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Math Manipulatives: fractions, integers, and variables

This is a sampling of some of the excellent ideas for using manipulatives in middle and secondary school math. One obvious problem presenting the information is that I have no manipulatives to demonstrate via cyberspace. I've scanned some sketches that will hopefully help.

1. Low tech -- your 'digital compter' -- your fingers and the times tables.

A. The"nines times tables trick."

I was unsuccessful teaching this to kids the first few times I tried. I succeeded when I focused on the kinesthetic part of it -- the 'motor memory' -- and physically walked and talked through the process several times, focusing on processes my student knew already -- counting by tens and counting by ones.

Assign numbers to your fingers beginning with 1 at the left pinky, ending with 10 at the right pinky. Put your hands on the table in front of you. Take the number of the finger that you want to multiply by 9 and bend it under - it's the placemarker between the tens and ones. Before the empty space, count by tens; after the empty space, count up by ones (starting from where you finsihed counting by tens). So for "9 x 4," you'll count 10-20-30-.space....31, 32, 33, 34, 35, 36.

This is exactly the same manipulation of numbers as subtracting one from the 6 to get the amount of tens, and then figuring out how many more you'd need to add to that digit to get 9 (since all of the digits of things divisible by 9 will always add up to 9 or a multiple of 9).

Trick Two: "the upper tables"

You can do the 'upper products' with your fingers if you mentally number them "6-7-8-9-10" from the thumbs out on each hand. Then, say, if you wanted to multiply 8 X 7, you would grab the 6,7,&8 fingers on one hand and hold on to the 6 & 7 fingers w/ the other... five fingers holding on and the rest straight out still. The five fingers are the tens -- fifty -- and if you multiply the other pair of fingers - 2 x 3 - you get six... for fifty-six. Still not as fast as knowing "5678" -- 56=7 x 8.

It gets slightly tricky for 6 X 6 and 7 X 6, which give you "20 + (4 x 4)" and "30 + (4 x 3)" respectively, so you have to add a number more than ten.

2. Still pretty low-tech: manipulatives.

Fraction pieces

These colored sections of circles were translucent (made them work nicely on the overhead). Most math books have these kinds of examples of fractions at the beginning of the 'fraction' section; most students are pretty successful at figuring out how to name the fractions based on the parts shown. Bridging the understanding of the manipulative to the symbols is another task, though. One thing that helps is to write out the fraction in words and explain that three fourths is, actually, 3 of something... like three penguins or hot dogs or planets... so, since three hot dogs plus three hot dogs would be 6 hot dogs, 3 fourths plus three fourths is... not 6 eighths, but 6 fourths.

By placing different combinations on top of each other, a lot of discovery can be made about what kinds of fractions are less than or greater than each other, and how equivalent fractions do represent the same amount. 2/4 is going to fit exactly on 1/2.

Operations using fractions can be demonstrated and practiced using the manipulatives, including adding the ones with different denominators. It's more clear when you're holding these things of different sizes that 1 of the half things plus 1 of the third things gives you... 2 things, but not of the same size, so you need to do something about that.

Multiplication can be shown, too, and is better explained when the word "of" is taught to mean the same as "times." 1/2 of 20" is often a concept a student has, while 1/2 times 20 is ... a math problem, we can't do math problems... It's also a good way to expand the knowledge of those students who understand "half," but not "thirds" or "fourths."

Division is better verbally understood (and demonstrated) by rephrasing "1 7/8 divided by 3/8" into "how many 3/8 are there in 1 7/8?"

On the whole, giving students lots of time to explore and translate pictures into symbols and symbols into pictures can be very helpful -- but often there isn't enough time to really cement the ideas in. Making the connections between the concrete manipulatives, the mathematical symbols, and understanding in the spoken language of the student is important for getting the most out of the manipulatives.

Integers

see also Teaching Positive and Negative Integers

A few ways of presenting integers were discussed, involving using blue and red translucent bingo chips.

First, the concept of zero and numbers greater and less than that can be explored with concrete examples. Several people at the session expressed a dissatisfaction with using number lines. One teacher had used elevators in large buildings with above and below ground levels as a 'concrete reference.' Another good example was football gains and losses: Iif you gain 5 yards but get a 5 yard penalty, you're where? Back where you started... that 5 yard penalty is "negative 5" because it cancels out 5 yards. Money found and spent was another reference -- if you find 10 cents and then spend it, where are you? Back where you started -- wherever that was.

Then the chips are brought out -- and placed on the overhead on an array:

You'd start with adding two groups of positive numbers; the goal is to relate what the student already knows to the new domain. Negative integers don't change the rules for positive integers -- they just help us explain things that positive integers don't do a good job with. The more students can tell *you* about the operations, and demonstrate them with the chips, the better.

For a verbal approach to making the shift to negative integers, you can announce that henceforth subtraction will be banned, and only "adding the negative of a number" will be permitted. That can be demonstrated with the chips. Instead of actually taking away two positive chips, adding two negative chips would mean that two of the positive ones were 'canceled out' as in the examples of football yards and money. Students can be guided to discover that "subtracting a negative number" would therefore be adding the negative of that negative number.... the positive one.

Here's what happens when the you add a bigger negative number; see if the students can discover that it's the same thing as saying 3 - 4.

Here is how you would do that problem as a "subtraction" problem. Since you've run out of things to physically take away, you take a plus and minus, since they cancel each other out, and "add zero." Then you can take away four -- and you're left with the one negative chip.

Multiplication as repeated addition can be shown with the chips, too:

Other manipulatives: A balance scale can also be used to demonstrate what the equals sign is all about, and the idea behind "subtracting the same thing from both sides," which so many students struggle with. If two piles of stuff are the same, and you take something away from one side, how can you get them back in balance?

You can use the scale to demonstrate solving for unknowns -- your unknown, X, and two weights, balances out 8 weights on the other side of the scale. How can you figure out what x is? How would you say that in mathematical terms?

Stay tuned... if I can figure out how to get that on a scanner, or make my own graphics, I will...