Maryland Math A.SSE.B.3.e

MathGrades 9–12SEEING STRUCTURE IN EXPRESSIONS

The Standard

A.SSE.B.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Translate between the standard and general form, Ax² + Cy² + Dx + Ey + F = 0, of a second degree equation.

Maryland College and Career Ready Standards for Mathematics (2025) · Interpret the structure of expressions.

Teacher's field guide

What Students Need to Do: Students group x and y terms and complete the square to rewrite an expanded equation in a form that shows its shape. They expand and combine like terms to go back.
What Mastery Looks Like: Given 4x² + 9y² - 16x + 18y - 11 = 0, a student gets (x - 2)²/9 + (y + 1)²/4 = 1 and names the center (2, -1). They expand their answer to check it.
Common Misconceptions: When completing 4(x² - 4x), students may add 4 to the other side instead of 16. They may also read x - 2 as a center at -2 or forget to divide so the right side equals 1.
How to Assess It: Exit ticket: Rewrite x² + y² - 6x + 4y + 9 = 0 in standard form. State the center and radius, then expand your answer to check it.

Lesson moves

On graph paper, plot an ellipse centered at (2, -1) with horizontal radius 3 and vertical radius 2; write both equation forms.
Ask students to explain in writing why completing the square in 4(x² - 4x) changes the other side by 16, not 4.
Run a relay: one student converts x² + y² - 6x + 4y + 9 = 0, and a partner expands the result to check.
A sprinkler reaches 4 meters from (3, 2); students write its circular boundary in both forms and explain what each shows.

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