Decimals and Fractions

The journal question, up on the board before the lesson began, was: "Find as many different ways as possible (at least three) to represent 0.40."

I put it up before the lesson began because kids need to know what it is they're expected to learn, just as adults do. Putting up what they're expected to be able to do is one way to accomplish this. I also put up the "practice" sign, to indicate that the journal in question was not being marked for report-card purposes (that would have had the "Game time" sign.) It's a chance for them to explore what they know and for me to see where the holes are in their understanding, before I grade them on it. It reduces a lot of stress when they know that mistakes don't "count," they're just chances to ask questions.

We had already reviewed decimal tenths and fractional tenths the day before. They were fine with that, in that they could model tenths with base-ten blocks and draw them on fraction strips, and write them as fractions and as decimals. The hundredths were new to them, so we began by asking about times when they'd seen numbers like that before. Money and measurement were both mentioned, giving them a context which we'd use later in the lesson.

On the overhead, I put up a 10x10 grid, with nothing shaded. We discussed the concept of an array, and linked the idea to multiplication; that a grid like this could come in any number on either side and was an excellent way to represent multiplication when they needed to. I also pointed out the concept of a square, where the numbers across the top and down the side were the same.

We went from there to asking the kids to identify the whole, and show me how they would write it. This seems so obvious that many teachers end up skipping it, but it's a crucial piece of knowledge for kids. If they're used to thinking of a flat base-ten block as having 100 pieces in it - that is, 100 whole numbers - it's a bit of a switch to start thinking about it as a single whole divided into parts.

They didn't have a problem representing ten parts out of a hundred, but it took them some work to realize that those ten parts were equal to one part out of ten; I used the sticks from the base-ten blocks to show that there were ten of them, and each stick was going to be written in the first spot after the decimal. We didn't spend a ton of time on equivalencies, just pointing out the concepts for now, because they were to be reinforced in group work.

The centre activities were engaging and easy to set up. For the first, kids needed a sheet of 10x10 grids and a six-sided die (sometimes called a number cube, to the horror of my gaming friends.) In pairs, they were to take turns rolling the die and deciding whether the number that came up was to be a tenth or a hundredth. Then they coloured that in on one grid. The goal was to get closest to one whole in exactly seven rolls, without going over. The solitaire version is to get between 0.9 and 1 in seven rolls. The kids loved it, especially since I gave them sparkly gamer dice to use.

The second required a bit more prep. My textbook set has photocopiable sets where each page has cards of a certain way of representing decimal tenths. One page has 0.1, 0.2, etc; the next page has 1/10, 2/10, etc; the third page has squares with ten strips, and a different fraction shaded for each card; the fourth is hundredths as fractions, then hundredths as decimals, and finally 10x10 grids with whole rows shaded. The kids had to first cut out all the cards, then make sure they knew which cards were representations of the same quantity (note: I did not phrase this as "meant the same thing," which can be confusing) and then play a memory game where they had to figure out the locations of two or more cards representing the same quantity. They could also play a Snap card game, where each kid gets half the cards, they turn over one card each at the same time, and if the cards match, the first person to realize it snaps them both up. Next week, I'll be adding in other fractional equivalents, like fifths.

The kids who did those activities mostly understood what was up with the journal question. The kids who did the naming-hundredths activity need to be rotated through one of the other ones before they really get it, so the journal is due on Tuesday. We'll be moving into metric measurements by next week because they make use of the base-ten system, hundredths, and introduce a third strand to this unit.