to your house, but not near each other. Which gas station would you go to for gas? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is to understand that the variability is worth paying attention to. If any students chose to compute the mean, median, IQR, MAD, or range for the two stations, ask them to share their results and how those values help them decide. Interestingly, the mean is the same for the gas stations. But any measure of variability shows that the prices at station B are more stable and predictable than the prices at station A. If no students suggest computing measures of center and variability and comparing them, be prepared to demonstrate.
Here are the summary statistics for each set of data:
For Station A:
Mean: 2.94
Five-number summary: 2.38, 2.68, 2.86, 3.26, 3.59
Range: 1.21
IQR: 0.58
MAD: 0.29
For Station B:
Mean: 2.94
Five-number summary: 2.84, 2.85, 2.95, 3.03, 3.05
Range: 0.21
IQR: 0.18
MAD: 0.07
If time permits, discuss which measures are helpful. Here are sample questions:
“Was the mean helpful in deciding the gas station from which you would purchase gas?”(No, because both means were the same.)
“How did the range (or IQR, MAD, five-number summary) help you decide which gas station you would go to?” (Whichever gas station had the highest range [or IQR, MAD] is the gas station I chose.)
“In the real world, how would you use the gas prices at each gas station to make a decision?” (I would think about which one generally had lower prices and choose that one, or I would see which one had more predictably cheaper prices and choose that one.)


correct_option_index: 0