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Distributive Property and Percents

May 10, 2012

It was a great class. We were examining how to calculate the taxes and discounts on goods. Pretty simple stuff that all of the students were familiar with. After all who does not go the store to buy things and find out they are a little short on cash due to some taxes they did not figure into the final price. We started with a problem that went something like this … “Jim goes to the store to buy an iPod. If the iPod is $150.00 and tax is 12% what is the final cost?”

Most students recognized the need to calculate the tax and add it to the cost to determine total cost.

Total Cost = Original Price + Tax

However there were some that recognized that you could simply multiply by 112% (1.12) to find the total cost.

Total Cost – Original Price (1.12)

I asked the students why this worked and everyone was slow to answer. Then suddenly from the back of the room one of the students said “Distributive Property.”

I asked if he knew why it worked and he was pretty honest, “No Mr. DeMerchant, it was just the thing I remembered.”

This has me thinking of the application of the distributive property [e.g. a(x+y) = ax+ay ] in elementary and middle school mathematics. Early multiplication models and the development of number facts start the journey of the distributive property through the math curriculum. Repeated addition is in itself an application of the distributive property (e.g. 7×4 = 7(3+1)=7(2+1+1)=7(1+1+1+1)=28) although it is not often talked about in these terms. The applications of this extend into geometry, algebra and other mathematical topics.

In Grade 4 of the Western and Northern Canadian Protocol students use arrays for multiplication of numbers:

5 x 12 = 5×10 + 5×2 = 50 + 10 = 60 or

4 x 341 = 4×300 + 4×40 + 4×1 = 1200 + 160 + 4 = 1364

Mental mathematics comes much easier when the distributive property is used. I was witness to this when the class was working out the iPod problem.

Calculating $150 x 1.12 can be difficult to do in your head.

However, if you change the problem using the distributive property it becomes much easier

$150(1+0.1+0.01+0.01)=$150+$15+$1.50+$1.50=$168.00

As I tell my students the rules of mathematics do not change and in a later post I hope to expand on other applications of the distributive property which occur in the curriculum. Analyzing mathematical problems and applying the distributive property can tie parts of mathematics curriculum together.

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2 Comments leave one →
  1. May 11, 2012 7:41 am

    Indeed when you do “regular” multiplication, you use the distributive property. When you “carry” the extra to add it on, you’re merely doing some addition mixed into the process.

    Thanks for sharing this!

    • May 11, 2012 6:26 pm

      Indeed. I had a much longer post but decided to split this one up. The more I looked at the distributive property the more I saw. Pretty cool. Thanks for the comment.

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