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: Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.

Use functions to model relationships between quantities.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Interpret expressions for functions in terms of the situation they model.

Build a function that models a relationship between two quantities.

Analyze functions using different representations.

Interpret functions that arise in applications in terms of the context.

Prove and apply trigonometric identities.

Understand the concept of a function and use function notation.

Define, evaluate, and compare functions.
problem_activity: Activity
This activity focuses on a specific aspect of the previous activity: combining two functions to make a new function. Here, students sketch the graph of the function defined as the sum of two original functions. To help students make generalizations about the process, in each question one of the starting functions is always the same while the other is either constant, linear, or quadratic (MP8).
Monitor for students who approach the activity in different ways based on how precise they make their sketch, such as:
making a rough sketch by observation
plotting a few points by summing together the
\(y\)
-values of the two functions and then connecting them with a curve
measuring the distance of each function from the
\(x\)
-axis in a few locations and then combining those heights to determine the
\(y\)
-value of the new function at specific values of
\(x\)
making a table of values for
\(E\)
and the other function, as in the previous activity, in order to find the sums before making a sketch of the new function
Also look out for possible incorrect approaches due to students not considering the numerical values of the functions and what it means to add, such as by:
sketching a curve that is halfway between the two given graphs
letting the horizontal asymptote dictate end behavior on the right regardless of the value of the other function
Ask students who tried either of these last two approaches if they would be willing to share during the discussion. If possible, identify students who first did a sketch incorrectly and then revised their work to share at the end of the discussion.
Launch
Arrange students in groups of 2–3. Tell students that in this activity they are going to sketch graphs of functions that are the sum of two other functions. Ask students to first complete the sketch of
\(S\)
on their own and then check their work with their group before continuing on to the other graphs. Depending on time available, assign each group 3–5 of the graphs to complete.
Since this activity was designed to be completed without technology, ask students to put away any graphing devices.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, students can use different color highlighters for each graph. Students can identify and label points on each graph. Remind students that they are sketching the sum of the two functions.
Supports accessibility for: Visual-spatial processing
Student Facing
Here are the graphs of two functions,
\(E\)
and
\(L\)
. Define a new function
\(S\)
by adding
\(E\)
and
\(L\)
, so
\(S(x) = E(x) + L(x)\)
. On the same axes, sketch what you think the graph of
\(S\)
looks like.
[IMG]
Sketch the graph of the sum of
\(E\)
and each of the following functions.
[IMG]
[IMG]
[IMG]
[IMG]
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?
Here are the graphs of two functions,
\(U\)
and
\(V\)
. Define a new function
\(W\)
by multiplying
\(U\)
and
\(V\)
, so
\(W(x) = U(x) V(x)\)
. On the same axes, sketch what you think the graph of
\(W\)
looks like.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to share how they made their sketches, focusing on how the numerical value of the new function is defined by the sum of the numerical values of the two original functions. The discussion begins with students sharing techniques that worked for them and then focuses on techniques that did not work and why, calling back to the strategies that did work in order to make sense of what went wrong.
Invite previously identified students who graphed correctly to share their process in the order listed in the Activity Narrative. Display student graphs and any accompanying work for all to see during the explanation. Since these were only sketches, students did not have to make a detailed table, but using a table illustrates clearly that adding two functions together means adding together their output values.
Next, invite previously identified students who took an incorrect approach to share in the order listed in the Activity Narrative. Where possible, identify what function the student did graph. For example, if a student sketched the curve halfway between the two starting graphs, the function they graphed is the average of the two starting graphs instead of the sum. If a student’s sketch appears to have a horizontal asymptote at
\(y=0\)
on the right side, it's possible they were thinking more of a function defined by the product of the two original functions.
If time allows, ask students to choose one of the questions and sketch the graph of a function defined by the
difference
between
\(E\)
and the other function.


correct_option_index: 6