Kiss those Math Headaches GOODBYE!

Archive for the ‘Subtraction’ Category

How to Combine Positive & Negative Numbers — Quickly and Easily

If you or someone you know struggles when combining numbers with opposite signs — one positive, the other negative — this post is for you!

To be clear, I’m referring to problems like these:

 – 2 + 7 [first number negative, second number positive], or

+ 13 – 20 [first number positive, second number negative]

To work out the answers, turn each problem into a math-story. In this case, turn it into the story of a tug-of-war battle. Here’s how.

In the first problem, – 2 + 7, view the – 2 as meaning there are 2 people on the “negative” team; similarly, view the + 7 as meaning there are 7 people on the “positive” team.

There are just three things to keep in mind for this math-story:

1)  Every “person” participating in the tug-of-war is equally strong.

2)  The team with more people always wins; the team with fewer people always loses.

3)  In the story we figure out by how many people the winning team “outnumbers” the other team. That’s simple; it just means how many more people are on that team than are on the other team. Example: if the negative team has 2 people and the positive team has 7 people, we say the positive team “outnumbers” the negative team by 5 people, since 7 is 5 more than 2.

Now to simplify such a problem, just answer three simple questions: 

1)  How many people are on each team?
In our first problem, – 2 + 7, there are 2 people on the negative team and 7 people on the positive team.

2)  Which team WINS?
Since there are more people on the positive team, the positive team wins.

3) By how many people does the winning team OUTNUMBER the losing team?
Since the positives have 7 while the negatives have only 2, the positives outnumber the negatives by 5.

Now ignore the answer to the intro question, Question 1, but put together your answers to Questions 2 and 3.




All in all, this tells us that:  – 2 + 7 = + 5

For those of you who’ve torn your hair out over such problems, I have good news …


But to believe this, it will help to work out one more problem:  + 13 – 20.

Here, again, are the common-sense questions, along with their answers.

1)  How many people are on each team?
In this problem, + 13 – 20, there are 13 people on the positive team and 20 people on the negative team.

2)  Which team WINS?
Since there are more people on the negative team in this problem, the negative team wins.

3) By how many people does the winning team OUTNUMBER the losing team?
Since the negatives have 20 while the positives have only 13, the negatives outnumber the positives by 7.

Just as you did in the first problem, put together your answers to Questions 2 and 3.




All in all, this tells us that:  + 13 – 20  = – 7

Now try these for practice:

a)  – 3 + 9

b) + 1 – 4

c)  –  9 + 23

d)  – 37 + 19

e) + 49 – 82

Answer to Practice Problems:

a)  – 3 + 9 = + 6

b) + 1 – 4 = – 3

c)  –  9 + 23 = + 14

d)  – 37 + 19 = – 18

e) + 49 – 82 = – 33

Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like the way Josh explains these problems, you will very likely like the Algebra Survival Guide and companion Workbook, both of which are available on  Just click the links in the sidebar for more information! 


“Math Cafe” Gives Students Options for Math Success

Sometimes when I tutor I tell students that they are “hanging the in Math Cafe.”

I explain that when I tutor, I like to offer students a “menu of math options.”

Math Cafe

Math Cafe, Open 24/7 = 3.42857 ...

Instead of showing students just one way to work a math problem, I try to present a “menu” of approaches, and I tell students that they need to listen so they can decide which approach works best for them.


Abbreviating the Order of Operations

My recent posts about “Dear Aunt Sally” have, I hope, shown how dangerous it is to teach the memory trick of Please Excuse My Dear Aunt Sally — at least without some additional explanation.

Today I propose a way to save Dear Aunt Sally, for those of you who still like her.

As you know, the memory sentence is often abbreviated PEMDAS, which stands for:  PARENTHESES, the EXPONENTS, then MULTIPLICATION, then DIVISION, then ADDITION, then SUBTRACTION.

The problem with PEMDAS is that it makes kids think they always multiply before dividing, and that they always add before subtracting.

For students attached to PEMDAS, I let them use it, but I have them write it a novel way, so they realize they must pay attention to the left-right orientation of the operation symbols.

In the new way of writing PEMDAS, I put M and D in the same place, separate by the word “or.” Then I do the same for the A and S. So the whole memory device looks like this:

Alternative for PEMDAS

My suggestion is that teachers who like PEMDAS try this and see if your students start making fewer mistakes with the order of operations.

For those of you who never liked PEMDAS on the first place, I recommend that you check out the order of operations as presently in a clearer way, as it is in my Algebra Survival Guide. To get a feel for that, you can download the chapter of the book on Positive and Negative numbers here. Just click where it says:  See Sample Chapters of the Algebra Survival Guide and Workbook.

Then you’ll want to get the Survival Guide for the chapter on Order of Operations, which you can do through, here. Sorry, I can’t make everything available for free … I do have a business to run, with new products I’d like to create and make available.

Best way to write PEMDAS

Addition and Subtraction: More Bad Behavior by Dear Aunt Sally

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
= 7

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
=  9

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

Cover of

Cover via Amazon



The problem:

It’s your friend’s 73rd birthday. You’ve put together a surprise party and baked a special coconut meringue cake. But at the last minute you realize that — golly gee! — you forgot to get candles.

Rummaging through your drawers with just five minutes before your friend’s scheduled arrival, you find that you do have 14 candles. And being a brilliant mathematician, you realize that you can represent the number 73 with these 14 candles, using every candle. How do you do it?

As a hint, here’s a model showing how to do a problem like this, if you are celebrating someone’s 44th birthday, when you have just 13 candles. Notice that each dot on the top row is one candle.


Note that you may use icing to create the symbols: +, –, x, ÷, and you may also put in exponents, using candles to show the value of the exponent.

Have fun!

Send answers to:

Make the Subject line: POTM

Please include your full name, where you live, and if you don’t mind, describe your connection to math and math education (for example: teacher, tutor, math enthusiast, etc.).

The first person to send in a correct answer receives a $20 gift certificate toward the purchase of any Singing Turtle Press products. I’ll fill the winner in on the details by email.

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