few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


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


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Understand ratio concepts and use ratio reasoning to solve problems.
problem_activity: Activity
In this activity, students explore two unit rates associated with the ratio, think about their meanings, and use both to solve problems. The goals are to:
Help students see that for every context that can be represented with a ratio
\(a:b\)
and an associated unit rate
\(\frac{b}{a}\)
, there is another unit rate
\(\frac{a}{b}\)
that also has meaning and purpose within the context.
Encourage students to choose a unit rate flexibly depending on the question at hand. Students begin by reasoning whether the two rates per 1 (cups of oats per 1 cup of water, or cups of water per 1 cup of oats) accurately convey a given oatmeal recipe. As students work and discuss, notice those who use different representations (a table or a double number line diagram) or different arguments to make their case (MP3). Once students conclude that both Priya and Han's rates are valid, they use the rates to determine unknown amounts of oats or water.
Launch
Some students may not be familiar with oatmeal; others may only have experience making instant oatmeal, which comes in pre-measured packets. Explain that oatmeal is made by mixing a specific ratio of oats to boiling water.
Arrange students in groups of 2. Give students 2–3 minutes of quiet think time for the first question. Ask them to pause and share their response with their partner afterwards. Encourage partners to reach a consensus and to be prepared to justify their thinking. See MLR 3 (Clarify, Critique, Correct).
After partners have conferred, select several students to explain their reasoning and display their work for all to see. When the class is convinced that both Priya and Han are correct, ask students to complete the rest of the activity.
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with their partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “I noticed _____ so I . . .”, “Both _____ and _____ are alike because . . .”, and “How do you know . . . ?”
Supports accessibility for: Language; Organization
Student Facing
Priya, Han, Lin, and Diego are all on a camping trip with their families. The first morning, Priya and Han make oatmeal for the group. The instructions for a large batch say, “Bring 15 cups of water to a boil, and then add 6 cups of oats.”
Priya says, “The ratio of the cups of oats to the cups of water is
\(6:15\)
. That’s 0.4 cups of oats per cup of water.”
Han says, “The ratio of the cups of water to the cups of oats is
\(15:6\)
. That’s 2.5 cups of water per cup of oats.”
Who is correct? Explain your reasoning. If you get stuck, consider using the table.
| water (cups) | oats (cups) |
|--------------|-------------|
|      15      |      6      |
|      1       |             |
|              |      1      |
The next weekend after the camping trip, Lin and Diego each decide to cook a large batch of oatmeal to have breakfasts ready for the whole week.
Lin decides to cook 5 cups of oats. How many cups of water should she boil?
Diego boils 10 cups of water. How many cups of oats should he add into the water?
Did you use Priya’s rate (0.4 cups of oats per cup of water) or Han’s rate (2.5 cups of water per cup of oats) to help you answer each of the previous two questions? Why?
Student Response
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Anticipated Misconceptions
Some students may think that Priya and Han cannot both be right because they came up with different numbers. Ask them to explain what each number means, so that they have a chance to notice that the numbers mean different things. Point out that the positioning of the number 1 appears in different columns within the table.
Activity Synthesis
Focus the discussion on students’ responses to the last question and how they knew which rate to use to solve for unknown amounts of oats and water. If not uncovered in students’ explanations, highlight that when the amount of oats is known but the amount of water is not, it helps to use the “per 1 cup of oats” rate; a simple multiplication will tell us the missing quantity. Conversely, if the amount of water is known, it helps to use the “per 1 cup of water” rate. Since tables of equivalent ratios are familiar, use the completed table to support reasoning about how to use particular numbers to solve particular problems.
Consider connecting this idea to students’ previous work. For example, when finding out how much time it would take to wash all the windows on the Burj Khalifa, it was simpler to use the “minutes per window” rate than the other way around, since the number of windows is known.
Leave the table for this activity displayed and to serve as a reference in the next activity.
Writing, Listening, Conversing: MLR3 Clarify, Critique, and Correct.
Before discussing students’ approaches to the final question. Present the following explanation: “I did not use either Priya’s rate (0.4 cups of oats per cup of water) or Han’s rate (2.5 cups of water per cup of oats) because they are not equal.” Ask students to critique the reasoning and identify the error(s), as they work in pairs to propose an improved response that details how either rate can be used to answer the question. If students can’t make the connections between these two rates, consider asking them what each number means. This will help students reflect on efficiently using a unit rate associated with the ratio, cups of water to the cups of oats.
Design Principle(s): Optimize output (for explanation); Cultivate conversation; Maximize meta-awareness


correct_option_index: 1