options: Solve systems of equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Multiply and divide within 100.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Use the four operations with whole numbers to solve problems.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Interpret the structure of expressions.

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

Understand the relationship between zeros and factors of polynomials.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Represent and analyze quantitative relationships between dependent and independent variables.

Gain familiarity with factors and multiples.

Perform arithmetic operations on polynomials.

Work with addition and subtraction equations.

Create equations that describe numbers or relationships

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

Reason about and solve one-variable equations and inequalities.

Write expressions in equivalent forms to solve problems.

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

Analyze patterns and relationships.

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

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

Represent and solve problems involving multiplication and division.

Represent and solve problems involving addition and subtraction.

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

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

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

Work with radicals and integer exponents.
problem_activity: Activity
This activity is an opportunity to put together the learning of the past several lessons: correspondences between tape diagrams, equations, and stories, and using representations to reason about a solution. The focus of this activity is still contrasting the two main types of equations that students encounter in this unit.
Launch
Keep students in the same groups. 5 minutes of quiet work time followed by sharing with a partner and a whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing
Reading, Representing: MLR6 Three Reads.
Use this routine with the first story to support students’ understanding of the situation and how to represent it with a tape diagram or equation. Use the first read to orient students to the situation (Lin and volunteers are hanging flyers at school). Use the second read to identify the important quantities (number of volunteers, number of flyers each). After the third read, ask students to brainstorm how the situation can be represented in a tape diagram or equation. This will help students make connections between equations and diagrams and understand the language of a situation that will distinguish which form of and equation would be an appropriate representation.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
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Story 1: Lin had 90 flyers to hang up around the school. She gave 12 flyers to each of three volunteers. Then she took the remaining flyers and divided them up equally between the three volunteers.
Story 2: Lin had 90 flyers to hang up around the school. After giving the same number of flyers to each of three volunteers, she had 12 left to hang up by herself.
Which diagram goes with which story? Be prepared to explain your reasoning.
In each diagram, what part of the story does the variable represent?
Write an equation corresponding to each story. If you get stuck, use the diagram.
Find the value of the variable in the story.
Student Response
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Student Facing
Are you ready for more?
A tutor is starting a business. In the first year, they start with 5 clients and charge $10 per week for an hour of tutoring with each client. For each year following, they double the number of clients and the number of hours each week. Each new client will be charged 150% of the charges of the clients from the previous year.
Organize the weekly earnings for each year in a table.
Assuming a full-time week is 40 hours per week, how many years will it take to reach full time and how many new clients will be taken on that year?
After reaching full time, what is the tutor’s annual salary if they take 2 weeks of vacation?
Is there another business model you’d recommend for the tutor? Explain your reasoning.
Student Response
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Activity Synthesis
For each story, select 1 or more groups to present the matching diagram, their equation, and their solution method. Possible questions to ask:
“How were the diagrams alike? How were they different?” (They have the same numbers and a letter. One has 3 equal groups and an extra bit, the other just has 3 equal groups, but each group is a sum.)
“How were the stories alike? How were they different?” (They were both about distributing 90 flyers. In one story, Lin makes a series of moves to each volunteer. In the other story, she gives each volunteer the same amount, but then there are some left over.)
“What parts of the story made you think that one diagram represented it?”
“Explain how you reasoned about the story, diagram, or equation to find the value of the variable.”


correct_option_index: 21