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?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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:
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?


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.

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, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Activity
The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. Students flip a coin to determine the number of heads that show up for several groups of 20 flips. Students should recognize that some variability is expected from the expected 10 heads that are predicted for 20 flips, but that some results are very unlikely even if they are possible. Students should begin to recognize when a statistical model is consistent with results and when the data is so unusual that a new model might be more appropriate.
In later lessons, students will need to recognize when a normal curve is an appropriate model for data so that the area under a normal curve can be used to get information about the data.
Digital
Launch
Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins in the applet and collect data from the class to be included in a dot plot.
Action and Expression: Internalize Executive Functions.
Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization
Student Facing
What is the probability that you will flip heads when using the coin in the applet?
Estimate the number of heads you will get when you flip the coin 20 times.
Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.
|  trial number   | 1 | 2 | 3 | 4 | 5 |
|-----------------|---|---|---|---|---|
| number of heads |   |   |   |   |   |
Create a dot plot that shows the number of heads in 20 flips using data from the class.
What is the least number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
Priya flips her coin 20 times and it lands showing heads twice.
Is it possible for this to happen with a fair coin?
Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins and collect data from the class to be included in a dot plot.
Action and Expression: Internalize Executive Functions.
Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization
Student Facing
What is the probability that you will flip heads when using the coin you have?
Estimate the number of heads you will get when you flip the coin 20 times.
Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.
|  trial number   | 1 | 2 | 3 | 4 | 5 |
|-----------------|---|---|---|---|---|
| number of heads |   |   |   |   |   |
Create a dot plot that shows the number of heads in 20 flips using data from the class.
What is the fewest number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
Priya flips her coin 20 times and it lands showing heads twice.
Is it possible for this to happen with a fair coin?
Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
This time instead of flipping the coin 20 times and counting how many heads result, consider how many coin flips it takes to get a heads.
Before flipping estimate the average number of flips it will take to get your fist heads.
Flip! Record how many flips it takes to get your first heads. Then start again and record how many flips it takes to get your next heads. Keep flipping and recording how many flips it takes to get successive heads until you get 30 heads. Create a dot plot that shows how many flips it took each time to get a heads and compute the mean number of times it took.
Based on the dot plot, describe a range of values that represent a reasonable number of coin flips it takes to get a heads.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may be uncomfortable with the less precise answers for the work in this unit and this activity in particular. Remind students that the actual world is messy and the work of statistics is to help make sense of that messiness. Math is not proving Priya has a fair coin or not, but can give us evidence to suspect that it is not and a decision can be made based on that evidence and anything else that is known about the situation.
Activity Synthesis
The purpose of this discussion is to talk about making and justifying claims based on data. Here are some questions for discussion:
“You performed 20 coin flips 5 times. Based on your data alone, what range of values would you predict for the number of times the coin would land on heads if it was flipped 20 times?” (I got 8, 9, 10, 12, 13 heads so I would predict that a coin would land on heads between 7 and 13 times. I included seven in my estimate because 13 is 3 above ten and 7 is 3 less than 10. The coin should land on heads an average of 10 times.)
“How does your prediction compare to the range of values you described from the class's data? Why do you think it was the same or different? Explain your reasoning.” (My range was smaller than the range from the whole class data. Since there was much more data for the whole class, I expected there to be a broader range of values since there was likely to be more variability.)
“If 1,000 people flipped a coin 20 times, how might that change your prediction from the rage for the class's data?” (If we had 1,000 data values, I would feel really confident about the range of values that I would expect from 20 coin flips. I know that it is possible to get between 0 and 20 heads, but I wonder how unlikely it really is to get 0 or 20. I think the 1,000 data values would help me to know a typical range of values.)


aligns: no