Josh here with an announcement.

Singing Turtle Press is poised to come out with a new product. It will be an eBook that I have written, and it deals with geometry and geometric figures.

This eBook makes a lot of use of color in teaching geometric concepts and figures. I have found the use of color a very handy tool for differentiating concepts in math education in general, and in geometry in particular.

I have two requests for comments to this post:

1)  Would anyone like to guess how I have used color in Geometry in this upcoming eBook?

2)  Would anyone like to share ways that you have used color in Geometry in your own teaching?

It would be fun and interesting to hear some of your ideas on these topics. Just add a comment to this post, or send an email to:  josh@SingingTurtle.com

Educational researchers today are studying what makes analogies such an effective teaching tool. They have found that the use of analogies is one of the best techniques for making concepts “stick.” By relating that which students need to know to that which they already do know, teachers create bridges in understanding, and those bridges give students a way to grasp a new and difficult concepts.

The same holds true in math class. If we teachers use powerful analogies that make concepts more memorable, students 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 students learn about our number system, on the one hand, and which also helps students work with fractions, on the other hand.

The analogy is to something I call the “Ladder of Primes.”

The Ladder of Primes is like a real ladder. It has a bottom, and it has rungs which you climb. By climbing you reach new heights of understanding and become capable of accomplishing things that you wouldn’t be able to do had you stayed on the ground. 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, 2, rising to whatever prime you need for any situation.

I always have students draw out a “Ladder of Primes” for all of the primes between 0 and 20. When they make it — often putting it on a post-it note — it looks like this:

"Ladder of Primes"

There are two main ways to use the Ladder of Primes, and I will touch on just one of those in this post. I’ll mention the other ways in a later post.

One of its most obvious and best uses — it helps us reduce fractions to lowest terms.

If students 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)  gain an algorithm that works for any fraction,

3)  learn about what the primes are and how they work.

I tell students that when they are reducing a fraction, they need to start on the lowest rung on the Ladder of Primes, and that they are to 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:  630/1386

Starting on the lowest run of the ladder, students first check to see if 2 divides into both numbers. Using rules of divisibility (see my posts on the rules of divisibility), students will easily see that 2 does divide into both numbers, leading to the more reduced fraction:  315/693

Ask if 2 goes in again. It does not. Students step up one rung, to 3. Ask if 3 divides evenly into both numbers. It does, leading to the more reduced fraction of 105/231. Ask if  3 divides in again. Since it does, we use it again. This is a key point. You move up to the next rung only if you have exhausted the usefulness of a prime. Dividing by 3 again, we get:  35/77. Ask about 3 again. Since it does not go into either number, we move to the next rung:  5. But 5 does not divide into both (and a prime must divide into both the numerator AND denominator in order to be used), so we move up to 7, which DOES go into both, leading us to the fraction:  5/11. Both 5 and 11 are prime, and whenever both numbers are prime, the fraction is fully reduced. So we are done. See how this works?

One thing to point out is that if students 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 exhaust 2, they do not need to test any of the multiples of 2:  4, 6, 8, or any other multiples of 2. 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 bring it up.

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 charges do. Feel free to let me know through a comment if you have any questions on how the Ladder of Primes works or should be taught.

Now try reducing these fractions using the Ladder of Primes. To add depth to the problem set, ask students 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).

I just happened to be looking at the NM Highway signs page online a couple of days ago when I saw this nice little list of signs, just below:

NM Highway Signs

I couldn’t help but notice that there are quite a few recognizable geometric figures on this page, and I thought, “This would be a cool thing to show to kids who either have studied, or are studying geometry.”

My suggestion: Show this to your children and ask them how many geometric figures they can recognize.

A partial list of shapes is listed below. You may want to use it to prompt your kiddos as they try to find the shapes.

You might give them a set of goals to shoot for, like this:

1 – 3 shapes noticed:  ”Road Scout”

4 – 6:  ”Highway Hawk”

7 – 9:  ”Eagle Eye”

10+:  ”Road Warrior”

GEOMETRIC OBJECTS TO FIND —

Sign:  Object

All four-sided signs:  quadrilateral / parallelogram / rhombus (diamond)

Five-Sided Signs:  non-regular pentagons

Think Black Line Bordering Signs:  concept of perimeter

Crossroads:  perpendicular lines

Signal Ahead:   rectangle / circles (ovals?) / space between rectangle and circles

Two-Way Traffic:  ray / parallel but opposite-directed rays

Right Lane Ends:  skinny rectangle, perhaps (object on the left) / adjacent segments (object on right side)

Pedestrian Crossing:  parallel segments or lines

Divided Highway:  parabola (or close to it) in the curved part of the barrier object

Low Bridge:  line (since it has arrows on both ends)

Hill:  right triangle, with its two legs and hypotenuse. / Also this diagram shouts out the concept of “slope” as it shows how slope that the truck is descending is composed of rise and run.

Did I miss anything? You tell me, by writing a comment to this post.

FINAL NOTE:   Students might find it fun to create geometry problems based on the shapes they find. For example, in Signal Ahead, a problem could be:  If the dimensions of the rectangle are 20 x 8, and if the radius of the circles is 2, what is the area of the region between the rectangle and the three circles? Have them give the project to you, or to one another, and then they can “grade” the answers.

So what, right?

Well, maybe not so fast …

If you have a mathematical/logical bent of mind, you might find that interesting.

Friday the 13th is generally considered a bad luck day. So if that is the case, you might wonder if Monday the 13th would be the logical opposite to Friday the 13th, a good luck day. Afterall, Friday is the end of the workweek, and Monday is the beginning of the workweek.

So in that sense, can it be said that Monday and Friday are opposites? And what might that imply.

So here is the challenge. Compose a logical argument as to whether or not Monday the 13th should be considered a lucky day.

That is the challenge for Monday, the 13th of June 2011.

HINT:  You may want to include information about the “truth value” (truthiness, as Steven Colbert likes to say) of statements and their converses.

REWARD:  The first person who presents a compelling logical argument, one way or the other, wins a $10 gift certificate toward the purchase of any Singing Turtle Press products. All comments must be posted by 1 a.m. on Tuesday, the 14th of June, this year.

Just as I was finishing my last post, on a new way to find the LCM for a pair of numbers, I discovered another way to do the same thing.

Coffee, Pi and More

I was looking at the problems at the end of my last post, these problems:

b)   15 and 20;  LCM  =  60

c)   18 and 20;  LCM  =  180

d)   24 and 28;  LCM  =  168, ….

… when I noticed something.

If, in the first problem, you take 15 and 20 and make it into a fraction, you would get 15/20, which reduces to 3/4.

Then if you take the 3 in the numerator of 3/4 and multiply it by 20, you get 60, the LCM.

Similarly, if you take the 4 in the denominator of 3/4 and multiply it by 15, you also get 60.

I thought about this a bit and realized that it leads to another way to find the LCM for any two numbers.

The steps work as follows:

1st) Write the two numbers as a fraction, with the smaller number as numerator. Then reduce the fraction to lowest terms.

2nd)  Multiply the original fraction by the reciprocal of the reduced fraction. The fraction that you wind up with has the LCM as both the numerator and denominator.

See how the two steps work with the next problem:  Numbers are 18 and 20.

1st)  18/20 = 9/10 (reciprocal = 10/9)

2nd)  18/20 x 10/9 = 180/180, so 180 is the LCM.

- – - – - – - – - – - – - – - – - – -

One more example, with the somewhat larger numbers of 24 and 44.

1st)  24/44 =  6/11 (reciprocal = 11/6)

2nd)  24/44 x 11/6  =  264/264, so 264 is the LCM.

- – - – - – - – - – - – - – - – - – -

It is that easy. If you’d like to “get a handle” on this project, try these practice problems using this technique.

PROBLEMS:

a)  6 and 8

b)  4 and 10

c)  9 and 15

d)  10 and 16

e)  14 and 21

f)  18 and 45

g)  24 and 28

h)  27 and 63

i)  32 and 48

j)  45 and 55

ANSWERS:

a)  6 and 8; RECIPROCAL = 4/3, LCM  =  24

b)  4 and 10; RECIPROCAL = 5/2, LCM = 20

c)  9 and 15; RECIPROCAL = 5/3, LCM  =  45

d)  10 and 16; RECIPROCAL = 8/5, LCM  =  80

e)  14 and 21; RECIPROCAL = 3/2, LCM  =  42

f)  18 and 45; RECIPROCAL = 5/2, LCM  =  90

g)  24 and 28; RECIPROCAL = 7/6, LCM  =  154

h)  27 and 63; RECIPROCAL = 7/3, LCM  =  189

i)  32 and 48; RECIPROCAL = 3/2, LCM  =  96

j)  45 and 55; RECIPROCAL = 11/9, LCM  =  495
Josh Rappaport is the author of the award-winning Algebra Survival Guide and several other supplemental math books. To check out these products, follow the links in the sidebar, or just visit the Singing Turtle Press website.

But never fear. I am not going to write another 375 pages on the topic.

What I do need to bring to your attention, though, is that there are two LCM situations that I did not take into account yesterday. So to present a complete picture, I need to explain (for those who have not already figured this out by themselves) how to use my new technique in those two situations.

Coffee, Pi and More

You will notice that in my write-up yesterday — and in the practice problems I provided — the gap always divided evenly into the smaller number. How convenient, right? In the first example, we had a gap of 3 dividing into 12; in the next, a gap of 4 going into 20. Of course this does not always happen. Consider a situation in which we want to find the LCM for 10 and 16. The gap of 6 (16 – 10 = 6) does NOT divide evenly into the smaller number, 10. So what would we do here?

In such a situation, we’d run through the multiples of the smaller number till we hit a multiple that the gap does divide into evenly. In this example of 10 and 16, we would look at the first multiples of 10:  20, 30, 40, 50, etc., till we find one that 6 does divide into evenly. Whoa! No need to go even that far, you point out … since 6 does divide evenly into 30. How many times? Five times. After getting this number, 5, follow the same procedure  I laid out yesterday … multiply this quotient, 5, by the larger number, 16, to get the LCM. So the LCM would be 80, since 5 x 16 = 80.

For practice, look at one more situation: find the LCM for 25 and 35. The gap, 10 (35 – 25 = 10), does NOT divide evenly into the smaller number, 25. So we would check out the multiples of 25:   25, 50 … BINGO! 10 does divide evenly into 50. How many times? 5. So we would just multiply 5 by the larger number, 35, getting 175. And that is the LCM for 25 and 35.

Note that in both of these number pairs: 10 and 16, and 25 and 35, there is a common factor for each pair. For 10 and 16 the common factor is 2; for 25 and 35, the common factor is 5. The fact that such a common factor exists for each pair is what allows us to use this approach. The common factor is also what makes it work out that the LCM is less than the mere product of the two numbers.

But as you may recall from studying this topic in other places, whenever two numbers do NOT have a common factor, the numbers are said to be “relatively prime.” And whenever two numbers are relatively prime, their LCM is nothing less than the product of the numbers.

Example:  9 and 16. Each number is composite, yet they share no common factors. The non-trivial factors of 9 are 3 and 9; the non-trivial factors of 16 are 2, 4, 8, and 16. No factors overlap, so 9 and 16 are relatively prime. That being the case the LCM for 9 and 16 is simply the product:  9 x 16 = 144.

And this is indeed the case whenever two numbers are relatively prime.

For our purposes, what we need to keep in mind is this: we can use the new technique I have presented yesterday and today whenever two numbers have a common factor. But when the two numbers do NOT have a common factor, we cannot use this technique. Instead, we just multiply the two numbers together, and the product we get is the LCM.

And now, some practice problems to help you nail down this method for finding the LCM.

For the following problems,  first determine whether or not the numbers are relatively prime. If they’re relatively prime, multiply them to find the LCM. If they’re not relatively prime, use the method outlined in this post to find the LCM. The answers follow.

PROBLEMS:

a)   6 and 10

b)   8 and 15

c)   8 and 14

d)   12 and 20

e)  27 and 35

f)   15 and 25

g)   32 and 55

h)   28 and 36

i)   64 and 77

j)   48 and 58

ANSWERS:

a)   6 and 10 — LCM = 30

b)   8 and 15 — Relatively prime, LCM = 120

c)   8 and 14 —  LCM = 56

d)   12 and 20 — LCM = 60

e)  27 and 35 — Relatively prime, LCM = 945

f)   15 and 25 — LCM = 75

g)   32 and 55 — Relatively prime,  LCM =  1,760

h)   28 and 36 —  LCM = 252

i)   64 and 77 — Relatively prime,  LCM =   4,928

j)   48 and 58 — LCM  =  1,392

Josh Rappaport is the author of the award-winning Algebra Survival Guide and several other supplemental math books. To check out these products, follow the links in the sidebar, or just visit the Singing Turtle Press website here.

While there are several techniques that “work” — by which I mean techniques we can teach to students and have them learn quickly — I’ve known of no technique that makes good intuitive sense. In other words, I’ve known no technique whose underlying principle felt obvious.

Feeling frustrated, I started looking for a technique that would have that undeniable “ring of truth.”

Coffee, Pi and More

And so, after playing around in my “sandbox of numbers” for quite a while,  I’m happy to report that I’ve finally found what I had been looking for.

In today’s post I will show you a way to find the least common multiple that makes sense, at least to me. I hope it will make sense to you as well.

This method takes a bit of time and space to explain — as you can see by the length of this post. But once you understand the method, you’ll find it fairly quick and easy, as you’ll see by looking at the brevity of Examples A, B, and C at the bottom of this post. And to test your understanding, you’ll get a chance to practice this technique at the end, too.

To describe the method, though, I do need to define a few terms and explain a few ideas. I’ll do that now, using, as an example, the numbers 12 and 15. To follow along, suppose that we’re trying to find the LCM for these numbers: 12 and 15.

DEFINITIONS:  We’re finding the LCM for a pair of numbers. To make it clear which of the two numbers we’re referring to at any given moment, we’ll either refer to the smaller number or to the larger number. In our example, 12 is the smaller number, 15 the larger.

One thing that this new method pays attention to is the difference between the two original numbers. For 12 and 15, the difference is 3, for the simple reason that 15 – 12 = 3. For any pair of numbers, we call this difference between the initial numbers the gap.

The new method also takes a look at the first multiples (Ms, in my shorthand) of the two numbers.  By multiples, I mean what we get when we multiply both of the original numbers by the early Natural Numbers of 2, 3, 4, etc.  So …

2nd Ms means second multiples — 24 & 30 in our example, since 12 x 2 = 24; 15 x 2 = 30.

3rd Ms means third multiples — 36 and 45 in this example.

4th Ms means fourth multiples — 48 and 60, and so on.

We write these labels —  2^nd Ms, 3^rd Ms, 4^th Ms, etc — as column headers in the chart below.

To distinguish the rows in the chart, Ms/S stands for multiples of the smaller number; Ms/L stands for multiples of the larger number.

Another thing that this method pays attention to is the difference between the numbers in each column. For example, the second multiples of 12 and 15 are 24 and 30. So in the 2^nd Ms column, the difference is 6, since 30 – 24 = 6. We write these differences in the chart’s bottom row, with Ds being my shorthand for differences. Note that we write the gap in parentheses right after Ds.

With all of this in mind, here’s our chart for 12 and 15:

You might be wondering why we’re paying attention to the differences. It turns out that the differences help us see when we find common multiples in general — and when we find the all-important LCM in particular. Here’s how this works …

As noted, 12 and 15 start out with a gap of 3.

Moving to the 2^nd Ms column, the difference becomes 6. In the 3^rd Ms column, the difference grows to 9. See the pattern? Because the gap is 3, the difference grows by 3 as we move each additional column to the right.

Then, when we get to the 4^th Ms column, something interesting happens. First, notice that the difference in this column is 12, which just happens to equal the smaller number. Looking up in the chart, notice that the multiple of 12 in 4^th Ms is 48. Then notice that, since the difference between the smaller and larger number here is 12, the number below 48 is 12 more than 48, which it is, as it is 6o. But then also notice that when you look just to the right of 48, the next multiple is also 12 more than 48, again 60. That is true because as you move to the right in the 12s row, each new number is 12 more than the previous number, since these are the successive multiples of 12.

So this 48 in the 4^th Ms column is in a sense “the magic number.” Whether you look below this 48 or you look to the right of it, you get a number that is exactly 12 more than 48. Those two numbers — the upper being a multiple of 12, the lower being a multiple of 15 — therefore must be the same. And they are. They are both 60.

So 60 must be a common multiple for 12 and 15. Not only that, but since this is the first time this happens in the chart, 60 is the Least Common Multiple, or LCM. Here is an updated chart that shows these relationships:

The general principle, as far as the chart goes, is this: when we find the column in which the difference equals smaller number, the multiple of the smaller number in that column is the “magic number.” When we look below this “magic number” and to the right of it, we see the same number, and this number is the LCM. The reason, in general, is this: with a difference equal to the smaller number, the number below the “magic number” must be greater by the value of the smaller number. And the number to the right of the “magic number” must also be greater by the value of the smaller number because that row simply lists the multiples of the smaller number.

In a way this is so simple and obvious, it’s almost hard to see (like trying to look out and see your own nose — without a mirror!).

To make it easier, let’s look at one more example. This time we’re seeking the LCM for 20 and 24. So now the smaller number = 20; larger number = 24; gap = 4. Chart looks like this:

Scan the chart to see where the difference equals the value of the smaller number, 20. This happens in the 5^th Ms column. So the “magic number” is 100, and looking below and to the right of 100, we find the LCM, 120.

You might be wondering, though, if it is possible to mentally figure out in which column the difference will equal the smaller number. For if we could do this, we could find the LCM WITHOUT a chart, through this technique.

Turns out there is an easy way to do this. Figuring it out relates to the way that the gap makes the difference slowly but steadily increase as we read the chart left to right.

Look at the bottom row to see how the differences increase as we move from left to right:  4 — 8 — 12 — 16 — 20 …

These differences grow by 4 with each new pair of multiples. And that’s because the gap is 4.

How then do we figure out when that difference will be 20? This is like a 4th grade word problem. We just divide the smaller number, 20, by 4, and we get 5. This means that for this problem, the difference will be 20 in the 5th Ms column. And of course this is the case.

And this holds true in general. To find out in which column the difference equals the original smaller number, just divide the smaller number by the gap. The quotient you get gives the column number where this occurs.

So the process of finding the LCM, using this technique, boils down to just two math thought steps.

1^st)  Figure out in which column the difference will equal the smaller number.

2^nd)  Find the value of the larger number in that column.

And here are the corresponding two math steps:

1st)  Noting the gap, divide the smaller number by the gap, and get the quotient. (The quotient tells us the column number where we find the “magic number.”)

3rd)  Multiply the quotient by the larger number. This product is the multiple of the larger number found just BELOW the “magic number.” This will be the LCM.

Let’s try a few more so you can get the hang of this.

- – - – - – - – - - – - – - – - – - - – - – - – - – - - – - – - – - – -

EXAMPLE A —

Problem:  Find LCM for 16 and 20.

1st)  (Smaller #) ÷ (Gap)  =  16 ÷ 4 = 4, so Quotient = 4.

2nd)  (Quotient) x (Larger #) = 4 x 20 = 80 = LCM

Answer:  LCM for 16 and 20 is 80.

- – - – - – - – - - – - – - – - – - - – - – - – - – - - – - – - – - – -

EXAMPLE B —

Problem:  Find LCM for 21 and 28.

1st)  (Smaller #) ÷ (Gap)  =  21 ÷ 7 = 3, so Quotient = 3.

2nd)  (Quotient) x (Larger #) = 3 x 28 = 84 = LCM.

Answer:  LCM for 21 and 28 is 84.

- – - – - – - – - - – - – - – - – - - – - – - – - – - - – - – - – - – -

EXAMPLE C —

Problem:  Find LCM for 60 and 72.

1st)  (Smaller #) ÷ (Gap)  =  60 ÷ 12 = 5, so Quotient = 5.

2nd)  (Quotient) x (Larger #) = 5 x 72 = 360 = LCM.

Answer:  LCM for 60 and 72 is 360.

-       – - – - – - – - - – - – - – - – - - – - – - – - – - - – - – - – - – -

PRACTICE:

Find the LCM for these pairs of numbers using the two-step process shown above.

a)   12 and 18

b)   15 and 20

c)   18 and 20

d)   24 and 28

e)   12 and 14

f)     14 and 21

g)   30 and 36

h)   36 and 45

i)     24 and 32

j)     36 and 48

ANSWERS:

a)   12 and 18; LCM  =   36

b)   15 and 20;  LCM  =  60

c)   18 and 20;  LCM  =  180

d)   24 and 28;  LCM  =  168

e)   12 and 14;  LCM  =  84

f)     14 and 21;  LCM  =  42

g)   30 and 36;  LCM  =  180

h)   36 and 45;  LCM  =  180

i)     24 and 32;  LCM  =  96

j)     36 and 48;  LCM  =  144

FINAL NOTE:  I should mention before signing off today that there are two types of situations that I have not yet accounted for. In other words, there’s one kind of situation where we have to do one additional step to find the LCM. And there’s another situation in which we get the LCM in an even easier way. Fortunately, though, this is not terribly complicated. And I’ll explain these two situations in tomorrow’s post.

During this transition between the end of the school year and the start of summer, I will be taking a break from blogging. The break will be either one week or two weeks … I’m not quite sure yet. In any case I will be back in touch in the early-to-mid part of June.

Best,

—  Josh

I was looking through the links to movies with math themes, and a question came up.

On the site showing the movies, the text says that there are “mathematical themes and patterns motivated by math” in the introduction scene for the James Bond movie, Casino Royale, this clip:

I’ve watched the clip a few times, and I have my own ideas as to mathematical themes and patterns.

But I thought I’d open it up. Watch this clip and comment on what math themes and patterns you can find in the introduction. It will be fun to hear the various things that people find. If you are a teacher or parent, feel free to ask your students or children what mathematical themes and patterns they see. It could be anything.

Just add a comment to this post, and I’ll share any comments so we can all take a look  …

This gives us all a good chance to think about how mathematical patterns appear in popular culture.

So recently I stumbled upon a memory trick that helps students tell which of two numbers is greater and which is less.

Let's Reduce Mistakes in Algebra!

You might be thinking:  greater and less?! Why would any student have trouble with that? Well, before students hit negative numbers and absolute value, there is generally little trouble. The greater numbers are the larger numbers, the lesser numbers are the smaller numbers. And kids basically know what we mean by larger and smaller whole numbers, when they are dealing with positive numbers and zero.

But when students encounter negative numbers, some things change.
While 10 > 5,   – 10 IS not > – 5. Instead:  – 10 < – 5.

As if that were not enough, absolute vale comes along and makes things still more confusing, since it takes the value of any number and makes it positive. So now:

abs. value of – 10 > abs. value of – 5

With the concepts of negatives and absolute value “muddying” the once clear water, students do get confused, especially with the negative numbers.

If you want proof, try this. Talk to three students in Algebra 1. First ask them what is larger:  10 or 5. Talk to a few kids, and you’re likely to get either a straight correct answer or that classic teen look of :  ”What, you think I’m stupid?”

But then ask the same kids what is LARGER:  – 10 or – 5. Do this a few times, and you’ll observe a discernible pause before you get an answer; not infrequently, the answer will be – 10!

The trick that helps students straighten this out deals with the number line. Fortunately, on the number line, things are actually still simple when it comes to greater and less. If you are looking at any two numbers, the number that is to the right is greater, the number to the left is less.

This holds true whether both numbers are positive, both are negative, or one is positive and the other negative. And it holds true if either number is zero. That pretty much covers all cases!

So what is my big trick? It is right there, in the words greater and less.

The word “LESS” share the letters “LE”  in the word “LEFT.”

The word “GREATER” share the letters “RT”  in the word “RIGHT.

So the greater number is to the right; the lesser number is to the left.

When I point this out to students, they generally nod their head and say, “Oh …. yeah … “

I point out that whenever they need to figure out which of two numbers is greater or less, they just need to visualize the two numbers on the number line (some students need to make a “thumbnail” sketch of the two numbers, along with zero). Then all they do is visually check and see which number is to the left, and that number is LESS. Check and see which number is to the right, and that number is GREATER. No more pondering over the impact of negative signs or absolute value symbols. As in elementary school, the answer could not be more plain.

A simple trick, but an easy, helpful one.