Crossing Paths Slides Crossing Paths
Finding and Verifying Intersection Points
Algebra I Intervention
Today's Mission
1
Find where a line crosses a curve (parabola or circle) on a graph.
2
Verify that our "visual guess" is mathematically correct using algebra.
"Where do they touch?"
What is an Intersection?
2 Points
The line passes through the curve.
1 Point
The line "kisses" the curve at one spot.
0 Points
The lines never meet.
The "Look and Check" Strategy
1
LOOK
Find the \( (x, y) \) coordinate where they touch.
2
PLUG IN
Substitute the numbers into BOTH equations.
3
VERIFY
Does it make BOTH statements true? \( (5=5?) \)
Algebra is just a way to prove what your eyes already saw!
Example 1: Parabola
Step-by-Step
EQUATIONS:
\( y = x^2 \)
\( y = x + 2 \)
Let's LOOK:
At what coordinates do they cross?
Think Aloud:
If I put 2 in for \(x\), do I get 4 for \(y\) in both math sentences?
Look at points!
The Algebra Check
Testing Point \( (2, 4) \):
\( y = x^2 \)
\( 4 = (2)^2 \)
\( 4 = 4 \) YES!
\( y = x + 2 \)
\( 4 = 2 + 2 \)
\( 4 = 4 \) YES!
Since both are True , \( (2, 4) \) is a valid intersection point!
Example 2: Circle
EQUATIONS:
\( x^2 + y^2 = 25 \)
\( y = 3 \)
A circle centered at (0,0) and a flat horizontal line.
"Where does the line \(y=3\) cut through the circle?"
0, 0
Your Turn!
Open your activity sheets.
"Look, Plug, Verify!"
Crossing Paths Worksheet Crossing Paths
Algebra I Intervention: Systems of Equations
Name:
Date:
Today's Objective
Find the intersection points \( (x, y) \) on a graph and verify them algebraically using substitution.
1
Visual Detective
A. Locate the Intersection Points
Where do they cross?
Point 1:
(___, ___)
Point 2:
(___, ___)
B. Locate the Intersection Points
Where do they cross?
Point 1:
(___, ___)
2
The Algebra Proof
Let's verify the points you found in Problem A above.
System of Equations:
\( y = x^2 - 1 \)
\( y = x + 1 \)
Reminder
Substitute your \(x\) and \(y\) values into both equations. If both results are equal, your point is correct!
Check Point 1: (___, ___)
Equation 1: \( y = x^2 - 1 \)
Equation 2: \( y = x + 1 \)
Check Point 2: (___, ___)
Equation 1: \( y = x^2 - 1 \)
Equation 2: \( y = x + 1 \)
3
Challenge: Predict and Draw
A parabola is already graphed below for \( y = x^2 \).Task: Draw the line \( y = 4 \) on the graph. Where do they meet?
Predicted Points:
1. (___, ___)
2. (___, ___)
Show your verification here:
Crossing Paths Exit Ticket Exit Ticket: Intersection Check
Name:
The Goal:
Identify the intersection points between the circle and the line, then verify one of them using the equations provided.
1 Find the Points
Intersection A:
(___, ___)
Intersection B:
(___, ___)
2 Verify Your Work
Equations
\( x^2 + y^2 = 25 \)
\( y = 4 \)
Show the math check for one point:
Self-Check: How confident do you feel about finding intersection points?
🤔
Still confused
👍
Getting it
🎯
I've got this
Crossing Paths Teacher Guide Teacher Guide
Lesson: Crossing Paths (Systems Intervention)
Standard
CO HS.A-REI.C.7
Instructional Purpose
This Tier 2 intervention is designed for students who struggle with the abstract nature of systems of equations. By grounding the algebraic verification in a "Visual Detective" approach, students build a mental model of what a solution actually is (the physical point where two shapes touch) before engaging in the formal algebra.
Common Misconceptions
Coordinate Swap: Students may swap \(x\) and \(y\) during substitution.Single Solution Bias: Students may find one point and assume they are finished, missing the second intersection of a parabola or circle.Algebraic Disconnect: Seeing the math check as a "chore" rather than a proof of their visual estimation.
Lesson Flow
05m
Hook & Visual Intro
10m
Guided Example 1 (Parabola)
10m
Guided Example 2 (Circle)
15m
Partner/Small Group Practice
05m
Exit Ticket
Facilitation Prompts
"Before we look at the equations, look at the graph. How many times does the line cross the curve?"
Purpose: Encourages students to predict the number of solutions (0, 1, or 2) before starting the algebra.
"If we find the point (2, 3), what does that 2 represent? What does that 3 represent?"
Purpose: Reinforces coordinate geometry and prepares them for accurate substitution into the \(x\) and \(y\) variables.
"The math says \(4 = 4\). Why is that a good thing for our 'Visual Detective' work?"
Purpose: Connects the identity \(a = a\) to the physical intersection on the graph.
Differentiation Strategies
Support (Scaffolding)
Provide a Substitution Template where circles and boxes are pre-drawn for \(x\) and \(y\).
Use Colored Highlighters : Highlight all \(x\)'s in one color and \(y\)'s in another on the worksheet.
Physical Trace: Have students use their finger to follow the path of the line until it "bumps" into the parabola.
Extension (Challenge)
Tangent Challenge: Ask students to find a horizontal line that only touches the circle at exactly one point (tangent).
