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

Use random sampling to draw inferences about a population.

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

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

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

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

Calculate expected values and use them to solve problems.

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

Understand and evaluate random processes underlying statistical experiments.

Interpret linear models.

Draw informal comparative inferences about two populations.

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

Develop understanding of statistical variability.

Investigate patterns of association in bivariate data.
problem_activity: Activity
Following the concept developed in the warm-up, students will continue to use the idea that outcomes from previous experiments can help inform the likelihood of an outcome when the experiment is repeated. In this activity, students play a game while collecting data. Bags of colored blocks are set up with different probabilities to win and students use random chance to collect points when the color of the block they choose matches the color written on the bag. In the discussion, students will use the data they have collected in three rounds of the game to make a decision about which bag might be most likely to produce a winning block. As an opening activity, this is meant to motivate students into thinking about likelihood of events. As such, it is not important to resolve the questions in the discussion into perfect agreement.
Students are arranged in groups of 4 and given one of 3 different bags labeled with colors. Create enough bags so that each group will have a turn with each of the different color bags after three rounds.
Launch
Arrange students in groups of 4. Explain to students how to play the game:
There will be 3 rounds of the game. Their group will get a different bag of blocks for each round.
They are not allowed to look into the bag or take more than one block out of the bag at a time.
During a round, each person in the group gets 4 turns to take a block out of the bag.
If the block matches the color that is written on the outside of the bag, the person scores (1 point during round 1, 2 points during round 2, and 3 points during round 3). When the block is any other color, they do not earn any points.
They put the block back in the bag, shake the bag to mix up the blocks, and pass the bag to the next person in between each turn.
At the end of a round, they record everyone's points from that round and wait for a new bag of blocks.
Distribute bags of blocks. Each group gets one of these bags:
A bag labeled "Green" that contains 9 green blocks and 3 of another color.
A bag labeled "Blue" that contains 8 blue blocks and 8 of another color.
A bag labeled "Red" that contains 4 red blocks and 10 of another color.
Engagement: Provide Access by Recruiting Interest.
Begin with a small-group or whole-class demonstration of how to play the game.  Check for understanding by inviting students to rephrase directions in their own words.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Your teacher will give your group a bag of colored blocks.
Follow these instructions to play one round of the game:
Everyone in the group records the color written on the bag in the first column of the table.
​Without looking in the bag, one person takes out one of the blocks and shows it to the group.
If they get a block that is the same color as the bag, they earn:
1 point during round 1
2 points during round 2
3 points during round 3
Next, they put the block back into the bag, shake the bag to mix up the blocks, and pass the bag to the next person in the group.
Repeat these steps until everyone in your group has had 4 turns.
At the end of the round, record each person's score in the table.
|         | What color bag? | person 1's score | person 2's score | person 3's score | person 4's score |
|---------|-----------------|------------------|------------------|------------------|------------------|
| round 1 |                 |                  |                  |                  |                  |
| round 2 |                 |                  |                  |                  |                  |
| round 3 |                 |                  |                  |                  |                  |
Pause here so your teacher can give you a new bag of blocks for the next round.
Repeat the previous steps to play rounds 2 and 3 of the game.
After you finish playing all 3 rounds, calculate the total score for each person in your group.
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?
Tyler’s class played the block game using purple, orange, and yellow bags of blocks.
During round 1, Tyler’s group picked 4 purple blocks and 12 blocks of other colors.
During round 2, Tyler’s group picked 11 orange blocks and 5 blocks of other colors.
During round 3, Tyler forgot to record how many yellow blocks his group picked.
For a final round, Tyler’s group can pick one block from any of the three bags. Tyler’s group decides that picking from the orange bag would give them the best chance of winning, and that picking from the purple bag would give them the worst chance of winning. What results from the yellow bag could have lead Tyler’s group to this conclusion? Explain your reasoning.
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 groups may choose a bag for the bonus round based on the number of points they got in the earlier rounds. Since points are equal in the bonus round, help them see that the blocks picked out is a more important thing to consider than the points for the round.
Activity Synthesis
Tell students that there is a bonus, fourth round. In this round, a block that matches the color on the bag is worth 25 points. Each person will only have one chance to draw a bonus block. In this round, each group will get to choose which bag they would like to draw from. At least 3 people in the group must agree on the bag they will use.
Give students 5 minutes of small-group discussion to agree on which bag the group will use. Tell students that, after the bonus round, they will be asked to explain their reasoning for choosing the bag. It is ok for more than one group to select the same color bag. Identify groups that use the results from the previous rounds to determine which bag will be more likely to draw a winning block.
Allow each group to use the bag they have selected to play the bonus round and total their points.
After all students have played the bonus round, select groups to share their reasoning for choosing which bag to use. Select previously identified students to share their reasoning as well.
Ask students, "Was one bag more likely to give you a winning cube than the others? Explain your reasoning."
Open one of each color bag to reveal the contents. Ask students, "Based on what you see now, does this change your answer? Explain your reasoning."
The green bag should provide the best chance to win points, but it is not essential for the class to come to this understanding at this point.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to support whole-class discussion as groups share which bag they would choose during the bonus round. Display sentence frames for students to use as they respond to each group: "I agree because ….” or "I disagree because ….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making


correct_option_index: 12