And they did it without calculators

The next time you think about telling your math teacher you can’t add 14 and 35 without a calculator, think about the ancient people who handled their Base 12 math system without anything more than their fingers.

You can read how this fascinating system (that permeates our modern-day lives) was most likely established and worked. Or you can try to figure out on your own how to do it on your own hands and check yourself!

Crazy how much math you can do with a little creative thinking! (And my middle school and high school students can count on having this thrown in their face every time they tell me they need a calculator to figure out how many times 6 goes into 36!)

A neat way to look at solving word problems

I keep reading articles about using number bars to solve problems over at Let’s Play Math, and I love them! I’ve tried to figure out how to incorporate them into the very structured curriculum where I work because I think it would help so many of our struggling students!  (I may smuggle it in covertly, anyway.)

To see this techinique in use and why it’s a great method, check out this post on pre-algebra for the third grader. If students could visually see what’s going on in the problem, I think we’d see far fewer students frustrated by story problems.

What’s better is that this technique is good not only for those just learning how to interpret word problems, but I think even my high school students who struggle with word problems would benefit from this approach with some of their work.

It’s not enough to say, “This is how you’re going to run into the math in the real world.” We have to arm students with a means for dealing with problems, starting with a simple concrete approach like these number lines and helping them move on to more deductive, less visual means for tackling word problems.

The Transitive Property

Have you ever seen those Everybody Loves Raymond commercials? According to the commercial, everybody loves Raymond, and Everybody Loves Raymond was showing on Wednesday, therefore everyone loves Wednesdays.

The transitive property essentially works the same way. If you can prove something is equal to a second thing, and that the second thing is equal to a third thing, then the first and third things are equal.

Mathematically, this looks like:

If a = b and b = c, then a = c

This does show up in geometric proofs, but it’s actually a deductive reasoning skill. Mastering the transitive property can help you figure out problems you’re stuck on.

That wraps up our week on properties! Feel free to suggest topics for next week. We love hearing from you!

The Identity Property

So far, we’ve covered properties that work for both addition and multiplication. Today, we’re going to look at the identity property. Both addition and multiplication have an identity property, but it works differently for each operation.

The idea behind the identity property, regardless of whether you’re adding or multiplying, is that you can do something to a number and its value won’t change.

In addition, the identity property adds something to the number without changing the number. The only number you can add to anything without changing  is 0, so the identity property for addition looks like:

a + 0 = a

When you add 0, you literally add nothing, and so the number remains the same.

For multiplication, we have to find a number to multiply numbers by that won’t change the value of the numbers. Well, we know from our math facts that anything times 1 is itself, so the identity property for multiplication looks like:

a * 1 = a

These both may seem obvious now, but as you get into higher levels of math, both can make complex problems a lot simpler to tackle.

Tomorrow, we’ll look at the transitive property.

The Commutative Property

Yesterday, we looked at the associative property,, which held true for both addition and multiplication. Today, we’re going to tackle the commutative property, which also works for both addition and multiplication.

In order to remember the commutative property, I’ve found it helps to remember what it means to commute. Have you ever heard someone talk about their long or easy commute to work? This means they’re going from one place and going somewhere else. That’s the basic idea behind the commutative property. The numbers go different places.

Mathematically, the commutative property looks like this:

a + b = b +a

The a and b might commute to another spot, but it’s still the same problem.

Tomorrow, we tackle the identity property!

The Associative Property

One of my students was studying for a test last week, and she reminded me (as I jumped about the table like a half-crazed loonatic) that I haven’t attempted to cover any of the properties yet, so I thought I’d dedicate this week to covering a number of them.

Properties are useful problem-solving tools. Often, they can allow you to see computations you can make that might be readily obvious, or they can help you deduce correctly. Today, we’re going to start with the Associative Property.

Let’s say I have three friends- Anna, Ben, and Casey. One day, I notice Ben sitting with Anna. The next day I notice he’s sitting with Casey. They’re all still my friends, but they’re sitting in a different way.

The first time, Ben was associating with Anna, but Casey was still part of the group. The second time, Ben was associating with Casey, but Anna was still part of the group. Mathematically, this might look like:

(a +b) + c   and a + (b + c)

The groups may look different because of the parentheses, but it’s still the same three numbers. This is the associative property. It says my answer is the same, regardless of how I add three numbers together. The property formally looks like this:

(a + b) + c = a + (b + c)

This property holds true for both addition and multiplication.

Tomorrow, we’ll take a look at the commutative property.

Reading well is connected to math

This has become a very common conversation at work, but it’s amazing how many people don’t realize how true it is.

If a student cannot comprehend what they’re reading, then they have diminished chances of doing well in math.

It sounds cruel, but it’s true. These days, about half of all math classes is dealing with word problems. The most important step in solving any word problem is understanding what information you’ve been given, and what the problem is asking for. Because this is how students will encounter math in the real world, we put a lot of emphasis on it in math classes. So often, though, the student gets frustrated by the problem because they don’t understand what it’s saying at all. It’s more than just, “Oh my gosh, it’s a word problem. I can’t do this!” It’s literally, “How many apples does Tom have? Well, Jane has 3 and they have 10 all together. But the problem doesn’t tell me how many apples Tom has, so I can’t solve this.” (Yes, I’ve really run into this with students. A lot of my students, actually.)

While math class can prepare students to deal with solving equations, being able to comprehend what the question is asking and then being able to successfully translate that into a solvable equation requires a little help from reading class. (Math is an interdisciplinary subject. Who knew?) Unfortunately, because math is still “scary”, even the student who does extremely well at reading comprehension can sometimes be thrown by a word problem simply because they’ve told themself math is scary.

Quick math tricks

I’ve covered some of these with my own students or here in the blog, but here’s a list of great tricks to help you calculate quickly.

(These could come in handy for those of you facing down the PSAT or SAT over the next couple of months!)

Ancient math

Hey, everyone. I guess you’ve noticed Dead Bunny is a boit quiet lately. Things have been crazy around work, and that’s eaten into my time to blog and to work on the other projects I’m trying to prepare for the site. I’m trying to get everything back under control, but I’m losing the battle at the moment.

Hopefully, you will all forgive me while I share interesting math articles I’m reading. It’s not the same as a math lesson, but some of them are pretty interesting.

This first one focuses on a hobby of mine: cultural anthropology. Denise over at Let’s Play Math has been moving her newsletters online, and one of them had an interesting lesson on multiplying in Ancient Egypt. This is actually part of a rather inspiring story series. I highly recommend tracking down the rest of them.