More algebra with manipulatives

Subtracting 2 from both sides

by Kathy Kuhl

Last time, I began explaining bean algebra—a way to solve linear equations using beans. Manipulatives are simply objects you handle to help you understand a concept. The containers represent the variable x. I labeled mine with post-its.

Last week, I showed you how to use beans to model equations like 14 =2x. Click here if you want a refresher.

Algebra manipulatives show 12=3x
We divide 12 beans into three groups to figure how many beans are in each x container.

But why bother building models? I’ve found multisensory mathematics to be the best way to get mathematical concepts to stick.

Multisensory Math isn’t a curriculum, it’s a very effective teaching approach based on the Multisensory Structured Langage (MSL) or Orton-Gillingham principles of teaching reading. It turns out that what helps dyslexics learn to read can help anyone learn math concepts.

algebra manipulatives
Build, draw, and write to grasp and remember concepts.

Build it, Draw it, Write it

One key principle of Multisensory Math is that students need to build, draw, and write illustrations in order learn and remember. For example, above is an example of how to teach the concept of three-fourths:

  1. On the lower left, first we build 3/4s of a square, show 3/4 of a rod, and 3/4 of a circle.
  2. Then we draw (at top of the illustration). Our drawings don’t have to be precise like this. A rough sketch is fine. Sketching or drawing helps improve memory.
  3. The third stage is writing the numerals or the equations. Notice above we’ve written 3/4 as a fraction and in words. Notice two ways to write the fraction: vertically as in the illustration, and horizontally, with a slash* as I’ve done in this paragraph.

In educational terms, we say we are asking kids to show us a concept through Concrete examples, Representation, and Abstraction (CRA). But I prefer words of one syllable: “Build it, draw it, write it.” [Thanks to the Alabama chapter of the International Dyslexia Association for permission to use their slide.]

Last time, I showed you how to build 12 = 3x. How would you draw it?

My drawing might be as simple as this. It’s not to scale, not fancy, just a sketch.




Now, let’s make the puzzle more interesting

Let’s try add some beans that aren’t in a container to both sizes of the ruler.

For example, take 12 = 3x and add 2 beans to each side.

That shows 14 = 3x + 2.

Showing 14 = 3x + 2
Showing 14 = 3x + 2

One for you to solve:

Next, I am going to build a different equation in bean algebra, without telling you what my new x is. I’ll choose a new value (“secret number”) for x and put that many beans in each container. Let’s see if you can figure this one out:

22 = 4x + 2
22 = 4x + 2

On the left, I have 22 beans. On the right, four x containers and 2 more beans.

So you can can see the equation I built is 22 = 4x + 2.

How do we solve this kind? Remember that both sides of the equation must stay equal. So what we do to one side, we must do to the other.

It’s easy to rid of those loose extra beans on the right first. If I remove two loose beans on the right, I must remove two beans on the left, too.

Subtracting 2 from both sides
If you remove two beans from the right side, you must also remove two from the left.

Now we have 20 = 4x. We need to divide the beans on the left into four equal groups.

20 = 4x
20 = 4x

Dividing into 4 equal groups gives us groups of 5 beans. Each group of 5 beans is equal to one container. So x = 5. You or your child should conclude I’ve put 5 beans in the container.

Practice and play bean algebra

Build lots of problems for each other to practice with. Here are some tips:

  • Remember to empty out all the containers after each problem.
  • When creating a new problem, decide first what x will be and put that many beans in each container.
  • Don’t remind your children of the steps, just remind them to do the same thing to each side.
  • Give your children lots of time to experiment. Let them play. Let them build problems for you or others to solve. Have fun!

Soon we’ll finish the bean algebra series with problems including negative integers, those powerful anti-matter beans I described in this earlier post. This 4-part series concludes here. 

How do you use manipulatives to teach beginning algebra? Please enter your ideas in the comment section below?

*Today’s vocabulary word is “virgule,” meaning a slash like this: “/” I think it’s more fun to say than “slash,” and more elegant.

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One Comment

  1. How I wish manipulatives had been around when I took algebra. I never really understood it until using them for math lessons when I taught 3rd and 4th grade. They really work. Thanks for adding this to’s Tuesday link up.