kitchen table math, the sequel: help desk - percent chart

Thursday, December 20, 2007

help desk - percent chart

Is there an all-around mode of charting the values in percent problems similar to the charting taught for distance problems?

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15 comments:

Instructivist said...

[Is there an all-around mode of charting the values in percent problems similar to the charting taught for distance problems?]

I don't understand the question.

The best way to do percent word problems is to translate them into symbols and turn them into an equation.

For example:

What percent of 420 is 7?

"Of" means multiply and "is" is = (no need to debate what the meaning of "is" is).

Let's use p for what percent.

So

420p = 7

p = 7/420

Or

82 is 90% of what number?

Change 90% to a decimal first.

Trnalste:

82 = 0.9n

n = 82/0.9

It works like a charm.

December 20, 2007 at 5:12 PM
Catherine Johnson said...

It does!

Definitely.

Saxon does a cool thing (of which I'm pretty sure You Math People disapprove) where he uses the initials of the English words as the variable.

If the question is "What number is 6% of 10" he writes:

WN = 6/100 x 10.

"WN" stands for WN.

If the question is What percent of 60 is 15? he writes:

WP x 60 = 15

December 20, 2007 at 5:32 PM
Catherine Johnson said...

My question is about solving percent word problems.

One of the difficulties with percent word problems is that there are so many....varieties.

Take price & tax problems (or price & discount).

types:

* original price & percent discount given, find final price

* original price & final price given, find percent discount

* percent discount & final price given, find original price

Then you can also get into two-step versus 1 step manipulations (i.e. translating a 10% deduction into a 90% price....)

C. does so incredibly well with word problem charts that it strikes me I should have a chart for percent problems.

Saxon teaches a terrifically useful chart for doing percent increase and decrease problems.

December 20, 2007 at 5:36 PM
Instructivist said...

A simplified way to do tax poblems is to tack on the decimal to "1".

If an item costs $80 and the tax is 5% (unrealistic in my state) then multiply

80 x 1.05 = 84.

This eliminates the addition step.

[* percent discount & final price given, find original price]

This is a bit more challenging.

If an item was discounted, say, 30% from its original value, then the discounted price is 70% of the original price. That's the number to work with.

Let's say some nice lined leather gloves cost $80 at some fancy store after being discounted 30%, then what was its original price?

80 would then be 70% of the orignal price we will call x. So translated we have the equation:

80 = 0.7x or

x = 80/0.7 which gives you an unrealistic original price. I would choose better numbers.

I know regular KTM readers know all this. I am doing these exercises for the benefit of the school kid who might chance upon KTM.

December 20, 2007 at 6:22 PM
SteveH said...

"The best way to do percent word problems is to translate them into symbols and turn them into an equation."

Exactly! N equations in N unknowns.


This chart technique reminds me of the charts I had for mixture problems. I hated them.


"One of the difficulties with percent word problems is that there are so many....varieties."

This is a difficulty with all types of word problems. I prefer a technique that focuses on selecting the governing equation, drawing pictures (as needed), defining variables, and writing down any legal equations that come to mind. Many solution techniques seem to focus on ways to keep variables and equations to a minimum. The technique he uses for this one problem is NOT helpful for many other problems.

For this problem, you first have to know that the governing equation is D=RT. Second, you use this equation twice, once for each car, so the variables are not the same! If we use 1 and 2 as variable subscripts for both cars, then you can quickly write down these two equations:

D1 = R1*T1

D2 = R2*T2

Already, I would give you some partial credit. The goal is to get more equations. I have two equations and 6 unknowns. I need 4 more equations or I have to eliminate some variables.

Like the video, if you draw a picture, then you will (should) see that

D1 + D2 = 520

One equation down, 3 to go.

Both cars travel the same time, so:

T1 = T2

Two equations to go. Oh yeah...

R1 = 55

R2 = 75

I would give you at least 80% of the credit for the problem at this point.

I guess my point is that many word problem solution techniques seem to be the sort of solution that you might come up with after you've done the problem. It's what you get after you've cleaned up all of the missteps.

Back when I taught computer programming, I used to pose problems (a function or subroutine) to do in class that I had never seed before. I would show the students how I solved the problems. I would explain exactly what I was thinking. I did this for this same reason. Too many explanations are the cleaned up perfected solutions, not the first shot solution.

If your problem does not fit the chart, you're lost. However, if you use the technique of selecting the governing equation and then start defining variables and ANY legal equations, you're going in the right direction.

December 20, 2007 at 6:48 PM
Catherine Johnson said...

A simplified way to do tax poblems is to tack on the decimal to "1".

That was a complete and total revelation to me back when I first learned it from C's 5th grade teacher.

I'm not sure he has it yet....one thing I am seeing is that percent-percent-percent is probably a very good way to develop number sense, etc.

That's percent as opposed to fractions. I spent a huge amount of time on fractions, but percent is actually more confusing challenging if only for the Wickelgrenian reason that "it all looks alike" - that you're doing so much with so little, with so few symbols.

I'll dig up that reference & post.

December 20, 2007 at 7:10 PM
Catherine Johnson said...

We're slowly working our way through all of the Primary Mathematics Enrichment practice percent problems.

December 20, 2007 at 7:11 PM
Catherine Johnson said...

I prefer a technique that focuses on selecting the governing equation, drawing pictures (as needed), defining variables, and writing down any legal equations that come to mind.

interesting

I'm not sure that's different from the charts C has been taught, which correspond to the equation.

(Is it different?)

They right the equation on top of the chart; each variable in the equation represents one column.

December 20, 2007 at 7:12 PM
Catherine Johnson said...

Actually, that's what I'm asking for - a percent chart representing the standard equation....

December 20, 2007 at 7:13 PM
Catherine Johnson said...

I found an interesting study saying kids do MUCH better learning percent when it's taught as a ratio.

I was taught percent as a proportion & I've remembered, easily used it, and understood it for my entire adult life.

So now I have to make sure C. can do percent as a proportion.

December 20, 2007 at 7:15 PM
SteveH said...

"I'm not sure that's different from the charts C has been taught, which correspond to the equation."

But what if you have a different variation or use of the governing equation? What if one car stops at a restaurant for an hour? What if a car passes through a 25 mile per hour zone? Fixed charts (as a simulation of real equations) fail.

Real equations don't have that limitation. ALL problems are approached the same way; find the governing equation, draw pictures, find N equations in N unknowns, turn the crank. This applies even when you get to things like Bernoulli's equation.

Many early algebra word problems describe complex solution methods because they are desperately trying to avoid multiple variables and equations. Bar models are good at eliminating variables graphically, but like charts, they fail (or get complex) for many problems.

I distinctly remember feeling quite liberated when one teacher told us not to worry about too many variables or equations. He told us just to start writing down legal equations and searching for N equations in N unknowns.

Let the math do the work.

December 20, 2007 at 7:58 PM
Doug Sundseth said...

"Saxon does a cool thing (of which I'm pretty sure You Math People disapprove) where he uses the initials of the English words as the variable."

"WN" stands for WN."

I do disapprove, because it has a real potential to cause confusion: in normal algebraic notation, "WN" is the same as "W * N". If you're used to seeing two symbols represent one variable, seeing those same two symbols represent the multiplication of two variables is likely to cause pernicious difficulties.

If you need multiple characters to create understandable variables, the standard is to use subscripts.

ps. Tried to use subscripts here, but that html doesn't seem to be allowed in comments.

December 20, 2007 at 9:09 PM
Doug Sundseth said...

For "percent off" problems, I'd be inclined to be more explicit about what's actually happening:

Problem: The regular price is $20. The item is on sale at 30% off. What is the final price before tax?

Solution: The final price (F) is the regular price minus 30% of the original price.

"is" is "=", "of" is "*"

F = $20 - (30% * $20)

30% = 30/100 ("per" is division, "cent" is 100)

F = 19.95 - (30/100 * $20)

F = (1 * $20) - (30/100 * $20)

F = (1 - 30/100) * $20
@ Distributive property

F = (1 - 3/10) * $20
@ Reduce fraction

F = (10/10 - 3/10) * $20

F = 7/10 * $20

F = $14

This expanded process shows why "30% off" is the same as "70% of".

December 20, 2007 at 9:25 PM
Catherine Johnson said...

Saxon may use subscripts - let me see if I can check quickly.

I didn't worry about it for me, and I didn't use them enough with C. to have to think about it ---

[pause]

Yup.

He uses subscripts exactly as you suggest, I think.

If he's using "WN" for "What number" both letters are in caps & the 'N' is a subscript.

I did use these equations some both with C. & his friend M. one time when they had to cram percent for a test -- it was amazing. The direct translation of the words into these equations worked like magic.

December 21, 2007 at 11:15 AM
Catherine Johnson said...

Many early algebra word problems describe complex solution methods because they are desperately trying to avoid multiple variables and equations. Bar models are good at eliminating variables graphically, but like charts, they fail (or get complex) for many problems.

I distinctly remember feeling quite liberated when one teacher told us not to worry about too many variables or equations. He told us just to start writing down legal equations and searching for N equations in N unknowns.

Let the math do the work.

Interesting.

This makes sense.

I'm probably sounding all over the place on Saxon at this point, but as I think about it I would say that Saxon is in the Steve/Doug/et al camp.

I didn't learn charts in Saxon but, rather, a chart-like mode of writing equations....

I wonder if I can illustrate this.

yup, I can.

I'll type up his standard mode of handling distance problems in a bit.

When I say chart-like I mean that he has you learn a specific order for writing equations (or implied equations .... I don't think I was using an equals sign when I first started using his approach...)

December 21, 2007 at 11:17 AM

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