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Q&A #494

Teachers' Lounge Discussion: Multiplication tables

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From: Ray M <raypublk@san.rr.com>
To: Teacher2Teacher Public Discussion
Date: 1999012608:05:45
Subject: COMPREHENDING multiplication

Multiplication tables are an abstraction of many different concrete
operations.  Not only do you want to be able to comprehend that but
you want to be able to recognize new applications where the use of
multiplication is appropriate.  On a day to day basis, look for ways
to see "times" in everyday activities.
If I plant 20 beans in this row and I do that 5 times, how many...?
If I pour a two cup measuring cup three times, how many.....?
If I give you four jelly beans at a time (and do it), and I do that 3
times...?  (Judy is absolutely right about edible manipulatives.  All
animals, including humans have found it very important to remember HOW
they got their last food.  I would NEVER use TV as the reward.  TV is
already very seductive by design and is often a big part of the
problem.  How about, "If you'll learn these ten problems we can go for
a hike"? or read a book tonight or play Lego or almost anything else )

Another activity :At age 6, we went over the multiplication of mixed
sign integers using a sidewalk chalk number line.  The rules are
simple and make sense.  To multiply two positive numbers, you count
paces and groups of paces.  If the first number is negative, you pace
backwards.  If the second number is negative, you turn around and face
the negative numbers before you begin.  If both numbers are negative,
you face the negative numbers AND pace backwards so that you end up
traveling toward the positive numbers.

Another activity:  Drive a spike in a dirt lot or a concrete nail in a
crack in pavement.  Use a long string and a stick or sidewalk chalk to
draw various size circles.  Count paces across each circle.  Your
paces will be a different size and number than your child's so make
two tables.  On each table, record paces across and paces around. 
With "circles" that are about 10 meter "diameter",  a pattern will
emerge:  Around is a little bigger than 3 times "across" on both
tables.  If you want to get fancy, you can use a spreadsheet like 123.
Quatro, or Excel to plot the tabular data and even insert least square
fit lines.   The average of the slopes of the lines should be about
3.14....    If you need help with the spreadsheet, send an email.  If
you live in a supportive district, the circles can be homework over a
weekend with a request that parents and siblings contribute data too.

Finally, do some activities where the word times is used but the
multiplication table is inappropriate.  Surprises also help long term
understanding.
Fold a sheet of paper in half, fold that thing in half 4 more times. 
Unfold.  Count the rectangles marked by fold lines.  Its 2^n NOT 2*n.

Cut a sheet of paper in half.  Cut one of the halves in half.  Do that
m times.  Count the pieces.  It's only m; each cut generates one new
piece.  If you don't like that one change the rule slightly: cut one
of the pieces into three pieces 4 times. 

Take four distinct objects.  Choose one and put it in a cup.  Do that
three times.  How many different ways can you do that?  It's not 4
times 3.  You can do this as a class activity with small pieces of
red, blue green, and yellow construction paper and write a table (RGB;
RGY; RBY;....)  of what different kids put in their cup on the board, 
finish the table as needed, and finally show the result as 
4 ways to pick the first one times
3 ways to pick the second one times
2 ways to pick the third one


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