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

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Solve equations and inequalities in one variable.

Use polynomial identities to solve problems.

Write expressions in equivalent forms to solve problems.

Multiply and divide within 100.

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

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

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

Work with addition and subtraction equations.

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

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Solve systems of equations.

Represent and analyze quantitative relationships between dependent and independent variables.

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

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

Represent and solve problems involving addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Work with radicals and integer exponents.

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

Represent and solve equations and inequalities graphically.

Gain familiarity with factors and multiples.

Add and subtract within 20.

Write and interpret numerical expressions.

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

Perform arithmetic operations on polynomials.

Interpret the structure of expressions.

Use the four operations with whole numbers to solve problems.

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

Use properties of operations to generate equivalent expressions.

Analyze patterns and relationships.
problem_activity: Activity
In this activity, students match a card with two equations to another card describing the move that turns the first equation into the second. The goal is to help students think about equations the same way they have been thinking about hangers: objects where equality is maintained so long as the same move is made on each side. Additionally, this is the first activity where students encounter equation moves involving negative numbers, which is not possible when using hangers.
Launch
Review with students what we know about equations based on reasoning about hangers:
We can add the same quantity to each side, and the equation is still true (the hanger is still in balance).
We can subtract the same quantity from each side, and the equation is still true.
We can double or triple or halve or third the things that appear on each side, and the equation is still true. More generally, we can multiply the number of things on each side by the same number.
Tell students that hanger diagrams are really only useful for reasoning about positive numbers, but the processes above also work for negative numbers. Negative numbers are just numbers, and they have to follow the same rules as positive numbers. In fact, if we allow negative numbers into the mix, we can express any maneuver with one of two types of moves:
Add the same thing to each side. (The “thing” could be negative.)
Multiply each side by the same thing. (The “thing” could be a fraction less than 1.)
Arrange students in groups of 2. Give each group 12 pre-cut slips from the blackline master. Give 3–4 minutes for partners to match the numbered slips with the lettered slips then 1–2 minutes to trade places with another group and review each other’s work. Ask partners who finish early to write down on a separate sheet of paper what the next move would be for each of the numbered cards if the goal were to solve for
\(x\)
. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Your teacher will give you some cards. Each of the cards 1 through 6 show two equations. Each of the cards A through E describe a move that turns one equation into another.
Match each number card with a letter card.
One of the letter cards will not have a match. For this card, write two equations showing the described move.
Student Response
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Activity Synthesis
The goal of this discussion is to get students using the language of equations and describing the changes happening on each side when solving. Ask:
“What is a move you could do to the equation
\(7=2x\)
on card 1 that would result in an equation of the form
\(x=\underline{  }\)
? What is another move that would also work?” (Multiply each side by
\(\frac12\)
. Divide each side by 2.)
“Which numbered card was the most challenging to match?” (Card 2, because it at first I only looked at the
\(x\)
-terms and thought the move involved a change of
\(8x\)
.)
“Does anyone have a value for
\(x\)
that would solve one of the numbered cards? How did you figure it out?” (
\(x=2\)
is a solution for card 5. I added 3 to each side and then multiplied each side by
\(\frac14\)
.)
End the discussion by inviting groups to share the equations they wrote for card 6 and describe how they match the move "add
\(3x\)
to each side."
Speaking: MLR3 Clarify, Critique, Correct.
Display the statements: “When we add to both sides, it is the same.” and “When we multiply both sides, it stays the same.” Ask students to clarify or improve these statements in a way that is more specific. Prompt students to think about positive and negative numbers as well as fractions. This will help students to use the language of equations to explain why you can add (or subtract) and multiply (or divide) each side of an equation by an expression involving rational numbers and still have an equivalent equation.
Design Principle(s): Optimize output (for generalization)


correct_option_index: 7