The “Algebra Survival” Program goes totally electronic!
Singing Turtle Press is delighted to announce that the companion Workbook for the Algebra Survival Guide is now available in eBook format.
Certain areas of algebra are like pebbles in your shoe: looked at closely they’re tiny. And yet they are “oh-so-bothersome!”
As a tutor, I’ve long felt this way about a negative sign before parentheses. It’s a small thing, and it seems simple to grasp to those who get it. Yet students make so many mistakes when facing this situation, so to them it’s extremely irritating!
And there I was again, trying to help a girl understand how to simplify this expression:
– (– 5x + 3y – 7)
However this time I came up with something different, the word “opposite.”
I talked for a moment with my tutee about the idea of opposites, and then I started out like this:
Q: So, what’s the opposite of black?
She replied: White (with the teenage “that’s-totally-obvious-what are you-doing?-insulting-my-intelligence? accent)
I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she were to find the opposite, as in:
Q: opp (tall, happy)
She wrote: (short, sad), still wondering …
And I continued:
Q: opp (heavy, up)
She wrote: (light, down), sighing.
Then I explained that in math we express the idea of “opposite” with nothing more than the negative sign.
Then I gave her some problems with the negative sign:
Q: – (cold, left)
A: (hot, right)
Q: – (under, near)
A: (over, far)
She was still giving me that “this-is-so-easy-I-could-die” kind of look. When I thought about that, I realized it was good!
Next I explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. For example, the opposite of 5 is – 5; opposite of – 3/4 is 3/4; opposite of – 3x is 3x; opposite of y is – y, and so on.
Then I gave her these problems:
Q: – (+ 2x, – 5)
Still she was with me: – 2x, + 5
Q: – (– 4y, + 3x, – 6)
A: + 4y, – 3x, + 6
The sighing was slowing down, finally. Then I simply told her that we’re going to “lose” the comma (how’s that for modern slang!), both in the original expression and in their answer. Then I gave her a new problem:
Q: – (5a – 3a – 9)
This puzzled her a bit. So I explained that she needs to mentally group the term with the sign that lies to the left. And that if no sign is showing, as for leading positive terms, she needs to mentally insert the invisible positive sign: 5 becomes + 5; 2a becomes + 2a. Once she got that, she was able to proceed:
Q: – (5a – 3a – 9)
A: – 5a + 3a + 9
And so on … one success after another. The concept was sticking. And best of all, she had a conceptual framework — the concept of opposite — that she could “lean against” any time she got stuck.
The longer I tutor the more I realize that this kind of conceptual framework — a story or concept we know from everyday life, which relates to the algebra in a direct way — is a big key to helping students grasp algebra. I use these kinds of stories in my book, the Algebra Survival Guide, providing stories we know from everyday life, which serve as analogies that show how the math works. For example, in the Guide I use a “tug-of-war” analogy to show how you solve problems like: – 3 + 8.
I’ve had so much success with this “story”-approach to algebra that I am working on an eBook that provides a whole litany of stories that work for algebra. It is fun to work on, and kids like this approach because it gives them a new way — an everyday way — to relate to the math.
So in any case, my suggestion is that when you teach or review the concept of negative signs before parentheses, you might just try the “opposites” approach and see how it works with your students.
Attention: Dear Aunt Sally may not be fit for teaching students algebra!
A problem has been discovered in sweet Aunt Sally’s little memory trick: Please Excuse My Dear Aunt Sally.
Actually, make that two problems.
The first, revealed in my 9/9 post below, is that Aunt Sally wrongly makes students think they’re supposed to multiply before dividing. That’s because the word My (standing for MULTIPLY) comes before Dear (standing for DIVIDE).
Countless students have been deceived into thinking they’re supposed to multiply before dividing [See the 9/9 post for the full run-down on this problem.]
Today I want to point out another problem, and offer two solutions.
The second problem is that, since “Aunt” (standing for ADD) comes before “Sally” (standing for SUBTRACT), countless other students have been led to think they are supposed to ADD before they SUBTRACT.
Well, what are students supposed to do?
First of all, students need to realize that adding and subtracting are at exactly the same level of hierarchy as each other. But if that’s true, how can students ever decide which to do first.
Easy! Same solution as with multiplying and dividing. We simply look to see which of these operations is written first as we read the problem left to right.
Example: in the expression 8 + 3 – 4, the addition symbol precedes the subtraction symbol, so here we add before subtracting. And we simplify the expression like this:
8 + 3 – 4
= 11 – 4
But in the expression 8 – 3 + 4
the subtraction symbol is written before the addition symbol, so here we subtract before we add, and we simplify the expression like this:
8 – 3 + 4
= 5 + 4
It’s really that simple. Pay attention to which operation sign comes first as you read the problem from left to right. Then do the operations in the correct order based on that.
One other solution: in my book, the Algebra Survival Guide, I get away from the Please Excuse My Dear Aunt Sally approach, as I create my own memory trick, one that involves Strawberry Mousse. If you want to take a look at this approach, check out my book at this site.
From the homepage, click the link that says: View Sample Chapters of the Algebra Survival Guide, and download the chapter on Positive and Negative
A reader named Michelle said she enjoyed my post on Memorizing the Times Tables. And then she asked if I have any tips on teaching students to SUBTRACT INTEGERS.
It turns out that the answer is “yes,” and there are two places where I model this topic.
The first is an excerpt from my Algebra Survival Guide, an excerpt about subtracting integers that you can check out now.
First click on this link, then scroll down to read pages 43-46 (you can enlarge the print size by increasing the percentage located in the top bar):
But there’s more:
I wrote an entire book on the topic of combining integers, PreAlgebra Blastoff! We created foam manipulatives to get across the idea of integers. There’s a piece of a foam with a hole in the center, the NEGATon, which stands for – 1. There’s a piece that fills the hole, the POSITon, and that manipulative stands for + 1. When you put the POSITon inside the NEGATon, you get 0, and that is a piece called a ZERObi.
Using these manipuatives you can model and teach students how to combine integers, and add and subtract integers.
To learn more about this system and to see how the manipulatives work, go to this website:
Let me know if you have any questions on this topic. It’s certainly an important issue.
Have you ever wondered how students can make mistakes with equations that appear extremely simple to solve, equations as basic as: 9 – x = 11, or: – 9 + x = 11?
To me this was rather baffling until I started to see these equations through students’ eyes, thanks to some tutorees who told me why they were making some mistakes here.
Through this experience, I’ve come to realize that I should be careful before I label any equations “simple equations.”
Here’s a case in point, which I’ll call Example #1:
9 – x = 11
If you ask students how the 9 in this problem is connected to the left side of this equation, a distressingly LARGE NUMBER will say that the 9 is connected by SUBTRACTION. If you just think about it, it’s a very understandable mistake. There’s a big glaring negative sign to the the right of the 9. When students look at that negative sign, many think it tells them that the 9 is connected by SUBTRACTION. Using this analysis, and armed with the knowledge that you do “the opposite operation,” these students will blithely ADD 9 to both sides of the equation, but they’ve just gotten off on the wrong track, getting: 18 – x = 20.
Similarly, given an equation as “simple” as this, Example #2:
– 9 + x = 11
many students will tell you that the – 9 is connected by addition. Why? Because of the big glaring + sign to the right of the – 9. . Reasoning thus, these students will gleefully do the opposite operation and subtract 9 from both sides of the equation, landing in an equally sticky puddle of wrong thinking, getting: – 18 + x = 2
How can we help students avoid these common algebraic pitfalls? Here’s what I’ve found helps.
Tell students that if a number is connected to a variable by an ADDITION or SUBTRACTION sign, you have to look to the LEFT of the number — not to the right — to determine how it is connected to that side of the equation.
To see how this works, let’s look back at Example #1: 9 – x = 11
A student who understands the correct process knows she must look to the LEFT of the 9 to see how it is connected to this side of the equation. Since there’s NO VISIBLE SIGN to the LEFT of the 9, that means that there’s an invisible + sign, so the problem can be re-written, for sake of clarity as: + 9 – x = 11
Now the student is getting somewhere. Using the idea of looking to the LEFT, she looks to the left of the 9 and sees this + sign. This tells her that the 9 is connected by ADDITION, not by subtraction. Following the rule of inverse operations, she now knows to SUBTRACT 9 from both sides of the equation, to get: – x = 2
From there it’s just a matter of multiplying both sides by (– 1) to get the variable alone in its positive form, obtaining the answer: x = – 2
Similarly, Example 2 can be solved using the Look to the LEFT rule of thumb.
This problem is: – 9 + x = 11
The student must ignore the + sign to the right of the 9, and instead look to the LEFT of the 9, finding a – sign. That – sign tells the student that the 9 is connected to the left side of the equation by SUBTRACTION. Using the rule of inverse operations, he knows to add 9 to both sides of the equation, getting: x = 20 And that is the final answer.
As I think about this area of confusion, I think it occurs because we teachers sometimes tell kids that they must “get rid of the number that is connected to the variable.” I know I’ve been guilty of talking this way (but I am trying to reform myself).
When a number is multiplying or dividing a variable, it is perfectly appropriate to tell kids to “get rid of the number next to the variable.” For example, in: 3x = 21, students need to get rid of the 3 which is connected to x through multiplication. Likewise, in x/5 = 6, students must “get rid” of the 5 that is connected through division.
But my recent algebra epiphany is that when numbers are connected through addition or subtraction, as in these problems:
10 – x = 23
x + 7 = – 5
– 8 + x = 13
– 2 – x = 8
we need to be more subtle in our instructions. We must tell students to Look to the LEFT of the number on the same side of the variable. And then tell students to let the sign they see to the LEFT guide them as to how to get rid of the number term that is in the way.
Just a little self-confession that I thought I’d pass along.
Good news: I’ve already started using this “Look to the Left” approach, and it is clearing up lots of confusion for students.
Who says you can’t have fun with integers?
For a number of years I’ve been using a fun story to explain the concept of combining integers with different signs. While I used to struggle at getting this concept across to certain kids, now, thanks to this story, there’s virtually no one who cannot grasp the concept when it is taught this way.
I’d like to share it with you now so you can use it, too.
The idea was first presented on p. 41 of my Algebra Survival Guide, along with a cartoon picture. The basic idea is that you can conceive of problems like: – 3 + 9, + 6 – 14, – 9 + 4, etc. as a tug-of-war.
Here’s the situation — there are two teams, a positive team and a negative team. To see this kind of problem as made of two teams, you have to look at it in a certain way, a way that may be new to some of you. Take a problem like – 3 + 9. You DO NOT view this as “negative 3 plus 9.” Rather, you look at it as make up of two parts: a – 3 part, and a + 9 part. [This has always made more sense to me anyhow.]
So … the “– 3” part tells us there are three people pulling on the negative team, while the “+ 9” part tells us there are nine people pulling on the positive team.
To avoid false assumptions, tell students that all people pulling are equally strong. In other words it would be impossible, for example, for 3 on the negative team to beat 4 on the positive team. Whichever team has more people pulling must win. [I’ll discuss the situation with equal numbers of people on both teams at the end of this entry.]
Then tell students they need to ask and answer just two simple questions to find the answer:
Q#1: Which team wins? [In – 3 + 9, the positives win because they have more people pulling.]
Q#2: By how many people does the winning team outnumber the losing team? [In – 3 + 9, the positives outnumber the negatives by 6, since they have 9 pulling compared to the negatives, who have just 3 pulling.]
When students put their answers together, they get the answer to the problem: + 6. Amazingly simple, huh? All it takes is looking at things in this fun new way.
I’ve put together a little template that you can reproduce and use to teach this rule in this way.
What follows is an example that shows in step-by-step fashion how students would input the data that leads them to the correct answer.
First we’ll show this for the problem: – 6 + 2
The empty template:
Next students make tick marks to show the number of people pulling on the each team, 6 marks on the Negative Team side; 2 marks on the Positive Team side:
Next students answer the two questions, right on the template.
Finally students write in the answer, based on the answers to the questions.
Here’s another model of how students would use the template, this time for a problem whose answer is positive the problem: – 4 + 9
So that’s all there is to it. If you find anyone who cannot learn it this way, let me know. I’ll be amazed.
I suggest using the template for a few days, and then, once students have the idea down cold, let them go off the template. The nice thing is that even after they go off the template, if they get a wrong answer, a really wrong answer, like: – 3 + 5 = – 8, you can ask them to think this out as a tug-of-war, and they will virtually always get it right at that point. Cool, huh?
In a case where there are equal numbers of people on the positive and negative teams, the answer will be zero. Example: – 4 + 4 = 0. In terms of the tug-of-war, you might say this is a situation where neither team wins, and it ends as a tie. So a tie in the real world is a bit like 0 in the math world.
Feel free to leave comments on the blog on how well this works for you in your teaching.
And finally, if you don’t yet have my Algebra Survival Guide, it is loaded with analogies and metaphors just like this one. Teachers, parents, homeschoolers all enjoy and use this book.
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You’ll be able to download a sample chapter that includes the concepts in this post at SingingTurtle.com or Amazon.com
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