Saturday, June 8, 2013

Extending Multiplication to Larger Numbers



The transition from additive thinking to multiplicative thinking is very difficult and abstract for some kiddos.  Think about it, for as long as they've been working with numbers, the number 2 has represented two things.  Two ducks, two cookies, two linking cubes, etc.  Now, with multiplication, 2 x 8 means 2 groups of 8.  That's huge.  So, of course, it's important that students get LOTS of concrete practice with the concept of multiplication--building 2 groups of 8.  And then they should represent that same idea by drawing it (The representational stage of CRA).

Once students have a strong understanding of the meaning of multiplication, we can extend that understanding by bringing in place value concepts and using multiples of 10.  This is an essential understanding that must be in place prior to moving to multi-digit multiplication.

A great way to start this journey is with a version of Count Around the Circle from the book Number Sense Routines, by Jessica Shumway.  Students sit or stand in a large circle.  Start out skip-counting by a single digit number, for example 2.  Go around the circle, and each student says the next multiple of 2--2, 4, 6, 8, 10, etc.  Next, do the same routine with students counting by 20s instead.  You want them to see that it's the same pattern.  Feel like a challenge?  Try 200s.  Make this a regular routine in your classroom.  It's a great way to practice multiplication facts and to stretch students to think about larger numbers.

Next it's time to bring all of your CRA stages together.  I created a little work mat with number cards that you can use for small group instruction.

I printed the first page of numbers, which are the factors, on pink card stock, and the last two pages, the products, on yellow card stock.  There are a LOT of numbers, so you'll want to pick out the set of numbers you want to work with.  In the example above, I'm using the variations of 8 x 2.  Have students build a model of 8 x 20 with base-10 blocks and skip count by 20s to find the product.  You also want them to draw a representation (see the picture below).  The work mat provides the abstract number sentence, so you  have all three stages of CRA--concrete (manipulatives), representational (drawing), and abstract (symbols). 
Here I turned the factors around and had the student build (and draw) 20 x 8.  Once the student has built the model with units, encourage him to group them into tens for easier counting.
This photo shows the representations of 8 x 20 and 20 x 8.
Here's a shortcut way to draw representations of larger numbers.
Now let's bring in some problem solving.  Notice here that I've given the student one factor and the product, and the students has to determine the missing factor.  Encourage the student to draw a model to solve it!
This one will be a little trickier.  If students have realized the pattern of zeros from their previous experiences, they will  know that 200 goes in the blank--three extra zeros in the factors means three extra zeros in the product.  Be sure to have students justify their solution and explain their thinking!
Ooh, what a great class challenge.  Let students work in pairs to come up with  as many solutions as possible!
Ready to try it?  Download the work mat and number cards here.



10 comments:

  1. Wow, this is so great!!! I've been involved in a staff development already this summer that is all about concrete -> representational -> abstract and this is great to add to what I've been learning! I am going to have to check more of this out! Thank you for all of the links!

    Amanda
    Collaboration Cuties

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    1. CRA is so important, Amanda! Glad to hear it's been included in your staff development.

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  2. Thank you for the multi mat & number cards... You have tons of fab ideas!

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  3. Thank you very much for sharing your creativity! I'll be using this activity in my Math Centre next year!

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  4. As a homeschooling Mum, I LOVE your explanations and products. Thanks so much for taking the time and effort to create such great blog posts, freebies and products; it is very much appreciated.

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    1. So glad that you find my blog useful! Thanks so much for your sweet comment. :)

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  5. This is amazing! I just linked to this resource in my Pins of the Week post.

    Thanks for sharing your great ideas!

    Mary Kate
    Windy City Learning

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  6. Any suggestions for a student that always counts one extra when adding, subtracting,and even multiplying? We've tried to model "keep or in your heart" then count up but to no evail.

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Comments make me smile! :)

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