Remember the ol’ “if A equals B and B equals C, then A equals C” deal? At parties it’s a great line to drop. In math, it’s officially called … cue music…
The Transitive Property
Saying it is fun, teaching it is curious, learning it can be weird.
Grownups think it’s intuitive. But to a kid, it isn’t. It takes experience and experimentation to learn all the bits that we think are “common sense.”
The transitive property is really thinking things through. Starting from one place and moving along through another and then arriving at a third place.
There are many ways to help kids with this learning. Word problems simulate thinking stepping stones. But they can be rather stressful. If you do it through play, you reduce the stress that they face and give them skills they need to tackle advanced thinking, forever.
This video shows a nifty “toy” from Discovery Toys that can get kiddos using those brain stepping stones.
Notice the flow is
Choose the number tile with the question number.
Read and answer the question.
Correspond the answer to the letter in the answer box.
Put the number tile with the question number in the corresponding letter box.
Thinking through from question number to answer letter while avoiding the pitfalls is the challenge.
Have you played with these? How do you train your kids’ brains for the transitive property?
Want to give your kiddos a jump start on multiplication and division? What to help the ones struggling with division to grasp it better?
The Discovery Toys Measure Up Cups can do just that. They are built as a curriculum tool, in the proper ratios, so that the #6 cup holds exactly twice as much as the #3 cup. This allows for engaging and beneficial play that gives kids a grasp on how numbers relate to reality.
For example, in this video, kids can compare the numbers 3, 6 and 9 to see how they relate:
You don’t have to say out loud, “three plus 6 is 9” or, “9 divided by 6 is one with three left over.” But these concepts are ingrained into the child’s brain as they see this work.
I was at a seminar yesterday with world famous (and awful fun) Thiagi. He keeps his participants on their toes with small activities called jolts. Jolts are defined at interactive experiential activities and:
…force participants to re-examine their assumptions and revise their habitual practices.
One jolt we did yesterday got me to thinking about math. Okay, many of them had me thinking about math. But this one was about the multiplication principal of counting and sets.
The multiplication principal says that if you have 7 ways to do the first thing and 3 ways to do the 2nd thing, then you’ll have 3 * 7 = 21 ways to do them together. Assuming you pick one of each.
So if I want a sandwich (tuna, ham or turkey) and a side (chips, onion rings, fruit or fries) for lunch, I will have the option of 3 * 4 = 12 different lunches.
The Thiagi Circles Jolt
Here is the Circles Jolt that Thiagi offered: Draw this figure without ever lifting your pencil or retracing over any lines (or curves):
It isn’t hard, as you probably see. It takes anywhere from 10 to 45 seconds to figure it out. The question is, “Is there another way to do it than the one I thought of?”
So here are the ways I immediately thought of in the Thiagi Workshop:
The fancy pants teacher’s pet, Mark, was asked to demonstrate the answer. And he did it in an even different way! So I saw that there were 8 more ways to do it:
Trying to trump Mark and be the new teacher’s pet, I announced that there were indeed 16 ways to accomplish Thiagi’s goal.
There are 16 ways because of the Multiplication Principal of Counting.
Notice there are four questions to answer when solving this:
Do you start from left or right side of the drawing?
Do you start by drawing the outer circle or go through and start at the inner circle?
Do you go up or down on the first circle you draw?
Do you go up or down on the second circle?
Each question has two different options. So there are
2 sides to start from * 2 circles * 2 directions to go in the 1st circle * 2 direction in the 2nd circle
2 * 2 * 2 * 2 = 16 different ways to draw the diagram!
Teach this with experiments.
Although you now know how many there are, don’t teach it this way. Instead, just give the jolt to the kids. Once they solve it, show them “your way,” which should be any way that is different than theirs.
Then tell them that for ever unique solution, you’ll give them $1. (Thiagi gave out $1 bills yesterday, too!)
Once they come up with all 16, offer the extra $4 for some way they can show the number 16 with numbers (logically based on their solutions). They might come up with 2 * 8 or 4 * 4, as long as they can group their drawings in logical bundles.
I discovered this video about the philosophies of teaching math from the blog Learning to Teach, The Empty Set of Education. Having degrees in math and not teaching I found it very interesting from a theoretical point of view.
And I totally love his conclusion – the 5th philosophy!
What is your philosophy for teaching math? Share your thoughts in the comments.
I previously posted about prime factors and about using them in multiplication. Now we’ve got fractions to handle.
Before we get in too deep, first let me note the main reason why we do this with arithmetic. In algebra, students will be asked to take a rational expression, factor it and reduce it like this:
If they are familiar with how factoring works with numbers, this will be normal to them.
I also find that reducing factors in this neat and clean way helps a lot. In this video I showed the first few steps of reducing the fraction a sloppier way after I did it the “neat” way:
Here’s one that’s a little more complex:
Will be helpful to show your kids? Is there something I’m missing? Share with us in the comments.
This post is by request from @corrincross on Twitter.
Exponent rules are hard. Well, they are hard to remember, anyhow. But there’s an easy way that won’t make your brain ooze – and that’s doing a mini-experiment each time you have to deal with them.
First remember what an exponent really is. It’s a shortcut for multiplication. Check it out:
Notice that there are really only two rules that get you screwed up: when to multiply the tiny buggers and when to add them. Here’s how I teach this (and how I remember it myself!):
So the new rules are and .
And then what about those negative exponents? Again, you can’t remember a rule unless you remember what they heck is really going on. We go back to adding in this video to explain the similarity between the two shortcuts of multiplication and exponents:
Can you use this in your classrooms? Share your thoughts in the comments.
Thanks to Corrin Cross (@corrincross on Twitter) for requesting this post. Corrin is a Secondary Ed student with a math major and music minor at the University of Regina in Saskatchewan, Canada.
Alright, you might not win friends with this, but the more you can do arithmetic in your head… well… okay, you can’t influence people with it either. Regardless, it’s handy to know and helps with multiplication.
Here’s one that’s a little more challenging. The key is that you can combine the bits to something you’re more comfortable doing. I don’t like multiplying nines, so I avoid those.
Can you use this? Why or why not? Ha ha, just kidding – no essay questions here! – but if you do find a use for this, or if this annoys you, let me know in the comments.
Factoring is traditionally considered an “f-word” for students of math. But it sometimes has its good sides (which I’ll show over the next couple of days).
In the meantime, I’ll show the two ways (that I know) to factor numbers.
This method’s called a factor tree:
You don’t have to put the primes in order for it to be “right.” But it helps for later things.
Here’s another tree:
Here’s another way to factor using an upside division bar-type thing:
Remember, we don’t factor for the sake of factoring. This is only a tool to use when we do other stuff. You’ll see some of this in tomorrow’s post. Here’s one last example:
The way you factor is personal preference. Pick the one you like, or flip-flop. There’re no rules!
Share your preference (or a different way) in the comments!
This is a trick that everyone should learn in the 3rd grade. If not, then at least by the 9th grade.
If you’re in the dark on this one, no big deal. Now’s the time to learn. If you’re a parent – this trick will win you points like crazy. Especially if you get to it before the teacher does.
So here’s how it works. For both 3 and 9, start by adding up the digits in the number. If the result is divisible by 3, then so’s the original number. If the result’s divisible also by 9, then… you guessed it… so’s the original number.
If your result is way to big to tell, do it again. Check out the videos for both:
Questions or comments? Leave them in the comment section below.
