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.

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!

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.

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!

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.

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!)

As I said yesterday, today we’re looking at multiplying decimals and the sort of wacky rules it plays by.

Remember on Monday when I said decimals can be represented as fractions? Let’s start by reviewing what happens when we multiply fractions. It’s important to understand something that happens when we multiply decimals.

Let’s say we want to multiply 16.53 and 4.07. Remember that 0.53 is really 53/100 and 0.07 is really 7/100. If we multiplied these two fractions together, we’d end up with a denominator of 10,000.

If we multiply 16.53 and 4.07 together, we don’t actually line up the decimal points. We line up the far right numbers.

Correct: 16.53                                 Incorrect: 16.53

* 4.07                                                 * 4.07

———–                                                 ———-

We then multiply the numbers as if they were really 1,653 and 407, giving us an answer of 672,771. But we still have that decimal point to place. Above, I said multiplying the fractions would give us a denominator of 10,000, which says this number would end in the ten-thousandths. That means it would end four places after the decimal point, and would become 67.2771.

You could also remember this little shortcut: Count up how many numbers are to the right of all the decimal points, and then count that many numbers from the right in the answer.

Tomorrow, we should be taking a look at dividing decimals.

Continuing on our theme of decimals, today let’s look at adding and subtracting them.

If you know how to add and subtract whole numbers, then adding and subtracting decimals will be simple. The most important thing to remember when adding and subtracting decimals is that you must line up the decimal points when you write the problem vertically. It’s like normal addition and subraction; you’re aiming to line up the place values.

Correct:  16.85                                                   Incorrect: 16.85

-  4.91                                                                   – 4.91

———-                                                                  ———–

The normal rules about carrying and borrowing apply when you’re adding and subtracting decimals. The above problem, when worked correctly, will involve the 8 borrowing from the 6, and will give us back an answer of 11.94. The decimal point stays lined up with the other numbers in the problem.

Tomorrow, we’ll look at multiplying decimals, which plays by different rules.

Like the fraction, a decimal is just part of a number. In fact, we can represent a decimal as a fraction. For example, seven tenths can be represented in decimal form as 0.7 and in fraction form as 7/10. Both represent the exact same value.

Unlike fractions, though, decimals have a set of very specific names based on place value: tenths, hundredths, thousandths, etc. Fractions can be halves, thirds, fourths, thirty-sevenths, etc. We name decimals by saying the number represented and the place value of the digit farthest to the right. For example, 0.82 is read as “eighty-two hundredths”. 1.387 would be read as “one and three hundred eighty-seven thousandths”.

All decimals can be rewritten in fraction form (this is sometimes easier when working certain problems). You simply write the number in the numerator and the place value in the denominator. For example, 0.67 (sixty-seven hundredths) becomes 67/100. 2.45 becomes 2 45/100. Then, you reduce the fraction is necessary or possible. In the second example, the fraction simplifies, leaving us with 2 9/20.

Decimals aren’t any more scary than fractions. Just remember they’re just part of a number.