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

Add and subtract within 20.

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

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

Create equations that describe numbers or relationships

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

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

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

Solve equations and inequalities in one variable.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Use polynomial identities to solve problems.

Generate and analyze patterns.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Work with radicals and integer exponents.

Write and interpret numerical expressions.

Interpret the structure of expressions.

Represent and solve problems involving multiplication and division.

Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

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

Represent and solve equations and inequalities graphically.

Analyze patterns and relationships.

Gain familiarity with factors and multiples.

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Multiply and divide within 100.

Perform arithmetic operations on polynomials.

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.
problem_activity: Activity
The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number.
For students who are accustomed to viewing the equal sign as a directive that means “perform an operation,” tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, “The expressions on either side have the same value.”
Launch
Before students start working, it may be helpful to demonstrate how someone would figure out whether or not an equation is true without evaluating each expression. For example:
Is
\(4^2=2^3\)
true? Well, let's see. We can rewrite each side like this:
\(\displaystyle 4 \boldcdot 4 = 2 \boldcdot 2 \boldcdot 2\)
Then we can replace one of those
\(2 \boldcdot 2\)
's with a 4, like this:
\(\displaystyle 4 \boldcdot 4 = 4 \boldcdot 2\)
Now we can tell this equation is not true.
These problems can also be worked by directly evaluating expressions, which is fine, as it serves as practice evaluating exponential expressions.
Representation: Internalize Comprehension.
Begin with the demonstration as described in the launch to support connections between new situations involving evaluating exponential expressions and prior understandings. Use color or annotations to highlight what changes and what stays the same at each step.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Decide whether each equation is true or false, and explain how you know.
\(2^4=2 \boldcdot 4\)
\(3+3+3+3+3=3^5\)
\(5^3=5 \boldcdot 5 \boldcdot 5\)
\(2^3=3^2\)
\(16^1=8^2\)
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
\(\left( \frac12 \right)^4=\frac{1}{8}\)
\(8^2 = 4^3\)
Student Response
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Activity Synthesis
Invite students who evaluated the expressions and students who used structure or the meaning of exponents and operations to present their work. Compare and connect the strategies by noting where the use of structure might prove more efficient, where evaluating might be simpler, where thinking about the meaning of exponents and operations can make the true or false determination simpler.
Some guiding questions to highlight the meaning of exponents:
“Can we switch the order with exponents like we can with addition and multiplication—specifically, are
\(a^b\)
and
\(b^a\)
equivalent? How do you know?” (No, you can try different values of
\(a\)
and
\(b\)
or use the meaning of exponents to see that
\(a\)
multiplied
\(b\)
times is not always the same as
\(b\)
multiplied
\(a\)
times.)
“What change can we make to the equation
\(3+3+3+3+3=3^5\)
to make it true?” (Change addition to multiplication on the left, or change the exponent to multiplication on the right.)
“Your friend claims the equation
\(\frac12 \boldcdot  \frac12 \boldcdot  \frac12 \boldcdot  \frac12 = 4 \boldcdot \frac12\)
is true. What do you think they are misunderstanding? How can you convince them it is false?” (Possibly by saying that the left side shows multiplying
\(\frac12\)
4 times, but the right side shows multiplying 4 by
\(\frac12\)
, which means
adding
4 copies of
\(\frac12\)
, not multiplying them. You can show similar examples with other numbers, but the best way to convince them is to talk about what exponents and multiplication mean.)
“Can we show that
\(8^2 = 4^3\)
is true without evaluating both sides? What understanding about the meaning of exponents and operations can help us?” (
\(8^2\)
means
\(8\boldcdot 8\)
or
\(4\boldcdot 2\boldcdot 4\boldcdot 2\)
, which also equals
\(4\boldcdot 4\boldcdot 2\boldcdot 2\)
or
\(4\boldcdot 4\boldcdot 4\)
. Another way to write
\(4\boldcdot 4\boldcdot 4\)
is
\(4^3\)
. We are using the understanding that we can multiply in any order, or the commutative and associative properties of multiplication.)
Speaking, Writing: MLR8 Discussion Supports.
Revoice language and push for clarity in reasoning when students discuss their strategies for determining whether the equations are true or false. Provide a sentence frame such as “The equation is true (or false) because ______.” This will strengthen students’ mathematical language use and reasoning when discussing the meaning of exponents and operations that can make the equivalence of expressions true or false.
Design Principle(s): Maximize meta-awareness


correct_option_index: 28