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

Use random sampling to draw inferences about a population.

Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.

Understand and evaluate random processes underlying statistical experiments.

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

Investigate patterns of association in bivariate data.

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

Draw informal comparative inferences about two populations.

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

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

Summarize and describe distributions.

Develop understanding of statistical variability.

Interpret linear models.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.
problem_activity: Activity
The mathematical purpose of this activity is for students to create data displays and calculate statistics using technology. Students plot the survey data they collected from a statistical question in a previous lesson.
Launch
Arrange students in groups of two. Tell them that they will be using technology to create data displays and calculate statistics for data they collected from a survey question in a previous lesson.
Conversing, Representing, Writing: MLR 2 Collect and Display.
As pairs are working, listen for and collect vocabulary, phrases, and methods students use for creating dot plots, boxplots, and histograms using technology, such as data set, skewed distribution, median, the box plot for the IQR. Consider grouping language for each data display in different areas, and be sure to include the spreadsheet formulas. Continue to update collected student language throughout the entire activity and the entire unit. Remind students to borrow language from the display as needed. This will help students to expand their notation of recursive definitions.
Design Principle(s): Support sense-making; Maximize meta-awareness
Engagement: Develop Effort and Persistence.
Provide prompts, reminders, guides, rubrics, checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, create an exemplar dot plot highlighting specific features, such as the scale and labels, as well as calculations for the mean and median.
Supports accessibility for: Attention; Social-emotional skill
Student Facing
Use the data you collected from the numerical, statistical question from a previous lesson. Use technology to create a dot plot, boxplot, and histogram for your data. Then find the mean, median, and interquartile range for the data.
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?
A stem and leaf plot is a table where each data point is indicated by writing the first digit(s) on the left (the stem) and the last digit(s) on the right (the leaves). Each stem is written only once and shared by all data points with the same first digit(s). For example, the values 31, 32, and 45 might be represented like:
\(\displaystyle \begin{array}{r|l l} 3 & 1 & 2\\ 4 & 5 \end{array}\)
Key: 3 | 1 means 31
A class took an exam and earned the scores:
86, 73, 85, 86, 72, 94, 88, 98, 87, 86, 85, 93, 75, 64, 82, 95, 99, 76, 84, 68
Use technology to create a stem and leaf plot for this data set.
How can we see the shape of the distribution from this plot?
What information can we see from a stem and leaf plot that we cannot see from a histogram?
What do we have more control of in a histogram than in a stem and leaf plot?
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
Students may lose one data display when they begin to create the next one. Explain to students that it is important to copy their solutions into a more permanent place so they can refer to it later.
Activity Synthesis
The goal of this activity was for students to create graphs and find statistics using technology. Here are some questions for discussion.
“What were some challenges that you faced using technology and how did you overcome them?” (I was not sure what buttons to press to get to the spreadsheet. I checked with my partner and figured it out.)
“What width did you use for your histogram? Why?” (I used 5 because my data set has values ranging from 1 to 42. I could have used 10 but then I would have only had 5 bars.)
“What is the appropriate measure of center for your data set?” (The median was appropriate because my data set has a skewed distribution.)
“Which display allows you to calculate the IQR the most easily?” (The box plot because it displays Q1 and Q3.)
“Can you find the median using your histogram?” (No, the data is grouped into intervals, so a histogram cannot be used to find the middle value for the median.)


correct_option_index: 0