Homeschool Lesson Plans: Math Puzzle: How can this be true?

How can this be true?

I saw this math puzzle in my Facebook newsfeed. I've seen a similar one before regarding splitting up a bar of chocolate. Similar, in some ways, but different. First, the problem:

Some students might be clever enough to solve this without any help. Wonderful to have such a keen mathematically savvy mind. For those who don't, we don't want to frustrate them. The page from which I've drawn this puzzle provides a hint. If that is not enough, they also provide a more specific second hint.

However, if we are using this puzzle as a homeschool math assignment, we might want to milk it for educational value.

What is the name of this shape?

Presumably, the child will say it is a triangle. (Which is wrong, but we'll address that later.)

What are the dimensions of the red triangle?

3 by 8

The green?
 2 by 5

Are the upper and lower red and green triangles the same? 
The dimensions are the same. The upper and lower red and green triangles are coincident.

Are the orange and lime green irregular shapes also coincident?

Yes.

How many square units does the orange shape contain?

7.

How many square unites does the lime green shape contain?

8.

How do you calculate the area of a rectangle?

A rectangle's area, the number of squares, is the height times the length.

How, then, do you calculate the area of a triangle?

A triangle's area is half the area of a rectangle with the same length and height. Basically, a triangle is a rectangle split evenly in half along the diagonal.

What is the area of the red triangle?

3 * 8 /2 = 12 cubic units. (Could be feet or inches, we don't know what the boxes represent.)

The green?

2 * 5 / 2 = 5 cubic units.

What would the top shape, ABC's area be if we used the formula for the area of a triangle?

13 * 5 / 2 = 32.5

What would the top shape, DEF's area be if we used the formula for the area of a triangle and subtracted one cubic unit for the gap?

13 * 5 / 2 = 32.5 - 1 = 31.5

Add together the individual areas previously calculated previously. What is the true total area?

Orange = 7
Lime = 8
Red = 12
Green = 5
Total: 32

The Associative Law says that it doesn't matter in what order we add number together. Therefore, it shouldn't matter how we shuffle the shapes around. The actual area of the shapes as determined by adding the areas of the component parts is 32. Both of our calculations for the area on top and on bottom in the diagram are off by .5 cubic units, one greater and one lesser.

Assuming the formula to calculate the volume of a triangle is correct, and it can be shown to be correct by logic, we have a situation where the correct formula for the volume of a triangle does not result in the value calculated for the area. Therefor, the shape is not a triangle even though it looks like one.

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. ~~ Sherlock Holmes, written by Arthur Conan Doyle  

The name of the diagonal line of a triangle is called the hypotenuse. We can express the hypotenuse as a ratio of the values of length and height. Therefore the hypotenuse of the red triangle is 3 high / 8 long. The hypotenuse of the green triangle is 2 high by 5 long. Dividing it out you get a ratio of 0.375 for the hypotenuse of the red triangle and 0.4 for the green triangle.

Confirmation Bias causes people to see what they expect to see. In the case above, we saw two shapes which were nearly triangular and assumed they were triangular. In actuality, the "hypotenuse" bows downward in the top figure and upward in the second every so slightly. This explains where the "hole" in the second figure comes from.

Of course the website from which I've pulled the puzzle provides their own solution, but I think it is nice to draw the children through the logic slowly. :) Hope your students enjoy! Now that they've worked through this logic problem, they might want to visit the infinite chocolate bar problem referenced previously.

Problem

Solution