options: Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

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

Multiply and divide within 100.

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

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

Write and interpret numerical expressions.

Add and subtract within 20.

Represent and solve equations and inequalities graphically.

Create equations that describe numbers or relationships

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve problems involving addition and subtraction.

Interpret the structure of expressions.

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

Reason about and solve one-variable equations and inequalities.

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

Represent and solve problems involving multiplication and division.

Use the four operations with whole numbers to solve problems.

Gain familiarity with factors and multiples.

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

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

Perform arithmetic operations on polynomials.

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

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

Use properties of operations to generate equivalent expressions.

Work with addition and subtraction equations.

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Use polynomial identities to solve problems.

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

Write expressions in equivalent forms to solve problems.

Work with radicals and integer exponents.

Solve systems of equations.
problem_activity: Warm-up
Students extend their understanding from the previous lessons to recognize the structure of a linear equation for all possible types of solutions: one solution, no solution, or infinitely many solutions. Students are still using language such as “true for one value of x,” “always true” or “true for any value of x,” and “never true.” Students should be able to articulate that this depends both on the
coefficient
of the variable as well as the constant term on each side of the equation.
Launch
Give students 2–3 minutes of quiet think time followed by a whole-class discussion.
Student Facing
Consider the unfinished equation
\(12(x-3)+18=\,\underline{\qquad \qquad}\)
. Match the following expressions with the number of solutions the equation would have with that expression on the right hand side.
\(6(2x-3)\)
\(4(3x-3)\)
\(4(2x-3)\)
one solution
no solutions
all solutions
Student Response
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Activity Synthesis
In order to highlight the structure of these equations, ask students:
“What do you notice about equations with no solution?” (These equations have equal or equivalent coefficients for the variable, but unequal values for the constants on each side of the equation.)
“What do you notice about equations that are always true?” (These equations have equivalent expressions on each side of the equation, so the coefficients are equal and the constants are equal or equivalent on each side.)
“What do you notice about equations that have exactly one solution?” (These equations have different values for the coefficients on each side of the equation and it doesn’t matter what the constant term says.)
Display the equation
\(x=12\)
for all to see. Ask students how this fits with their explanations. (We can see that there is one solution. Another way to think of this is that the coefficient of
\(x\)
is 1 on the left side of the equation, and the coefficient of
\(x\)
is 0 on the right side of the equation. So the coefficients of
\(x\)
are different, just like the explanation.)


correct_option_index: 24