CCSS.Math.Content.HSG-GPE.A.3
The standard
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Common Core State Standards for Mathematics
What this standard means
Students need to connect a geometric definition to an algebraic equation. They should use distance formula from a point to each focus, set up a constant sum or constant difference, and simplify into the standard form for an ellipse or hyperbola. They also need to identify center, orientation, vertices, foci, and the meaning of a, b, and c.
Mastery looks like starting from a diagram or focus coordinates and producing a correct equation with clear reasoning. Students often mix up ellipse and hyperbola relationships, lose signs while squaring radicals, or forget that c is measured from the center to a focus, not from a vertex.
Ways to teach it
- Have students use string, pins, and graph paper to trace an ellipse, then measure distances to the two foci at several points.
- Ask students to explain why a constant sum makes a closed curve, while a constant difference makes two separate branches.
- Give two foci and one vertex, then ask students to write the equation and label a, b, and c in five minutes.
- Show how satellite dishes, whispering galleries, or GPS error regions use focus based curves, then match each situation to ellipse or hyperbola.
Plan a lesson for CCSS.Math.Content.HSG-GPE.A.3
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Related standards
- CCSS.Math.Content.HSG-GPE.A.2
Derive the equation of a parabola given a focus and directrix.
- CCSS.Math.Content.HSG-SRT.D.10
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
- CCSS.Math.Content.HSN-CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
- CCSS.Math.Content.HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.