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: Use probability to evaluate outcomes of decisions.

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

Summarize and describe distributions.

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

Investigate patterns of association in bivariate data.

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

Interpret linear models.

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.

Develop understanding of statistical variability.

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

Understand and evaluate random processes underlying statistical experiments.

Use random sampling to draw inferences about a population.
problem_activity: Optional activity
In this activity, students use the data from the previous activity and draw a scatter plot and a line that fits the data. The given data show a clear linear association, so it is appropriate to model the data with a line. Students can use a piece of dried linguine pasta or some other rigid, slim, long object (for example, wooden skewer) to eyeball the line that best fits the data. (The line of best fit has a variance of
\(R^2=0.94\)
.) Even though different answers will have slightly different slopes and intercepts, they will be close to each other.
Launch
Students in same groups of 3–4. Provide access to pieces of dried linguine pasta.
Representation: Internalize Comprehension.
Provide a range of examples and counterexamples for a best fit line. Consider displaying charts and examples from previous lessons to aide in memory recall.
Supports accessibility for: Conceptual processing
Student Facing
Make a scatter plot of the data.
[IMG]
Describe any patterns of association that you notice.
Draw a line that fits the data. Write an equation for this line.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If a student is stuck on making the scale on the graph, remind them to look at the range from the previous problem.
Activity Synthesis
At the start of the discussion, make sure students agree that there seems to be a negative association that looks like a line would fit the data nicely before moving on.
Invite several groups to share the equations they came up with and how they found them. They will likely have slightly different slopes and intercepts. Ask students to explain where those differences come from. (Not everyone chose the same points to guide the trendline they drew for the data.) The differences should be small and the different models give more or less the same information. In the next activity, students will be using the model (equation and graph) to make predictions.
Representing, Conversing: MLR8 Discussion Supports.
After students complete their scatter plots from the data, they should meet with a small group of 3–4 students to share and compare. While each student shares, circulate and encourage students to look for commonalities and to discuss their differences in displays. Tell students to generate an agreed on response for the equation and the visual display of the line even though they may have differences. Tell students that one member will share out, but they won’t know who, so they all need to be prepared to share and explain. During the whole-class discussion, amplify use of mathematical language such as “slope”, “intercept”, “equation”, “graph”, and “negative association”. Use this to help students develop their explanations of associations and differences that they see in how the data is interpreted.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 4