[Numeracy 135] Re: Teaching Fractions etc...
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Wed Feb 10 08:39:08 EST 2010
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Fractions are an area I think there's a whole lot of potential for bridging that cognitive breach between "real life thinking" and the fearful math problems.
I did a little Camtasia presesntation with fractions - the only problem is that there's this little teeny typo so, if I remember right, it says boldly that 1/4 is greater than 1/2. While I was ecstatic that the seriously struggling student I was showing it to took one look and said "That ain't right!!" ... I havent' had time to go back and reconstruct (can't find the original powerpoint I recorded).
However, one thing I did learn from delving into what other people have done: the absolute necessity of teaching the *least* common denominator is questionable if students are still developing the fraction concept.
Prime factorization is, absolutely, an invaluable skill set and an awfully good foundation for lots of algebra. However, if students are still wrestling with the concept of division and divisibility, then comprehension is going to be limited. If students are still trying to figure out fractions and why they need a common denominator at all, then I find throwing prime factorization into the mix pretty much puts the concepts in a blender and fragments them. I found that the state of California agrees with me.
My thinking is that some time spent building the division and divisibility concepts before leaping into fractions is one way to bridge the concrete-abstract chasm. But that's my next email...
Susan Jones
Academic Development Specialist
Center for Academic Success
Parkland College
Champaign, IL 61821
217-353-2056
sujones at parkland.edu
Webmastress,
http://www.resourceroom.net
http://bicyclecu.blogspot.com
>>> Maureen Carro <mcarro at lmi.net> 2/9/2010 2:20 PM >>>
On Feb 8, 2010, at 6:16 PM, George Demetrion wrote:
>
> This brings up to my mind the importance of:
> Sequencing skill development from basic to more advanced
> Maximum possible simplicity as a critical scaffolding strategy in
> its own right
> Incorporating mathematical meaning making and inquiry as a critical
> part of the ongoing work
> Individual and collaborative scaffolding
> Sequencing skill development from basic to more advanced
This IS the essence of it all! When teaching math concepts, it is
important to do a "task analysis" to determine what the student needs
to have already mastered before proceeding with the instruction. This
applies to all instruction.
1.So many students trip up on fractions simply because they do not
have "fluid recall" of multiplication/division facts. This is
necessary for finding the "factors...including GCF", and
"multiples...including LCM", and "common denominators...including LCD"
in order to add or subtract fractions, and in the end, simplify
(reduce) answers to lowest terms. There is a constant mental
manipulation of multiplication/division facts. Sorry.... but "finger
tricks" and multiplication charts..... etc are a huge distraction to
the fluid mental thought process necessary to carry out all the steps
of fraction algorithms. We practice math facts as part of our "warm
up skills drill"... until most know the facts fluently. They need to
become mental... with " fingers and tricks" used "only as a last
resort crutch" when fatigue sets in.
2. Of course they need to know addition and subtraction facts as well.
What else do they need to know?
3. Prime numbers.... if you talk about "prime factorization" to
determine LCM and thus LCD, they need to know what prime numbers are.
These come in handy to recognize lowest terms fractions.... ie, we
know 3/5 is in lowest terms because they are both prime numbers.....
the only common factor being 1. I use a 100's chart and do the
"Sieve of Eratosthenes" to learn the prime numbers up to 100 ( I have
included this at the end).
Eventually when this is part of a "warm up skill drill"...... the
students instantly recognize prime numbers.
4. Fluid execution of short division: with larger fractions, or
improper fractions, one needs to do short or mental division in order
to simplify answers. I find most students never heard of "short
division" . When I show them and we practice, they are eternally
grateful! We do 2 or 3 short division problems as part of our "warm
up skills drill".
5. Rules of divisibility help as well. As part of a "warm up skills
drill", I might dictate a number and ask if it is evenly divisible by
3, or 6... Most students can tell if it is divisible by 2, 5, and 10 ,
but there are more.
Here is the Sieve of Eratosthenes:
The Sieve of Eratosthenes
Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover
prime numbers. A sieve is like a strainer that you use to drain
spaghetti when it is done cooking. The water drains out, leaving your
spaghetti behind. Eratosthenes's sieve drains out composite numbers
and leaves prime numbers behind.
To use the sieve of Eratosthenes to find the prime numbers up to 100,
make a chart of the first one hundred positive integers (1-100):
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Cross out 1, because it is not prime.
Circle 2, because it is the smallest positive even prime. Now cross
out every multiple of 2; in other words, cross out every second number.
Circle 3, the next prime. Then cross out all of the multiples of 3; in
other words, every third number. Some, like 6, may have already been
crossed out because they are multiples of 2.
Circle the next open number, 5. Now cross out all of the multiples of
5, or every 5th number.
Continue doing this until all the numbers through 100 have either been
circled or crossed out. You have just circled all the prime numbers
from 1 to 100!
> Maximum possible simplicity as a critical scaffolding strategy in
> its own right
> Incorporating mathematical meaning making and inquiry as a critical
> part of the ongoing work
Using everyday examples to develop fraction concepts, especially
bridging the concept that a fraction is a division problem. ( If we
divide 1 candy bar evenly between two people, how much does each one
get?) I write 1/2 as I state this simple scenario. What I have
written is the answer....1/2.
Thanks to others who have offered some fine examples of this as
well......
> Individual and collaborative scaffolding
Absolutely.... students should contribute as to what to include in
"warm up drills".....for example.... eventually some things are no
longer needed, and others emerge! For success in higher math, there
is a huge knowledge base that builds, must be retained, and trained to
automaticity.
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