Divisible by Three - Simple Math Manipulatives

We made units, tens and hundreds and discovered if the sum of the digits in a number are  divisible by three, then the number is divisible by three.

More than five years ago I constructed these units, tens and hundreds from card stock paper by simply measuring centimeters and cutting. They are one of the most useful visual math manipulatives we have.

They have been extremely helpful in learning addition with carrying numbers, subtraction with borrowing, and place value. Most recently they were used to discover a trick in determining whether a number is divisible by 3.

First, several two and three digit numbers known to be divisible by 3 were displayed using the units. In the picture above moving from right to left, bottom to top the numbers 12, 15, 18, 21, 24, 27, 30, 33, 36, 120 and 141 are represented.

Since no pattern was initially obvious, I asked my son to count the number of pieces of paper in each pile ignoring the fact they represented ones, tens or hundreds. Moving from right to left, bottom to top the numbers of pieces of paper shown are 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, and 6.

Using this process my son noticed a theme and upon further thinking was able to deduce that if the sum of the digits in the number were divisible by three, then the number was divisible by 3. 
My son is a child who learns best by figuring things out for himself. Little information is retained when I explain something, especially in regards to mathematics. Although I lead him in this activity, he discovered the rule making this activity very effective for his learning style.

See our math page for more hands-on math ideas and activities.

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  1. This is the best explanation for divisibility by three that I've come across. Tiger would love to learn like this. I'm bookmarking this exercise for later. Thank you for sharing, Julie. What a fantastic way to learn!

    1. Thank you and I hope Tiger has fun discovering.

  2. I am pinning this for when my kids are older. So glad I found your blog! :)

  3. I LOVE this!! My other passion is math and you teach exactly the way I think math should be taught!!! I can only imagine his excitement as he started noticing the pattern. Have you ever heard of number clocks? You draw a circle and place the digits 0-9 evenly around the circle. Then choose a number and begin connecting the digits based on the ones digit of each of that number's multiples (i.e. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33) would be (3, 6, 9, 2, 5, 8, 1, 4, 7, 0, 3) Kids love seeing the patterns in this way and then trying to predict the patterns of new numbers. (If my explanation makes no sense, let me know and I'll try to find a way to send you a visual.)

    1. Yes I've heard of number clocks. In fact I did a post on them, but called them multiplication circles. Here's the link. http://highhillhomeschool.blogspot.com/2013/08/multiplication-circles.html

      We also did an activity making Christmas Stars that sounds even closer to your description. I will write about that in November or December as we get closer to Christmas. (I like seeing the star patterns as they emerge too.)

  4. Love that you used different colors for the ones, tens, and hundreds. Great for those visual earners!
    A Tall Drink of Water

  5. This is great! Kudos to you for making the manipulatives yourself too. This is exactly the way math needs to be explored.
    Fourth Grade Flipper

  6. Love your manipulatives! I can definitely see myself making something like them one day =) Thanks for the great ideas!

  7. What a great set of easy to make Manipulatives - I need to make some very soon as we are starting to look at 10's and 100's. Not quite at the divisible by yet.....

    Thank you for linking up to Tuesday Tots and just letting you know that I will be featuring this over on Rainy Day Mum this week


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