kitchen table math, the sequel

A friend of mine is a special ed teacher at a highly selective science-oriented magnet school. She has observed a number of incoming freshman who are unable to handle beginning (9th grade) algebra. I'm guessing that some (most? all?) of them have no inherent math disability, but have merely been poorly instructed (most come from elementary schools that use Everyday Math/Investigations and from middle schools that use Connected Math).

Anyway, when she asked me what I knew about math remediation programs that might help prepare these kids for algebra, I realized I had absolutely no ideas and should turn to ktm for help. Any suggestions?

I'm working with a boy in a neighboring town who can solve equations with positive values like the following:

3 + 2(x + y) = 9

He is having difficulty solving equations that require him to distribute a negative:

3 - 2(x + y) = -3

I remember C. having trouble distributing a negative, and I remember stumbling over minus signs myself when I was a kid. At some point, I solved my problems by deciding to treat minus signs as either a -1 or the addition of a negative, depending on the expression I was dealing with.

Thus -x became (-1)(x) and x-8 became x + (-8).

I don't think anyone ever told me to translate expressions in this manner. Quite the contrary; I have vague memories of reasoning it out for myself on more than one occasion.


Here's the way a sheet I have from Glencoe says to teach distribution of the negative:

Use the Distributive Property to write each expression as an equivalent algebraic
expression.
a. 3(w – 7)
= 3[w + (-7)] Rewrite w – 7 as w + (-7).
= 3w + 3(-7) Distributive Property
= 3w + (-21) Simplify.
= 3w – 21 Definition of subtraction

Unfortunately, this sequence doesn't solve the problem. My student can simplify 3(w-7); what he can't do is simplify 3–2(x+y).

Today I tried having him draw huge brackets around 2(x+y), then simplify the 2(x+y), and then simplify the remaining expression:

3–2(x+y)
3–[2(x+y)]
3–[2x+2y]
3-2x-2y

In effect, I was turning the problem into two distributions: first the 2, then the negative sign.

This approach always worked for me, but the logic of it wasn't obvious to my student.

One more thing: this student probably had Everyday Math in elementary school, and his current class seems to be intensely procedural. The only textbook his teachers are using seems to be a NY state test prep book.

I'm eager to hear any thoughts you have both about procedural teaching (including mnemonics) and about how I might help this student make some sense of the math he's learning. Moreover, and I hate to say this, but if I'm going to help him make some sense of it, I have to do it on the fly. Our time together is extremely limited.

If anyone knows of a good set of "instructional worksheets," that would be fantastic. I'm combing through my own collection.

Last but not least, what do you think of this video?

from Practical Math: Success in 20 Minutes a Day, page 2

42. Maria works at a clothing store as a sales associate, making d dollars every hour, plus 20% commission on whatever she sells. If she works h hours and sells a total of s dollars of clothing, what will her pay for that day be?

a. d + 20
b. d + 0.20s
c. ds + 0.20
d. d + 20s

grades 6, 7, & 8

More from the NSF's Algebra Cubed Project:

Using Ratios to Taste the Rainbow (pdf file)

Goal: At the end of the activity, the students will know that the actual ratio of colored skittles is not what the Mars company claims. They will also be able to calculate different ratios and percentages associated with the number of colored skittles given to their group.

KY Standards: MA-7-NPO-S-RP1
Students will apply ratios and proportional reasoning to solve real-world problems (e.g., percents, sales tax, discounts, rate).

The Mars company has been lying to us about the ratio of colored skittles?

Just spoke to C. who says he got the extra credit problem right on the test except he messed up the minus sign.

Messing up the minus sign is a chronic issue around these parts.

In the old days, C. would say things like, "I just forgot the minus sign!" Then I would say things like, "Well, the difference between plus-70 degrees and minus-70 degrees is the difference between life and death" and we'd go from there.

The other day I came across this post to Math Forum that made me think back fondly on those days:

I have wondered if anyone knows of a notation for negative numbers
> > that would help students avoid the crippling error of thinking that
> > they are just slightly flawed positive numbers. Have you not heard
> > a student say somethings like this," the solution of 2x = -6 is
> > 3 --oh, I mean negative 3.

Math Forum

Any more C. seems to take minus signs more seriously. But he's still losing track of them somewhere between Point A and Point B.

Any suggestions?

OK. Maybe it's me, but here is what's in my son's Glencoe Pre-Algebra book (2008 ed.), section 2-2 (Adding Integers).

- - - - - - - - -

Key Concept Number 1

Words:

To add integers with the same sign, add their absolute values.

The sum is:

- positive if both integers are positive
- negative if both integers are negative


Examples

-5 + (-2) = -7

6 + 3 = 9

- - - - - - - - - - - - -

Key Concept Number 2

To add integers with different signs, subtract their absolute values.

The sum is:

- positive if the positive integer's absolute value is greater
- negative if the negative integer's absolute value is greater

- - - - - - - - - - - - - - -

It seems to me that you can't make it much more complicated.


And he hasn't even gotten to subtracting integers, which states:

"To subtract an integer, add its additive inverse."

(Then use one of the rules above.)



It sounds like they are defining a foolproof method, but how about

4 + (-4) = ????

Neither integer is greater.

and how are these rules going to help when you get to variables, like

X +(3-2X) = ?????

You can't combine

X + (-2X)

because you don't know the absolute value of X?

Perhaps they will get new rules when they introduce variables.

And what about decimals? The sections only talk about integers!


- - - - - - - - - -

The simple rules I taught my son during the summer are:

1. When adding or subtracting, you often get two signs in a row, like:

5 + (-2)

If the signs are the same, combine them into a '+'

If the signs are opposite, combine them into a '-'

This works for numbers or variables. I told him that there is no difference between a sign and an operation, since: -5 is the same as (0 - 5).


2. When multiplying, you often get two signs that get multiplied together, like:

(-5)(-2)

Remember that this is the same as (-1)(+5)(-1)(+2) = (-1)(-1)(5)(2)

If the signs are the same, combine them into a '+'

If the signs are different, combine them into a '-'


3. When dividing, it's the same rule:

If the signs are the same, combine them into a '+'

If the signs are different, combine them into a '-'

[You just have to make sure that the signs are for factors. For (-3)/(-2+x), the sign in the denominator is a '+', not a '-'.]


The same rule applies to all three cases and for numbers or variables. It couldn't be easier. [I changed some flash cards to add minus signs to many of them.]


There are some things you have to watch out for and they need to understand variations, but it all boils down to one simple rule about combining signs.

- - - - - - - - - -
Things to feel comfortable about:

1. 5 = -(-5)

2. 2-5 = 2+(-5)

3. 5 = (+1)(+5)

4. -5 = (-1)(+5)

5. (-4)/(5) = (-1)(+4)/(+5) = -1(4/5) = 0 - 4/5


I told my son that there is a case where two positives equals a negative, and he said "Yeah, right!"

I'm sorry, I couldn't resist.

from the Comments thread to: What is 10%?


from Steve H:

"What is 10% off? issue."

I've just been going over this with my son. I keep asking him 10% of WHAT NUMBER, exactly? I want him to always know what that amount is. I don't call it the whole because it might be confusing for problems that ask for 10% off of 50% off. The classic problem is the store that tells you that you can get an additional 10% off of the 30% off discount price. The question is WHAT number, exactly, does the 10% refer to?

I also got done telling him that when you see something like 30% in a word problem, you never use the number 30 in the calculations. You have to use either .3 or 3/10. It seemed like a minor point, but I could see it sink in.

In the past, we've talked about 15% tips and how to calculate them, but he needs to see percent problems in all forms and see fractions and decimals as two different forms of the same thing.

Another good problem is to talk about the store owner who buys at the wholesale price and marks up by 100% to get the retail price. The store owner could then have a sale and mark down the goods by 30%. The question still has to be WHAT NUMBER, exactly, does the percent refer to?

Another issue I ran into was that he was thrown a little by things like 125% - percents greater than 100%.


from Joanne Cobasko:

I used many work sheets converting fractions to decimals (by doing the division-since a fraction is just a division problem) and then converting decimals to percents.

The repetition that a fraction is a decimal and a decimal is a fraction, and a percent under 100 is represented by a 2 digit decimal seems to be working well. I did this when I noticed that Saxon was not providing the simple algorithms to solve the fraction of a whole problems (and of course multiplication of decimals and fractions haven't been introduced so I taught that too). I have been teaching ahead of Saxons approach with the actual algorithms. My son hates drawing the pictures- but I have him draw to prove he understands. I taught him to write the equation first and solve the problem then draw the picture.

When Saxon began introducing problems looking for a fraction of a group I taught that the word "of" meant that you had to multiply the fraction (or decimal) by the whole number.

1/2 of 30 = 1/2 * 30/1 = 30/2 = 15
or
50% of 30 = .5 * 30 = 15.0

Start with the simple fractions 1/2, 1/4, 1/3, then work up to the others.

Memorizing that
1/2 =.5 = 50% or
1/4 = .25 = 25% or
1/3 =.333 = 33.3%

We are now approaching doing percents in our head such as
1/8 which is half of 1/4 so
1/8 =12.5% (1/2 of 25%) and that
2/5 = 40% because 1/5=20% so 2 of the 1/5's would be twice as much, so
2*20%=40%

I am hoping this familiarity with decimals, fractions and percents combined with memorization and mental math skills (which Saxon introduced in HS version of 5/4 and higher)will help my son to solve more advanced problems such as the ones you are now presenting to Chris.

[from Catherine: we are doing LOTS of these worksheets - and we need to do mental math, but C. isn't quite "up to that" yet...]


more from Steve H:

I see a clear difference in my son between before mastery and after mastery. Before mastery, he may be able to explain and do a problem eventually, but he doesn't fully grasp the subtleties and variations of what he is doing. After mastery, the process and understanding is automatic.

We've been working on combining plus and minus signs when you add, subtract, multiply and divide. Unfortunately, he is always trying to find a simple pattern that solves the problem. The fault with patterns is that they are based on nothing. There are lots of patterns that can be found and many of them are not helpful at all. I always try to explain things using the basic identities.

One thing we ran into the other day was where does the minus sign belong in a fraction. I told him that you can put the negative sign anywhere you want.

I told him to identify terms and always think of a term or number with a sign in front of it. If you don't see a sign, it's a '+'. I also told him that a minus sign is really a factor of -1.

if you have

3 - 1/2

Then the second term is

- 1/2

or it could be

(-1)(1/2)

or

(-1)/2

or

1/(-2)

You can put the minus sign in front of the number, like

-.5 or -(1/2)

or you can put it in the numerator or denominator. Since the fraction is just a number, you can think of the minus sign in front of everything, but you can also put it into the numerator or the denominator if you want.

He didn't like that idea.

I gave him this fraction.

(-2)/3

I then asked him what

(-1)/(-1)

equals. He hesitated and then asked, "One"?

I said OK, now multiply

(-2)/3 by (-1)/(-1)

to see what you get.

He knows how to multiply numbers with different signs, but he had to think about this. You could see the wheels turning.

I told him that whenever I look at a minus sign, I can see all of the different places I can put it or all of the different ways I can use it.

These things can't sink in without a lot of practice. Mastery provides understanding. It can't be rote. Understanding is not possible without mastery. Finally, mastery and understanding have little to do with pattern recognition.


from instructivist:

[We are now approaching doing percents in our head such as
1/8 which is half of 1/4 so
1/8 =12.5% (1/2 of 25%) and that
2/5 = 40% because 1/5=20% so 2 of the 1/5's would be twice as much, so
2*20%=40%]

This is a great way to learn mental math. I have been doing this instinctively.

Calculating tips of 15% or 20% (service mus really be good) menally should also be child's play. Ten percent of anything is easy. Add half of that and you get 15%. It's baffling that some kids struggle with this.

[1/3 =.333 = 33.3%]

There is a fancy, six-figure word that goes with repeating decimals (the bar on the repeating number or numbers): vinculum. Converting these repeating decimals to fractions is a nice algebra exercise. The number of numbers covered by the vinculum tells you if you need to multiply by 10x, 100x or whatever.

AND:

"Understanding is not possible without mastery."

That's a powerful statement. It should blow the constructivists out of the water who purport to seek "understanding" but disparage mastery with obnoxious phrases like "drill and kill."

AND:

It occurred to me that a calculator is of limited use when trying to figure out if certain fractions are repeating decimals when converted. The calculators I am familiar with do automatic rounding.

I tried 5/7 on my TI-30X IIS and get 0.71. No indication that a repeating decimal is involved. My TI-83 Plus gives me more but also rounds without showing the group of repeating numbers.

I see this as another reason why long division is important. How would calculator-dependent students see that the sequence 714285 repeats, I ask NCTM?

[Catherine again: I've informed C. that he will be doing long division worksheets shortly, and he will be doing them to fluency]


from le radical galoisien:

There is a rough method of deriving a fraction from any arbitrary decimal.
For example, 2/7 is 0.285714286 (etc.) 1 divided by that decimal is 3.5. That is 7/2, or the inverted form of 2/7 ...

hyperspecificity in autism
hyperspecificity in autism and animals
hyperspecificty in the rest of my life
hyperspecificity redux: Robert Slavin on transfer of knowledge

Inflexible Knowledge: The First Step to Expertise
Devlin on Lave
rightwingprof on what college students don't know
percent troubles
what is 10 percent?
birthday and a vacation

The whole family has gotten into the act on the What is 10% off? issue.

Even Ed is now writing math word problems.

This is serious.

Here are Ed's two from this afternoon:

1.
When you're training for a 10K race, you run a series of 1K laps. Your first lap takes you 4 minutes. Each subsequent lap is 10% slower than the last one.

What is your total time for the 10 laps?

2.
If on average you run 1K in 4 1/2 minutes, how long will it take you to run a 10K race?


I realize these two questions are logically contradictory, but I'm not going to worry about that for now. Ed says sports are a great source of word problems, and he's right.

I decided today to start giving C. the same problem written as a percent & as a fraction.

Then, having fixed on this plan, I decided to throw in a whole-part problem (something he's never done before) to boot:


1.
finding the parts when the whole is given [note: I labeled the problem with these words]

Christopher wants to buy a $50 video game for 20% off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.

2.
finding the parts when the whole is given

Christopher wants to buy a $50 video game for 1/5 off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.

3.
finding the parts when the whole is given

Christopher wants to buy a $50 video game for 1/5 off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.


This is one of those moments where you see exactly how valuable an experienced teacher at the top of his/her game is to kids learning math. Or to kids learning anything.

Because I've worked my way through so much of Saxon, Singapore, & "Russian Math," I have pretty good pedagogical content knowledge. For instance, I now know what "part-whole" versus "whole-part" problems are, a concept I'd never heard of before.

But I still lack "kids-learning-math" knowledge.

I don't have a good sense of the proper use of contrast and comparison in instruction (i.e. having C. do the same problem framed as percent and fraction - good idea or not?); nor do I have a sense of how long it should take for a student C's age to learn these things, which means that when C. doesn't seem to be learning what I'm teaching I can't tell whether he needs more practice or I need to teach differently or both.

I'm making all my mistakes with my own kid.

Still.

By the end of this summer - preferably by the end of tomorrow - he is going to know what 10% off is or I am going to die trying.


hyperspecificity in autism
hyperspecificity in autism and animals
hyperspecificty in the rest of my life
hyperspecificity redux: Robert Slavin on transfer of knowledge

Inflexible Knowledge: The First Step to Expertise
Devlin on Lave
rightwingprof on what college students don't know
percent troubles
what is 10 percent?
birthday and a vacation