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Archive for the ‘Creativity in Math’ Category

My WAGER (& DISCOUNT PLAN) to help ANYONE learn Algebra


A wager … and a plan.

I am making a wager that I can help ANYONE learn and deeply understand algebra. And I have a plan to do just that.

Algebra Tutoring Right Here!

I’ve been tutoring algebra for a long time (oh, just a bit over 30 years now), and I have developed many tips and tricks for this subject area. Not only that, but I’ve seen pretty much every mistake you can imagine. And I’ve learned how to explain why each mistake is incorrect and to help folks view each situation correctly.

So in the spirit of the Emma Lazarus poem on the Statue of Liberty, I say:

“Give me your confused, your bewildered, your frustrated students, yearning to comprehend, the befuddled refuse of your overcrowded classrooms. Send these, the despondent ones, your so-called failures to me. I will lift my lantern of algebraic clarity unto their puzzled eyes!”

And in fact, I am offering a special, now through the end of March. I will tutor anyone who wants algebra tutoring for the special rate of just $40/hour (+ tax if you live in New Mexico). I tutor by Skype or FaceTime, so this offer is open to anyone worldwide.

Also, for anyone who takes me up on algebra tutoring and who does three or more sessions with me, you will get copies of my Algebra Survival Guide and Workbook at a 25% discount.

To set this up, just send an email to:
josh@SingingTurtle.com
or send a text to:
505.690.2351

Remember this offer ends on 3/31/2020, so take advantage of it now!

An Unstated (but Useful) Algebraic Property


Imagine that you’re looking so intently for a “pot of gold” that you don’t see a “bowl of diamonds” dancing in front of your eyes.

That’s my sense of what happens in algebra when we talk about the Commutative Property of Addition. Focused on it, we fail to see a very important related property. In fact, this other property is so neglected that it has no name! At least I have never heard a name for it. If anyone has heard a name for it, please let me know, and I’ll spread the word.

Coffee, Pi and More

As a reminder, the commutative property of addition tells us the obvious fact that when we add two numbers, the order in which we add them makes no difference. For example, 5 + 3 = 3 + 5, and 21 + 13 = 13 + 21.

Often instructors will ask students:  does this commutative property also work for subtraction? So students will start to consider whether or not 8 – 3 = 3 – 8. No, they conclude. The commutative property does not work for subtraction. End of story.

But gosh, that should not be the end of the story. It should be the beginning of a new story! Why? Because there’s something very interesting about 8 – 3 and 3 – 8. Sure, the two differences are not equal. But take a look:  8 – 3 = 5, and 3 – 8 = – 5. The differences are opposites. In other words, it is starting to look as if:

a – b = – (b – a)

Well, let’s try another such problem to see if this opposites pattern happens again. How about now we try another pair of integers, one positive (8), the other negative (– 3).

Is 8 – (– 3)  =  – [(–3) – 8]  (?)

Well, 8 – (– 3) = 11, and –[(–3) – 8] = – [– 11]  = 11

So this has worked again.

One more time, to test all possibilities of positives and negatives. Let’s see if this also works if we start with two negative numbers:  (– 8) and (– 3).

Does (– 8) – (– 3)  =  – [(– 3) – (– 8)]

Well, (– 8) – (– 3)  = – 8 + 3 = – 5

And (– 3) – (– 8)  =  – 3 + 8 = + 5

So yes, once again the differences you get are opposites.

So this means that if we widen our vision beyond the classic commutative property, there’s another gem of a property to be learned and used. This property says that:

a – b =  – (b – a)

Since I’ve never heard a name for this property, I’ll just give it a name. I’ll call it the Opposite Differences Property.

In my next post, I’ll share some info on some of the nice ways we can use this property.