standard_description: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
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.)


aligns: no