Proofs/Formulas With A Memo Pad And Some Detective Work (all grades) Post 1 of 3
How much do we value information that is already packaged and ready for us? Obviously it depends but when it comes to formulas and proofs I believe your students will value their work much more if they value the formula/proof. One way to implement this is to take the formula away from them and make them derive it themselves. Lets start with something basic and then latter we can move on to something more challenging.
Since we have readers from all over the spectrum of mathematics, I would like to provide three examples of deductive reasoning for elementary, middle and high schools stretched out over three posts
The Right Triangle and the Rectangle:
The goal of this exercise is for your students to informally prove [to you] the area formula of a right triangle is true
What Each Student Will Need
- Two pieces (two colors) of construction paper
- Tape or glue
- Memo Pad (can be made from stapling scrap paper or using a post-it pad)
The Right Triangle and the Rectangle:
- Write At Top Of The Memo Pad “What Mr. [fill in the blank] Accepts To be True”
- Write the formula for area of a triangle on the board
- Ask the students to read the formula out loud
- Tell them you don't believe them. Ask them to prove it!
- For the first 5 minutes allow them to play, allow them to experiment ways to prove the formula is true. Ask them to share their ideas with each other and you. Applaud each idea but relay to them that it's still not enough proof. Tell the class that just like a detective doesn't always believe the witness, we will need to build up a case by starting with what we do believe.
- Using the materials above, ask each student to draw a 4 by 6 inch rectangle on both pieces of construction paper
- Ask them to calculate the area of the rectangle by multiplying its length by its width (4x6 = 24inches squared).
(Tell them that you accept that the area of a rectangle is the length multiplied by its width but you don't accept the area of a triangle is ½ the base times its height. We are going to add what it is you accept to be true in their memo pad)
- Add The Following To Your Memo Pad
- The area of a rectangle is its length multiplied by its width(L x W)
- Next, ask them to label the width and the length of each rectangle
- Now propose the question of what the area would be if you cut the rectangle in half (The answer should be 12 inches squared)
- Ask them to prove it by cutting one of the rectangles in half either horizontally or vertically through the middle of the rectangle as shown below and calculating the new area (will be either 6x2 or 4x3 = 12)
- Tell them you now believe them, that a rectangle cut in half will yield half the area and add this to your set of Axioms
- A rectangle cut exactly in half will yield two pieces, both with half the original area
- Now pose the question, is it possible to get a triangle from half a rectangle? Is it possible to get two triangles of equal size from exactly half a rectangle?
- Ask them to prove it.
- Ask them to cut the other rectangle (on the other piece of construction paper) in half as well but using the following criteria
- The rectangle must be cut in half
- Each piece must be the same shape and size
- The resulting shapes must be two triangles
- The students should quickly figure out that if they cut a rectangle in half diagonally they meet each of the criteria above.
- Add this to your set of axioms
- It is possible to get two triangles of equal size from half a rectangle
Now comes the hard part! Putting it all together and making the students see the solution. Let me show you how I might play-out this scenario if it were my classroom.
Me: “OK!, I get it!...you guys were right. You've proved the area for a right triangle is 1/2(b)(h). Let's move on to something else”
Class: “Wait, what?...What are you talking about? Show us”
Me: “Well, we agree that
- Area of a rectangle is Length x Width right?
(Draw a 4 x 6 rectangle on the board and label the area 24 inches squared)
Me: “We've also agreed that
II.A rectangle cut in half will yield two pieces, both with half the original area.
(Cut the rectangle in half diagonally and label each side 12 inches squared)
Me: “And finally, we agree that
- It is possible to get two triangles of equal size from half a rectangle.
Me: “Well, a Right triangle is nothing more than rectangle cut in half diagonally--There's the ½ part of the equation. And the base and height are simply new names for the length and width. Thus, it makes sense that ½(b)(h) is the area of a right triangle!