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: Work with equal groups of objects to gain foundations for multiplication.

Analyze patterns and relationships.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand properties of multiplication and the relationship between multiplication and division.

Use polynomial identities to solve problems.

Understand the relationship between zeros and factors of polynomials.

Interpret the structure of expressions.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Create equations that describe numbers or relationships

Use properties of operations to generate equivalent expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Write expressions in equivalent forms to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Write and interpret numerical expressions.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with addition and subtraction equations.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Add and subtract within 20.

Gain familiarity with factors and multiples.

Rewrite rational expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.

Use the four operations with whole numbers to solve problems.

Solve systems of equations.

Solve equations and inequalities in one variable.

Understand solving equations as a process of reasoning and explain the reasoning.

Reason about and solve one-variable equations and inequalities.

Perform arithmetic operations on polynomials.

Represent and solve problems involving addition and subtraction.

Work with radicals and integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and solve problems involving multiplication and division.
problem_activity: Activity
The purpose of this activity is to bring together identities and the work students did earlier with rational equations. For each question, students first rewrite the fraction
\(\frac{2}{15}\)
using the given formula and then prove that the formula is an identity.
The whole-class discussion at the end of the activity should focus on how students showed that the formula is an identity. Monitor for groups who use different methods to share during the discussion. For example, they may identify different common denominators when adding together the fractions in the expression on the right side of the formula.
Launch
Arrange students in groups of 2. Tell students to complete each problem individually, and then compare their work with their partner. Partners should discuss any similarities and differences in their methods, and decide which method they prefer.
If students need some practice adding fractions with unlike denominators, begin the activity by displaying the equation
\(\frac78 = \frac12 + \frac13 + \frac{1}{24}\)
for all to see and ask students how they can figure out without using a calculator that the equation is true. After some quiet work time, select 1–2 students to share their thinking about why the equation is true. The purpose of looking at this equation before the main activity is to make sure students recall that adding fractions requires a common denominator.
Speaking: MLR8 Discussion Supports.
Use this routine to support students in producing statements justifying their identity formulas. Provide sentence frames for students to use when comparing their methods and discussing which method they prefer, such as “I prefer _____ because _____ .”, or “We both have _____ which represents _____ .”
Design Principle(s): Support sense-making
Student Facing
[IMG]
In Ancient Egypt, all non-unit fractions were represented as a sum of distinct unit fractions. For example,
\(\frac49\)
would have been written as
\(\frac13 + \frac19\)
(and not as
\(\frac19+\frac19+\frac19+\frac19\)
or any other form with the same unit fraction used more than once). Let’s look at some different ways we can rewrite
\(\frac{2}{15}\)
as the sum of distinct unit fractions.
Use the formula
\(\frac{2}{d}=\frac{1}{d}+\frac{1}{2d}+\frac{1}{3d}+\frac{1}{6d}\)
to rewrite the fraction
\(\frac{2}{15}\)
, then show that this formula is an identity.
Another way to rewrite fractions of the form
\(\frac{2}{d}\)
is given by the identity
\(\frac{2}{d} = \frac{1}{d} +\frac{1}{d+1} + \frac{1}{d(d+1)}\)
. Use it to re-write the fraction
\(\frac{2}{15}\)
, then show that it is an identity.
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?
For fractions of the form
\(\frac{2}{pq}\)
, that is, fractions with a denominator that is the product of two positive integers, the following formula can also be used:
\(\frac{2}{pq}=\frac{1}{pr} + \frac{1}{qr}\)
, where
\(r = \frac{p+q}{2}\)
. Use it to re-write the fraction
\(\frac{2}{45}\)
, then show that it is an identity.
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 students may not be sure how to find a common denominator when the denominators all involve variables. Remind them that a common denominator is a multiple of all of the denominators. If needed, students can guess possible common denominators, and then test them by checking whether they're divisible by each of the denominators.
Activity Synthesis
The purpose of this discussion is for students to share how they proved the given formulas are identities. Ask 2–4 previously identified groups who used different methods to share how they each showed that the formula is an identity, recording their strategies for all to see.


correct_option_index: 22