Conic Intervention Guide Conic Intervention Guide
Small Group Tier 2 Intervention: Ellipses & Hyperbolas
Standard
G-GPE.A.3
Objective
Students will derive the equations of ellipses and hyperbolas centered at the origin given the foci by applying the geometric definitions (constant sum or difference of distances).
Pacing
Warm-up: Distance Formula Review (5 min)Instruction: Ellipse Derivation (15 min)Guided Practice: Hyperbola Setup (15 min)Assessment: Exit Ticket (10 min)
Intervention Focus
Tier 2 students often struggle with the algebraic stamina required for conics. Focus on the conceptual transition from "geometric distance" to "algebraic equation."
Common Pitfall: Squaring the binomial incorrectly during the radical removal process.
Support Strategy: Use color-coded radicals to help students track which part of the equation they are isolating.
Instructional Script
Phase 1: The Ellipse (The Sum)
"Imagine a string tied between two pins (foci). If we pull a pencil tight against that string, the total length of the string doesn't change. That constant length is \(2a\). Let's write that as an equation."
Prompt: "If the distance to Focus 1 is \(d_1\) and the distance to Focus 2 is \(d_2\), what is the total? How do we write \(d_1\) using the distance formula?"
Phase 2: The Hyperbola (The Difference)
"Now, instead of a constant sum, the hyperbola is about a constant difference . No matter where we are on the curve, the distance to one focus minus the distance to the other is always the same: \(|d_1 - d_2| = 2a\)."
Prompt: "How does the equation look different from the ellipse? Why do we use absolute value here?"
Progress Monitoring
Check Point Success Criteria Warm-up Correctly identifies \(x_1, y_1\) in distance formula. Setup Writes the radical equation \( \sqrt{} + \sqrt{} = 2a \). Algebra Isolates one radical before squaring.
Scaffolding Tips
MOD
Provide a pre-filled template for the radical expansion (e.g., \((x-c)^2 + y^2 = ...\)) to reduce cognitive load on basic algebra.
VIS
Use different colors for the 'x-movement' and 'y-movement' in the coordinate plane.
Conic Blueprints Slides Geometic Blueprint
Conic
Derivations
Building the equations for Ellipses and Hyperbolas from their geometric roots.
Project
G-GPE.A.3 Intervention
Target
Sum/Difference Definitions
The Ellipse Definition
Geometric Rule #1
The set of all points where the sum of the distances to two fixed points (foci) is constant .
\(d_1 + d_2 = 2a\)
Focus 1 Focus 2 P(x, y) d₁ d₂
The Algebraic Blueprint
The Variables
F Foci at \((\pm c, 0)\)
P Point \((x, y)\)
a Semi-major axis
Starting Equation:
\[ \sqrt{(x-c)^2 + y^2} + \sqrt{(x+c)^2 + y^2} = 2a \]
Next Step:
Move one radical to the other side before squaring!
Key Goal:
Eliminate both radicals to find the standard form.
The Hyperbola Difference
Geometric Rule #2
F₁ F₂ P d₁ d₂
The set of all points where the difference of the distances to two fixed points is constant .
\(|d_1 - d_2| = 2a\)
The equation setup looks the same, but with a minus sign between the radicals!
Critical Think
If the distance between the foci is \(2c\) and the constant sum is \(2a\)...
What happens to the ellipse if \(c\) gets closer to \(0\)?
A. It becomes a circle B. It becomes a line C. It becomes wider
Conic Derivation Blueprint Conic Blueprint Worksheet
Project: G-GPE.A.3 Intervention
Student:
Date:
1
Distance Review
Calculate the distance between point \(P(x, y)\) and Focus \(F(3, 0)\). Use the distance formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
2
Deriving the Ellipse
Definition: Constant Sum \(d_1 + d_2 = 2a\)
A
Write the equation using the distance formula for Foci \(F_1(-c, 0)\) and \(F_2(c, 0)\).
\(\sqrt{(x+c)^2 + y^2} + \) __________________________ \(= 2a\)
B
Isolate one radical. (Move the second radical to the right side).
\(\sqrt{(x+c)^2 + y^2} = \) ________________________________________
C
Square both sides and simplify. (Show your work below)
3
The Hyperbola Blueprint
A hyperbola is defined by a constant difference of distances.
Write the starting equation for a hyperbola with foci at \((\pm 5, 0)\) and constant difference \(2a = 8\).
What is the first algebra step you would take to solve this?
Conic Exit Ticket Exit Ticket: Conic Check
Deriving Equations from Definitions
Name
Date
1
An ellipse has foci at \((-4, 0)\) and \((4, 0)\) with a constant sum of distances \(2a = 10\).
Write the initial equation using radicals that represents this definition.
2
Identify the Error: A student is deriving a hyperbola equation and writes:
\[ \sqrt{(x+3)^2 + y^2} + \sqrt{(x-3)^2 + y^2} = 6 \]
Explain why this setup is incorrect for a hyperbola and what should be changed.
3
Reflect: Why do we isolate one radical before squaring the equation?
Mastery Check
How do you feel about today's goal?
Conic Blueprint Key Blueprint Answer Key
Conic Derivation Blueprint (Teacher Reference)
Subject
Geometry
1. Distance Review
Distance between \(P(x,y)\) and \(F(3,0)\):
\(d = \sqrt{(x - 3)^2 + (y - 0)^2} = \sqrt{(x - 3)^2 + y^2}\)
2. Ellipse Derivation Steps
A.
Setup Equation
\( \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a \)
B.
Isolate Radical
\( \sqrt{(x+c)^2 + y^2} = 2a - \sqrt{(x-c)^2 + y^2} \)
C.
Square Both Sides (First Expansion)
\((x+c)^2 + y^2 = 4a^2 - 4a\sqrt{(x-c)^2 + y^2} + (x-c)^2 + y^2\)
Cancel shared terms:
3. Hyperbola Setup
Starting Equation
\( | \sqrt{(x+5)^2 + y^2} - \sqrt{(x-5)^2 + y^2} | = 8 \)
(Note: Difference is \(2a=8\), Foci at \(c=5\))
First Algebra Step
Isolate one radical by adding it to the other side:
