Why+does+Inverting+&+Multiplying+Work

In order to help students understand why this algorithm works, they need a contextually rich and applicable problem. Some common examples are cake servings, or pieces of ribbons.


 * The Problem**

You like to bake a lot of cake, and like to serve large servings. You decide to bake 4 cakes, and to serve servings of 2/5 of a cake. How many servings will you have? (It will be more easily seen if the dividend is a whole number. I also like the quotient to be a whole number the first time.)


 * The Lesson**

Before students start to solve this problem, I like to have them draw it. It helps if we have spent some time drawing whole number division. A student may draw this by drawing 4 long narrow cakes, and then cutting them into fifths. They then would circle groups of 2/5 of a cake. They find there are 10 servings. Below is an example of this drawing.



Once students have the drawing we write the mathematical equation of what we are calculating on the board as a class. We label each part of the equation. In this case we write 4 cakes ÷ 2/5 cake per servings = 10 servings. The labeling is important for an extension later. We then calculate the quotient by using the invert and multiply algorithm 4/1 * 5/2. At this point they don't know why. I then ask some questions.
 * Looking at the picture, where did we multiply by 5? (//Answer-when we cut the cakes into 5 pieces each//)
 * Looking at the picture, where did we divide by 2? (//Answer-when we counted groups of 2 parts of the cake. Each cut represented 1/5 of a cake//)


 * Extension**

Once students begin to understand why they divide by the divisors denominator another similar problem can be given that does not include an even number of groups. (The quotient is a fraction.)

The problem can now be something like, "You make 3 more cakes an serve the same size serving as before (2/5 of a cake). How many servings do you get now? It is important that students draw the situation as before. The power in this exercise is they see the math they perform in the picture they draw. They draw 3 cakes, cut them into fifths, and count groups of 2. This time they have 1 of those partitions left over. As their answer many student will write 7 1/5, while others will write the correct answer 7 1/2. For me when I see this it is exciting! What a great conversation is going to follow. We eventually go back to the picture and write out the mathematical equation again 3 cakes ÷ 2/5 cake per servings = (7 1/2 or 7 1/5 servings?) I love it when I see it click for a student when they say, "Oh! You have 1/5 of a cake left over, but we are counting servings. 1/5 of a cake is 1/2 of a serving. So the answer is 7 1/2 servings." We also decide that you could say "You get 7 servings with 1/5 of a cake left over."




 * Assessment**

To see if students really understand this problem, give them another easily draw situation. Ask them to draw it, and write out the mathematical equation labeling what each number stands for. When they show their work for inverting and multiplying, have them answer where they see the multiplication in the problem, and where they see the division.

For example ask It takes 3/8 of a gallon of gas to mow a lawn. If the tank of lawn mower holds 5 gallons, how many times can you mow the lawn? You will look for the following to check students understanding.
 * They some how draw 5 gallons of gas divided into eights of a gallon.
 * They circle 3 groups of eighth of a gallon at at time.
 * The count the number of times the lawn can be mowed. (13)
 * They recognize they have an eighth of a gallon left over, or 1/3 of a mowing. (13 1/3 mowings)
 * They write out the division problem and label the parts. (5 gallons ÷ 3/8 gallons per mowing = 13 1/3 mowings)

If the concern is more having the students show they understand what is happening with the division and how it works, the next attached worksheet may be useful.