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?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

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?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Calculate expected values and use them to solve problems.

Understand and evaluate random processes underlying statistical experiments.

Draw informal comparative inferences about two populations.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Develop understanding of statistical variability.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Investigate chance processes and develop, use, and evaluate probability models.

Use random sampling to draw inferences about a population.

Summarize, represent, and interpret data on a single count or measurement variable.

Understand independence and conditional probability and use them to interpret data.

Investigate patterns of association in bivariate data.

Summarize and describe distributions.

Interpret linear models.

Use probability to evaluate outcomes of decisions.
problem_activity: Activity
The purpose of this activity is to let students investigate what happens to the standard deviation using different data sets. The goal is for students to make conjectures about what standard deviation measures and how relative size of the standard deviation can be estimated from the shape of the distribution. In particular, students should recognize that adding or subtracting the same value from each value in the data set will change the mean by the same amount, but the standard deviation remains unchanged. Multiplying or dividing each value in the data set by the same value scales both the mean and standard deviation by the same value.
Monitor for students who
Check values without much thought until the mean or standard deviation is correct.
Use a starting set of values, then modify a few of them up or down to match the statistics given.
Use symmetry to get the mean correct.
Adjust the data by multiplying all of the values by a number to change the spread of the distribution to get close to the standard deviation.
Identify students who create data sets with different types of distributions for “10 different numbers that have a mean of 1” and “10 different numbers with a standard deviation as close to 2.5 as you can get in one minute.”
Launch
Demonstrate how to find standard deviation and mean using the technology available. In Algebra 1, students are dealing with the entire population, not sampling, so the population standard deviation is used.
Open the Spreadsheet & Statistics tool from Math Tools, or navigate to
https://www.geogebra.org/classic/spreadsheet
. Enter the values in column A. Select all of column A, and then choose the button that looks like a histogram and “One Variable Analysis.” Click the button that says
\(\Sigma x\)
. The population standard deviation is labeled
\(\sigma\)
. Additionally, the spreadsheet command =SD(A1:A10) will compute the population standard deviation for data in cells A1 through A10.
Arrange students in groups of 2. Give students time to work through the first two questions, followed by a whole-class discussion.
Conversing: MLR 5 Co-Craft Questions.
Display only the first line of this task (“Use technology to find the mean and the standard deviation for the data in the dot plots”) and the dot plots, and ask pairs of students to write possible mathematical questions about the distributions. Then, invite pairs to share their questions with the class. This helps students produce the language of mathematical questions and talk about the relationships between the two quantities in this task (mean and standard deviation).
Design Principle(s): Cultivate conversation
Student Facing
Use technology to find the mean and the standard deviation for the data in the dot plots.
What do you notice about the mean and standard deviation you and your partner found for the three dot plots?
Invent some data that fits the conditions. Be prepared to share your data set and reasoning for choice of values.
Partner 1
​Partner 2
Dot plots:
[IMG]
[IMG]
[IMG]
Dot plots:
[IMG]
[IMG]
[IMG]
​​​​​​
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 6.
10 numbers with a standard deviation three times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 12.
10 numbers with a standard deviation four times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of this discussion is to understand that the standard deviation is a measure of variability related to the mean of the data set. The discussion also provides an opportunity for students to discuss what they notice and wonder about the mean and standard deviation.
Select previously identified students in the order listed in the narrative to share their distributions and methods for finding values that worked.
For at least one student’s data set that had a standard deviation close to 2.5, ask them how they might adapt the data so that it has a mean of 1. (Shift the data up or down so that it has a mean of 1.)
Here are some questions for discussion.
“What do you think that standard deviation measures? Why do you think that?” (I think that it measures variability because it behaved like MAD. When all the values were the same it was zero.)
“Why is the standard deviation the same for {1,2,3,4,5} and {-2,-1,0,1,2}?” (For each data set: 1) the values to the left of the mean are a distance of 2 and 1 from the mean and the values to the right of the mean are a distance 2 and 1 from the mean, and 2) the middle value is 0 away from the mean. This results in the standard deviation being the same.)
“Why is the standard deviation different for {-4, -2, 0, 2, 4} and {-4, -3, -2, -1, 0}?” (The values for the first data set are twice the distance from the mean as the values in the second data set. That makes the standard deviation for the first set greater than the standard deviation of the second data set.)
“When is the standard deviation equal to zero?” (It is zero when all the values are the same as each other or when there is no variability.)
“Was your mean the same as your partner’s mean in the fifth match?” (Answers vary. Sample response: My mean was much different from my partners. Mine was 5 and theirs was 10.)
“How did using technology help or hinder your mathematical thinking about standard deviation and mean?” (It really helped me because I did not get bogged down with the calculations and I could look for patterns in the data, the data displays, and the statistics. In particular, having to come up with my own data and then see what happened to the statistics was really helpful.)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer with standard deviation at the center, connecting to it all the related concepts mentioned during the discussion.
Supports accessibility for: Language; Organization


correct_option_index: 9