# 30 Different Ways to Say “I Love You” (Age 7)

## The Activities

1. Topic: Measurement: Book: Taro Gomi’s Playful Puzzles for Little Hands.  We only did a few puzzles towards the end of the book, most of them involved measurement and were pretty hard!
2. Topics: Geometry, Graphs: I made a set of Valentine’s Day themed arrow direction drawings, downloadable here.  The rules are, using graph paper (ideally with fairly small squares), you start at a vertex and have one of 8 directions and a distance.  I introduced something new this time, which is some of the instructions were in red, which means you moved your pencil but didn’t draw a line.
3. Topics: Counting, Graphs: I gave each kid a box of the kind of candy hearts that have messages like “Be Mine” or “Sweet On You” printed on them.  Each kid sorted their box by heart color, and then we made a combined graph with how many there were of each color.  Then we found as many distinct hearts (message + color) as we could.

## How Did It Go?

We had all five kids this week.  It was a high energy circle, partly because of candy and partly because four of the five kids had just been to Cirque du Soleil.  We spent five minutes at the beginning of circle so each kid who had been to the circus could say their favorite part, and then we got through the rest of circle without any mention of the circus!

#### Taro Gomi

One of the problems asked which of a bunch of hats was the shortest and tallest — we tried to find some kind of trick (e.g., number of stripes), but in the end all we could figure out was the measure.  Similarly, the next page had two different colored poles cut into pieces and asked which pole (when put together) was longest, which seemed really hard as well.

#### Arrow Drawings

The kids did pretty well on these, but there was a pretty big spread in ability.  Most of the kids made a small mistake from time to time, usually either going the wrong distance or not doing a diagonal at 45 degrees.  One kid was noticeably better, going faster and without mistakes.  I was worried the red instructions (pick up your pencil) would be confusing, but they understood it easily.

#### Candy Hearts

I was originally planning to have them sort by message and make a graph that way, but when we opened the boxes, it turned out that the printing quality on the hearts is really bad — probably at least 1/3rd of them have missing or unreadable messages.  Also, it turned out there are a TON of different messages (“Be Happy”, “Nuts 4 U”, …) — we counted 30 different ones — which would have made it hard to make a graph.  So we did color instead.  And then there was another surprise — there were FAR more oranges than anything else — 3 times as many as most of the other colors!  And it was consistent across boxes as well.  Seems like a pretty solid result that I’d expect to hold up across many boxes.  The kids were pretty excited to find all the different messages and laughed every time we found a new one.  The kids were also REALLY excited to eat some of the hearts, but as far as I know they listened to me and didn’t eat any until the end (they got three each).

# A Bag Full of Dice (Age 9)

## The Activities

1. Topics: Geometry, Three Dimensional Shapes: Book:  Sir Cumference and the Sword in the Cone by C. Neuschwander.
2. Topics: Geometry, Three Dimensional Shapes:  A while ago we bought 5 full sets of “D&D dice” (4, 6, 8, 10, 12, and 20 sided).  We counted the edges, faces, and vertices for each of these and made a chart like in Sir Cumference, showing that “Faces + Vertices – Edges = 2”.  I also pointed out the dual relationship between 6 & 8 and 12 & 20 sided polyhedra (i.e., 6-sided has 6 faces, 8 vertices, and 12 edges; 8-sided has 8 faces, 6 vertices, and 12 edges; you can switch between the two by putting a vertex in the middle of each face and connecting adjacent vertices).
3. Topic: Numbers: We did What’s the Secret Code? from youcubed.org.  There are some clues about what the secret number is like “The digit in the hundreds place is ¾ the digit in the thousands place.”  There is more than one answer which is cool.
4. Topics: Origami, Geometry: We did Paper Folding from youcubed.org.  There are a number of folding challenges like “Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.”

## How Did It Go?

We had four kids this week.  As usual some kids followed along better than others, but most people were engaged for both the dice activity and the paper folding.

#### Sir Cumference and the Sword in the Cone

The kids liked the book, they laughed at quite a few of the math puns.

#### Euler’s Polyhedron Formula

The kids definitely enjoyed making the chart.  They did a pretty good job staying on task (it was easy to get distracted and start rolling the dice).  Counting the edges on some of the dice was fairly tricky but was much easier with good grouping strategies.

#### What’s the Secret Code?

The kids did well on this except that they had trouble with the decimals.  They did find one of the decimal answers, because they knew that .5 = 1/2, but I believe there were other possible decimal answers as well.

#### Paper Folding

The kids solved all the tasks except the last one, which was making a non-diagonal 1/2 area square.  I figured out a pretty complicated way to do it (by transferring the side length of the diagonal answer onto a horizontal edge), they copied what I did but it was pretty tricky (see picture above).

# Odds & Ends (Age 7)

## The Activities

1. Topic: Probability: Book: A Very Improbable Story by E. Einhorn.
2. Topic: Probability:  First, I secretly put 2 red and 8 blue stones into a small drawstring bag.  Each kid took turns pulling one stone out, looking at it, and then putting it back.  The question was, are there more reds or blues?  I repeated it with 4 red / 6 blue, and also 5 red / 5 blue.  Finally, I made two bags, one with 10 red / 10 blue, and the other with 11 red / 9 blue, divided the kids into two teams, and asked them to figure out which bag had more reds.  I gave the kids paper and pencil and they decided to make charts to keep track of the results.
3. Topics: Numbers, Sorting:  I had about 20 different numbers on squares of paper, 0, 1, 3, 4, 6, 8, 12, 13, 100, 105, 1001, 1052, 1053, 1000000, -5, and -100.  First, I handed each kid one number and asked them to sort themselves.  We did this several times, starting simple and then using some of the trickier numbers.  Then, instead of handing them the numbers, I taped a number to each kids’ back, and without telling each other what the numbers were, they needed to sort themselves.  We did this a few times as well.
4. Topics: Tangrams, Geometry:  I gave each kid six different tangram puzzles.  For the kids who finished earlier, I had them work on the letter “A” from Tangrams: 330 Puzzles.

## How Did It Go?

We had four kids this week.  It was a good circle, a few of the kids got a little antsy when we were discussing the results of the bag counting, but otherwise they were all engaged the whole time.

#### A Very Improbable Story

The kids liked the cat on the head :).

#### Probability Bag

The kids immediately grasped the idea of looking for whichever color came out more often.  Not surprisingly, they were overconfident — once, after only 3 draws one kid concluded red was the winner and dumped out the bag, only to find out that there were 5 of each.

For the team activity, one of the teams delegated one person to pull the stones and the other to record, while the other was taking turns drawing out stones.  The former strategy was about 2x faster, so I suggested the other team use it as well.  It was very interesting to see the two charts (pictured above).  One was a standard tally chart, except with 6 instead of 5 in each group.  For the other, the kid started by writing a bunch of numbers, and then checking them off as stones were pulled out of the bag.  The results came out pretty nicely — exactly 50% for the 10/10 bag, and 55.6% for the 11/9 bag (expected 55%).  However, the kids were a bit confused by the fact that team 1 had counts of 15 red and 15 blue vs. 30 red and 24 blue for team 2 — at one point, one kid concluded that team 2 had more reds AND blues.  In fact, the only way I got them to conclude that team 2 had more reds was to ask them to guess what was in each bag.  Their guess for team 1 was 10/10, while their guess for team 2 was “6 more reds than blues” (not coincidentally, they had drawn red out 6 times more than blue).  I asked them how many reds there would be if there were 6 more reds than blues, and 20 total — this was actually quite hard for them and I had to help them a lot (the initial guess, 16, didn’t work).  Of course, 13/7 doesn’t match their observed results.  So, there’s clearly a lot more them to learn for the fine shades of probability!

#### Number Sorting

This activity was pretty easy for them, even with the numbers taped to their backs.  They had a lot of fun, particularly when I gave them negative numbers or really big numbers.  They did a great job not telling each other — the closest they came was saying one kid’s number was really low (when it was -100).

#### Tangrams

This group has done these puzzles before, but that wasn’t an issue, they didn’t remember the solutions.  They were better than last time, but the puzzles still definitely weren’t trivial.  The bonus puzzle is much harder because it wasn’t to scale, but they made a good effort and made progress.

# A Trick-or-Treat Circle (Age 8)

## The Activities

1. Topics: Proofs, Time, Logic:  I asked the kids to determine whether or not every year has at least one Friday the 13th.
3. Topics: Combinations, Combinatorics, Logic:  I had a list of 10 possible trick-or-treaters:
1. Evil Queen — Baddie, Girl
2. Bride of Frankenstein — Baddie, Girl
5. Princess — Goodie, Girl
6. Fairy — Goodie, Girl
7. Wizard — Goodie, Boy
8. King — Goodie, Boy
9. Alien — Neither, Neither
10. Slime — Neither, Neither

First, I asked how many ways there were to pick three trick-or-treaters.  Then I asked how many ways to pick three trick-or-treaters, with the requirement that there’s at least one Baddie, one Goodie, one Boy, and one Girl.  Note: Picking groups is much harder than picking ordered line-ups (where Evil Queen, Princess is different from Princess, Evil Queen).  If I were doing this again I would stick with ordered line-ups, it’s hard enough already.

## How Did It Go?

We had all five kids this week.  This was a pretty hard circle; 2 of the kids were engaged through-out, with one saying how they liked the hard problems; the other 3 were distracted a lot of the time.

#### Friday the 13th

This is a pretty tricky problem, it’s not immediately obvious how to do it even for adults.  The kids made some good progress and had some interesting ideas.  First, one kid figured out that for there to be a Friday the 13th, the 1st had to be a Sunday.  Another kid wrote down the years starting with 2000 (she wanted to check “all the years”).  I used my phone to look up the calendars for each year, and we checked which months had a Friday the 13th each year.  One kid was really excited to try to find a year with no Friday the 13th, because then they’d be done.  But there is indeed a Friday the 13th each year, so we didn’t find one :).  At this point, I gave them a hint, which is to draw a pie chart like in the picture above.  The idea is to go through an entire year starting with January, assume that the 13th in January is, say, a Sunday, and then figure out what day of the week the 13th is in each month.  If you do this, you’ll find that every single piece of the pie is filled, which is what you need to prove that there’s always a Friday the 13th.  Unfortunately, the kids were not good at doing the calendar arithmetic to figure out what day of the week Feb 13 is given the day of the week for Jan 13.  So, we didn’t get that far, and since we had already spent 25 minutes I moved on to the next activity.  Most of the time, two of the kids were working on the problem while the others were drawing, etc.

#### Trick-or-Treat Optimization

The kids liked the theme of optimizing trick-or-treating.  Unfortunately, I made an error in how I set up the problem.  My intention was that they should concentrate on how many blocks you’d have to walk, but I drew the houses big enough that they focused on visiting houses instead of walking along blocks.  The map I included above I redid afterwards to make it clearer that it’s about blocks, not houses.  The problem with houses is that if you have houses on the corners of streets, it makes the counting a lot messier.  And counting houses is a bit more intuitive, so that’s what they defaulted to.  The result of this was that about half the kids thought I meant that it took 10 minutes to visit three houses, when I actually meant it took 10 minutes to walk one block.  All the kids paid attention during this activity.

The kids figured out that you’d have to backtrack or at least revisit some blocks.  They were all pretty comfortable with figuring out how long it would take to visit all the blocks, but the idea of the best route wasn’t as compelling.  They did understand the idea of visiting as many as possible in 3 hours.  The final problem, about returning home each time, isn’t actually that interesting with the map I had, but they still had to think about it some to figure out how to do it.

#### Picking Trick-or-Treaters

This problem turned out to be harder than I expected.  I just forgot that they weren’t that comfortable with combinations yet.  Even if I had done the ordered line version, they still didn’t immediately remember how to do the multiplication to figure out the answer to the unconstrained version.  They did figure out this part, and we moved on to the constrained version.

I actually gave them a four person version that required 2 baddies instead of 1 — it turns out to be a lot harder than the three person version.  Also, the non-ordered version is a lot harder to think about.  With the three person version, it’s not so bad to reason along the lines of “Let’s pick the baddy first, and the goody second.  For each of the possible combinations (there’s only 4 distinct ones), we can figure out what the third person can be.”  The four-person version gets a lot more complicated, so I switched to the three-person version — we made some progress but didn’t solve it.

Again, two of the kids worked hard, while the other three were distracted.

# Impossible Flips (Age 6)

## The Activities

1. Topic: Puzzles:  Book: Taro Gomi’s Playful Puzzles for Little Hands.  Still haven’t quite finished, this time it was mostly mazes.
2. Topics: Number Line, Number Recognition.  We revisited higher/lower number guessing again, mostly from 1-100.  As usual, the theme of the game was a bear who wants to steal our picnic food.  But the bear print-out was missing so kids took turns standing next to the wall and using their finger as the bear.  I did a few numbers, and then each kid took turns thinking of their own number.  At the end, we had a discussion about what makes a good or bad guess, and then I did one from 1-1000.
3. Topics: Combinatorics, Geometry:  Using wooden pattern blocks, find as many ways as possible to make a 2×2 diamond.
4.  Topic: Logic:  We did the Seven Flipped activity from youcubed.org.  Starting with 7 shapes face-down (we used Scrabble tiles), you could flip 3 tiles at a time.  The goal is to flip all the tiles face up.  After they solved that, I switched to 7 tiles, flip 4 at a time (which is impossible) and then 5 tiles flip 2 (also impossible), and we discussed why it might be impossible.

## How Did It Go?

We had all five kids this week.

#### Number Guessing

There is a very wide range of abilities in this game.  By the end, three of the kids completely understood how the game worked, and during the discussion two of them worked together to figure out that they should guess half-way in between each time.  One of the other kids usually made proper guesses, but the final kid frequently made guesses outside of the current range (even when they were just reminded of what the current range was).  I also made a couple “illegal” guesses when I was playing, but was called out on it.  1-1000 is still pretty challenging even for the kids that get it.

#### Diamond Variants

The kids weren’t as in to this activity as I expected.  A couple of them went off task pretty quickly, building whatever they felt like.  One kid tried hard to use the skinny white diamonds, which doesn’t work.  Another kid was trying but kept building diamonds that were 3 units on the side.  One kid tried for a while to use squares, without success, but then eventually figured out a key insight for building different diamonds, which is that you can swap two adjacent triangles for a diamond, or vice-versa.  So that kid generated more than half of the variants we found.

#### Seven Flipped

The kids each had their own set of tiles.  There was lots of cheating, but it didn’t matter because I would just ask them to show me again.  At first the kids decided it wasn’t possible, but after a few minutes one of the kids figured it out.  Another kid watched them demonstrate, and then the two of them taught the other three.  Then I switched to 7/4.  There were lots and lots of claims of having done it, but it’s impossible :).  After a while, I asked them to try 5/2 instead.  A couple of the kids started to get the idea that it was impossible.  I myself made a bunch of moves on this problem with the kids watching, and we kept track of how many were face up.  With a hint the kids noticed is was only 0, 2, and 4.  I made a set of maybe 11 tiles with 6 flipped up, and then showed them all the possible moves (2 down -> 2 up, 2 up -> 2 down, and 1 up, 1 down -> 1 down, 1 up), and they saw that it could only be +2, -2, or 0.  One or two of the kids might have understood this proof that 5/2 is impossible.

# Unifix Estimating (Age 6)

## The Activities

1. Topic: Estimating. Book: Betcha! by Murphy. Two friends walk around town estimating the number of people, cars, and jelly beans they see.
2. Topic: Estimating, Counting. Predict how many Unifix cubes can fit in a small bowl. How many Unifix cubes tall are you? How many Unifix cubes tall am I?
3. Topic: Logic. A little boy rides the elevator alone to and from his 15th floor apartment. Whenever he goes down, he goes all the way down to floor one. Whenever he goes up, he takes the elevator up to the 7th floor, then the stairs up to the 15th. Why?
4. Topic: Geometry. How many rectangles are in various pictures? How many triangles?
5. Topic: Spatial Reasoning.  Cover a checkerboard with rectangular tiles that are two squares long. Are some boards impossible to cover? Why?

## How did it go?

This week we had four kids, after a couple weeks with just two kids per circle. The kids were all interested in the activity and stayed on task really well.

#### Unifix Estimating

First we each guessed how many cubes would fit in a cup. Then each kid tried to get as many as possible inside.

The guesses ranged from four to eight. At first everyone fit 9 in their cup (with the lid sealed). But I managed to fit 11 in.  After a lot of trying my son managed to squish in 12 cubes, much to his excitement.

12 cubes!!

Next we guessed how many cubes tall each kid was. We estimated by hold a stick of 10 cubes up to the kid’s body. A taller kid then decided to estimate his height by adding a few to the other kid’s height. The guesses were around 59 – 64 cubes. It was quite challenging to stick together that many unifix cubes, but the kids all stuck with it, and ended up with ~68 cubes per kid. We then guessed that I must be 100 cubes tall. I laid on the floor while kids made a very long unifix pole, and when we counted, it was 90 cubes long

#### The Boy in the Elevator

I got this story from Math from Age Three to Seven by Zvonkin. A little boy rides the elevator alone. When he goes down from the 15th floor, he goes all the way to the bottom. But when he goes up, he only goes to the 7th floor then walks up the stairs the rest of the way. Why?

The first suggestions were that maybe he wants exercise. Or maybe he doesn’t like the other buttons. At that suggestion, I drew them the buttons to see what they looked like:

I taped them up to the wall. No one had much to say about this, but then I asked one kid what would happen if her little brother pressed the buttons? She said he may be too short. Then another kid suggested maybe the boy was too short to reach the 15, and could only reach up to the 7. And on the way down, he can reach the 1 button easily.

#### Counting Shapes

In this activity, I showed the kids pictures of shapes I had drawn and we tried to find all triangles or rectangles in the picture.

At first the kids only see four rectangles in a picture like this. But after some looking, they noticed the big rectangle around the outside edge. Then later they noticed the long thin rectanble highlighted in green, and lastly the squareish rectangle in black. All the kids enjoyed this activity.

#### Tiling Checkerboards

I gave the kids a bunch of tiles that each would cover two squares on a checkerboard. Then I gave them increasingly interesting checkerboards to try to cover.

First they got a 4×4 checkboard which everyone easily covered.

Next was a 5×5 board:

Notice that one square is uncovered. The kids spent several minutes trying to rearrange the tiles to cover the last square. Eventually I suggested that maybe it’s impossible? If so, can you explain why? One kid suggested the tile is the wrong shape. Or maybe you should be allowed to let the square hang off the edge of the checkerboard?

Eventually, my son counted the squares on the board (5 on top, 5 down the side => 25 squares) and he said: “it’s impossible! 25 is odd, and the tiles can only cover an even number”. We checked it out with the other kids and eventually they were convinced.

My son said it should be possible because there’s an even number. But no one could do it. A couple kids suggested they would need to put the squares diagonally. I asked about the color of the remaining squares? We noticed it was always two white squares left. I asked if one tile can ever cover two white squares? The kids tried it and said no, but were not fully convinced.

The final board

This was the last board. Everyone immediately said it was impossible. One kid pointed out it would be possible if you could overlap the pieces, but no one had a clear explanation of how they were sure it was impossible otherwise.

# Should Have Done Something About Pokemon

## The Activities

1. Topics: Logic, Puzzles:  Book: Still More Stories to Solve by G. Shannon, stories 6-8.
2. Topics: Optical Illusions, Geometry:  We did several activities from The Usborne Optical Illusions Activity Book, by S. Taplin.  The first activity was about coloring a diamond grid — the well-known illusion about two different ways to see a pattern of cubes.  The second activity involved a pattern with several rows of arrows, odd rows point left and even rows pointing right — once colored, it can either look like, say, red arrows pointing right on a blue background, or blue arrows pointing left on a red background.  The third activity was a circle of dots which when connected in the specified way generated a circular hole in the middle in the shape of a circle.  I extended this activity by showing the kids how to draw a line drawing of a star: Draw a cross on a sheet of graph paper, and then draw a line from (0, X) to (12 – X, 0) for all X.

## How Did It Go?

We had all five kids this week.  I realized 15 minutes before circle we should have done an activity involving Pokemon, and indeed there was a lot of talk about Pokemon Go during circle while the kids were coloring.  We’ll definitely do something about Pokemon soon.

#### Still More Stories to Solve

The first puzzle was a variant of “This sentence is a lie.” — awfully hard for an 8-year-old to guess.  The second was about a king saying “Don’t do X until you see my face” (meaning, “until we meet again”) and then someone sees the king’s face on a coin so they do it earlier.  With some clues the kids realized that people’s faces were on money, and then they figured out the answer exactly.  The last one also went pretty well, they needed a lot of hints but they figured it out.

#### Optical Illusions

The first two only went ok, the kids were fine coloring the pictures, but then they weren’t impressed by the illusion at all.  For the cube one, I’m not sure if they were actually seeing it both ways, or if they were just uninterested; it’s very hard to tell the difference.  Most kids said something similar for the arrows: they said “They go both ways.”  One kid was quite sure that they were blue arrows going left, because the top and bottom row were red, so red looked like the background.  I added a row of blue arrows to the top, and then they said that the arrows didn’t go either way.

The third activity (circle of lines) was fine, not too hard and a nice-looking result — but still not that much excitement.  However, the star-shaped pattern was much more interesting to them.  It was tricky to do correctly — many of the kids repeatedly forgot to move one of the endpoints of the line.  In the end, all of the kids asked for me to make another cross on grid paper so they could take it home and try it again.  At first, I was doing the wrong thing — connecting (0, 12) to (1, 0) when I should have connected (0, 11) to (1, 0).  One very interesting thing about this activity is that there are several closely connected curves.  Besides the one we did, there’s also one where the length of the line you draw is constant (draw all possible lines of length 12 connecting the X axis to the Y axis), and there’s also connecting (0, X) to (1/X, 0), which makes a hyperbola.