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

Represent and solve equations and inequalities graphically.

Rewrite rational expressions.

Solve systems of equations.

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

Perform arithmetic operations on polynomials.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Multiply and divide within 100.

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

Gain familiarity with factors and multiples.

Work with addition and subtraction equations.

Generate and analyze patterns.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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.

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

Create equations that describe numbers or relationships

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

Use polynomial identities to solve problems.

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.

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

Analyze patterns and relationships.

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

Use the four operations with whole numbers to solve problems.

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

Solve equations and inequalities in one variable.

Represent and solve problems involving multiplication and division.

Represent and solve problems involving addition and subtraction.

Understand solving equations as a process of reasoning and explain the reasoning.
problem_activity: Activity
The purpose of this activity is for students to recognize that multiplying an equation by a value results in the same solution. This will be useful when transforming equations for the elimination method of solving systems of equations in the supported Algebra lesson. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive to the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner's work or explaining their reasoning.
Launch
Arrange students in groups of two. In each group, ask students to decide who will work on column A and who will work on column B. If students are unfamiliar with the row game structure, demonstrate the protocol before they start working.
Student Facing
Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time and check if your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.
Solve each equation.
| **          row         ** |         column A          |         column B          |
|----------------------------|---------------------------|---------------------------|
| **           1          ** |        \(3x-1=5\)         |        \(6x-2=10\)        |
| **           2          ** |   \(4(x+1) = 3x - 12\)    |     \(4x+4 = 3(x-4)\)     |
| **           3          ** |      \(14x+10=4x+6\)      |      \(7x+5 = 2x+3\)      |
| **           4          ** |        \(6x+3=33\)        |        \(4x+2=22\)        |
| **           5          ** | \(4x + 5y = 2 - 4x + 5y\) | \(4x + 9y = 2 + 9y - 4x\) |
| **           6          ** |     \(2x + 6y = 10\)      |     \(5x + 15y = 25\)     |
Student Response
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Activity Synthesis
The purpose of the discussion is to notice that equations that have been multiplied by the same value have the same solution.
Consider asking students to explain why their solutions might be the same for each row.
For each row, the two equations are equivalent. Sample responses for each row:
The equation in column B is 2 times the equation in column A.
In each column, one side of the equation is the distributed form of the same side of the equation in the other column.
The equation in column B is
\(\frac{1}{2}\)
of the equation in column A.
The equation in column B is
\(\frac{2}{3}\)
of the equation in column A.
The equation in column B has
\(4y\)
added to both sides of the equation from column A.
The equation in column B is
\(\frac{5}{2}\)
of the equation in column A.
While there may be other valid ways of connecting the equations (for example, in row 4, dividing the equation in column A by 3 and dividing the equation in column B by 2 result in the same equation), encourage students to recognize that there is a single multiple of one equation that results in the other equation.


correct_option_index: 19