# 2 1/2 Using Only Fives? (Age 7)

## The Activities

1. Topic: Fractions: Book: Fraction Action by D. Adler.
2. Topics: Arithmetic, Order of Operations: First I explained how to use parentheses in arithmetical expressions.  Then, using just the number 5, the four arithmetic operations (+, -, x, /), and parentheses, we made as many different numbers as we could (e.g., (5 + 5 + 5) / 5 = 3).
3. Topics: Maps, Spatial Reasoning: I had a floor plan of the first floor of our house, and using an iPad I took 7 different pictures in different places and directions.  Using an top-view of a person (with a big nose so you could see which direction he was looking), they had to show where I was standing and which way I was facing when I took the picture.  Then, I found a top down sketch of a room with a corresponding perspective drawing from one point in the room; using the paper person to show what perspective to use, the kids had to draw a picture of what the room looked like if you were standing there. Then we compared to the professional version. The original site is here, and a PowerPoint with the two pictures and the paper person is here.

## How Did It Go?

There were only three kids this week.  For whatever reason, the kids were less well-behaved than usual, particularly during the maps activity where there was a lot of (unhelpful) laughing.

#### Fraction Action

The kids paid attention but afterwards complained about the book a little.  Unlike most of the books we’ve been reading lately, it didn’t have a story.  Also, it has a number of hands-on activities that we didn’t actually do; you could spend at least half of circle on this book if you did all the activities, which were pretty good.

#### All Fives

Two of the kids already knew about parentheses, so I didn’t need to spend very much time explaining.  We went through a couple of examples, and then I gave them the 5’s problem.  They got 1 and 2 relatively quickly, and then got a bunch of multiples of 5.  The kids got 5 + 5 + 5 + 5 = 20 pretty early, but I mentioned you could do it with fewer 5’s and one of the kids got (5 X 5) – 5 after a while.  Each kid added their answer to a shared sheet (picture above).  But they had a tough time getting beyond that.  Eventually I said “If you can get 15, what else can you get?”  They didn’t get what I was hinting at, so I asked for each operation what happened when you applied it to 15.  Then they saw you could get 3 by dividing.  When the kid wrote down 5 + 5 + 5 / 5, they put the parentheses as (5 + 5) + (5 / 5).  I asked what this made, and they realized it was 11.  I pointed out that 5 + ((5 + 5) / 5) was also different.  One of the kids decided to write down a random sequence of operations, and with the parentheses they chose it was 5 – (5 X (5 / (5 + 5))), or 2 1/2.  The kids definitely didn’t realize the general patterns yet — for example, after we got 3, they didn’t see they could get 4 by taking 20 and dividing by 5.  So we should revisit this later and try to get more numbers.

#### Maps and Perspective

Unlike the little kids, the big kids were very good at knowing where various pieces of furniture were on the map, and wanted to a draw a bunch of them.  They could also take a point on the map and easily say where it was.  However, it turned out to be pretty hard for them to figure out where a picture was taken.  They could identify items in the picture, find them on the map, and decide that the person had to be nearby.  But what they had trouble doing was taking information about multiple different objects and combining them together to get the final position.  For example, in one picture, you could use the fact that the dinner table was between the camera and the beanbag to figure out where the camera was; but they wanted to put the person between the table and the beanbag.  Orientation of the camera was also somewhat difficult.  When we did the drawing activity, two of the kids drew overhead views similar to the picture I first showed them, while one of the kids drew a picture with approximately the right perspective (see above).