Maryland Math A.SSE.B.3.d
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.
Choose and produce an equivalent form of a second degree equation, Ax² + Cy² + Dx + Ey + F = 0, to reveal and explain properties of the conic section represented by the equation.
Maryland College and Career Ready Standards for Mathematics (2025) · Interpret the structure of expressions.
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students complete the square in x and y while keeping the equation balanced. They use the new form to identify the graph and describe its center, vertex, size, or direction.
What Mastery Looks Like
- Given x²−y²−4x−2y−1=0, a student rewrites it as (x−2)²−(y+1)²=4. They identify a hyperbola centered at (2, −1) that opens left and right, and check that the forms are equivalent.
Common Misconceptions
- Students may forget to add a square-completion term to both sides. Some call every graph with two squared variables a circle, or read the center from (x−3)² as −3 rather than 3. A negative right side in a sum of squares means there are no real points.
How to Assess It
- Exit ticket: Rewrite x²+4y²−6x+8y+9=0 by completing the square. Name the graph, then give its center and horizontal and vertical semiaxes.
Lesson moves
Ways to Teach It
Plot x²+y²−4x+2y−4=0 on graph paper, then trace (x−2)²+(y+1)²=9 on transparent paper and overlay the graphs.
Ask partners why x²−y²=9 opens left and right, while y²−x²=9 opens up and down; have them use intercepts to explain.
Match equation cards to completed-square cards for a circle, ellipse, hyperbola, and parabola; explain each match to a partner.
Sketch a satellite dish cross section modeled by x²−8x−4y+12=0; rewrite the equation to find its lowest point.
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