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Low & High Tech ideas for middle and secondary math
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.
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
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.
see also Teaching Positive and
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
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...