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

Use the four operations with whole numbers to solve problems.

Generate and analyze patterns.

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

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

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Gain familiarity with factors and multiples.

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

Write and interpret numerical expressions.

Reason about and solve one-variable equations and inequalities.

Solve equations and inequalities in one variable.

Represent and solve equations and inequalities graphically.

Work with addition and subtraction equations.

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

Represent and solve problems involving addition and subtraction.

Use properties of operations to generate equivalent expressions.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Add and subtract within 20.

Represent and solve problems involving multiplication and division.

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

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Rewrite rational expressions.

Interpret the structure of expressions.

Use polynomial identities to solve problems.

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

Work with radicals and integer exponents.

Multiply and divide within 100.

Solve systems of equations.

Analyze patterns and relationships.
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: 23