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Archive for the ‘Pre-Algebra’ Category

Subtracting Integers


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):

http://www.singingturtle.com/downloads/ASG%20sample%20chapter.pdf

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:

http://www.singingturtle.com/pages/ADMINISTRATORS3.html

Let me know if you have any questions on this topic. It’s certainly an important issue.

Happy teaching!

— Josh


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“Simple” equations? Not always so simple.


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.

Happy teaching!

— Josh


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Make Combining Integers a Lively Story



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:

Then students input the problem, – 6 + 2, and they write in how many people are on each team, like this:

 

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.

50% OFF Sale for both the Algebra Survival Guide and Workbook at my website: SingingTurtle.com

Algebra Survival Guide: Regular Price $19.95 — Sale Price $9.95

Algebra Survival Workbook: Regular Price $9.95 — Sale Price $4.95

ASG-Image-for-AD

 

 

You can also purchase both the Guide and Workbook at Amazon.com

You’ll be able to download a sample chapter that includes the concepts in this post at SingingTurtle.com or Amazon.com

You’ll find that the Algebra Survival Guide has 173 Amazon customer reviews, with a 4.5 star rating!

Happy teaching!

—  Josh


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Algebra mistakes due to Negative Signs? Use COLOR to EXPLAIN.


Have you ever noticed how much trouble students have figuring out the difference between these two kinds of expressions:

The key, I’ve found, is to use the Order of Operations to decode the meaning of each expression.

Ask your students to think through how many Order of Operation steps there are in each expression.

If they’re on the right track, they’ll see that the top expression has just one operation:  working out the exponent by multiplying (– 3) by itself twice. This looks like this:

As for the bottom expression, this is more difficult because of that “loose” negative sign, a negative sign that is not enclosed inside parentheses. Just like a “loose canon” can be a danger in the military, a “loose” negative sign can spell trouble on the algebra playing-field. Students will make all kinds of weird mistakes in trying to figure out what to do with this little varmint.

But if you tell students to view that “loose” negative sign as meaning (– 1) times the expression that follows it, they can use the Order of Operations again to do the right thing first. 

Specifically, they notice that there are two operations to be performed: multiplication and exponents, and they remember that they do exponents before multiplication, following the Order of Operations.

Following these steps, the simplification looks like this:

And while this is fine and correct, notice that you can “kick it up a notch” by using color to separate out the two operations. I use blue for decoding the negative sign and working it out; I use red for the 3-squared term. Check it out:


I’ve yet to meet a student who cannot follow this procedure, when color is used.

Try it out and see what kind of results you get.

Color Your Way to Integer Success


We all know that one of the trickiest subject for many students is the subject of combining integers.

I’ve hit on a new way to help students with this topic, a way that involves using color.

Using different colors helps students relate similar concepts and separate different concepts. Color works faster than underlining or drawing rings around numbers, it’s more attractive, and it makes a student’s page fun to look at, too.

Here’s just one example of how using color can help students make sense of those oft-bewildering positive and negative numbers.

Take a problem like:   + 3 – 7 + 6 – 9

Many students get confused by a problem like this because they don’t have the least idea of what to do first.

But when you use color, tell students that the first step is an easy step:  just re-write the problem, making the positive numbers red, and the negative numbers blue, like this:


The next step is equally easy … group the positives on the left and the negatives on the right. I use my handy-dandy “double-slash” divider to show the separation, like this:

After that, use the rule for combining integers with the same sign (here you can say “with the same color!”), like this:


At this point you depart from color, as you combine the integers with different signs, and you get the answer, like this:


All together, it looks like this:


 

I have found this approach extremely helpful for those students — and you know the ones I’m talking about — who just struggle endlessly with these rules. 

By the way, if you’d like to see the chapter of my Algebra Survival Guide, which explains the “Same-Sign” Rule and the “Mixed-Sign” Rule, just go to this page and scroll down about half-way down till you come to the link for downloading chapters:

http://singingturtle.com/pages/PARENTS3.html

Please try this out yourself and feel free to let me know how it goes. I’m always open to feedback.

Making Sense of Negative Signs


If you’ve ever seen students struggle with expressions that have a negative sign in front of parenthses (I sure do!), expressions like this:  – (8x – 5)

then this blog entry is for you!

I’ve developed a new way to help students get this concept right — and to remember the concept so they continue to get it right, week after week. I’ll explain the basic approach in this entry, then give more details in the next entry.

The big problem with the typical textbook presentation for this concept is that it is filled with math gobbledygook. I’ve found that you can cut out the gobbledygook and instead relate this kind of problem to everyday life. Once students learn it this way, they’ll never forget it!

 

MAIN IDEA: encourage students to think of the negative sign as the as the everyday word, OPPOSITE.

So start out by asking simple questions, but writing them in a pseudo-math format.

Example:

Write:  opp (black)

while you ask:  What is the opposite of black?

When students give the answer, what you write now looks like this:

opp (black)

= white

————————

Continue with other examples, like this:

opp (tall)

= short

————

opp (down)

= up

———–

Once students get the basic idea, expand on the idea by telling students they can take the opposite of two concepts, not just one. Show this by writing expressions like:

opp (white, short)

Students should answer:

opp (white, short)

= black, tall

and: 

opp (left, slow)

= right, fast

————-

Once students master this idea, extend the lesson to NUMBERS, first by pointing out that all numbers (except 0) and MONOMIALS have opposites. e.g., that the opposite of + 3 is – 3; the opposite of – 5 is + 5, the opposite of – 8x is + 8x, the opposite of 6ab is – 6ab, etc.

Now challenge students to do problems like this:

opp ( + 5x, – 7 )

They should get:

opp ( + 5x, – 7 )

=     – 5x, + 7

——————–

Then simply explain that in math, we express the idea of opposite by using a “–” sign in front of the (    ), and that we write the terms inside (    )  without commas.

Give students this problem:

– ( 7x + 4 )

See if they can get the answer, which should look like this:

–  ( + 7x – 4 )

= – 7x + 4

——————-

I hear a lot of: “Oh, that’s easy,” when I explain it this way. I think that’s because opposites is a concept students know “cold.” Plus, using the opposites concept connects the algebra to non-math concepts, and students often find that refreshing. I’m sure you’ve noticed that, too.

Then give students a set of problems, like these:

1)  – ( – 7a – 4 )

2)  – ( – 3x + 7 )

3)  – ( + 8y – 3 )

4) – ( 4p – 6a + 12 )

5)  – ( 9x + 4y – 3 )

Have fun with this lesson. If you think of any ways to extend it, or if you find any tricks that make it work especially well, feel free to share them by sending them to:  josh@SingingTurtle.com