Midpoint & Distance
Table of Contents
- Introduction (For Teachers)
- Purpose
- Relevant Standards
- Warm-Up (Students Start Here!)
- Warm Up
- Midpoint
- Midpoint Visualization
- Discovering the Midpoint Formula
- Midpoint Formula Applications Pt. 1
- Midpoint Formula Applications Pt. 2
- Midpoint in 3D
- Distance Formula
- Copy of Coordinate Plane: Distance Visualization
- Coordinate Plane: Distance Formula - Pythagorean Theorem
- Copy of Application problem: Resource Center
- Copy of Extension Application problem (Open-ended): Resource Center
- Copy of Distance in 3D
- Exit Ticket
- Distance Formula: Exit Ticket
Purpose
This book goes through a visual and algebraic discovery of both the midpoint formula and the distance formula. It contains review materials like the Pythagorean Theorem and finding a mean in the warm-up, and some closing questions in the exit ticket. The book also contains challenge problems that extend these topics to 3D. [br][br]We believe this book would work well for an advanced activity for 8th grade math, a "grade-level" activity for high school geometry, or a bit of review for students who are past standard high school geometry and more interested in how to apply these topics in 3D.
Warm Up
Previous Knowledge
[b]What do you remember about the mean? How would you calculate the mean of two numbers?[br][br][/b]Hint: sometimes you may have heard the "mean" be referred to as an "average".
Practice
[b]What is the mean of 3 and 11? [br][br][/b]Hint: use the applet below to drag the point in the middle around until it's halfway between 3 and 11.
Previous Knowledge
[b]What do you remember about the Pythagorean theorem? When can it be used & what is it used for? [/b]
In the applet above, the areas of the purple and the blue squares add up to the area of the red square. The side length AC is 4 and if we square that we get the purple area of 16. The side length AB is 3 and if we square that we get the blue area of 9. Lastly, the side length BC is 5 and if we square that we get the red area of 25. [br][br]The length of the legs (the shortest sides, those adjacent to the right angle) in a right triangle are A & B in the Pythagorean Theorem [math]A^2+B^2=C^2[/math]. C is the length of the hypotenuse (the longest side, opposite the right angle).
Practice
[b]Find the hypotenuse in a right triangle with legs of 5 and 6 inches. Round your answer to 2 decimal places.[/b]
Midpoint Visualization
[b][size=150]A midpoint is a point on a line segment that is equidistant from both endpoints. [/size][/b]
Move points A, M, and B around to create a setup where M is the midpoint of A and B. Then use the checkbox and slider to check your work. NOTE: DON'T USE POINTS THAT ARE ON A HORIZONTAL OR VERTICAL LINE - THE HINT WON'T WORK.
How you know that your choices of locations for A and B (above) are good? Justify, illustrate, or explain below.
Move points A, M, and B around to create an ENTIRELY DIFFERENT SETUP that works!
How you know that your choices of locations for A and B (above) are good? Justify, illustrate, or explain below.
Move points A, M, and B around to create an ENTIRELY DIFFERENT SETUP (from the two you made above) that works!
How you know that your choices of locations for A and B (above) are good? Justify, illustrate, or explain below.
Copy of Coordinate Plane: Distance Visualization
[b][size=150]In this activity, we are going to explore the movement of the two points on the coordinator plane. You could move the two black points anywhere and test the movement between the two points.[br][/size][/b][size=200][b][size=150]Press MOVE. Then CONTINUE.[/size][/b][/size]
[b][size=150]Problem 1-1: What do you notice? What do you wonder?[/size][/b]
Problem 1-2: In this app, position the two points to create a right triangle whose legs measure 5 units and 8 units.
Problem 1-3: Here, place the runner's initial point at (1,-4). Place the other point at (-5,3). Then press the buttons (like before) to create a right triangle.
[size=150][b]Problem 1-4: Based on the above activities, what do you notice about distance "[/b][i]d[/i][b]"? [/b][/size]
[size=150][b]Problem 1-5: What method or concept could you use to find the distance "[/b][i][b]d"? What is the related formula? [/b][/i][/size]
Distance Formula: Exit Ticket
[size=150][b] The formula gives the distance between two points A (x[sub]1[/sub], y[sub]1[/sub]) and B (x[sub]2[/sub], y[sub]2[/sub]),[br][/b] [math]AB=\sqrt{\left(\varkappa_1-\varkappa_2\right)^2+\left(y_1-y_2\right)^2}[/math][/size]
[size=150][b]If the distance between the points A (5, - 2) and B(1, a) is 5,[br][/b][list=1][*][size=150][b]Determine the value a, and show your work on the applet;[/b][/size][/*][*][b]Please locate points A and B on the coordinate plane.[/b][/*][*][b]Display the distance line between A and B on the coordinate plane;[/b][/*][/list][/size]