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: Extend the properties of exponents to rational exponents.

Perform arithmetic operations with complex numbers.

Reason quantitatively and use units to solve problems.

Perform operations on matrices and use matrices in applications.

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

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

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

Perform operations on vectors.

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

Use properties of rational and irrational numbers.

Use complex numbers in polynomial identities and equations.

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

Represent and model with vector quantities.
problem_activity: Optional activity
In this activity, students are building skills that will help them in mathematical modeling (MP4). Asking students to request needed information invites them to identify their variables and helps students recall how much information is needed to determine a scale factor and apply it to a known distance to find the scaled distance. This activity also allows students to practice calculating proportional relationships in a scale drawing. Monitor for students who:
calculate using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{Earth’s actual diameter}}=\frac{\text{planet’s scaled diameter}}{\text{planet’s actual diameter}}\)
calculate using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{planet’s scaled diameter}}=\frac{\text{Earth’s actual diameter}}{\text{planet’s actual diameter}}\)
find a unit rate and scale up or down
calculate by finding a scale factor and multiplying their planet’s actual diameter by that scale factor
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Tell students they will be making a scale drawing of the solar system, with accurately scaled sizes and distances from Earth for each planet. Display a circle with 2 cm diameter and explain that it will represent Earth. Show students where the class will (attempt to) display the scale drawing of the solar system. Explain that each group will be assigned a planet to draw and add to the scale drawing, at the appropriate distance.
Arrange students in groups of 3. Assign each group a planet. If there aren't enough groups to do all the planets pick some planets that are smaller than Earth and some that are larger than Earth. It is also fine to have multiple groups work on the same planet.
Pause the class after students have determined what information they need. Ask groups what information they need. Display the table and tell students this should be enough to figure out all the information they asked for. Note that any reference to the distance from a planet to Earth assumes that all planets are aligned perfectly in a line in order of orbital distance to the Sun.
| planet  | average diameter (km) | average orbit radius (km) |
|---------|-----------------------|---------------------------|
| Mercury |                       |                           |
|  Venus  |                       |                           |
|  Earth  |                       |                           |
|  Mars   |                       |                           |
| Jupiter |                       |                           |
| Saturn  |                       |                           |
| Uranus  |                       |                           |
| Neptune |                       |                           |
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed_____ so I . . .”, “Why did you . . .?”
Supports accessibility for: Language; Social-emotional skills
Student Facing
The class is going to make a scale drawing of the planets in the solar system and their distances from Earth. Your teacher will assign you a planet to draw and place on the scale drawing. A circle with a diameter of 2 cm represents Earth.
What information do you need in order to draw your planet and place it accurately on the class’s scale drawing?
Your teacher will give you some information. Record the information here. Calculate any additional information you need.
Draw the scale drawing of your planet on a separate sheet of blank paper. Label it with measurements. When you’re finished, place it the correct distance from Earth on the class’s scale drawing.
You probably weren’t able to place your planet the correct scaled distance from Earth. Why not?
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?
What has the greater quotient: the radius of Earth compared to the Sun or the radius of the Sun compared to the radius of the star Betelgeuse?
If a circle of radius 6 cm represents the star Betelgeuse, how large would the Sun be? How large would the earth be?
Draw the scale drawing of the Sun and Earth, or explain why you can't.
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
If students struggle to use the information about orbital radius to find the planet’s distance from Earth encourage them to sketch the situation and label the information they know. (Image not to scale.)
[IMG]
Activity Synthesis
The key point of this discussion is to make sure students recall methods for finding unknown values in proportional relationships, and understand that the measurements in the scale drawing must all be scaled by the same scale factor.
Invite multiple students to share the calculations they did to find the size of their planet, given Earth’s scaled and actual size. First, select students to share who calculated using the equivalent ratios
\(\frac{\text{Earth’s scaled diameter}}{\text{Earth’s actual diameter}}=\frac{\text{planet’s scaled diameter}}{\text{planet’s actual diameter}}\)
or
\(\frac{\text{Earth’s scaled diameter}}{\text{planet’s scaled diameter}}=\frac{\text{Earth’s actual diameter}}{\text{planet’s actual diameter}}\)
. Next, select students to share who calculated by finding a scale factor or unit rate. Invite students to share how the scale factor is included in all the calculations. (The quotients in the equivalent ratios are the same as the scale factor.)
Ask students:
What units are your answers in? (Scaled values are in cm, actual values are in km.)
Why didn't you need to convert from kilometers to centimeters? (The scale factor includes the units already.)
How many times bigger is Earth than our 2 cm model? (A scale of
\(2 \text{ cm}:12,\!756 \text{ km}\)
is equivalent to a scale of
\(1 \text{ cm}:6,\!378 \text{ km}\)
which is equivalent to a scale of
\(1 \text{ cm}:637,\!800,\!000 \text{ cm}\)
. So it's 637,800,000 times bigger!)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their strategies for calculating the size of their planet, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words that helped clarify the original explanation, such as “ratio,” “scale factor,” “scaled diameter,” and “actual diameter.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 2