Comparing Decimals

Today, Student A had a worksheet for comparing decimals. She was supposed to say which decimal in each pair was greater than the other, or if they were equal.

Because she originally completed the assignment by guessed (a pretty usual occurrence for A), I decided to give her some steps to follow in comparing decimals before we even began correcting the assignment.

First, we talked about how decimals are just like fractions – meaning they are just part of a number, instead of a whole number. With that in mind, we went on to our steps for comparing decimals:

Step 1. If there is a whole number (the part of the number that is to the left of the decimal point), compare the whole numbers first. If the whole numbers are different, then all you have to do is tell which whole number is greater. (for example, 12.3 and 14.72 – 14.72 is greater because 14 > 12, and it doesn’t matter how much “extra” you tack on to the 12, 12 is still smaller than 14). Otherwise, move on to step 2.
Step 2. Since the whole number parts are the same, we will need to compare the decimals. Remember that decimals are like fractions, so .1 = 1/10, and .01 = 1/100. Well, it’s hard to compare fractions when the bottoms are different (that is, it’s not very intuitive) and it is equally hard to compare decimals when we don’t have the same number of places, such as in 4.35 and 4.3.
To make things easier to “see” (instead of having to think too hard – mathematicians never like to think harder than they absolutely have to – that’s why we have “tricks” for everything) we are going to make sure each decimal has the same number of “places”. To do this, all we have to do is add zeros on the right hand side of the number. Going back to our example of .35 and .3, since .35 has two places and .3 only has one, we will add a zero to .3 and have .30. Now we are comparing .30 and .35 – well, that is just like comparing 30 and 35. It is clear that 35 > 30, so .35 > .30.
If you had .30 and .3, when you add a zero to .3 you will get .30 and .30 – in this case, the numbers are equal.

I was really impressed with Student A’s internalizing this concept. After five or six problems she was able to figure them out quickly, and explain her answers just as fast. She really understood the concept, and that was a huge success for us.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Times on the Table

One of the things I worked with a lot with student A over the summer was TIMES TABLES.

While I am not a supporter of a lot of rote repetition when it comes to learning math, I definitely support repetition when it comes to remembering math. But only after the initial learning has taken place.

It doesn’t do a student any good to memorize the times tables if the student doesn’t even understand what the times tables are. Such was the case with A. We spent a lot of time just learning about numbers in general, and then specifically multiplication, before we moved on to memorizing the times tables.

As part of “figuring out” the times tables, we learned a lot of tricks, and used some games, songs, and pictures to learn the times tables. We looked at patterns in the numbers, talked about sets, and played a lot with Cuisenaire rods to get a physical grasp of the concept.

Only then did we move on to drills.

Here are the posts describing how we learned each set of times tables.

Twos
Threes – Three is a Magic Number
Fours
Fives
Sixes
Sevens
Eights
Nines – The Great Nines
Tens
Elevens – Some Tricky 11’s – in each place
Twelves

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Combining Like Terms – Sorting Letters

Did you ever play sorting games as a little kid? If you fold your own laundry, then you should be a pro at sorting – pants, shirts, socks, sweaters.

Combing like terms is just like sorting laundry, or marbles (as illustrated in my sorry attempt at drawing marbles).

When we have “letters” (also known as variables) in an expression we often want to combine them so that we have an easier expression to look at.

Think of it like having a pile full of different color marbles (like in the illustration above). Let’s look at an example:

2x + 4y – 3xy + 7y – 5x + 2xy

Let’s look at the different “color” marbles we have (that is, the different “terms”)

We have x’s, y’s, and xy’s (the xy’s are their own type of term, because xy is not the same as x or y).

Now we combine the things that are “like” – We have 2x and –5x, so that is –3x. We have 4y and 7y, so that is 11y. Finally, we have –3xy and 2xy, which is –xy.

Our “simplified” expression is –3x +11y –xy.

Because of the commutative property of addition, we can write the expression in any order we want to.

-3x – xy + 11y
11y – 3x – xy

etc, etc

Here are some practice problems for combining like terms (also known as “simplifying expressions”)

Simplify the Expressions by combining like terms

1.

2.

3.

4.

5.

(Pre-Math Tip! Play sorting games with your preschooler. You may not realize it now when you are sorting blocks and dishes and beads, but your preschooler is building the foundation for basic algebra skills like being able to combine like terms!)

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.