CCSS.Math.Content.HSA-APR.D.6
The standard
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.
Common Core State Standards for Mathematics
What this standard means
Students need to rewrite a rational expression by dividing polynomials. They should see it like fraction division with remainders: numerator divided by denominator equals a polynomial quotient plus a smaller leftover fraction. They can use inspection for simple cases, long division for routine cases, and technology for messy ones.
Mastery means students can choose a method, carry out the division correctly, and check by multiplying back. Common trouble spots are missing zero terms, sign errors during subtraction, stopping before the remainder has smaller degree, and not understanding that different forms can represent the same expression.
Ways to teach it
- Hands-on activity: Give students polynomial division cards and have them match each rational expression to its quotient, remainder, and rewritten form.
- Discussion prompt: Ask, “How is polynomial long division like whole-number division, and where does the analogy stop working?”
- Quick assessment: Have students rewrite (x^2 + 5x + 6) / (x + 2) and explain how they checked the result.
- Real-world connection: Use revenue divided by items sold, modeled by polynomials, to discuss average price plus a leftover term.
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