by Josh Rappaport
Remember the best teachers, those who made their classes come alive? Do you recall how they used analogies —stories that related things we already understood to the difficult new concepts they were trying to help us learn?
Educational researchers are studying what makes analogies such an effective teaching tool. They have found that, by relating that which children need to learn to that which they already do know, educators create bridges to understanding, and that those bridges give children a handle for learning difficult concepts.
The same holds true in math lessons. If we parents use analogies that make math concepts more memorable, our children are more likely to enjoy the lesson, and as a result, they'll be more likely to remember what was taught.
I would like to present a quick-and-easy analogy that helps children learn about our number system, on the one hand, but that also helps students work with fractions, on the other hand.
The analogy is to what I call the "Ladder of Primes."
The Ladder of Primes is like a real ladder. It has a bottom, and it has rungs that you climb. By climbing you reach new heights of understanding, and the sweeping vision you gain when you climb lets you see and understand more about math. Unlike a real ladder, though, the Ladder of Primes is made up of rungs that are numbers, and those numbers are the prime numbers, starting with the smallest prime number, 2, and rising to whatever prime you need for any situation.
Let’s recall that a prime number is a whole number that has only one pair of numbers that divide into it evenly: itself and 1. For example, the number 7 is prime because the only numbers that divide into 7 are 1 and 7. Six, on the other hand, is not prime because there are two pairs of numbers that divide into six: 1 and 6, of course, but also 2 and 3. Numbers like six, which are not prime, are called “composite” numbers, or just “composites.”
The Ladder of Primes is a visual ladder that shows the first eight prime numbers, with the smallest prime, 2, on the bottom, and the largest of these eight primes, 19, at the top.
When children draw the Ladder of Primes — often putting it on a post-it note or index card — it looks like like this:
- 19 -
- 17 -
- 13 -
- 11 -
- 7 -
- 5 -
- 3 -
- 2 -
(Of course, the ladder should have side rails, but I don’t know how to make those on a word processor.)
There are two main ways to use the Ladder of Primes, and I will touch on one of those in this post. I'll mention the other way in a later article.
One of its most obvious uses: the Ladder of Primes helps us reduce fractions to lowest terms.
If children work their way up the Ladder of Primes when reducing fractions, they will:
1) end up with a fraction that is reduced to lowest terms,
2) learn a procedure that works for any fraction,
3) learn what the primes are and how they work.
I tell children that when they are reducing a fraction, they need to start on the lowest rung on the Ladder of Primes, and that they work their way up the ladder until the fraction is fully reduced. Here are the steps for using the Ladder of Primes to reduce a fraction.
Suppose you need to reduce the fraction: 330/495
Starting on the lowest run of the ladder, children first check to see if 2 divides into both numbers. Using rules of divisibility, children will see that 2 does not divide into both numbers. That means they need to move one rung up the ladder, to 3.
3 does divide into both 330 and 495, and when you divide both numbers by 3, you get the reduced fraction: 110/165. [FYI: there’s a quick trick for seeing if 3 goes into a number evenly. Just add up the number’s digits. If 3 divides into the sum of the digits, 3 divides evenly into the number. For example, for 495, 4 + 9 + 5 = 18. Since 18 is divisible by 3, 495 is divisible by 3.]
So now we’re at 110/165. Using the trick just described for 3, we can see that 3 does not divide evenly into 110. Even though 3 does divide evenly into 165, that does not help us; a number must divide evenly into both the numerator and denominator to help us reduce a fraction. So we move another rung up the ladder, to 5.
5 does divide into both 110 and 165, giving us the reduced fraction: 22/33.
It’s obvious that 5 does not divide evenly into 22 or 33, so we move one rung up the ladder, to 7.
Since 7 does not divide evenly into 22 or 33, we move up another rung on the ladder to 11.
11 does divide evenly into both 22 and 33, giving us the reduced fraction: 2/3.
Whenever the numerator and denominator are apart by 1, as they are in 2/3, that means the fraction is fully reduced. So therefore we have reduced the ungainly fraction 330/495 to 2/3. Of course, if we had really amazing number sense, we would have seen at the outset that 330 and 495 are both multiples of 165, and we could have reduced the fraction at one fell swoop from 330/495 to 2/3 at the start.
But c’mon! Most people don’t have that degree of number sense. That being the case, the Ladder of Primes is a great way to reduce fractions.
One thing to point out is that once children exhaust any given prime, they can be certain that no multiples of that number will go into the more reduced numerator and denominator. For example, if students use the number 2 as much as they can, they do not need to test any of the multiples of 2: 4, 6, 8, etc. This characteristic of primes resonates with the workings of the original tool for discovering the primes, the sieve of Eratosthenes. If your students have never seen that, now would be a good point to show them how it works. There are many websites devoted to the sieve.
In any event, the point of this lesson is that by using the primes, students can systematically and carefully test for all multiples of the early primes. That makes it safe to move to the next prime, allowing them to rest assured that they have missed no factors.
There is more to know about and to think about regarding the Ladder of Primes, but this is just meant to be an introduction. To help solidify the lesson, here are some practice problems you can have your children do.
Reduce these fractions using the Ladder of Primes. To add depth to the problem set, ask children to write down which primes they used, and how many times each was used, as in this example:
Ex: 60/96 = 5/8 — Used 2 (2x), 3 (1x).
a) 24/40
b) 9/24
c) 30/42
d) 30/70
e) 42/70
f) 120/268
g) 66/198
h) 104/364
i) 189/441
j) 68/153
Answers:
a) 24/40 = 3/5 — Used 2 (3x).
b) 9/24 = 3/8 — Used 3 (1x).
c) 30/42 = 5/7 — Used 2 (1x), 3 (1x).
d) 30/70 = 3/7 — Used 2 (1x), 5 (1x).
e) 42/70 = 3/5 — Used 2 (1x), 7 (1x).
f) 120/268 = 30/67 — Used 2 (2x).
g) 66/198 = 1/3 — Used 2 (1x), 3 (1x), 11 (1x).
h) 104/364 = 2/7 — Used 2 (2x), 13 (1x).
i) 189/441 = 3/7 — Used 3 (2x), 7 (1x).
j) 68/153 = 4/9 — Used 17 (1x).
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
Remember the best teachers, those who made their classes come alive? Do you recall how they used analogies —stories that related things we already understood to the difficult new concepts they were trying to help us learn?
Educational researchers are studying what makes analogies such an effective teaching tool. They have found that, by relating that which children need to learn to that which they already do know, educators create bridges to understanding, and that those bridges give children a handle for learning difficult concepts.
The same holds true in math lessons. If we parents use analogies that make math concepts more memorable, our children are more likely to enjoy the lesson, and as a result, they'll be more likely to remember what was taught.
I would like to present a quick-and-easy analogy that helps children learn about our number system, on the one hand, but that also helps students work with fractions, on the other hand.
The analogy is to what I call the "Ladder of Primes."
The Ladder of Primes is like a real ladder. It has a bottom, and it has rungs that you climb. By climbing you reach new heights of understanding, and the sweeping vision you gain when you climb lets you see and understand more about math. Unlike a real ladder, though, the Ladder of Primes is made up of rungs that are numbers, and those numbers are the prime numbers, starting with the smallest prime number, 2, and rising to whatever prime you need for any situation.
Let’s recall that a prime number is a whole number that has only one pair of numbers that divide into it evenly: itself and 1. For example, the number 7 is prime because the only numbers that divide into 7 are 1 and 7. Six, on the other hand, is not prime because there are two pairs of numbers that divide into six: 1 and 6, of course, but also 2 and 3. Numbers like six, which are not prime, are called “composite” numbers, or just “composites.”
The Ladder of Primes is a visual ladder that shows the first eight prime numbers, with the smallest prime, 2, on the bottom, and the largest of these eight primes, 19, at the top.
When children draw the Ladder of Primes — often putting it on a post-it note or index card — it looks like like this:
- 19 -
- 17 -
- 13 -
- 11 -
- 7 -
- 5 -
- 3 -
- 2 -
(Of course, the ladder should have side rails, but I don’t know how to make those on a word processor.)
There are two main ways to use the Ladder of Primes, and I will touch on one of those in this post. I'll mention the other way in a later article.
One of its most obvious uses: the Ladder of Primes helps us reduce fractions to lowest terms.
If children work their way up the Ladder of Primes when reducing fractions, they will:
1) end up with a fraction that is reduced to lowest terms,
2) learn a procedure that works for any fraction,
3) learn what the primes are and how they work.
I tell children that when they are reducing a fraction, they need to start on the lowest rung on the Ladder of Primes, and that they work their way up the ladder until the fraction is fully reduced. Here are the steps for using the Ladder of Primes to reduce a fraction.
Suppose you need to reduce the fraction: 330/495
Starting on the lowest run of the ladder, children first check to see if 2 divides into both numbers. Using rules of divisibility, children will see that 2 does not divide into both numbers. That means they need to move one rung up the ladder, to 3.
3 does divide into both 330 and 495, and when you divide both numbers by 3, you get the reduced fraction: 110/165. [FYI: there’s a quick trick for seeing if 3 goes into a number evenly. Just add up the number’s digits. If 3 divides into the sum of the digits, 3 divides evenly into the number. For example, for 495, 4 + 9 + 5 = 18. Since 18 is divisible by 3, 495 is divisible by 3.]
So now we’re at 110/165. Using the trick just described for 3, we can see that 3 does not divide evenly into 110. Even though 3 does divide evenly into 165, that does not help us; a number must divide evenly into both the numerator and denominator to help us reduce a fraction. So we move another rung up the ladder, to 5.
5 does divide into both 110 and 165, giving us the reduced fraction: 22/33.
It’s obvious that 5 does not divide evenly into 22 or 33, so we move one rung up the ladder, to 7.
Since 7 does not divide evenly into 22 or 33, we move up another rung on the ladder to 11.
11 does divide evenly into both 22 and 33, giving us the reduced fraction: 2/3.
Whenever the numerator and denominator are apart by 1, as they are in 2/3, that means the fraction is fully reduced. So therefore we have reduced the ungainly fraction 330/495 to 2/3. Of course, if we had really amazing number sense, we would have seen at the outset that 330 and 495 are both multiples of 165, and we could have reduced the fraction at one fell swoop from 330/495 to 2/3 at the start.
But c’mon! Most people don’t have that degree of number sense. That being the case, the Ladder of Primes is a great way to reduce fractions.
One thing to point out is that once children exhaust any given prime, they can be certain that no multiples of that number will go into the more reduced numerator and denominator. For example, if students use the number 2 as much as they can, they do not need to test any of the multiples of 2: 4, 6, 8, etc. This characteristic of primes resonates with the workings of the original tool for discovering the primes, the sieve of Eratosthenes. If your students have never seen that, now would be a good point to show them how it works. There are many websites devoted to the sieve.
In any event, the point of this lesson is that by using the primes, students can systematically and carefully test for all multiples of the early primes. That makes it safe to move to the next prime, allowing them to rest assured that they have missed no factors.
There is more to know about and to think about regarding the Ladder of Primes, but this is just meant to be an introduction. To help solidify the lesson, here are some practice problems you can have your children do.
Reduce these fractions using the Ladder of Primes. To add depth to the problem set, ask children to write down which primes they used, and how many times each was used, as in this example:
Ex: 60/96 = 5/8 — Used 2 (2x), 3 (1x).
a) 24/40
b) 9/24
c) 30/42
d) 30/70
e) 42/70
f) 120/268
g) 66/198
h) 104/364
i) 189/441
j) 68/153
Answers:
a) 24/40 = 3/5 — Used 2 (3x).
b) 9/24 = 3/8 — Used 3 (1x).
c) 30/42 = 5/7 — Used 2 (1x), 3 (1x).
d) 30/70 = 3/7 — Used 2 (1x), 5 (1x).
e) 42/70 = 3/5 — Used 2 (1x), 7 (1x).
f) 120/268 = 30/67 — Used 2 (2x).
g) 66/198 = 1/3 — Used 2 (1x), 3 (1x), 11 (1x).
h) 104/364 = 2/7 — Used 2 (2x), 13 (1x).
i) 189/441 = 3/7 — Used 3 (2x), 7 (1x).
j) 68/153 = 4/9 — Used 17 (1x).
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
