# Finishing the Fives (Age 8)

## The Activities

1. Topic: Logic. Book: Sideways Arithmetic From Wayside School, by Sachar. Chapter 2.
2. Topic: Arithmetic, Patterns. Make the numbers 1 – 60 using only fives. For example, 26 = (5×5) + (5/5).

## How did it go?

We had three kids for circle. Everyone was very focused, and followed my rule that we cannot draw Pokemon during Math Circle.

#### Sideways Math

The kids love the story of this book, and the problems were great for sideways thinking. First we reviewed the chapter from last week, since David led that circle. The kids were able to quickly explain why elf + elf = fool.  The rules are that each letter stands for one number 0..9, and each letter is a different number. The key to this one is that fool has an extra digit than elf, so we know e + e results in a carry. This means f must be a 1, which can be used to solve all the other letters too.

After this review, we read chapter two. In this chapter, Sue gets very upset because she thinks it’s weird to add words. Instead you should add numbers! Like 1 + 1 = 2!

The teacher writes that problem on the board as:

one + one = two

Sue says no! You should put the numbers there, not words! The teacher says, what numbers? Sue says “1 and 2!”. The teacher laughs: but there are no 1s or 2s in the answer!

Then we worked together to solve this problem, knowing that none of the letters stands for a 1 or 2.

We figured out that o must be < 5 because it shouldn’t cause a carry. o can’t be 1 or 2. It can’t be 0 because that would force ‘e’ to also be 0.  o can’t be 3 because there’s no number such that e + e = 3. So o must be 4.

At that point we got stumped because e + e = 4, but e is not allowed to be 2. We couldn’t figure out our mistake, so I checked the solution at the back, and realized, of course, that e + e must be 14. Then we quickly solved the rest of the problem.

Sue, in the book, then starts shouting out a bunch of math facts like: one + two = three, four + seven = eleven, and all the kids laugh at her. The teacher laughs to and says it’s impossible. So our next task was to prove that those problems are impossible.

one + two = three.  We quickly realized that there are no three digit numbers that add to a five digit number.

four + seven = eleven. This one was much trickier. Our intuition was that too many numbers have to be zero for this to work, e.g. u + e = e, o + v = v, f + e = e. But we had trouble proving it was impossible because what if there were carries involved? In fact, I thought I had proved it, until a kid explained that maybe u + e = e because r + n caused a carry. So really 1 + u + e = e + 10, which is possible if u is 9 and e is 7, for example. So we didn’t quite get a satisfying proof.

five + two = seven. I did most of this one myself. First I figured out that i + t must carry to the f, so that f + 1 = se. That means f = 9, s = 1, e = 0.  But we also know e + o = n, but that’s impossible if e is zero, because e + o must then equal o.

At this point the kids started to get a bit antsy. Some kids wanted to read the next chapter because the story is so funny, but no one really wanted to work on any more problems, so I ended the activity here.

#### Fives Chart

Two weeks ago, the kids filled out about half of chart where you compute the number 1 – 60 using only fives. For example, twenty is (5 x 5) – 5. We promised them a small prize if they could get 40 of the numbers completed, and another prize if they could fill them all in. This week, one of the kids realized that if the chart contains the answer for a number like 40, you can easily compute 39 and 41 by adding or subtracting 5/5.  This allowed them to quickly finish the whole chart. They still were pretty interested in using smaller numbers of 5s when possible, recognizing that it is not very elegant to write 5/5 fifty eight times to get 58.

Here’s a part of their chart:

# Taking a Finger Walk (Age 6)

## The Activities

1. Topic: Puzzles: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We’re most of the way through now, probably one circle left of puzzles.
2. Topic: Logic: I printed sheets with 6 uncolored flowers on one side, and 9 on the other.  There were two puzzles: For 6 flowers, “There are more red flowers than purple, and more yellow flowers than red.  For 9 flowers, “There are more red flowers than purple, more blue than red, and the same number of blue and yellow.”
3. Topic: Spatial Reasoning:  Corey and I built a number of models out of Legos.  The kids each picked a model and had to copy it exactly.  They could pick it up and look at it from any angle.  Each kid copied several models.
4. Topic: Attributes, Games:  We played a couple rounds of Set with just the solid cards.

## How Did It Go?

We had four kids this week.  One kid had been gone for a couple months, but now everyone is back from summer trips.  This circle went well, the kids were all interested in all the activities.

#### Playful Puzzles

We spent quite a bit of time on a puzzle with two kids at either end of a very windy path, with the question “Where will they meet?”  We measured by placing coins from either end.  They also enjoyed a page where you were supposed to “take a walk” with your finger by tracing a path and following various instructions along the way (e.g., “Take a rest here” or “Go around this corner really fast”).  Every kid did it once.

#### Flower Logic

The kids figured out the answers pretty quickly.  Interestingly, different kids figured out the second one from the first.

#### Lego Models

Different kids definitely had different skill levels on this one.  One kid breezed through a whole bunch, while others took quite a bit longer.  The trickiest ones were the dinosaur, because it was irregular and a bit complicated internally, and an offset colored square because it was tricky to get the right pattern of blocks on the bottom row (the kid working on it initially had the colors going the inverse rotation).

#### Set

We’ve played before with this circle, which the kids remembered, but not all kids remembered the rules.  Our son has played a lot, so after he got a few I said he had to let other kids get sets.  I was happy because all the other kids got at least one set on their own.