options: Write expressions in equivalent forms to solve problems.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Multiply and divide within 100.

Represent and analyze quantitative relationships between dependent and independent variables.

Use properties of operations to generate equivalent expressions.

Represent and solve problems involving addition and subtraction.

Perform arithmetic operations on polynomials.

Add and subtract within 20.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use polynomial identities to solve problems.

Use the four operations with whole numbers to solve problems.

Interpret the structure of expressions.

Work with radicals and integer exponents.

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

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

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

Represent and solve problems involving multiplication and division.

Write and interpret numerical expressions.

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

Understand the relationship between zeros and factors of polynomials.

Create equations that describe numbers or relationships

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

Generate and analyze patterns.

Gain familiarity with factors and multiples.

Work with addition and subtraction equations.

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

Solve equations and inequalities in one variable.

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

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

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

Solve systems of equations.

Analyze patterns and relationships.

Represent and solve equations and inequalities graphically.
problem_activity: Activity
In this activity, students examine equations with one variable to determine the number of solutions. In the associated Algebra lesson, students will find the number of solutions for a system of equations. In the discussion after this activity, students are asked to look for ways they can connect the number of solutions to the equation without solving.
Launch
Arrange students in groups of 2.
Demonstrate that there are three options for the number of values that make a linear equation true.
One. For example,
\(x+1 = 3\)
is only true when
\(x = 2\)
and not true for any other value of
\(x\)
.
Zero. For example,
\(x + 1 = x\)
is not true for any value of
\(x\)
.
Infinite. For example
\(x + 1 = x + 1\)
is true for any value of
\(x\)
.
Tell students that they do not need to solve any of the equations, although rewriting the equations in other forms may help students recognize the number of solutions.
Student Facing
How many values of
\(x\)
make each equation true?
\(3x + 1 = 10\)
\(2x + 12 = 2x + 10 + 2\)
\(2x = x + 2\)
\(3(x+4) = 3x + 4\)
\(\frac{2x+6}{2} = x + 6\)
\(0 = 0\)
\(x + 3x - 4 = 7(x - \frac{4}{7})\)
\(0 = 6\)
With your partner, discuss what you notice about the equations based on the number of solutions they have.
Student Response
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Activity Synthesis
The purpose of the discussion is to recognize ways to determine the number of solutions from an equation without solving. Select students to share their answers as well as what they noticed during their discussions.
To continue the discussion, consider asking:
"What do you notice about equations that have one solution?" (On each side of the equation, there are different number of
\(x\)
variables.)
"Is there a way to know whether there will be infinite or no solutions based on the original equation?" (When each side of the equation has the same number of
\(x\)
variables and the constants are also equal, there will be infinite solutions. When each side of the equation has the same number of
\(x\)
variables and the constants are not equal, there will be no solution.)


correct_option_index: 15