# [Numeracy 635] Re: That old thinking style thing

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tjdclaire at cox.net tjdclaire at cox.net
Tue Dec 21 23:14:24 EST 2010

I teach solving equations by telling students that the person who made up the equation already knew the answer (certainly when I make up an equation, that is so). What they are to do is to undo what the person who made the equation did. (There is a lot more to my explanation, but this discussion made me stop and think...) Perhaps it would help if students were assigned a number that x had to be equal to and then asked to write true equations using that x. Change their perspective on what is happening.
Claire Ludovico

---- ROBERT G STEINKE <steinkedb at q.com> wrote:

> the type X - 3 = 2 and X/2 = 5".

>

> Here is where the idea of "only 2 kinds of problems" comes in (see the May

> 2008 Focus of Basics article I wrote).

>

> In addition and multiplication equations (3 + 2 = x or 5 * 2 = x) the WHOLE

> is by itself on one side of the equal sign.

> In equations using subtraction and division signs, (X - 3 = 2 and X/2 =

> 5) the WHOLE is the first number in the equation.

> When you are using the two mats that you describe, the WHOLE is in one

> location, the PARTS are in the other, regardless of the operation.

> You don't solve X - 3 = 2 by subtraction; you solve it by addition because

> it is a "find the WHOLE" problem.

> You don't solve X/2 = 5 by division; you solve it by multiplication because

> it is a "find the WHOLE" problem.

>

> Dorothea Steinke

> Dorothea at numberworks4all.com

>

>

> -----Original Message-----

> From: numeracy-bounces at lincs.ed.gov

> [mailto:numeracy-bounces at lincs.ed.gov]On Behalf Of Susan Jones

> Sent: Tuesday, December 21, 2010 9:41 AM

> To: The Math and Numeracy Discussion List

> Subject: [Numeracy 633] Re: That old thinking style thing

>

>

> I'm hoping this discussion doesn't get lost in the holidays and that people

> will chime in when they come back if they're festivating...

>

> One of our Math teachers does something really similar to this (I don't

> think he uses mats -- but I'll suggest it, since visual boundaries have a

> peculiarly positive effect on many), and this is the kind of project I may

> try in a workshop setting.

>

> I think that once the idea is grounded, with dedicated explicit practice in

> translating the manipulative experience into "regular person" language as

> well as "math language," and then a few more times practicing... then we

> could show it visually ... say with something put together on the ol'

> computer... maybe a Flash exercise. (Which I have yet to learn. Maybe

> start this break :)) Or, draw it on the board and make an "old fashioned"

> video of it.

>

> Subtraction... let's see... that idea that something *was taken away* is

> hard to convey, but I like the six pack with the empty places idea. And,

> I think, it's a grand place to work in the "normal people language" and make

> up situations to learn that "x - 2 = 3" is a different way of saying "what

> is three two less than?" and working to make that statement *mean*

> something. "Three is two less than what?" they'd probably get... show 'em

> that we're "just" turning some words around...

>

> I think that stuff coudl be conveyed with some creative graphics. If we can

> get that "whole to part" concept, then we'll be conveying "that X in 'x -

> 3'? That's the whole thing that you *started* with, before you yanked

> three away." (I'm trying to think on behalf of the language-oriented

> learners as well as the visually-oriented ones.)

>

> http://www.hewlett.org/programs/education-program/open-educational-resources

> says that in Mid-December they'll accept letters of inquiry for the

> development of Open Source Materials. Their wording talks about fairly big

> attempts -- but other things I've been reading suggest that they would

> really like to do some focus on "deeper learning" in basic math for people

> at community colleges. I wonder if some collaborative work could be

> funded... some lessons in Flash or getting somebody who already knows to

> put things together...

>

>

>

>

> Susan Jones

> Parkland College

> Champaign, IL 61821

> 217-353-2056

> sujones at parkland.edu

> Webmastress,

> http://www.resourceroom.net

> http://www.bicycleuc.wordpress.com

>

>

>

> >>> "Kate Nonesuch" <Kate.Nonesuch at viu.ca> 12/20/2010 2:26 PM >>>

> 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/>

> http://www.bicycleuc.wordpress.com <http://www.bicycleuc.wordpress.com/>

>

>

>

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