Factoring Money

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We had a great kick off to Mathletes this year with the children very excited about math, their relative mastery of number, and the possibilities for growth this year.

We walked through the different denominations of Mathlete Dollars and the famous mathematicians that are on each. We even looked at a 50 Dollar note that had a picture of one of our students. She solved a problem better than I could last year. The other children are excited about this prospect.

Counting money with the same denomination is a perfect way for them to be introduced to factoring and multiplication. We did not focus on the language of "factoring" and "multiplication" yet for the younger groups because students who do not "think" that they know these concepts will reject the ideas. Next week, once they show some success, I let them know that they have been factoring and multiplying all along. 

NOTE: SOME OF THE KINDERGARTENERS AND 1ST GRADERS MAY ONLY WANT TO FOCUS ON FINDING FACTOR PAIRS FOR $10, $20, $50, AND MAYBE $100. THE OTHERS MAY WANT TO GO TO $500, $1,000, OR EVEN $10,000. AND OF COURSE, SOME WILL PUSH THEMSELVES BEYOND.

THE IMPORTANT THING IS THAT THEY WORK AT A COMFORTABLE PACE.

We start with building $10. Of course, we can have:

 I recommend that you stick with whole numbers if your child is not ready for quarters, dimes and nickles. Of course, they will be soon as we will be exploring fractions in almost every class.

The children noticed that for each factor pair (for example, 2 and 5) we also have the reverse (5 and 2). I did not let them know that this is a very important property of mathematics called "commutative property." I will introduce that vocabulary next week when they have mastered the concept. This property holds for addition and multiplication, not subtraction and division.

The children also noticed the following concepts when looking at the change:

<!--[if !supportLists]-->•  <!--[endif]-->since there are four quarters in a dollar, simply multiply 4 by 10 to get 40

<!--[if !supportLists]-->•  <!--[endif]-->since there are five nickles in every quarter, we simply multiply the number of quarters by five or add five of those numbers to get 200.

<!--[if !supportLists]-->•  <!--[endif]-->since there are two nickles to every dime, they were able to take half of the nickles to figure out the number of dimes, 100.

<!--[if !supportLists]-->•  <!--[endif]-->of course, some children approached this problem differently, and they were very proud of their methodologies.

 My method of factoring which I will teach them for all whole numbers is simply based on doubling and halving the factor pairs. For example, looking at factoring $20, you simply double the first number and take half of the second number.

If you are factoring 100:

Now, we notice that if we doubled 4 to 8 and took half of 25 which is 12 and 1/2, that would still work, but we are no longer dealing with whole numbers. Mixed numbers like 12 and 1/2 are not factors. However, they found it fascinating that 8 times 12 and 1/2 is still 100, 16 times 6 and 1/4 is still 100 and so on.

Some children quickly picked up on missing factor pairs like 5 20s. When you double the 5 and take half of the 20, you get 10 10s. They also picked up on the fact that using the reverse property (commutative) would not add a factor pair. So the whole number factors of 100 and their reverse pairs are:

The non-whole number pairs are:

 Some students noticed that these numbers were exactly 10 times the number of quarters, dimes and nickles of the $10.

 I have attached pdfs of the worksheets handed out in class as well as money sheets which have 12 of the same type of bill per page. There are denominations of 1,2,5,10,20,50, and 100.

The additional challenge this week is to add up all of the bills and give the total dollars. Some children will add each denomination and then multiply by 12; for the younger children, it may be better to have them add up each page first, then add the totals (for example, 12 + 24 + 60 + 120 .....). For the kindergarteners and some 1st graders, it would be a great challenge just total each page and write the sum on the bottom of each page.

I highly recommend that each child cut out these bills individually to have 84 separate bills to use to play math games. Of course, some children will want to print out multiples of each page. Buy lots of ink for these children.

Enjoy!!!

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