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Q&A #2454


Decimals, even numbers, fractions, and order

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From: Jenny (for Teacher2Teacher Service)
Date: Nov 17, 1999 at 18:29:20
Subject: Re: Decimals, even numbers, fractions, and order

>How would you explain to children the following:

Lori - most of the patterns you mention below are results of the structure of
our number system, so if you help students understand number concepts,
they'll see these patterns as part of the number's "attributes".  Look below
each question for some ideas on how you could help students develop their
understanding of them.  Also remember that having students write about their
understanding and discuss their observations will help them understand it
better.

>*Even numbers always end in 2, 4, 6, 8, and 0.

** Have your students use a hundreds chart to mark every other number
starting at 2 - they'll see the pattern that the numbers end in 2, 4, 6, ...
These are called the even numbers.  Mathematically, we define even numbers to
be numbers which can be divided by 2 (you will also note that these are all
multiples of 2).  If your students have worked with rectangular arrays to
show multiplication, have them build "2 by" rectangles for the even numbers -
2 by 1 for 2, 2 by 2 for 4, 2 by 3 for 6, etc. "2 by" rectangles can all be
divided by 2 and any even number can be represented as a "2 by" rectangle.
(Also, numbers divisible by 3 (or multiples of 3) are "3 by" rectangles,
numbers divisible by 4 (or multiples of 4) are "4 by" rectangles, etc.)

>*You can decide which of two fractions is greater by finding equivalent
 fractions with the same numerator and comparing them.

** Give students some fractions and ask them to determine which are bigger.
Don't tell them how to do it.  Then have them share their strategies.  You'll
find some use equal denominators rather equal numerators to compare - others
will make a visual picture.  You can provide fraction bars, graph paper, or
have them make the fraction kit from Family Math.

>*You can stick zeroes at the end of a decimal amount and not change its
 value, ex, 3.45 = 3.45000000.

** If you let a square = 1, then cut it into 100 equal parts and select 45 of
them, you have 0.45.  If you take that same square and cut it into 1000 equal
parts and select 450 of them, you will take the same part of the square.
Have your students do it - start with tenths.  Take centimeter grid paper and
have them mark out several squares 10 squares by 10 squares.  Make sure they
agree all squares have the same area.  Divide one into ten equal pieces
(tenths), one into 100 equal pieces (hundredths).  Then have them color in 3
tenths (.3) of the square divided into tenths.  Then use the other square and
color in 30 hundredths (.30) of the square divided into hundredths.  Don't
have them divide the other square into 1000 parts, but if they did, how many
would they need to color to get the same part of the square?  What would they
call it?
>
>*You can add the same amount to both numbers when your subtracting them and not change the answer.

** Try mental math with this one.  See how some of your students mentally
find 199 - 15.  If your kids are like the ones I've worked with, at least one
will say "I found 200 - 16".  Mathematically, this is a very potent skill -
what you're really doing is adding 0.  200 = 199 + 1 and 16 = 15 + 1.  Since
you're subtracting, you're adding 0 (1 - 1).  Try it also with counters.
Start with 8 counters and take out 3 (8-3).  What if you have 8 counters and
put one more in?  If you take out 3, you don't have the same number left.
How many more would you need to take out to have the same number left?

>*You can halve one number and double another when you multiply and not
 change the answer.

** This is the same principle - only now it's multiplication, so you're
multiplying by 1.  Again, rather than telling students they can do this, give
them problem situations and have them focus on the similarities.  6 x 8 is
hard for some kids, but if they think of 12 x 4, it's easier.  Also, have
them look at rectangles.  Build a 6 by 8 rectangle.  It uses 48 tiles.  Now,
what happens when you build a rectangle using the same number of tiles, but
change the 8 side to 4?  To make it work, all the extra tiles have to go on
the 6 side.

   xxxxxxxx                     xxxx
   xxxxxxxx                     xxxx   what do you do with the others?
   xxxxxxxx                     xxxx   because they're half, you can
   xxxxxxxx  halve one side     xxxx   just add them to the end of this
   xxxxxxxx                     xxxx   rectangle and make it a 4 x 12
   xxxxxxxx                     xxxx
                                XXXX
                                XXXX
                                XXXX
                                XXXX
                                XXXX
                                XXXX

 -Jenny, for the Teacher2Teacher service

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