Activity 1: The Guess (conjecture)

Draw a picture with a line segment AB representing the river.

Anywhere along one side of the river place any point labeled J, representing where John is standing.

Anywhere along the same side of the river, as John, place any point labeled T, representing the burning tent.

Think about the path John will run. He must first run to the river then to the burning tent.

Mark the point along the river that you think would make the shortest distance for John to run.

Label this point R.
Q1. What measurements could we make to be more accurate?
Measure and . Record the measurements of each angle on the board.
Q2. Make a conjecture on their relationship based on the class set of data.
Activity 2: The Mathematical Approach.

First draw the image of T by reflecting T in the line AB. Label the image of T as T’.

Now draw the line JT’ with a dotted line. Label the intersection of and as E.

Measure the total distance from John to this intersection point, to the tent (that is measure the distance from J to E and add that to the distance from E to T).

Measure the angles . Record the measurements.
Q3. What is the relationship of the angles?
Q4. Make a conjecture as to what you think point E is?
Measure the path John takes to E to the tent. How was your first guess?
The Shortest Mirror Problem
What’s the shortest mirror you’d need on the wall in order to see your full reflection from your toes to the top of your head?
Activity 3: Constructing Paths Light Travels

Draw two vertical parallel line segments to represent you and a mirror.

Label the endpoints of one line segment with “Top of Head” and “Toes”.

Place a point below the top of your head and label it “Eye”.
Draw One Beam of Light by following the steps below:

Reflect the point representing the top of your head across the line segment representing the Mirror.

Draw the line segment from the image to your eyes.

Label the intersection of this line segment with the Mirror as T.

Draw the path a beam of light will take from the top of your head to your eyes.
Draw A Second Beam of Light by following the steps below:

Reflect the point representing your toes across the line segment representing the Mirror.

Draw the line segment from the image of your toes to your eyes.

Label the intersection of this line segment with the Mirror as B.

Draw the path a beam of light will take from your toes to your eyes.
Q5. What will you see between T and B when you look in the mirror?
Measuring

Measure the distance between T and B. Record it on the board.

Measure and record the distance between the top of your head and your toes (in your picture—not for real).
Examine the class data.
Q6. What relationship is there between a person’s height and the distance between T and B?
Generalization
What is the shortest mirror a person n feet tall will need in order to see their full reflection?
Share with your friends: