Mathematics » High School: Algebra » Seeing Structure in Expressions

Interpret the structure of expressions.

• 1. Interpret expressions that represent a quantity in terms of its context.^★

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  • Interpret parts of an expression, such as terms, factors, and coefficients.

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  • Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P.

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• 2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).

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Write expressions in equivalent forms to solve problems.

• 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.^★
  

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  • a. Factor a quadratic expression to reveal the zeros of the function it defines.

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  • b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

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  • c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1/12)^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

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• 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.^★

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