kitchen table math, the sequel: the hard kind of percent problem

## Thursday, February 14, 2008

### the hard kind of percent problem

The hard kind of percent problem is this one:

John paid \$52.50 for a shirt including tax of 5%.
What was the price of the shirt before tax?

I imagine this problem is a cinch (cinch?? sp?) for the gifted, but for the non-gifted, this problem is HARD.

I was thinking this problem is an example of a partitive word problem, but now, re-reading Carolyn's original post on the subject, maybe not.

variations on a theme: 3 kinds of percent problems

As far as I can tell, every percent problem comes in three forms. (PLEASE correct me if I'm wrong.)

price of shirt: \$50
tax: 5%
price of shirt including tax: \$52.50

1. A shirt sells for \$50. Sales tax is 5%. What will the total cost be?
2. A shirt sells for \$50. After tax, the shirt sells for \$52.50. What is the tax rate?
3. A shirt sells for \$52.50 including a tax of 5%. What was the original price of the shirt?

Problem number 1 is easy.

Making the move to problems 2 and 3 is hard for many students.

But 3 is the killer. (I think.)

If you're having to teach or reteach percent to your children, be sure to include lots of number 3s in the mix.

........................

Do math textbooks call numbers 2 and 3 "work backwards problems" problems these days?

They may.

........................

Liping Ma deals with these problems in her chapter on division by fractions.

She describes asking her group of U.S. elementary teachers to create a word problem representing 1 3/4 ÷ 1/2:

Imagine that you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be a good story or model for 1 3/4 ÷ 1/2?

She goes on to say:

...division by fractions is an advanced topic in arithmetic. Division is the most complicated of the four operations. Fractions are often considered the most complex numbers in elementary school mathematics. Division by fractions, the most complicated operation with the most complex numbers, can be considered as a topic at the summit of arithmetic.

The summit of arithmetic!

I love it!

The U.S. teachers didn't fare well.

Of the 23 U.S. teachers, 212 tried to calculate 1 3/4 ÷ 1/2. Only nine (43%) completed their computations and reached the correct ansewr. For example, Mr. Felix, a beginning teacher, gave this explanation.:

I would convert the 1 3/4 to fourths, which would give me 7/4. Then to divide by 12/2, I would invert 1/2 and multiply. So, I would multiply 7/4 by 2 and I would get 14/4, and then I would divide 14 by 4 to get it back to my mixed number, 3 2/4 or then I would reduce that into 3 1/2.

[snip]

Tr. Bernadette, the experienced teacher who was very articulate about the rationale for subtraction with regrouping, tried a completely incorrect strategy:

I would try to find, oh goodness, the lowest common denominator. I think I would change them both. Lowest common denominator, I think that is what it is called. I do not know how I am going to get the answer. Whoop. Sorry.

Tr. Bernadette sounds a little like me trying to do a simple percent change problem the other night. C. and I looked at the same problem last night and we both said: how did we make this so complicated?

That is the \$40,000 dollar question.

The \$40,000 dollar answer is: failure to transfer.

Ma goes on to discuss the word problems U.S. teachers came up with to represent 1 3/4 ÷ 1/2, and there we see a clean sweep: only one of the 23 teachers produced a correct model of the problem, and that one model was "pedagogically problematic."

I am proud to say that when I read Ma I was immediately able to produce a mathematically correct word problem representing 1 3/4 ÷ 1/2. As I recall my problem went something like this:

Catherine has two dogs, and each dog eats 1/2 can of dog food every morning. If she has 1 3/4 cans of dog food left, how many servings is this?

Six of the U.S. teachers confused dividing by 1/2 with dividing by 2.

The Chinese teachers created two genres of problems:

• quotitive division, which Ma calls the measurement model
• partitive division - finding a number such that 1/2 of it is 1 3/4

Which brings me back to my hard percent problem.

At least, I think it does.

the "measurement" model

The measurement model is more obvious to me. Obvious meaning easy to understand. My own word problem falls in the measurement category: how many 1/2s are in 1 3/4?

How many 1/2s can 1 3/4 be divided into?

How many 1/2-can servings of dog food are there in 1 3/4 cans of dog food?

If I'm measuring the number of 1/2s in 1 3/4, I divide 1 3/4 by 1/2.

1 2/3 ÷ 1/2 = 3 1/2.

I have 3 1/2 servings of dog food.

the "partitive" model

Chinese teachers gave the measurement model short shrift:

Among more than 80 story problems representing the meaning of 1 3/4 ÷ 1/2, 62 stories represented the partitive model of division by fractions--"finding a number such that 1/2 of it is 1 3/4":

Division is the inverse of multiplication. Multiplying by a fraction means that we know a number that represents a whole and want to find a number that represents a certain fraction of that.

After all this time, I still find this passage mystifying.

I was even more mystified when I read the story problem Tr. S came up with:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station A to Station B?

[isn't this written wrong? shouldn't the numbers be reversed?]

I'm going to reverse them:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station A to Station B. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station B to Station A?
update: see below

Here's another:
The mom bought a box of candy. She gave 1/2 of it which weighed 1 3/4 kg to the grandma. How much did the box of the candy originally weight? (Ms. M.)

For a lot of us, the candy problem will be most obviously similar to the "hard" percent problem.

To solve the candy problem:

let x = original weight

1/2x = 1 3/4
1 3/4 ÷ 1/2 = x
3 1/2 = x

The original box weight 3 1/2 pounds.

The hard percent problem has the same form. We know the final price; we know the tax rate. We need to know the price of the shirt sans tax.

let x = price of shirt
1.05x = 52.50
52.50 ÷ 1.05 = x
50 = x

price of shirt = \$50

I have to include this for people like me: the reason you take 1.05x is that it is a shortcut.

I spent years of my life taking 5% of \$50 to get the sales tax (\$2.50), then adding the sales tax amount to the price of the shirt.

Then C's 5th grade teacher explained to me that instead of doing these two steps I could simply multiply the price of the shirt by 1.05 and get the whole thing over with.

The "1" in the 1.05 ensures that the price of the shirt is part of the final value.

That was a revelation.

are partitive problems harder than measurement problems?

I still don't know what kind of language to use in describing these problems (if I'm not going to use the term "partitive," that is, and for the time being I am not).

Can we say that the shirt problem is a part - part - whole problem in which the whole and a part are known and we have to find the other part?

hmmmm.....

Not exactly.

unknown part: price of shirt
unknown part: sales tax in dollars
part: sales tax rate
whole: final price

I have no idea what to call these problems or even how to describe them.

All I know is that they are difficult for kids to learn, and they don't "come naturally" once a student has learned how to find the price of a shirt given the original price and the sales tax rate.

Any thoughts?

....................................

update (from above)

I'm thinking there is a translation problem with the train problem as printed in Ma's book:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station A to Station B?
As it stands, this problem does not model 1 3/4 ÷ 1/2.

This problem models 1 3/4 x 1/2.

I think the problem should read:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of [the time it takes to go from] Station A to Station B. How long does the train take going from Station A to Station B?

let x = time it takes to go from Station A to Station B
time it takes to go from station B to Station A: 1 3/4 hours

1/2x = 1 3/4
x = 1 3/4 ÷ 1/2
x = 3 1/2

It takes 3 1/2 hours to go from Station A to Station B Anonymous said...

I have lots of thoughts, but your post covered a lot of ground...

They are NUMBERS. And they behave like NUMBERS behave, and their rules are sound.

Let's try to understand what the normal rule for division of natural numbers is:

for whole numbers M and N, and M is a multiple of N and N is nonzero, then M divided by N, or M / N = Q means that M = Q * N.

Let's try that again:

When we say M divided by N = Q, that means M = Q * N. So it is best to understand division as just multiplication, but shown differently.

So what does the above mean?

We are saying that M is the total of Q mulitples of N. Every time you aren't sure about division, you go back to writing it as multiplication.

If this is really, truly clear in your head and your son's head, then it isn't so hard to solve the percent problem. But if it's NOT clear, not really, truly clear, then it's hard.

So now let's go to division of fractions. They behave JUST LIKE NATURAL NUMBERS, so the rules for division still apply.

3/5 divided by 10/9 is the same as

3/ 5 * 9/10. why?

Because M/N = Q is the same as M = Q * N. okay, this will still take time, but it matters.

3/ 5 is the same as 3/5 * 9/10 * 10/9.

Why? Because 10/9 * 9/10 is 1. Is that obvious and believable? Is it to your son? Does he know that is true, and believe it, because you just multiply the tops and the bottoms and they are the same?

but then you can regroup. (this is why association and distribution matter.)

3/5 = (3/5 * 9/10) * 10/9
But NOW we are in the M = Q * N form.

Call 3/5 M. Call (3/5 * 9/10) Q. Call 10/9 N.

We know that division of M/N = Q is just the same as M = Q * N.

So we now know that 3/5 = (3/5 * 9/10) multiples of 10/9. And so

And we can now see that 3/5 divided by 10/9 is 3/5 * 9/10. because M/N = Q is the same as M = Q * N.

That's what Ma's partitive division is saying.

(cont) Anonymous said...

So, to the original problem:

you first have to see that the "including" means
"I paid for the shirt, and I paid tax, and the tax is found from the price of the shirt.

The total = price of shirt plus tax.

This is a classic Singapore bar problem, no? Does that make it easier to see?

You paid X for the shirt. You paid T(X), the tax on the price of the shirt. How much was the tax on the price?

That is just the same as your #1 formulation, but you don't have the price as a number just yet. But if you can see that it's the same as if you had the price, you're golden. And that's what you need to see.

Percents are just fractions whose denominators are ALWAYS 100. Fractions are numbers, so we can handle this as numbers.

So, the T(X), the tax on the price of the shirt is 5%. What does that mean? 5% means 5/100. 5% tax on the price of the shirt means 5/100 * the price of the shirt. Call the price of the shirt X.

You KNOW how to figure the tax, right? you said so yourself: you can do version 1:

shirt sells for X, sales tax is 5%, so the total cost is ...

So now, that's the bar part of your problem.
you've got X + 5/100 * X = 52.50,

That's 105/100 * X = 52.50
X is gotten by dividing both sides by 105/100.

(cont) Anonymous said...

I didn't see the translation problem in the train problem. I immediately saw the "time of that from " to mean exactly how you expanded it. But I do see how that could confuse someone, ESPECIALLY a child, into thinking it was supposed be backwards, ot something.

but the problem is unnatural, in that the most Obvious way to explain it is in the reverse direction: something is TWICE as long as something else. If you saw it that way, the fiction of "dividing by 1/2" would go away as you'd immediately see you were just multiplying by 2, and the problem would be easier to solve. If you saw that you just needed to double 1 3/4, would you have been so confused?

Again, would doing this as a bar problem have helped you?

N is the time it takes to go from B to A.

This is 1 3/4 hours.

The time it takes to go from B to A is HALF the time it takes to go from A to B.

So draw the picture:
........ = 1 3/4
................. is long enough that the first picture is HALF of the second.

OR: the SECOND picture is TWICE the first.
Then the problem solves itself.

And there is how we get back to Q = M / N is the same as M = Q * N.
M = Time from A to B. N = Time from B to A.

Time from A to B = 2 * N. And N is 1 3/4.

Catherine Johnson said...

One more "data point": at this point in the class the kids are learning to solve percent problems as proportions for the first time ever.

That complicated the "hard" problem because instead of finding 5% sales tax by setting up this proportion:

5/100 = X/\$50

you have to find one hundred and five percent - you have to remember that when you add tax to the price of the shirt you've got a percent increase

Catherine Johnson said...

I read once that a fraction is "the comparison of two numbers via division"

(it's possible that the word used wasn't "fraction," but ratio -- I don't know)

is that wrong (in your view)?

Is it contradictory to say that a fraction is a number and to say that a fraction is a comparison of two numbers?

Catherine Johnson said...

Every time you aren't sure about division, you go back to writing it as multiplication.

right -- and I still haven't been able to get that across, I think Anonymous said...

so back to the original difficulty:

why is the 3rd formulation hard to solve?

I think it's this:

when you do a standard "x is the price, what is the tax", the price is given. you dont' even need to READ or THINK, you just look for the shortcut of multiply the tax by the price.

but in the third formulation, the PRICE you're given isn't THAT SAME price. You're now given the TOTAL price, and you don't compute the tax by multiplying the prercentage on this price. so the immediate "dont' read the problem carefully" answer is just flat out wrong: 52.50 * .05 is not correct.
but now, the confusion sets in: wait, do I take the tax on the TOTAL price? or some other price? What price? is this the price I use, or not? since you never understood what was the underlying THING as an abstraction, it's hard to keep track of it. 5% of something I don't know?!?!?!? how does THAT work? And figure out something SUCH THAT SOMETHING ELSE is true?????

Every watch the Price is Right? To this day, no one can easily explain the check game, where you need to write a check for an amount such that THAT VALUE, when added to the value of something ELSE, is less than 1000 but not exceeding 1100, or something. it's too many symbols to hold in your head unless you are comfortable understanding that symbols represent things.

Your translation problem is the same representation issue: "it is only 1/2 time of that from": you had to EXPAND THAT "that" to see the problem clearly. It's hard for kids to expand the percent problem to see it.

Catherine Johnson said...

Why? Because 10/9 * 9/10 is 1. Is that obvious and believable? Is it to your son?

I would say no.

I bring this up a lot....and the problem may be failure to transfer as opposed to not believing that a non-zero number divided by itself is 1. Anonymous said...

--- fraction is "the comparison of two numbers via division"

Is it wrong? Let's stick with this: it is NOT HELPFUL.

How does a student add, subtract, or multiply "comparisons of numbers" let alone divide them? it makes it sound like fractions are aliens, and the arithmetic rules dont' apply, but new, unreasonable, arbitrary rules must be learned. NO NO NO NO NO . We must help students to see the continuity of the number system.

A fraction is a NUMBER. Now, it is a number SUCH THAT it is the SAME as a "comparison of numbers via division", but you need to know what "comparison via division" means, and that's vague unless you ALREADY knew what a fraction was! Anonymous said...

It's not helpful to say it's a number AND a comparison, because what are these magical "comparisons" ? What are the rules for them? Do comparisons have distributive properties, associative properties, identity rules? how do you add comparisons, etc. When you say something is a number and not a number, you lose the simplicity of the fact that numbers can all be worked on with the same rules.

Instructivist said...

I read Carolyn's explanations of the partitive/quotitive distinction in the old KTM and the difference between the math knowledge of Chinese and U.S. teachers.

Elementary teacher training in ed schools is a travesty. Drastic changes are needed. Prospective teachers should be required to take rigorous courses in various academic disciplines and demonstrate mastery. Fewer theory and vacuous methods courses, more academic subject matter courses.

I was amused by this post in Bridging Differences on the math capabilites of a teacher:

I would contend, that was the problem with local school control prior to education reform and the standards movement. There were no defined school/district/state/federal curricula for teachers to follow. They taught, pretty much, whatever they wanted to the exclusion of a more comprehensive curriculum. I can't tell you how many elementary teachers I worked with who taught language arts six hours a day, 180 days a year because that was what they were most comfortable doing. They intentionally avoided math, science and history because they were NOT as comfortable with those disciplines. They didn't do well with them when they were in school so they avoided teaching them when they became teachers. Thank goodness for ed reform. As an aside, these were the same people, who every time we got a raise, came running to my room asking how much they were going to be making next year because they couldn't figure out what a five percent raise was on top of their fifty two thousand dollar existing salary. God's honest truth.

What kind of well-rounded academic year could those kids be in store for with that teacher’s mission statement?

Posted by: Pau Hoss | February 14, 2008 12:11 PM

Catherine Johnson said...

PISSED OFF TEACHER has a post on geometry teachers in h.s. I've been meaning to get a link to. Anonymous said...

ARRGH! I made a mistake, and probably confused you more!

See, this is why ALL teaching should be Direct Instruction, with SCRIPTS that you don't deviate from.

Here goes:

fractions are NUMBERS. They exist on the number line.

Decimals are a special type of fraction: they have a 100 in their denominator.

PERCENTAGES are numbers, represented as a decimal, but they are only used to refer to a MULTIPLE of a number. they are "ths" of something. so 5% of x is 5/100ths of x.

5% means 5% OF SOMETHING. A decimal is just .05, but 5% refers to .05, or 5/100ths OF something.

Now, what is multiplication? It's grouping a bunch of numbers together conveniently so you don't have to add them.
So, 24 = 3 * 8 is the same as 8 + 8 + 8. We say that there are 3 8's. The three is the number of multiple 8s we've got.

And when we say we've got "3 of these 8s", we see the OF, and we think , oh right, we could add 3 groups of 8, or we could just multiply 3 * 8.

same with percentages. they are multiples of something. 5% of 100 is saying "we've got 5/100ths of 100" and we know that when you've got a number of somethings, you can multiply to figure out the total count of the somethings.

SteveH said...

Wow! That might be your longest post.

I haven't had a chance to read it all, but I am interested. I vaguely remember having some trouble with percents, but after some point, I never had trouble again. I will think about it some more, but it seems that all of my problems went away when I focused on figuring out what number is the 100% number.

"A shirt sells for \$52.50 including a tax of 5%. What was the original price of the shirt?"

This is difficult because the 100% number is not given. But, hopefully, you see that:

1.05 * X = 52.50

Remember that I like to use formulas, not proportions or charts. Maybe it's because I automatically see an added 5% tax as a multiplier of 1.05 times the 100% number. Depending on the problem, I either see the percentage all by itself or added/subtracted from one.

If you think of the equation above, you could have a problem statement where any one of the three numbers is unknown.

We got into this a little bit a while back on KTM. You can also have trickier problems, like:

A store discounted a \$100 jacket by 30%, but it didn't sell. They decided to take 20% more off of the price. What is the final sale price?

Of course, the big question is what is the 100% number for the second 20% off, the original retail price, or the discounted 70% price?

Catherine Johnson said...

This is difficult because the 100% number is not given. But, hopefully, you see that:

1.05 * X = 52.50

Right -- that's one of the things that is VERY hard to see.

I'm guessing this goes back to the "oneness" of 100 percent....as well as to the fact that in daily life we rarely talk about percentages greater than 100%. (At least, I rarely do.)

Chris is starting to see this reasonably often. (As I say, I didn't see it until his 5th grade teacher taught it to me!)

Remember that I like to use formulas, not proportions or charts. Maybe it's because I automatically see an added 5% tax as a multiplier of 1.05 times the 100% number. Depending on the problem, I either see the percentage all by itself or added/subtracted from one.

Right - last night I was trying to get C. to "see" this --- to see that we're talking about a formula or an equation, not a chart or a proportion.

I'm also TRYING to get him to see that the formula is the next step in the proportion.

Needless to say, he's hung up on cross-multiplication.

I've got to get Allison's explanation pulled together & put up front.

Catherine Johnson said...

Cross-multiplication is incredibly useful but it is a real "conversation stopper."

Instructivist said...

[let x = price of shirt
1.05x = 52.50
52.50 ÷ 1.05 = x
50 = x

price of shirt = \$50

I have to include this for people like me: the reason you take 1.05x is that it is a shortcut.]

I don't think 1.05x = 52.50 is an example of the shortcut method. You need to write it that way to make the equation true and to solve this percentage problem algebraically.

The shortcut method would apply to problems like this. A shirt costs \$25.00. The sales tax is 12%. What is the total cost. 25 x 1.12. Anonymous said...

maybe you should let me EDIT it first! :) at least to put together the decimal and percentage bits... Anonymous said...

what is cross multiplication? Anonymous said...

oh, back to the "does he know 9/10 * 10/9 is 1"

does he know that 90/90 is 1?

does he know that if he computes it? Does he know that 9/10 * 10/9 =
9*10 / 10* 9, and 9* 10 = 90, and 10*9 = 90, and wow, 10*9=9*10 because it's always true in the natural numbers that A*B = B*A?
Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?

Doug Sundseth said...

"I'm guessing this goes back to the "oneness" of 100 percent"

I find it useful to emphasize that "percent" is an abbreviation of "per centum" or "per one hundred".

Thus 5% is "five per one hundred", which is often written as "5/100". To cement this, you might want to discuss "mill levies" in property taxes. 19 per mill is the same as 1.9 percent.

This helps to illustrate that the definition of "percent" is not a unique thing, but that it exists on a spectrum. Specifically, it means that a percent is just a fraction.

ps. There is a character for "per mill" -- ‰.

Catherine Johnson said...

I find it useful to emphasize that "percent" is an abbreviation of "per centum" or "per one hundred".

You know what??

That is a great idea.

Thank you.

I find this whole realm fascinating (obviously).

I've been reading Vicki Snider's book advocating the development of a science of teaching (which would mean actually paying attention to the science of teaching we already possess).

She says that to the best of her knowledge no one knows how to teach "critical thinking," "higher order thinking," and, I assume, "knowledge transfer" and "generalization."

I think she's right (although I have yet to read Arthur Whimbey's books. Whimbey claims you can teach people to think.)

This is why I must get around to writing up the cumulative practice study.

That study set out to see if mathematical problem solving could be taught.

Catherine Johnson said...

does he know that 90/90 is 1?

yes

definitely

does he know that if he computes it? Does he know that 9/10 * 10/9 =
9*10 / 10*9

yes

Interestingly, I find that the commutative property seems to be somewhat "natural" or "obvious" to students....

I think.

The identity properties aren't so obvious (that's my impression).

90/90 = 1 is "less obvious" than 90*10 = 10*90.

Catherine Johnson said...

I don't think 1.05x = 52.50 is an example of the shortcut method. You need to write it that way to make the equation true and to solve this percentage problem algebraically.

The shortcut method would apply to problems like this. A shirt costs \$25.00. The sales tax is 12%. What is the total cost. 25 x 1.12.

That makes sense.

I can say the same of 25 x 1.12, though.

Until C's 5th grade math teacher showed me this I'd never thought of it.

SteveH said...

"...he's hung up on cross-multiplication."

Warning! Warning! Warning!

There is no such thing as cross-multiplication!!! It's not an identity.

OK. I'm getting carried away. We've been here before. You know I think it causes way too many misunderstandings. The classic is when a student tries to solve:

X + 2/3 = 4/5

Students see those two fractions on either side of an equals sign and their brain screams CROSS-MULTIPLICATION!

SteveH said...

Let me add that it's OK to have "cross-multiplication" come out of your mouth if your brain is really thinking "identity".

Catherine Johnson said...

Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?

I'm going to say 'no' to that.

I remember spending a VERY long time wrestling with slightly more elaborate versions of this back when I first started teaching C. math....

But I'm afraid I can't reproduce those traumas now.

Let's see.... I really had problems (and may still have problems) understanding why fraction multiplication means that you multiply the numerator by the numerator and the denominator by the denominator -- what is going on there????

I felt that invert-and-multiply as the means of dividing a fraction by a fraction worked by magic.

That finally got cleared up when I came across a homeschool page showing the division of a fraction by a fraction as a manipulation of a complex fraction.

Obviously, to change the denominator to 1 you had to multiply the denominator fraction by it's reciprocal.

That was a revelation.

SteveH said...

"Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?"

I remember thinking that the basic identities were all pretty trivial and stupid.

a*b = b*a

1*a = a

Big Deal!

However, you need to see these identites in use. You need to see them over and over again in real applications. Everything has to be seen in terms of the basic identities.

Just the other day I had to have a talk with my son who had the following to solve:

15/4 * 32/5

He was going to multiply 15*32 and 4*5. He couldn't see that he could rearrange it to 15/5 * 32/4, reduce, and then multiply.

a*b = b*a

It IS a big deal, and it's not trivial. Anonymous said...

She says that to the best of her knowledge no one knows how to teach "critical thinking," "higher order thinking," and, I assume, "knowledge transfer" and "generalization,

But that doesn't stop them from trying. In the second grade.

This is off topic, but if you need a laugh about now, try this link. It's worth it. I can't watch it without tears.

Susan S. Anonymous said...

--.... I really had problems (and may still have problems) understanding why fraction multiplication means that you multiply the numerator by the numerator and the denominator by the denominator -- what is going on there????

let's solve the first part first.

2/3 * 4/5
is the same as saying
(2 divided by 3) * (4 divided by 5) =
(2 / 3) * (4 / 5)
don't think of these as fractions, think of them as numbers with operations in between. We could re group this as
((2 / 3) * 4) / 5
and we could regroup this again as
((2 * 4) / 3 ) / 5

so that's the same as seeing that first we multiplied the top of the fractions together and then we divided by one bottom and then divided by the other.

that's why you multiply the tops--because the associativity property and commutative properties let you do this.

And once you see that you've got
((2 * 4) = 8, what's remaining is
8 / 3 / 5
(cont). Anonymous said...

So now we've got (8 /3 ) /5.
That is: (8 divided by 3) divided by 5.

So how do we handle this?

We go back to the earlier discussion of division:
q = m /n means there is some number q where m = q * n.

So 8 divided by 3 means there is some number q where 8 = q * 3. q is 8 /3.

And now, we divide that q by 5.
so NewQ = q / 5.
But that just means q = 5 * NewQ.

So the original division of 8 by 3 means there was some q that when multiplied by 3 gives you 8. And now we find that there's some newq that when multiplied by 5 gives you 8 / 3.

So 8 = q * 3. And q = 5 * NewQ.
So 8 = 5 * NewQ * 3. Rearranging,

8 = 5 * 3 * NewQ.
8 = 15 * NewQ.

NewQ = 8 /15.

We could also have done it as
Q = (8/3) / 5.
(8 /3 ) / 5 means there is some number q such that
8/3 = Q * 5.

And 8/3 is a quotient that we can rewrite too.
Q_2 = 8/3.
That just means there is some number Q_2 such that
8 = Q_2 * 3.

So we could have said:
Q_2 = Q * 5
and
8 = Q_2 * 3 = Q * 5 * 3.

so 8 = Q * 15.

What's Q again? It's the quotient of 8/ 3 /5.

That's the same as saying
8 = Q * (3 * 5).

this is not easy stuff, and I probably could have done it better. Anyone else want to try?

Hypatia said...

I probably shouldn't weigh in here because I just got back from teaching a four hour math class, but I can't resist it.

Remember your algebra 1/b *a = a/b = a(1/b)
With numbers on a more elementary level, consider 3/4.
With a ruler, draw a line 3" long.
Without mmeasuring try your best to divide that line into 4 equal part. Now measure each part with the ruler. Each part will meaure 3/4 of an inch. So 1/4 of 3 is 3/4.

Now draw a line 1" long and divide it into 4 parts. 3 of those parts is 3 * 1/4 or 3/4. That's the second way of looking at the fraction 3/4.

Either way 3/4 is a definite number on the number line.

As for the 3 problems, I would say they all very nicely can be set up as proportions and solved the same way.

Karen A said...

SusanS--

I just watched the video. Thank you for a much-needed laugh. My heart is breaking as a result of the latest campus shootings at Northern Illinois; it hit just a little too close to home. Anonymous said...

Karen,

I know. I saw it before all of the Northern IL stuff started happening. It might not be as funny now, but I think all of us have wanted to do that to our computers. Odd how all of the observers just watch and even go back to work.

I'm in the Chicago area, too. It makes you want to keep the kid home from college, as well. Jeez.
What is going on? They're reporting that the shooter was a "stellar student."

Susan S. bky said...

I would like to add an increment to the discussion of teaching percent problems.

(1) The discussion of partitive and so on seems like jargon on the not-helpful side, but I have not read all the old posts, so ....

(2) but still, I think it is a mistake to think that there are really three kinds of percent problems, at least for beginners. For beginners I like to start at the beginning every time (which obscures the difference among the types of problems until the end game), which is a setup like this:

Let P be the price of a shirt. The total cost is P plus tax, which is a fraction of P. We know the total is 52.50. We also know the fraction, the tax rate, to be 0.05, so the tax is P x 0.05. The situation is then

total = 52.50
= P + P x 0.05
= P x ( 1 + 0.05 )
= P x 1.05

and now P = 52.50 divided by 1.05, which I'm guessing will be done with a calculator ...?

This is what I call starting from the beginning. If you knew the price P, but not the total, the total is thus calculated. (It seems like the hardest problem would be to find the tax rate if that were not known, since you have to remember to subtract 1 -- you want 0.05, not 1.05).

There seem to be three critical prerequisites. First, do students understand how and when to introduce a variable ("Let P be the price of the shirt"); second, do they understand distribution, which is critical in going from

P + P x 0.05

to

P x 1.05 ;
and third, do they understand that undoing a product requires division?

If any of these three issues are shaky, it would seem that you need to do everything you can to shore up that understanding immediately. You can use percent problems as a setting for all three issues, but your priority should be teaching use of variables, distributivity, and dividing to solve T = P x 1.05 for P.

The poster, Catherine Johnson, set up the problem like this:

"let x = price of shirt
1.05x = 52.50
52.50 ÷ 1.05 = x
50 = x"

I think that introducing 1.05 directly (aka the shortcut) should only be done when students have done so many problems, and seen the use of distribution yield 1.05 from 1 and 0.05 enough times that they do it themselves at the outset.

Karen A said...

SusanS--the shooter was a graduate of NIU, and was attending graduate school at the University of Illinois. The media is reporting that he had been "off his meds" and was behaving erratically for the last several weeks. No real motive has yet emerged as to why he targeted NIU in general, and that class in particular.

Our neighbor is a senior at Northern; he's okay, but his mother is (naturally) pretty badly shaken up. So many of our university's students have friends and family at NIU, so our student body is pretty shaken up as well.