options: Use properties of operations to generate equivalent expressions.

Write and interpret numerical expressions.

Rewrite rational expressions.

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

Solve systems of equations.

Interpret the structure of expressions.

Use the four operations with whole numbers to solve problems.

Create equations that describe numbers or relationships

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write expressions in equivalent forms to solve problems.

Generate and analyze patterns.

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

Represent and solve problems involving multiplication and division.

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

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

Multiply and divide within 100.

Use polynomial identities to solve problems.

Represent and solve problems involving addition and subtraction.

Add and subtract within 20.

Reason about and solve one-variable equations and inequalities.

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

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

Solve equations and inequalities in one variable.

Represent and solve equations and inequalities graphically.

Gain familiarity with factors and multiples.

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

Work with addition and subtraction equations.

Analyze patterns and relationships.

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

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

Represent and analyze quantitative relationships between dependent and independent variables.

Work with radicals and integer exponents.

Understand the relationship between zeros and factors of polynomials.

Perform arithmetic operations on polynomials.
problem_activity: Warm-up
This warm-up prompts students to compare four lines. It invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about lines and their properties. To allow all students to access the activity, each figure has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., line
\(t\)
has a different color) and find reasons based on geometric properties (e.g., a slope triangle of line
\(u\)
is not similar to the slope triangles of the other three lines).
So far, we have only considered lines with positive slopes. The purpose of this warm-up is to suggest similarities (same vertical and horizontal lengths of slope triangles) and differences (since they are not parallel, there is something fundamentally different going on here) between lines whose slopes have the same absolute value but opposite signs.
As students share their responses, listen for important ideas and terminology that will be helpful in the work of this lesson. Students may:
Identify lines that have the same slope.
Identify points where lines intersect.
Distinguish between lines with positive and negative values for slope and be able to articulate the difference clearly.
Launch
Arrange students in groups of 2–4 and provide access to geometry toolkits. Display the image of the four lines for all to see. Ask students to indicate when they have noticed one line that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask each group to offer at least one reason a particular line doesn’t belong.
Student Facing
Which line doesn’t belong?
[IMG]
Student Response
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Activity Synthesis
After students have conferred in groups, invite each group to share one reason why a particular line might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question asking which shape does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to use mathematical terminology (parallel, intersect, slope) correctly. Also, press students on unsubstantiated claims. For example, a student may claim that
\(u\)
does not belong because it has a different slope. Ask how they know for sure that its slope is different from the other lines. Demonstrate drawing a slope triangle and computing slope.
Based on the work done up to this point in the unit, students are likely to assume that the slope of
\(v\)
is
\(\frac13\)
. In the discussion, solicit the idea that there is something fundamentally different about line
\(v\)
compared to the others. You could use informal language like “uphill” and “downhill,” or “tilt direction.” The expressions positive and negative slope do not need to be introduced at this time.


correct_option_index: 20