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: Use polynomial identities to solve problems.

Create equations that describe numbers or relationships

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

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

Perform arithmetic operations on polynomials.

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

Represent and solve problems involving multiplication and division.

Solve equations and inequalities in one variable.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Represent and solve problems involving addition and subtraction.

Use properties of operations to generate equivalent expressions.

Rewrite rational expressions.

Reason about and solve one-variable equations and inequalities.

Use the four operations with whole numbers to solve problems.

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

Analyze patterns and relationships.

Work with addition and subtraction equations.

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

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

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

Solve systems of equations.

Gain familiarity with factors and multiples.

Write and interpret numerical expressions.

Multiply and divide within 100.

Write expressions in equivalent forms to solve problems.

Work with radicals and integer exponents.

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Interpret the structure of expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.
problem_activity: Activity
Continuing their work from the previous lesson, the purpose of this activity is for students to write a rational equation to model a situation and then use it to answer questions. The difference between this activity and previous ones in which students solved rational equations is that now students must solve a rational equation in which the denominators of the two expressions on each side are different expressions involving the variable
\(r\)
.
Monitor for students who either graph each expression to find the point of intersection or set up an equation that they solve to share during the class discussion.
Making graphing technology available gives students opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Tell students to read the activity and answer the first problem. After quiet work time, ask students to compare their responses to their partner’s and reach agreement on the two expressions. Select 2–3 students to share their expressions with the class, recording them for all to see. Once students are in agreement on the expressions, allow them to continue to the next problem.
Conversing: MLR8 Discussion Supports.
Use this routine to help students explain their expression to their partner. Students should take turns sharing their expressions and explaining their reasoning to their partner. Display the following sentence frames for all to see: “_____ represents _____”, “Where does _____ show . . . ?”, and “That could (or couldn’t) be true because . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about modeling situations with rational expressions.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Noah likes to go for boat rides along a river with his family. In still water, the boat travels about 8 kilometers per hour. In the river, it takes them the same amount of time
\(t\)
to go upstream 5 kilometers as it does to travel downstream 10 kilometers.
[IMG]
If the speed of the river is
\(r\)
, write an expression for the time it takes to travel 5 kilometers upstream and an expression for the time it takes to travel 10 kilometers downstream.
Use your expressions to calculate the speed of the river. Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
Students may not know how to use the information that the boat goes 8 kph in still water. Tell them that if a boat is going downstream, then the river is pushing it forward, so it will travel at its speed plus the river’s speed. If the boat is going upstream, then the river will push against it and slow it down.
Activity Synthesis
The purpose of this discussion is for students to share how they solved a rational equation in which the denominators of the rational expressions are not the same. During the discussion, students should make connections to how they solved rational equations in the previous lesson in order to reason that they can strategically multiply by a common denominator in order to get an equation without variables in denominators.
Select previously identified students to share how they answered the last question, starting with any who used graphs. The main focus of the discussion should be on how students identified a solution to the equation
\(\frac{5}{8-r}=\frac{10}{8+r}\)
. Here are some questions for discussion:
“What are some possible first steps to solving the equation?” (Multiplying by
\(8-r\)
or
\(8+r\)
.)
“Is it better to multiply each side by
\(8-r\)
first or
\(8+r\)
first?” (No matter the order, multiplying by one then the other results in the linear equation
\(5(8+r) = 10(8-r)\)
.)
If time allows, display a graph of
\(y=\frac{5}{8-r}\)
and ask, “What is the meaning of the vertical asymptote
\(r=8\)
in this situation?” (If the boat goes 8 kilometers per hour and the river goes 8 kilometers per hour, then the boat does not go anywhere.)


correct_option_index: 1

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