Treasure Hunt
The Midpoint Formula
OBJECTIVES
In this lesson you will:
 Derive the Midpoint Formula.
 Use the Midpoint Formula.
KEY TERMS
 midpoint
 Midpoint Formula
Ms. Lopez is designing a treasure hunt for kindergarten children. The goal of the treasure hunt is for children to learn direction (right, left, forward, and backward) and introduce distance (near, far, in between). The treasure hunt will be on the school playground.
PROBLEM 1
Plotting the Treasure Hunt
Ms. Lopez drew a model of the playground on a grid as shown. She uses this model to decide where to place items for the treasure hunt, and to determine how to write the treasure hunt instructions. Each grid square represents one square foot on the playground.
 1. What are the coordinates of:

 a. the merrygoround


 The merrygoround is located at (11, 2).


 b. the slide


 The slide is located at (3, 2).


 c. the swings


 The swings are located at (3, 12).

 2. Determine the distance, in feet, between the merrygoround and the slide. Show all your work.

 11 3 8

 The merrygoround and the slide are eight feet apart.
 3. Ms. Lopez wants to place a small pile of beads in the grass halfway between the merrygoround and the slide. How far, in feet, from the merrygoround should the beads be placed? How far, in feet, from the slide should the beads be placed?
 The beads should be placed four feet from the slide and four feet from the merrygoround.
 4. What should be the coordinates of the pile of beads? Explain how you determined your answer. Plot and label the pile of beads on the coordinate plane for Questions 1 through 4.
 The pile of beads will be located four feet to the right of the slide, so its coordinates will be (7, 2).
 5. How do the coordinates of the pile of beads compare to the coordinates of the slide and merrygoround?

The \begin{align*}y\end{align*}
y− coordinates of all three items are the same. The \begin{align*}x\end{align*}x− coordinate of the pile of beads is 4 units greater than the \begin{align*}x\end{align*}x− coordinate of the slide and is 4 units less than the \begin{align*}x\end{align*}x− coordinate of the merrygoround.  6. Ms. Lopez also wants to place a pile of kazoos in the grass halfway between the slide and the swings. What should the coordinates of the pile of kazoos be?
 Explain your reasoning. Plot and label the pile of kazoos on the grid.
 Distance between swings and slide: 12 2 10
 Distance between slide and pile of kazoos: 5

The pile of kazoos will be 5 feet greater on the \begin{align*}y\end{align*}
y− coordinate than the slide, so its coordinates will be (3, 7).  7. How do the coordinates of the pile of kazoos compare to the coordinates of the slide and swings?

The \begin{align*}x\end{align*}
x− coordinates of all three items are the same. The \begin{align*}y\end{align*}y− coordinate of the pile of kazoos is 5 units greater than the \begin{align*}y\end{align*}y− coordinate of the slide and is 5 units less than the \begin{align*}y\end{align*}y− coordinate of the swings.  8. Ms. Lopez wants to place a pile of buttons in the grass halfway between the swings and the merrygoround. What do you think the coordinates of the pile of buttons will be? Explain your reasoning. Plot and label the pile of buttons on the shown grid.

The coordinates of the pile of buttons should be (7, 7). The \begin{align*}x\end{align*}
x− coordinate is halfway between the \begin{align*}x\end{align*}x− coordinates of the slide and merrygoround, and the \begin{align*}y\end{align*}y− coordinate is halfway between the \begin{align*}y\end{align*}y− coordinates of the swings and slide.  9. How far, in feet, from the swings and the merrygoround will the pile of buttons be? Show all your work and explain how you determined your answer.
 Round your answer to the nearest tenth if necessary.

Let \begin{align*}d\end{align*}
d be the distance between the swings and the merrygoround. 
\begin{align*}(3 \quad 11)^2 \quad (12 \quad 2)^2 \quad d^2\\
( \ 8)^2 \quad 10^2 \quad d^2\\
64 \quad 100 \quad d^2\\
164 \quad d^2\\
12.8 \quad d\end{align*}
(311)2(122)2d2( 8)2102d264100d2164d212.8d  The distance between the pile of buttons and either piece of equipment will be about 12.8 2 6.4 feet.