# [Numeracy 630] Re: That old thinking style thing

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Kate Nonesuch Kate.Nonesuch at viu.ca
Mon Dec 20 15:26:52 EST 2010

Hi Susan,
I'm intrigued by your ideas of making videos--I should make a video of the following idea--it would be easier than writing it out.

Here's a visual and kinaesthetic way I've used to demonstrate solving equations involving addition and multiplication:

You need two mats, or two spaces, one for each side of the equals sign.

On the left, a closed envelope represents X (or E), with some visible counters as necessary. On the right are some counters. The total number of counters on each mat is the same.

For example: To show X + 3 = 5, on the left hand side, the envelope has 2 counters inside it and there are 3 counters lying in the open. The right hand side has 5 counters and no envelope.
The question to learners is: The number of counters on each mat is the same. How many counters in the envelope? How do you know? What operation did you do to find out?

Example 2: To show 3X = 12, on the left hand side there are three envelopes, each with four counters inside. On the right hand side are 12 counters lying in the open.
The question to learners is: The number of counters on each mat is the same. Each envelope has the same number of counters inside. How many counters in each envelope? How do you know? What operation did you do to find out?

I encourage students to play with the counters on the right hand side to help them figure it out, e.g., in example 1, to separate out three counters on the right hand side to isolate the number hidden in the envelope.

And, as you said in your post, the thinking is the important part--after I've done a few, I ask students to make up some problems for the other students. Sometimes they set it up with the envelope first, then have to write the equation based on what they have set up; sometimes they write an equation first, and then have to figure out how to represent it with envelope and counters. Either way, lots of thinking about what an equation represents, and the difference between 2X and X+2 comes through loud and clear!

However, this method is not so useful in showing equations of the type X - 3 = 2 and X/2 = 5. Anybody have ideas for that?

Kate Nonesuch
Victoria, BC
kate.nonesuch at viu.ca

________________________________

From: numeracy-bounces at lincs.ed.gov on behalf of Susan Jones
Sent: Fri 12/17/2010 1:19 PM
To: kabeall at comcast.net; The Math and Numeracy Discussion List
Subject: [Numeracy 627] That old thinking style thing

... take all that stuff with the x's and show LOTS of examples of putting numbers in there for the x's. For students who do have some number sense, plugging in numbers and *testing* adding 3x + 2x and discovering that it's 5x no matter what you call x... as long as you call it the same thing... could really help. (I'm also trying to figure out ways to do this that *aren't* entirely depending on language and discussion... some of my guys get *so* much smarter when they can see things...)
..........
I'm curious -- what are other folks doing to engage students in that elusive comprehension aspect?

Susan Jones
Parkland College
Champaign, IL 61821
217-353-2056
sujones at parkland.edu
Webmastress,
http://www.resourceroom.net <http://www.resourceroom.net/>
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