standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Activity
The purpose of this activity is to get students thinking about strategically solving equations by paying attention to their structure. Distribution first versus dividing first is a common point of divergence for students as they start solving.
Identify students who choose different solution paths to solve the last two problems.
Launch
Arrange students in groups of 2. Give students 2 minutes quiet think time for problem 1, then 3–5 minutes partner time to discuss problem 1 and complete the other problems. Follow with a whole-class discussion.
Conversing: MLR7 Compare and Connect.
Display Noah’s and Lin’s solution methods side by side. Once students have determined that they are both correct, ask students to explain the differences between the approaches. Amplify mathematical language students use to distinguish between distributing or dividing. This will help students reflect on and produce mathematical language to explain why they choose to take either approach when answering the remaining questions.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Noah and Lin both solved the equation
\(14a=2(a-3)\)
.
Do you agree with either of them? Why?
Noah's solution:
\(\begin{align*} 14a&=2(a-3) \\ 14a&=2a-6 \\ 12a&=\text-6 \\ a&=\text-\frac12 \\ \end{align*}\)
Lin's solution:
\(\begin{align} 14a&=2(a-3) \\ 7a&=a-3\\ 6a&=\text-3\\a&=\text-\frac12 \end{align}\)
Elena is asked to solve
\(15 - 10x = 5(x + 9)\)
. What do you recommend she does to each side first?
Diego is asked to solve
\(3x -8 = 4(x + 5)\)
. What do you recommend he does to each side first?
Student Response
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Student Facing
Are you ready for more?
In a cryptarithmetic puzzle, the digits 0–9 are represented with letters of the alphabet. Use your understanding of addition to find which digits go with the letters A, B, E, G, H, L, N, and R.
HANGER + HANGER + HANGER = ALGEBRA
Student Response
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Anticipated Misconceptions
Some students may not distribute or collect like terms before performing the same operation on each side.
Activity Synthesis
Have previously identified groups share the different solution paths they chose for solving the last two questions.
To highlight the different strategies, ask:
“What are the advantages of choosing to distribute first? To divide first?” (Answers vary. Distributing first eliminates confusion about which terms can be subtracted from each side. Dividing first makes the numbers smaller and easier to mentally calculate.)
“What makes it easier to distribute versus divide first on the last question?” (Dividing by 4 before distributing will result in non-integer terms, which can be harder to add and subtract mentally.)
“Is one path more ‘right’ than another?” (No. As long as we follow valid steps, like adding or multiplying the same thing to each side of an equation, the steps are right and will give a correct solution.)


aligns: no