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: Create equations that describe numbers or relationships

Perform arithmetic operations on polynomials.

Generate and analyze patterns.

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

Represent and solve problems involving multiplication and division.

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

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

Write and interpret numerical expressions.

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

Work with radicals and integer exponents.

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

Multiply and divide within 100.

Solve systems of equations.

Write expressions in equivalent forms to solve problems.

Interpret the structure of expressions.

Add and subtract within 20.

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

Analyze patterns and relationships.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Gain familiarity with factors and multiples.

Use polynomial identities to solve problems.

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

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Work with addition and subtraction equations.

Represent and solve equations and inequalities graphically.

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

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

Solve equations and inequalities in one variable.

Represent and analyze quantitative relationships between dependent and independent variables.
problem_activity: Activity
This task is analogous to a previous activity with positive exponents. The number line strongly encourages students to think about how to change expressions so they all take the form
\(b \boldcdot 10^k\)
, where
\(b\)
is between 1 and 10, as in the case of scientific notation. The number line also is a useful representation to show that, for example,
\(29 \boldcdot 10^\text{-7}\)
is about half as much as
\(6 \boldcdot 10^\text{-6}\)
. The last two questions take such a comparison a step further, asking students to estimate relative sizes using numbers expressed with powers of 10.
As students work, look for different strategies they use to compare expressions that are not written as a product of a number and
\(10^{\text-6}\)
. Also look for students who can explain how they estimated in the last two problems. Select them to share their strategies later.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time, followed by partner discussion and whole-class discussion. During partner discussion, ask students to share their responses for the first two questions and reach an agreement about where the numbers should be placed on the number line.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the visual display of rules for exponents.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students as they describe how they compared and estimated the difference in magnitude between the pairs of values
\(29⋅10^{\text-7}\)
and
\(6⋅10^{\text-6}\)
or
\(7⋅10^{\text-8}\)
and
\(3⋅10^{\text-9}\)
. For example, “First I ____ , then I ____ .” or “_____ is (bigger/smaller) because ______ .”
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Label the tick marks on the number line.
Plot the following numbers on the number line:
A.
\(6 \boldcdot 10^{\text -6}\)
B.
\(6 \boldcdot 10^{\text -7}\)
C.
\(29 \boldcdot 10^{\text -7}\)
D.
\((0.7) \boldcdot 10^{\text -5}\)
Which is larger,
\(29 \boldcdot 10^{\text -7}\)
or
\(6 \boldcdot 10^{\text -6}\)
? Estimate how many times larger.
Which is larger,
\(7 \boldcdot 10^{\text -8}\)
or
\(3 \boldcdot 10^{\text -9}\)
? Estimate how many times larger.
Student Response
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Anticipated Misconceptions
Students may have trouble comparing negative powers of 10. Remind these students that, for example,
\(10^\text{-5}\)
is 5 factors that are
\(\frac{1}{10}\)
and
\(10^\text{-6}\)
is 6 factors that are
\(\frac{1}{10}\)
, so
\(10^\text{-5}\)
is 10 times larger than
\(10^\text{-6}\)
.
Students may also have trouble estimating how many times one larger expression is than another. Offer these students an example to illustrate how representing numbers as a single digit times a power of 10 is useful for making rough estimations. We have that
\(9 \boldcdot 10^\text{-12}\)
is roughly 50 times as much as
\(2 \boldcdot 10^\text{-13}\)
because
\(10^\text{-12}\)
is 10 times as much as
\(10^\text{-13}\)
and 9 is roughly 5 times as much as 2. In other words,
\(\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50\)
Activity Synthesis
Select previously identified students to explain how they used the number line and powers of 10 to compare the numbers in the last two problems.
One important concept is that it’s always possible to change an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, we can think of
\(29 \boldcdot 10^\text{-7}\)
as
\((2.9) \boldcdot 10 \boldcdot 10^\text{-7}\)
or
\((2.9) \boldcdot 10^\text{-6}\)
. Another important concept is that powers of 10 can be used to make rough estimates. Make sure these ideas are uncovered during discussion.


correct_option_index: 9