Sorting and Unsorting (Age 7)

The Activities

1. Topic: Units. Book: Dinosaur Deals by Murphy. In this book, a boy wants to get a T-Rex trading card. He finds a girl who will trade it for 3 Allosaurus cards, but the boy only has 1 Allosaurus. How can he get the T-Rex.

2. Topic: Sorting, Teamwork. We repeated the sorting activity from a few weeks ago, sorting the cards 1 – 104. This time we started by discussing possible strategies based on what went well last week. Then I timed the kids to see how fast their sort was, so we can try to get faster in the future.

3. Topic: Programming. We played the game Robot Turtles, with a few rules changes to make it more cooperative (in past circles some kids have gotten upset if their turtle falls behind):

• All turtles are trying to get to the same jewel.
• Turtles can walk on top of each other.
• Each person gets only one ‘laser’ card, so sometimes you have to work together to rescue a friend trapped behind ice blocks.

 

How did it go?

Sorting

First we discussed what worked well last time, and what strategies we could use this time. One kid said that sorting goes slowly when people hold cards in their hands (they spend a lot of time rummaging through the cards), so someone proposed laying the cards out on the ground so everyone can see.  Then another kid suggested sorting the cards into groups of 10 at the start (1-9, 10-19, etc), then doing the big sort. I made labels for each group of ten cards, and put them around the table. Then I gave each kid 1/4th of the deck, and started the timer.

Sorting into decades went very smoothly, everyone was working together, and in parallel. There were some mistakes, for example someone misread 72 as 27, but overall progress was quick.

Sorting into piles of 10

Next, two of the kids took the 1-9 pile and started sorting it into the final spot on the ground. The other two kids picked up some random piles. One kid laid out the 100s, 70s, and 90s on the ground, but ended up mixing them all together. The other kid took just the forties, and laid them out in order 40 -49. Then he picked up the 50s and laid them out in order under that 40s.

Sorting the 40s and 50s (soon to be undone by a friend)

Meanwhile the other two kids got up to the 40s. One of them came over and scooped up the row of 40s, completely mixing them up, and then resorted the cards into the final positions. This wasn’t too too slow, but seemed suboptimal 🙂

The sorting really slowed down once we got to the mixed up 80s, 90s, and 100s. Three kids had a bunch of random cards in their hands, and one kid was distractedly counting the already sorted cards.  I pointed out that it seemed slow, and that people were holding cards, so they laid the cards down and eventually finished.

The final time was 16:30, which is not terrible, but definitely can be improved.

Afterward, I asked the kids what went well, and what could have gone better. One kid said it would be better if I didn’t take out some cards. At the start of the activity I randomly pulled out 7 cards from the deck, and told the kids. In practice this speeds up the sort a lot, because they don’t get stuck trying to find one card forever, and just move on, assuming that it must be one of the removed cards.

In this discussion, I demonstrated how one kid had sorted the 40s, and then the work was lost when the friend scooped them up. The kids then suggested picking the cards up in order. We tried this, and found that it was indeed quicker to lay down a sorted pile of ten cards than a shuffled one.

I also pointed out that the beginning was really fast because everyone was able to help at once, but no one had any strong ideas about how to make the full search parallelizable.

Robot Turtles

Most of the kids had played this before. Some groaned for some reason, when they saw it, but everyone seemed excited. There was a bit of extra energy left over from sorting, so this was a wild 10 minutes, but we did finish a couple puzzles. The tricky parts were that kids wanted to move their turtles while laying down their programming cards, and also, they would mix up the two turning directions without noticing. But overall they were much better at this than I expected. Their favorite part was using the lasers to rescue their friends.

 

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).

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.
   

   10 red / 10 blue

   

   11 red / 9 blue
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.

Picking Pasta (Age 7)

The Activities

1. Book: Alice in Pastaland: A Math Adventure by A. Wright.
2. Topic: Combinations: Inspired by Alice in Pastaland, I asked the kids to figure out how many ways there are to choose two different kinds of pasta from ten different choices.
3. Topics: Tangrams, Geometry: We did a number of tangrams from Tangrams: 330 Puzzles by R. Read.
   

   

How Did It Go?

We had four kids this week.

Alice in Pastaland

The kids were very excited to finish reading Alice in Pastaland.  It doesn’t have all that much math content, but it does mention numbers frequently.  It also inspired the next activity…

Choosing Pasta

We’ve investigated this problem before, but this time I wanted them to come up with the general formula.  When I gave them the problem, they had no idea how to proceed, and I’m sure they wouldn’t have gotten anywhere unless I got them started.  I said that a good problem solving strategy is to look at simpler versions of the same problem.

First, I asked how many ways if there were two kinds of pasta; they quickly got one as the answer.  Next I asked if there were three kinds.  They were able to demonstrate all three, using the picture of plates of pasta I drew (above).  Next I asked four kinds of pasta.  After some wronger guesses they settled on five, by finding five different answers.  I then wrote the table in the picture above: 4 columns of 3 rows each, AB/AC/AD, BA/BC/BD, CA/CB/CD, DA/DB/DC.  The highlight of circle was that they noticed the duplicate pairs themselves: AB vs BA.  They didn’t notice that every combination had one and only one match, but when I asked why there were the same number of originals as cross-outs, one of them realized they were in matched pairs.  For 4, they simply counted to 6.  Next we did 5, and one of the kids wrote out the entire table of 20 combinations (including duplicates), counted it, and divided by two.  I asked if there was a faster way to do this, and they saw they could use multiplication.  From here, with just a bit more help, one of the kids was able to answer the full problem.  Out of the four kids, two were pretty involved and (I think) understood the answer at the end, the others probably not.

For a reward, they all got a (very small) prize at the end of circle.

Tangrams

Unlike previous circles, we just did the puzzles straight out of the book.  This is quite a bit harder because they don’t have an outline to put their shapes in, and it’s quite a bit harder to understand the scale of the various parts.

The kids have varying abilities at Tangrams.  One interesting difference is that some of them still have trouble copying a completed Tangram.  They can get the shapes in the right general location but sometimes have problems with exactly orientation or positioning.

I started by giving each kid a different puzzle from the same page.  It turns out that none of them are ready yet to solve a puzzle without help.  So, I switched to having everyone work on the same puzzle.  The thing I tried to teach them is that they should first look for where the two big triangles are.  Some of the kids could solve some of the problems once they knew where the big triangles went.  Working as a group was a pretty good way to teach them strategies for solving tangrams, but the disadvantage is that it’s now a direct race to finish, so one of the kids got frustrated when they were slower than the others.  We ended up doing about six different puzzles as a group.

Self-Portrait in Blocks (Age 7)

The Activities

1. Topic: Arithmetic. Book: Alice in Pastaland, by Wright. This book is a re-telling of Alice in Wonderland with a light math theme. The kids loved the theme of the book, and were happy to work out the various math problems. It’s a longer book, so we only read the first half.
2. Topic: Programming. We brought back our old pen-and-paper programming language.   Here are the programs.
3. Topic: Geometry. We gave each kid a strip of paper with a word on it, like cat or crab or rainbow. The kid then tried to make that object using wooden pattern blocks. When done, the other kids tried to guess what it was.
   

 

How did it go?

We had 4 kids this week. It was a very focused and fun circle. Everyone stayed on task the whole time.

Programming

Three of the four kids immediately remembered how the language worked. They zoomed through the programs quickly, making progress on their own.  There were still a few mistakes, but very few.  The trickiest program we had today had nested loops:

Do 4 times {

Do 2 times {

Print “B”

}

}

One kid figured this out completely independently. Two others got it with help.

The fourth kid was less comfortable with programming. They understood assignment and printing, but still didn’t fully understand the concept of a loop.  For example, which lines should you repeat? What’s inside the loop and outside?  With some one-on-one help they completed five or six programs, but they still need more practice.

Pattern Block Pictures

Everyone loved this activity. The kids quickly made surprisingly good pictures of cats, crabs, tree, flower, star, airplane.  The hardest ones were: pizza, dragon, and car.

We had a few minutes left after we finished all the words I had prepared. Three kids decided to write their own cards for me and the other kid to make.  One of the cards I got said “Corey” (my name).  So I made a lovely self-portrait out of blocks:

 

 

Arranging Angles (Age 7)

The Activities

Corey was gone this week, so two parents graciously volunteered to lead circle this week.

1. Topic: Money:  Sold!  A Math Adventure by N. Zimelman and B. Barnard.
2. Topic: Geometry: This activity explored the fact that the sum of the angles of a triangle is 180 degrees, and a (convex) quadrilateral 360 degrees.  Starting with various triangles and quadrilaterals, the kids cut off the corners and then arranged them next to each other.  They also tried to measure the angles using a protractor.
3. Topic: Building: Using our collection of Magformers, the kids tried to build as tall a tower as possible.

How Did It Go?

There were some challenges this circle.

Adding Angles

The goal was to get them to discover that the sum of the angles was 180 or 360 degrees.  Apparently, this didn’t work out too well, and the kids ended up spending a lot of time cutting without much discovery.  Afterwards, several of them said (repeatedly) that “they didn’t know what the point was.”  This is a very nice activity, so we’re going to try it again later; the parents who led it suggested that it might work better if the shapes were smaller so it took less time to cut them out.

Tower Building

The two parents had a contest with the kids to see who could build a taller tower.  Unfortunately, one of the kids knocked over the parents’ tower partway through.  Tower building with Magformers works pretty well, but at a certain point the towers tend to collapse in on themselves.

Spending a Million Dollars (Age 7)

The Activities

1. Topics: Multiplication, Division: Perfectly Perilous Math, Challenge 3 — if you spent 50 cents every second, how many days does it take to spend 1 million dollars (to nearest day).
2. Topic: Combinations: We continued the pumpkin activity from last week.  Last week, they had made a bunch of pumpkins but hadn’t checked for duplicates.  This week, we tried to come up with strategies for finding all the duplicates.  Also, I started with the simplest version of the problem (circular eyes, nose, and mouth), and then gradually added elements (non-symmetric parts, multiple choices), computing how many different combinations there were each time.
3. Topics: Logic, Measurement: Perfectly Perilous Math, Challenge 4 — if you have one bucket that holds 3 quarts, and one that holds 5, how can you measure exactly 4 quarts using just those two buckets?  We worked as a group to solve this problem using various containers and beads (instead of water).

How Did It Go?

We had all five kids this week. All the activities this week were group activities, which turned out to be a problem. Most of the time, only two or three of the kids were actually working on solving the problem; the others were drawing, making paper airplanes, or otherwise not paying attention. Kid A grabbed and crumpled another Kid B’s paper, and later in circle (unrelatedly) Kid B threw a container of beads at Kid A, which we all had to help pick up. I think another issue was that the problems were all pretty hard; the kids working on them did a good job, but it discouraged the other kids.  One of the kids who drew during the Million Dollars activity said they couldn’t help because they didn’t know how to do multiplication, and then was very attentive for the remaining two problems.

Spend a Million Dollars

One of the kids is a lot better at large multiplication than the others, so that kid did most of the computations.  Another kid helped figure out what the right numbers to multiply were.  The kids figured out how many seconds were in a day, but then they were planning to multiply by 50 (cents) — they didn’t realize that 50 cents was half a dollar.  They also needed help figuring out how to divide 1 million by 43,200 — they know the method of repeated addition/multiplication, but I think their number sense suffers a lot past 1,000.  They got pretty close to getting the right answer, but I had to a help a lot for the final few steps.

Pumpkin Combinations

One of the kids suggested sorting the mouths by orientation, and then by type of mouth.  A different kid decided to implement this, but changed it to have 1 column for each of the combinations of 4 mouths.  Some of the other kids helped sort (there were about 35 pumpkins total), and eventually the pumpkins were all in columns.  There were still 8-9 pumpkins in each column, and they mostly stopped making progress at this point, besides finding a few duplicates ad hoc.  I suggested putting the circle eyes above the triangle eyes, but they didn’t take to this idea.

Then I switched gears a bit and started with a problem with circle eyes, nose, and oval mouth.  This meant there was only one possible pumpkin, which the kids figured out right away.  Then I made the nose triangular.  There were answers of both 2 and 4; the 4 answer was reasonable because they showed how you could put the straight edge in any of the 4 directions (although two of them make pretty weird noses).  I said we should only allow the two directions, and then added triangular eyes.  They got 4 right away.  Then I added a smiley/frowny mouth, and it got much harder.  I got answers of 4, 6, and 8.  6 is an interesting answer, because you could think of that if you considered varying each dimension one at a time.  I had the kids try to write out all the combinations — I really should have had them to all 4 for the previous problem first, because the key insight here is that if you group the faces by smiling vs. frowning, you have the same set of 4 faces from the round mouth case, once for each mouth.  In the end, one kid was able to figure out that there should be 32 different answers for the previous week’s problem, but wasn’t able to figure out which faces were still missing.

Tricky Buckets

The beads worked okay, although one of our containers was more like 2.6 than 3, so the measurements didn’t quite come out right.  Also, we ended up picking up a bunch of beads because of the bead-throwing incident.  One of the kids figured out a solution right away using a third bucket; and then they were able to come up with the 2-bucket solution with only a bit of help.  I asked a follow-up question about a 5 and 7 bucket (trying to make each of 1, 2, 3); the kid who had done the best on the first part was able to make progress and solved 2 and 3 but not 1.

Zombies and Torture Chambers (Age 7)

The Activities

1. Topic: Multiplication, Story Problems. Book: The Book of Perfectly Perilous Math by Connelly. Chapter 1: The Pit and the Pendulum. In this chapter you are tied to a table as a sharp pendulum swings back and forth, lowering with each swing. Meanwhile a rat is one minute away from gnawing through the ropes that hold you. Will you escape before the pendulum kills you?
2. Topic: Pendulums. Each kid had a spoon pendulum. We experimented with how many times the pendulum would swing back and forth in a certain amount of time.
   1. All 5 pendulums were the same length, but dropped from different heights.
   2. 5 different length pendulum.
   3. One pendulum that we all counted together, with different length strings.
      

      Spoon pendulum.

3. Topic: Logic, Story Problems. Chapter 13: The Rope Bridge, from the Perfectly Perilous book. Zombies are twenty minutes away from you. You and four friends have to cross a rope bridge. Two people can go at a time, and  you have to share one flashlight. Each person takes a different amount of time to cross: 1, 2, 3, and 8 minutes. Can you all cross in time?
   

   Our picture of the problem, complete with a Minecraft zombie.

4. Topic: Combinations. How many different jack-o-lanterns can you make with a set of eyes, nose, and mouth?  There are 2 different eyes, 1 nose, and 2 mouths. Each feature can be right side up or upside down.
   

   Our messy pumpkin patch.

How did it go?

We had all 5 kids this week. Halloween was yesterday so we had spooky, Halloween themed activities.

The Pit and the Pendulum

The kids instantly loved this book. They wanted to look at all the chapters at once, and were excited (and slightly scared) by the short stories.  The book says the pendulum swings back and forth every 7 seconds, and drops one inch per second. The pendulum is 15 inches above you. How long till it hits you? The kids quickly figured out that the answer would be 15 X 7. One girl computed this using long-hand multiplication. I had the other kids check her work by computing: 10*7 + 5*7, which they could all do.

So the pendulum would drop in 105 seconds, but the rat will untie you in 60 seconds, leaving 45 seconds to escape.

Pendulum Spoons

I gave each kids a pendulum spoon. I wanted each pendulum to be 12 inches long so I had a measuring tape. All the kids wanted to measure their own pendulum, which slowed us down.  Eventually, they were all set. Everyone released their spoon at the same time, and counted their swings for 20 seconds. Three kids got 20 swings, and two got 18 swings, so it was pretty close.

Next each kid chose a different length of pendulum. We had some more trouble because at first everyone wanted a long pendulum, and then as soon as I convinced a couple kids to use a short one, then everyone wanted a short one. Finally I got 2 kids with long strings, and 3 with short ones. We tried to time them for 20 more seconds, but spoons were hitting each other, and everyone was laughing and forgetting to count.

I corralled everyone together again, and we used just one pendulum and counted it together, with a long, medium and short string. We did the long string twice to make sure our result was repeatable. We found that in 10 seconds, the long pendulum swung 6 times, the medium 8 times, and the short 12 times.

I asked why that might be. One girl suggested that different parts of the rope may be pulling on the longer pendulum. My daughter suggested that the longer pendulum travels further on each swing.

The Rope Bridge

Again, the kids loved the themes in this book. They all wanted to draw and talk about zombies, and also to solve the problem to see if the kids lived. Two kids quickly came up with the same idea: the fastest person should go with a friend across the bridge, and then run back with the flashlight to get the next person.

Everyone liked that plan, but one girl was worried it would be too slow. So, we worked it out together. First the 1 minute and the 8 minute person crossed. There were some interesting ideas about how long it would take them to get across. Someone said it would take 9 minutes. Someone else said it should take 1 minute, because the fast person could carry the slow one on their back. We finally decided it should take 8 minutes, becuase the 1 minute would have to slow down to the speed of the slower person.

We worked out it should take: 8 + 1 + 3 + 1 + 2 = 15 minutes to cross, which gave us one extra minute before the helicopter arrived to rescue us from the zombies.

Pumpkin Combinations

My daughter was really excited to do this one, because the older circle doesn’t work with pictures and glue sticks very often any more. This time, I decided not to sort the pumpkins for the kids. I would make them decide if there were any duplicates.

At first everyone just made pumpkins, but it quickly became difficult to tell if the pumpkin was new. A couple kids volunteered to sort the pumpkins. One kid wanted to sort them first by whether the nose was up or down. Another kid wanted to make rows for each kind of eye.

We ended up making rows for each eye, but then the kids couldn’t figure out how to sort within each row. My daughter wanted to make the columns have the same nose and mouth, but many pumpkins were missing, and there were many duplicates which complicated it.

Also, having 3 kids try to sort at once, without a clear strategy was pretty chaotic. The first kid independently tried to come up with a strategy by drawing different types of pumpkins on her paper. The second kid drifted between the sorting and the pumpkin making without really being engaged. The third made pumpkin after pumpkin without checking if there were duplicates. The fourth delivered pumpkins to the floor where we were sorting.  My daughter and I tried somewhat unsuccessfully to sort the pumpkins together. Then we ran out of time.

So…we don’t know how many different pumpkins we made!  Follow up work for next time.

Lots of Bunnies (Age 7)

The Activities

1. Topics: Fibonacci, Sequences: Book: The Number Devil by H. Enzensberger, night 6, which covers Fibonacci numbers.  We explored a number of the properties of Fibonacci numbers (such as that fact that the squares of two consecutive Fibonacci numbers is also a Fibonacci number), and drew “Bunny diagrams” to see how Fibonacci could arise in nature.
2. Topics: Graphs, Coordinates: The kids each did another worksheet from worksheetworks.com.  This time the coordinates included negative values.
3. Topics: Games, Addition: We played Clumsy Thief, which involves finding pairs of numbers that add to $100.

How Did It Go?

We had only 3 kids this week.

Fibonacci

The kids were able to write out the Fibonacci sequence until about 1000.  The kids varied in how interested they were in the various properties; one kid was “bored”.  One of the kids really understood the bunnies diagram and did quite a few rows.

Coordinate Pictures

As usual, the kids enjoyed doing the pictures.  One of them had brought back their homework from a previous week (another coordinate picture) and got a prize.  The negative coordinates confused them for a while, but still the actual difficulty is knowing whether the first coordinate is horizontal or vertical.  By the end they were doing pretty well, and they all finished one picture.

Clumsy Thief

Two of the kids liked the game, the other wasn’t so sure; I think this corresponded roughly with how quickly they were able to find the pairs that added up to $100.

Losing at My Own Game (Age 7)

The Activities

1. Topic: Triangular Numbers.  Book: The Number Devil by Enzenberger, Chapter: “The Fifth Night”, pages 89 – 101.
2. Topic: Triangular Numbers. Compute the 100th triangular number.
3. Topic: Big Number Guessing. Think of a number between 1 and 1,000,000, how many guesses does it take to get it.

How Did it Go?

Number Devil

My daughter whined when she heard we were reading this book, but a couple other kids cheered, so it’s a mixed review.  This chapter was about triangular numbers.  Here is wikipedia’s picture of the first six triangular numbers: 

In the book, Robert learned that he could compute the 12 triangular number (1 + 2 + 3 + 4 + …+ 12) as:

1	2	3	4	5	6
12	11	10	9	8	7

Each column in this table adds up to 13, so the 12 triangular number is 13 * 6 = 78.

The kids all enjoyed seeing the pattern of how to compute the next triangular number from the previous (i.e. the 12 triangular number is the 11th + 12 more).

The 100th Triangular Number

After the book, I asked the kids what they thought the 100th triangular number was. The kids said the 10th triangular number was 55, so the 100th should be 55 * 10 = 550.  I said we should compute it similarly to the the way they figured out the 12th in the book.

The kids then suggested I should group the numbers into sets that add up to 13.  Instead I said we should pair up the smallest and the largest numbers.

1 with 100, 2 with 99, etc.  I asked how many pairs of numbers we would have? The kids at first said 100, or said they couldn’t figure it out.  Then I showed how there had been 6 pairs when there were 12 numbers, and one girl said “Oh it should be 50!” and explained that she divided 100 by 2.

Next we checked how much each pair added up to: 101.  So the 100th triangular number should be 101 * 50.  One girl tried to compute this long-hand, but got immediately frustrated because she had only done 2 digits times one digit in school. She felt that this would be impossible until her school taught it.

I suggested first calculating 100 * 50. Eventually someone said that was 5000.  But we were supposed to compute 101 * 50, so how many more 50s do we need to add? My daughter said one more, so the 100th is 5050.

Big Number Guessing

First we warmed by up having the kids guess my number between and 1 and 20. This was easy. Then I had one kid write down a number between 1 and 1,000,000. He chose 456, 276.  The other kids thought it would take 800 guesses to find his number.  I gave them 25.  They got down to 456,267…456,299 before they ran out of guesses.  This was definitely NOT easy for them to think of a number between 461000 and 447,000.

Next I claimed that I could guess their number between one and one million in just 25 guesses. They wrote down a secret number, and I started guessing, doing binary search.

It went well at first, but I wasted a few guesses by getting mixed up with whether they had said higher or lower. Then when I had 10 guesses left, I got mixed up again and wasted about 6 guesses going the wrong direction.  I hit 25 guesses and did NOT get their number.  My daughter kindly gave me two extra guesses, and I got it, but I totally lost.  The kids enjoyed that 🙂

This was a good activity because it works on their sense of big numbers. But it was hard enough that some kids (including my daughter), didn’t really want to participate, so I had 2 – 3 very interested kids and a couple more who were busy drawing.