options: Work with radicals and integer exponents.

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

Reason about and solve one-variable equations and inequalities.

Add and subtract within 20.

Interpret the structure of expressions.

Understand the relationship between zeros and factors of polynomials.

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

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

Represent and solve equations and inequalities graphically.

Solve equations and inequalities in one variable.

Generate and analyze patterns.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Write and interpret numerical expressions.

Solve systems of equations.

Represent and solve problems involving multiplication and division.

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

Perform arithmetic operations on polynomials.

Gain familiarity with factors and multiples.

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

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

Use polynomial identities to solve problems.

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

Use properties of operations to generate equivalent expressions.

Write expressions in equivalent forms to solve problems.

Analyze patterns and relationships.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Create equations that describe numbers or relationships

Multiply and divide within 100.

Represent and analyze quantitative relationships between dependent and independent variables.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.
problem_activity: Warm-up
Students reason about the relationship between distance, rate, and time to solve a problem. The purpose is to reactivate what students know about constant speed contexts, where constant speed is represented by a set of equivalent ratios associating distance traveled and elapsed time. In the longer activity that follows, students represent a constant speed context using a table, equations, and graphs.
As students work, watch for different representations used (particularly tables) as well as for students who calculate each person's speed in miles per hour or each person's pace in hours per mile.
Launch
Consider starting off by having students close their books or devices, and display the following situation. Ask students, "What do you notice?" "What do you wonder?"
Lin and Jada each leave school at 3 p.m. to walk to the library. They each walk at a steady rate. When do they arrive?
Give them 1 minute to think of at least one thing they notice and one thing they wonder.
Students might notice that they leave at 3 p.m., that they walk at a steady rate (also called a constant speed), and that there is not enough information given to answer the question.
Students might wonder many things, but in order to answer the question, they would need to know:
How fast do they each walk?
How far away is the library?
Ask students to open their books or devices and use the additional information to solve the problem by any method they choose. If desired, remind students of tools that may be appropriate including double number lines or tables of equivalent ratios.
Student Facing
Lin and Jada each walk at a steady rate from school to the library. Lin can walk 13 miles in 5 hours, and Jada can walk 25 miles in 10 hours. They each leave school at 3:00 and walk
\(3\frac14\)
miles to the library. What time do they each arrive?
Student Response
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Activity Synthesis
Invite students who used different representations and lines of reasoning to share. If no student mentions it, demonstrate how to represent on person's trip using a table of equivalent ratios with columns representing distance and time. Ask a student to explain how to use the table to reason about the distance traveled in 1 hour and the time it takes to travel 1 mile.
One way to reason is to notice that Lin can walk 26 miles in 10 hours, so she walks slightly faster than Jada (who can complete 25 miles in 10 hours) and should arrive a bit sooner. Both of these ways of reasoning are in preparation for the following activity.


correct_option_index: 31