Kiss those Math Headaches GOODBYE!

Archive for the ‘Addition’ Category

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 Amazon.com, 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
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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
Numbers.

Cover of

Cover via Amazon

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NOVEMBER PROBLEM OF THE MONTH


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.

birthday-candles-potm-image

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:

info@SingingTurtle.com

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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Exploring Addition Facts


QUESTION:

I have been asked to help my grandson memorize the addition tables. If I ask him what is 8 + 5, I watch him doing it in his head. He gets the right answer but takes a long time, which may hurt him later when he is introduced to multiplication. Do you have any suggestions?

MY REPLY:

Great question.

Without knowing your grandson, it’s impossible to know what he is doing when he takes some time to figure out addition facts.

But I’d suggest is that you just ask him — in a happy and curious way — what thought steps he is going through.

You can tell him that you’ve heard that students do mental math in different ways, and that you’d be interested to know how he is doing it. You might also want to reassure him that there is no “wrong” way to do math in your head.

[Be aware: some children have trouble verbalizing what they’re “doing” when they do math mentally. If your grandson has trouble telling you, you might prompt him by asking if he’s using either of the strategies I describe below.]

It may be that your grandson is trying to retrieve a memorized fact, but it’s more likely that he is using some kind of mental operation to arrive at the answer.

For example, it may be that he is “counting up” 5 from 8, to get to the answer. (If so, it would be commendable that he can count up 5 in his head — without using his fingers. Not all students can do this.) It’s also possible that he is taking 5 from the 8, and giving it to the first 5 to make 10, and then tacking on the extra 3, to get to 13 (an advanced strategy).

The main point, though, is that you can’t know till you ask him.

And the other point is that it’s often fascinating to open up a dialogue like this with kids, to find out how they do mental math.

Once you get the dialogue underway, I’d suggest that you just follow wherever it leads. For example, if your grandson is using the second strategy I mentioned (making 10 and adding on), ask him if he can extend the process a bit, and do problems like 18 + 5, 28 + 5,
etc.

If, on the other hand, he is “counting up” 5 from the 8, see if he can use the second method, too.

Essentially, what you have here is a great opportunity to find out how your grandson does addition, and to explore the operation with him. And have fun doing it.

Please feel free to write back if you do open up this kind of dialogue. I’d be curious to know what happens.

And to get to the heart of your question, I would say: Yes, you do want your grandson to develop speed or “fluency,” as teachers like to say. But when he is first learning the facts, it’s critical that he think about the operation, not just memorize facts.

To help him gain speed, I would suggest that you use flash cards or fact worksheets (just google math addition worksheets, and you’ll find loads of them).

And to help him develop a range of good strategies to help him learn the facts with understanding,
I suggest the Facts That Last series by Creative Publications.

https://www.creativepublications.com/productfamily.html?PHPSESSID=4c695f197ef3330314c21e50c8b2d70e&familyid=27