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: Perform operations on vectors.

Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

Perform operations on matrices and use matrices in applications.

Apply and extend previous understandings of numbers to the system of rational numbers.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Reason quantitatively and use units to solve problems.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Extend the properties of exponents to rational exponents.

Represent and model with vector quantities.

Compute fluently with multi-digit numbers and find common factors and multiples.
problem_activity: Activity
In this activity, students interpret equations that represent situations (MP2). The purpose is for students to see that equations of the form
\(x + p = q\)
can be solved by adding the opposite of
\(p\)
to the equation, regardless of whether
\(p\)
is positive or negative. Students also see that equations of the form
\(px = q\)
can be solved by multiplying the equation by the reciprocal of
\(p\)
. Through this work, students see that the structure of equations can be used to reason about a path to a solution (MP7) even when negative values are included or when a variable can represent a negative number.
Launch
Give students 5–6 minutes of quiet work time followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Reading: MLR6 Three Reads.
Use this routine to help students interpret the representations of the situation in this activity. Use the first read to help students comprehend the different stages of situation (e.g., hiking a mountain, temperature falls, cost of the trip) without using numbers. Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., increased elevation 290 feet, now at 450 feet; temperature fell 4 degrees, now is at 32 degrees; 3 times as many students this year, now 42 students; cost is
\(\frac23\)
of last year’s, cost
$
32 this year). For the third read, ask students to brainstorm possible strategies to answer the given questions.
Design Principle(s): Support sense-making
Student Facing
The Hiking Club is on a trip to hike up a mountain.
The members increased their elevation 290 feet during their hike this morning. Now they are at an elevation of 450 feet.
Explain how to find their elevation before the hike.
Han says the equation
\(e + 290 = 450\)
describes the situation. What does the variable
\(e\)
represent?
Han says that he can rewrite his equation as
\(e=450 + \text-290\)
to solve for
\(e\)
. Compare Han's strategy to your strategy for finding the beginning elevation.
The temperature fell 4 degrees in the last hour. Now it is 21 degrees. Write and solve an equation to find the temperature it was 1 hour ago.
There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.
Explain how to find the number of students that came on the hiking trip last year.
Mai says the equation
\(3s=42\)
describes the situation. What does the variable
\(s\)
represent?
Mai says that she can rewrite her equation as
\(s=\frac13 \boldcdot 42\)
to solve for
\(s\)
. Compare Mai's strategy to your strategy for finding the number of students on last year’s trip.
The cost of the hiking trip this year is
\(\frac23\)
of the cost of last year's trip. This year's trip cost
$
32. Write and solve an equation to find the cost of last year's trip.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other rational numbers
\(a\)
and
\(b\)
.
[IMG]
Decide which of the following numbers are positive and which are negative.
\(a-1\)
\(a-2\)
\(\text-b\)
\(a+b\)
\(a-b\)
\(ab+1\)
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
Students may be misled by words to add or multiply (or subtract or divide) by the wrong numbers. For example, the word "increased" in question 1 may lead students to simply add the numbers they see, while the words "three times as many" in question 4 may lead students to multiply the numbers in the problem. Encourage students to make sense of the situations by acting them out or using visual diagrams, which will help them understand the actions and relationships in the stories.
Activity Synthesis
Tell students: “We learned four things about the hiking trip in this activity: the students were climbing, the temperature was falling, there were more students this year than last, and the cost of the trip was less this year than last.” Then ask them:
“Think about how you knew what operation described the rise in elevation, fall in temperature, rise in number of students, and fall in the cost. How did you know whether the situation used adding or multiplying?”  (This conversation can highlight the problem with relying on “key words” . For example, when students see “times as many” , they might want to multiply the numbers they see in the problem. Encourage students to make sense of the situations by acting them out or drawing diagrams.)
“How did you decide how to solve for the unknown quantity?”
“What are some ways to know that a situation involves negative values?”


correct_option_index: 5