# Invert and Multiply…but WHY?

Most of my posts so far have addressed high school level math.  This one is for younger students (and their teachers).

At another blog that I follow, Kitchen Table Math, there has been recent discussion about the “invert and multiply” rule for dividing fractions and whether it is important and/or appropriate to teach the reasoning behind this rule.

You know I am going to vote “yes”.  But what kind of derivation should we use here?

The rule can be derived algebraically.  But that doesn’t really help the grade-school students who are just learning it.  It can also be derived in a algebra-ish way.  A commenter at Kitchen Table Math pointed to the explanation found here:

http://www.moveitmaththesource.com/realfractions/mrnovaksproof.html

Mr. Novak’s explanation is elegant and concise. And it certainly answers the question: why do we invert and multiply in order to divide fractions?  But it would be nice if students also had an intuitive feel for why this works.

We can build the rule step-by-step in a manner that helps that intuition to develop.

Step 1: Division is for Sharing

You have 12 cookies – each a pretty good size —  and you need to put them in individual serving bags to hand out at a party.  By deciding how many to put in each bag, you also determine how many serving bags you can make.  For example, you could put 12 cookies in each bag, but then you would only have enough for 1 bag.  On the other hand, it you put 6 cookies in each bag, then you can fill 2 bags because 12/6 = 2. That seems obvious, but it establishes a method: we are going to divide the total number of cookies by the amount in each bag to find how many bags we can fill.  You soon notice that if you put fewer cookies in each bag, you can fill more bags.  For example:

 If we take 12 cookies… And put 12 in each bag… 12 ÷ 12 = 1 We can fill  1 bag If we take 12 cookies… And put 6 in each bag… 12 ÷ 6 = 2 We can fill  2 bags If we take 12 cookies… And put 4 in each bag… 12 ÷ 4 = 3 We can fill  3 bags If we take 12 cookies… And put 2 in each bag… 12 ÷ 2 = 6 We can fill  6 bags If we take 12 cookies… And put 1 in each bag… 12 ÷ 1 = 12 We can fill  12 bags

So yes, dividing by a smaller number gives a bigger answer.  And with 12 cookies, 12 bags is the most you can fill…or is it?  I mentioned that these were pretty big cookies.  Hmmm…

Step 2: Dividing by Fractions (of the “one over” variety)

You decide to put make the serving size a ½-cookie.  Since that serving size is even smaller, you expect to fill more bags.  Though you have not been taught how to divide fractions, you can do this one by thinking:

If I am putting ½ in each bag, then each cookie is enough for 2 bags.  So 12 cookies fill 24 bags.   I guess that must mean 12 ÷ ½ = 24.

Need to fill more bags?  Use smaller portions!  Give everyone 1/3 of a cookie…

You can see where this is going.  Let’s continue the chart…

 If we take 12 cookies… And put 1/2 in each bag… 12 ÷ (1/2) = 24 We can fill  24 bags If we take 12 cookies… And put 1/3 in each bag… 12 ÷ (1/3) = 36 We can fill  36 bags If we take 12 cookies… And put 1/4 in each bag… 12 ÷ (1/4) = 48 We can fill  48 bags If we take 12 cookies… And put 1/5 in each bag… 12 ÷ (1/5) = 60 We can fill  60 bags

The trend has continued.  Dividing by a smaller number gives a bigger answer.  And even if you don’t know the rule for dividing by fractions, you can certainly divide by these fractions!

To divide by “one over anything”, just multiply by the denominator.

STEP 3: Any fraction at all!

Suppose you have those 12 big cookies and you decide to give everyone (2/3) of a cookie.  How many servings can you make?

We can figure this out using the reasoning we followed above, with one extra twist.

We have already seen that if you want to give everyone just (1/3) of a cookie, you will have enough to fill 12 × 3 = 36 bags.  But now you decide to give everyone TWO thirds of a cookie.  But you have already filled all those bags – why not just give everyone two of the bags?  You will have enough for 36 ÷ 2 = 18 servings.

Now go back and look at how we got that answer.  We don’t have to know an official rule to divide the 12 cookies into servings that each have 2/3 of a cookie.  We can just multiply by 3 and then divide by 2, which is the same as multiplying by 3/2.

Let’s try another:

Suppose you have 36 pizza pies to sell and you want to give 3/5 of a pizza to each customer.  (No one cuts pizzas into fifths, but don’t worry about that.)  How many customers can you serve before running out of pizza?

Well, if we give everyone ONE fifth of a pie, we can feed 36 × 5 = 180 customers.  Giving everyone THREE fifths means that we can only serve 180 ÷ 3 = 60 customers.

So to find 36 ÷ (3/5) we multiply by 5 and then divide by 3 which is the same as multiplying by 5/3.

Which derivation is “better”?

If I could only show students one explanation, I would go with Mr. Novak’s.  But if I had time to explore and play, I would start with what you see above.  I like that it encourages number sense and intuition.  Also, it introduces an idea that students will see again many times: that existing patterns can be extended to cover new situations.

# Still Feeling Mean

Here’s a quick addition to that last post on various types of means.  By coincidence, yesterday, a question came up in my AP physics class about the meaning of the average radius of an ellipse.

We were looking at the laws that govern circular orbits.  But the orbits of planets are actually elliptical with the Sun at one focus.   The average orbit radius is the length of the semi-major axis. [When I first read that, it seemed wrong to me: how can the semi-major axis be the average distance — shouldn’t it be the greatest distance? But average orbit radius is average distance to the Sun, not to the center of the ellipse.  The Sun is at one focus so the planet distance is sometimes more than the length of the semi-major axis and sometimes less.  But that’s not the same thing as the average radius of the ellipse, which is the question that was asked.]

I’m not sure how useful it is to find the average radius of an ellipse, but it is interesting — and better with a picture!

The distance from the center of a circle to any point on the circle is a constant — that’s the length of the radius.

The distance from the center of an ellipse to a point on the ellipse is not constant.  The longest it can be is the semi-major axis, a.  The shortest it can me is the semi-minor axis, b.  But what do we mean by the average radius?  One possible interpretation is that the average radius is the arithmetic mean of the two semi-axes.  But here’s another interpretation:

What is the radius of the circle that would have the same area as the ellipse?

Clearly, that radius would have to be more than b and less than a.  So maybe it is their mean…but which kind of mean?

To answer this, you need to know two formulas:

Acircle = πr2

Aellipse = πab

So if the two areas are two be equal, we need r2 = ab.

In other words…

For a circle to have the same area as an ellipse, the length of its radius must be the geometric mean of the lengths of the ellipse’s semi-major and semi-minor axes!

I didn’t know that.  And I was surprised to see a geometric mean pop up in class so shortly after my last post.  Also, I should add: I don’t know how to show that the semi-major axis is the average orbit radius.  If any of my fine students can come explain that to me, it is certainly worth a little extra credit.

# MEAN, MEAN, MEAN…

One of my very favorite students greeted me this new year with a question:  what are the different kinds of means?  This is a fun question to explore.  We are going to look at three kinds of means, each associated with a type of sequence.

Suppose we start with two numbers.   Their mean is a number that lies between them.  But exactly where it lies depends on what type of mean you are talking about.

If left unspecified, the term “mean” probably refers to the “arithmetic” mean.  So let’s start with that one.  (And by the way, helpful note for beginners:  “arithmetic” here is an adjective.  Read with emphasis on 3rd syllable.)

ARITHMETIC SEQUENCES AND ARITHMETIC MEANS

A sequence of numbers is called arithmetic if the difference between any two consecutive terms is a constant.  For example:

2, 5, 8, 11, 14… has a common difference, d = 3.  (We calculate the current term minus the previous term.)

Say we have been given two numbers A and B.  We want to insert a number between them so that the three of them together form one of these arithmetic sequences. We are looking for the “arithmetic mean.”

We have A, x, B and we want the differences to be constant.

That means we can write: x – A = B – x

From that we get: 2x = A + B

And then: x = (A + B)/2

This probably does not surprise you.  The arithmetic mean is also just the average.

If A = 30 and B = 60 we get an arithmetic mean of 45, thus forming the sequence:

30, 45, 60

Then, if you check, you find the difference between consecutive terms is constant:

45 – 30 = 60 – 45 = 15. So the sequence is arithmetic.  45 is the arithmetic mean between 30 and 60.

Sometimes, this is the kind of mean we care about.  For example…

You drive for two hours at 30 mph and for two hours at 60 mph.  What is your average speed for the trip?

When you solve this, you will discover that when you drive at different speeds for equal time periods, your average for the trip is the arithmetic mean of the two speeds.

ANOTHER SEQUENCE, ANOTHER MEAN

A geometric sequence is one where the ratio between every two consecutive terms is constant.  For example:

1, 3, 9, 27, 81… is a geometric sequence with a common ratio of 3. (Now we calculate the current term divided by the previous term.)

So again we have been given two numbers A and B.  And now we want to insert a number between them so that the three of them together form one of these geometric sequences.  We are looking for the “geometric mean.”

We have A, x, B and we want the RATIOS to be constant.

That means we can write: x /A = B/ x

Cross multiply to get: x2 = AB

And then: x = √(AB)

For example, if A=9 and B=16 then the geometric mean x = √(9×16) =√144  = 12.

We should check our work to see if the ratios are constant:

12/9 = 16/12 = 1.333… so yes, the sequence is geometric.  12 is the geometric mean of 9 and 16.

Sometimes, this is the kind of mean we care about.  For example, here is a problem you may have seen in geometry:

In the triangle above, if x = 9 and y = 16 determine the height, h.

You can use similar triangles to show that the height of that altitude is equal to the geometric mean of the lengths of the two segments on the hypotenuse.

ONCE MORE, WITH HARMONY

A sequence is called “harmonic” if there is a common difference between the reciprocals of consecutive terms.  For example  (in fact, the classic example):

1, 1/2 , 1/3, 1/4, 1/5… is harmonic.  Its reciprocals are 1, 2, 3, 4, 5… which is clearly arithmetic with a common difference of 1.

And one more time, we have been given two numbers A and B.  Now we want to insert a number between them so that the three of them together form one of these harmonic sequences.  We are looking for the “harmonic mean.”

We  have A, x, B and we want the difference of the reciprocals to be constant.

That means we can write:

So suppose A = 40 and B = 60. Their harmonic mean is:

To check our work, we have to look at the sequence of the reciprocals:

1/40, 1/48, 1/60… or, expressed over a common denominator,

6/240, 5/240, 4/240…sure enough, the reciprocals form an arithmetic sequence.

Sometimes, this is the kind of mean we care about.  For example, here is an SAT classic:

You drive to work at 40 mph and then you drive back home at 60 mph. What is your average speed for the trip?

You can solve this by calling the length of each leg d, finding the time for each leg of the journey, adding to get the total time and then using total time and total distance to find average speed.  When you are done, you are in for a surprise.  When you travel equal distances at two different speeds, your average speed for the trip is the HARMONIC mean of the separate speeds.

I’ll close with a question: why is this called “harmonic”?  Does it have something to do with music and harmony?

Happy New Year!