# 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:

# Birthday Treasure Hunt (Age 6)

## The Activities

1. Topic: Multiplication. Book: Too Many Kangaroo Things To Do, by Murphy. This book is about friends planning a surprise party for Kangaroo, using multiplication along the way. The kids all enjoyed the book, taking turns computing the simple multiplication (1×1 up to 4×4). One kid proudly predicted that the animals must be planning a surprise party.
2. Topic: Various, Story Problems. I made a grid of hexes that were hidden at first. The goal was to find the hex with a diamond printed on it. Each turn the kids got to move their piece to uncover a new hex and then solve a different type of math problem for each picture type. Here are the hex pictures you need, and the full list of problems is below. We worked as one team, and I asked each kid to try each problem. If someone solved it faster than the others, then they were supposed to whisper the answer in my ear instead of shout it out. As soon as the jewel was uncovered, all 4 kids got to pick a prize from our treasure box.
1.  Firefly – square numbers:
1. First square bigger than 0.
2. First square bigger than 5.
3. First square bigger than 10.
4. First square bigger than 20.
5. First square bigger than 30.
6. First square bigger than 40.
7. First square bigger than 50.
8. First square bigger than 60.
9. First square bigger than 70.
2. Unicorn – fractions:
1. Divide a circle in half, then split each piece into 3 pieces.  How many pieces do you have?
2. Divide a circle in half, then split each piece in half, then split each piece in half. How many pieces do you have?
3. Divide a circle in four pieces. Then split each piece in 3 pieces. How many pieces do you have?
4. Divide a circle in half. Then split each piece into 3. Then split each piece into 2. How many pieces do you have?
3.  Dragon – money:
1. A diamond ring costs \$100. How many rings can Hans buy with \$125?
2. Diamond earrings cost \$20. How many earrings can Olaf buy with \$207?
3. A diamond necklace costs \$11. How many necklaces can Marshmallow buy with \$110?
4. Elsa bought 20 diamond rings that each cost \$10. How much money did Elsa spend?
5. Sven bought 4 bracelets that each cost \$32, and 3 rings that each cost \$14. How much money did Sven spend?
6. Anna spent \$60 on 5 necklaces. How much did each necklace cost?
7. Hans spent \$39 on 3 bracelets. How much did each bracelet cost?
4. Troll – story problems:
1. A troll had 12 muffins. He ate some of them. Now he has 7 muffins. How many did he eat?
2. There are 20 muffins. Some trolls came. Each troll ate 4 muffins. How many trolls are there?
3. 4 trolls brought muffins to a party. Each brought the same amount. There are 24 muffins at the party. How many did each troll bring?
5. Witch square – codes: Figure out what the coded word is by subtracting the given number from each letter. For example, DBU -1 = CAT
1. -1:  DBU
2. -2: DTQQO
3. -1: QPJTPO
4. -2: JCV
5. -1: TQFMM
6. Maze – patterns:
1.  1 5 9 13 __   __
2.  1 2 2 3 3 3 4  __  __  __  __
3. 91 82 73 64 __   __   __
4. 11 22 33 __  __  __  __
5. 1 1 2 3 5 8 __  __  __
6. 1 2 4 8 __  __

## How did it go?

We had four kids today and they were all very motivated by wanting to earn a prize in honor of my son’s upcoming birthday. We played the game with 37 hexes, and the kids got unlucky and didn’t find the jewel until they had uncovered 30 hexes. Toward the end I started letting them move 2, 3, or 4 hexes without solving the problems, just to make sure we found the jewel.

All four kids worked hard on the game questions. My son is quite far ahead of his age in calculation and story problems but he did a really good job not telling the other kids the answers. The other kids stayed involved though, and we made sure to work out each answer as a group, using Base Ten blocks or counting on our fingers if necessary. One kid got bored after 30 minutes but didn’t distract the others. Another kid especially enjoyed problems the required counting by 4, 20, or 11. At first he didn’t think he could count by 11s, but quickly he saw the pattern and took the lead.

The fourth kid is the least comfortable with the number line but he got really excited by square numbers and solved all three square problems before anyone else (smallest square above 0, smallest square above 5,  smallest square above 10). We used Base Ten Blocks to do this. I showed the kids how 9 is a square number because you can make a square out of 9 unit cubes, and he then spent some time making other squares out of unit cubes. He also solved this pattern: 1, 2, 2, 3, 3, 3, 4, _, _, _, _ first.

Everyone enjoyed decoding the witch’s code and trying to sound out the trickier words…pasta? pesto? poh-aye-son? Ooohhhh: poison!

The unicorn fraction problems turned out to be tricky. All the kids could follow the instruction: draw a circle and divide it in half. But “Now divide each piece into three pieces” was tricky. Only my son figured out how to divide each half into three equal pieces. The other kids ended up drawing straight lines and getting three very uneven pieces. Most kids also forgot to divide *each* half, so they would get ‘4’ as the answer instead of 6.

We finally uncovered the jewel, and celebrated. Then everyone picked a prize and ran around outside to get rid of their pent up energy. A very successful circle!

# Time for Tessellations (Age 6)

## The Activities

1. Topics: Logic, Puzzles:  Book: Playful Puzzles for Little Hands, by Taro Gomi.  This is the third time we’ve done puzzles from this book.
2. Topics: Tesselations, Geometry, Patterns:  Each kid made a square-based tessellation by starting with a 3 inch square of poster-board, drawing an inset on two adjacent sides, cutting it out, and taping to the opposite side.  They each kid filled a 8.5 x 11 sheet of paper with the tessellation and colored it.

A rather ambitious tessellation

## How Did It Go?

We had three kids this week.  We took it pretty easy this week after the month long break.

#### Playful Puzzles

This continues to be a great book, lots of interesting fine details in the “What’s different?” puzzles.

#### Tessellations

I started by showing them how to make a tessellation and how to trace it.  Two of the kids made fairly simple tessellations, one made a very complicated one.  I cut them all out and taped them.  The two kids with easier tessellations started getting distracted and chatting part way through, so all three kids ended up nearly finishing.  Two of the kids picked rainbow colors.

# A Triangular Circle (Age 8)

## The Activities

1. Topic: Logic: Book: Still More Stories to Solve, by G. Shannon.  We read a few stories picking up where we left off last time.
2. Topics: Geometry, Proofs:  I started by exploring the properties of angles resulting from a line A crossing two other parallel lines B and C.  Each kid drew a picture and measured the angles, and noticed that the “Z” angles are the same, as well as the angles “translated” along line A.  I told them that this is a property of parallel lines, and then we used it to prove that the sum of the angles of a triangle is 180 degrees.
3. Topics: Patterns, Sequences:  I made a very large Pascal’s triangle (20 rows) on a sheet of poster board.  The kids looked at it and noticed some patterns (symmetry, rise then fall across a row, counting numbers, triangular numbers), I pointed out the “dog-leg” property, and then we covered all the even numbers with pennies, resulting in the pattern below (which is a discrete approximation of Serpinski’s triangle, a famous fractal).

## How Did It Go?

#### Angles

The hardest part of this was getting the abstract idea that this would hold for ANY picture, not just the one they drew.  This was particularly true for the triangular proof — it’s tricky to keep track of the difference between the name of the angle (e.g., A) and the number of degrees in its measurement.  Once you get the idea, mathematicians often gloss over this point and use the same variable to apply to both, but for the kids this was confusing.  Some of the kids understood the triangle proof, I think.

#### Pascal’s Triangle

The kids did a pretty good discovering the various properties.  I gave them some hints for the up/down property, but the only two things I directly suggested were the dog-leg property and covering the even numbers.  The pattern is really cool!

# The Case of the Missing Blink Cards (Age 8)

## The Activities

1. Topic: Measurement. Book: Measuring Penny by Leedy. Lisa gets a homework assignment to measure something in a variety of ways. She decides to measure her dog Penny.
2. Topic: Sorting, Patterns, Charts. We have a card game called Blink, which has cards of six different colors, with six different shapes, and five different numbers (1 – 5). Here are some sample cards.

We calculated that there should be 6 * 6 * 5 = 180 unique cards. However, the Blink deck only contains 60 cards.  I asked the kids to figure out which cards are missing, and if there’s any pattern.

3. Topic: Protractors, Measurement, Triangles. Each kid got a protractor, and  triangle I had drawn before circle. We measured each angle, and then added them up to see what we got.

## How did it go?

We had 4 kids this week. Overall, it was a fun, focused circle.

#### Measuring Penny

We had read this book over a year ago. Some kids remembered it, but 3 out of 4 kids wanted to hear it again. This time I took several different breaks to discuss the book. For example, when they talked about ‘nonstandard units’, I measured one of the kids’ hair in number of “Corey Hands”. We also measured everyone’s ears in centimeter. The kids had a great time with the book, and stayed focused and interested.

We had told the kids about the missing cards several weeks ago, and they all remembered that there are supposed be 180 cards.  We recalculated it again, just to be sure.  Then I asked if we could figure out which cards are missing?  I started them off by sorting through the cards and showing that there were only two cards that had red lightning bolts on them.

The kids took over from there. At first they just randomly picked a color and shape, and looked for the matching cards. Soon this became unmanageable, so one of the kids suggested moving to the floor, and making a separate row for each color, and use columns for the shapes. This resulted in the following chart:

I then asked the kids several questions about the cards, which were easy to answer with this chart. How many of each color are there? 10.  How many of each shape? 10.

How many of each number? This one was trickier because the chart is not sorted by number. One kid wanted to rearrange the chart, but instead we went row by row looking for each number. We found there are 12 of each. We also found that for each color, there are two of each number.  During this time, two kids counted the attributes, and two kids were keeping notes.

One kid’s notes.

Next I asked how many cards are missing from each row? The kids looked at their chart, and said two are missing from each row. We then calculated there must be 12 missing cards total. But that would make only 12 + 60 = 72 cards, not 180, like we calculated.

I should have asked the kids where the other cards were, but instead I just showed them how to update the chart to sort by number too.  So we had a row for each color, and a column for each number/shape combination. Two kids helped me fix the chart:

The new chart.

Then we counted the missing cards in each row. This time we found there are 20 cards missing in each. 20 * 6 = 120 + 60 = 180!

#### Triangles and Protractors

I handed out several big triangles I had drawn with sharpie before circle. The kids used protractors to measure the three angles, and then add them up.

Using the protractor was still challenging for the kids, but they all made progress when I helped them. The kids added angles up to 180 four or five separate times. We also got 182  and 183 a couple times. Weirdly, when I did it myself, I got 188 for a triangle…Not sure why.  But the kids actually noticed that it was near 180 all the time, so we may be almost ready for the proof that they must always be 180.

My daughter measuring a triangle.

# The Parallelogram and the Pendulum (Age 8)

## The Activities

1. Topics: Logic:  Still More Stories to Solve, by G. Shannon.  We read and discussed the first two stories.
2. Topics: Spatial Reasoning, Tangrams:  We did the same set of tangrams from a few weeks ago (letters, numbers, and things from Cinderella).
3. Topics: Physics, Experiments:  Inspired by the Galileo chapter of Mathematicians are People Too from a few weeks ago, I hung a makeshift pendulum from the ceiling — a roll of tape suspended from an 8′ thread, hanging from a sticky hook attached to the ceiling.  I had pre-marked the 2′, 4′, 6′, and 8′ points away from the center of the roll of tape.  We released the pendulum twice at each length, varying the height that we released it at, and timed how long it took to go 20 swings.
4. Topics: Sorting, Patterns:  We have a card game called Blink — basically a racing version of Uno.  Each card has some number of symbols 1-5, one of six colors, and one of six shapes.  There are 180 possible combinations, but only 60 cards in the deck.  After we figured out there should be 180, I asked the kids to find out which ones are missing.

## How Did It Go?

We sat on the floor this week to make room for the pendulum; this tends to make them a little crazier since they can easily roll around on the floor.

#### Still More Stories to Solve

I wasn’t crazy about the first puzzle, but the second one, about two brothers having a contest to see whose horse would get somewhere LAST, was nice.  The kids figured it out with some hints.

#### Tangrams

Corey and I discovered that I’m better at Tangrams than she is :), so unlike last time, where Corey AND the kids were stuck I was able to help them solve the puzzles.  The main thing I tried to teach them was to figure out where the big triangles go first; it’ll be interesting to see if next time we do Tangrams they remember this.

#### Timing a Pendulum

As you can see from the chart above, we had really reproducible results.  I believe we were actually only counting 19 swings (we started on 1 as we let go and then stopped when we said 20, when we should have let it swing again).  Anyway, I had incorrectly remembered from physics long ago that the time was linearly proportional to the length of the pendulum, so I was initially worried about the timings we were getting — but once they were all in, it become obvious (to me, not the kids) that the time is proportional to the square root of the length.  I asked how long the pendulum should be to get 15 seconds; and also, how long would a 32′ pendulum take.  They were comfortable assuming a linear relationship, but when I pointed out that 8′ was four times 2′ while 60 s was only two times 30 s, they couldn’t really use that information — one kid did guess 1/2 foot for the 15 seconds question, but they didn’t stick to their answer so I think it was just a guess.

One thing that worked out well is that the pieces of tape served as resting spots for the thread so that it would stay at the right length (I didn’t cut the string, we had looped it over the hook like a pulley).  If I hadn’t had the tape sticking out, it would have been hard to maintain a constant length.

I only had time to explain the problem and figure out how many cards there should be before circle ended.  We’ll probably do the main activity next week.

# Flowers, Stars, and Crabs (Age 5)

## The Activities

1. Topic: Comparisons: Book: Anno’s Math Games II by M. Anno, Chapter 2.  This chapter has a number of side-by-side similar pictures, and you have to find the similarities and differences.
2.  Topic: Logic: We did “Boole Says” again — I said something like “Stand up if you are a boy OR are wearing socks.”  This time, the kids each got to come up with a command, and we also did some commands using “not” (“stand up if you are NOT wearing socks”).
3. Topics: Patterns, Geometry: We used pattern blocks to draw pictures.  I named 2 or 3 different kinds of shapes (e.g., triangles and squares), and then each kid made a picture using those shapes.  Each time, we each went around and said what we had made.  We did about 5 rounds total.
4. Topics: Scale, Astronomy: We explored The Scale of the Universe 2 web app for a while.  This is a visualization of the universe from the smallest scales (Planck length) to the largest (size of the observable universe).  Starting at human scale, we scrolled both bigger and smaller and discussed what we saw.

## How Did It Go?

This circle went really well, the kids were very interested in all the activities.

#### Anno’s Math Games II

We spent a while finding all the differences in the most complicated of the activities.

#### Boole Says

The kids enjoy anything involving standing up and sitting down over and over :).  They did a pretty good job following the commands, a few of the kids got confused by AND vs. OR sometimes, but the other kids helped them.  When the kids were giving their own commands, I think every single one of the commands included themselves; and several of them ONLY included themselves.  One of the commands was “Everyone wearing flowers stand up”, and one of the kids didn’t realize they had flowers on their tights until everyone else pointed it out.

#### Pattern Block Pictures

The kids enjoyed this activity a lot.  There were a lot of flowers.  One kid’s pictures were noticeably different from the others — for example, one picture was a pretty good crab.  There were a few pictures that the kid didn’t know what it was.

#### The Scale of the Universe

The kids voted to go bigger first.  One of the kids has done a lot of astronomy things in the past, and was able to identify a bunch of the objects, such as the Lunar Lander, Apollo rocket, etc.  Smaller turned out to be less interesting — pretty quickly all you’ve got is particles, which aren’t too exciting.  The kids’ favorite thing was scrolling through the whole thing really fast, which looks like going through a tunnel because the way there are smaller and smaller concentric circles.