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.

Interpret linear models.

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

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

Develop understanding of statistical variability.

Use random sampling to draw inferences about a population.

Investigate patterns of association in bivariate data.

Use probability to evaluate outcomes of decisions.

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

Draw informal comparative inferences about two populations.

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

Calculate expected values and use them to solve problems.

Understand and evaluate random processes underlying statistical experiments.

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

Summarize and describe distributions.
problem_activity: Activity
In this activity, students use their understanding of spreadsheets to compute useful statistics from data sets. In the associated Algebra 1 lesson, students will learn alternative methods for computing these values.
Launch
If students need a reminder about how to use a spreadsheet as a calculator, demonstrate how to do basic calculations such as =A1+A2, =SUM(A1:A10), =(A1+A2)/A3, and =COUNT(A1:A50) or =LENGTH(A1:A50) in GeoGebra.
Student Facing
Here is a list of the number of pages for fiction books on a shelf.
15
243
426
175
347
186
236
394
170
412
242
479
185
254
186
278
277
278
486
207
378
251
458
360
440
181
349
482
382
Describe a method to find the mean of these values. (Do not calculate the value yet.)
Input the data into a spreadsheet in the same column. In another column, write a spreadsheet formula that will compute the value of the mean for the values in the column. What is the mean number of pages in these books?
The first book on the list with 15 pages was a recording error and should have been 158 pages. You need to compute the new mean using 158 instead of 15. Would you rather compute the new mean by hand or use a computer? Explain your reasoning.
What is the new mean number of pages in these books with the updated value of 158 pages for the first book in the list?
Another book is found and added to the shelf. This new book has 519 pages. How would you update your spreadsheet formula to include this new value?
Student Response
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Activity Synthesis
Select students to share their responses and methods for finding the mean from a list of values in a spreadsheet. The purpose of the discussion is to point out the advantage of using a spreadsheet to calculate statistics, especially for very large sets of data. For further discussion, consider asking these questions:
"Which part of the process for finding the mean for these values took the most time?" (Typing the values into the spreadsheet and making sure they are correct.)
"Often large data sets are collected electronically and automatically added to a spreadsheet. When there are thousands of data values in a spreadsheet, what other advantages do you think might be helpful when using technology to compute the mean?" (It may be difficult to understand the data when there are so many values, so using the spreadsheet to calculate the mean can help in understanding what is typical. It would definitely be faster to use the spreadsheet to calculate the mean than doing it by hand.)
"What other statistics would be useful and easier to calculate using a spreadsheet?" (If the data set is large or the numbers are difficult to understand at a glance, the median, quartiles, interquartile range, mean absolute deviation, maximum, and minimum should all be easy for a computer to find from a list.)


correct_option_index: 14