Towers for Treasure

The Activities

  1. Topic: Measurement: Book: Measuring Penny by Leedy.
  2. Topic: Combinatorics: Given a chain of 5 circles, how many different ways can you color in 0 circles? 1 circle? 2 circles? How can you know if you’ve found them all? Do you notice any patterns? Here is the Powerpoint file for the circle coloring chart.

    The wall chart with all the different ways to color 0 – 5 circles out of 5.

  3. Topic: Engineering, Building: Build the tallest tower you can out of 20 Keva blocks.

    Working on the 20-block tower.

How did it go?

We had all 6 kids at circle again this week. Overall, it was a really successful circle…Most of the kids were engaged most of the time.  All the kids especially enjoyed the building competition.

Measuring Penny

This is a book about a girl measuring different aspects of her dog Penny and other dogs. She uses lots of different units (inches, centimeters, # of dog biscuits), and dimensions (height, nose length, how high the dog can jump, weight, etc.).  The kids really loved this book. It combined story-telling and math in a really nice way.

They especially loved when the book discussed how much money Penny was worth. $1000 as a burglar alarm, $20/day in entertainment, and 1 million dollars in love.  Several kids commented afterward about how much they had enjoyed the book.

Circle Combinatorics

We did this activity a 1.5 years ago, but I wanted to see if they could go a bit deeper this time.

First I asked how many different ways there are to color in zero circles.  All the kids saw there was only one way…not coloring anything. So on the wall chart I wrote 0 at the top, and 1 at the bottom for the number of ways.

Then we moved on to coloring in 1 circle out of 5.  Kid #1 and Kid #2 immediately said there were 5 ways. I let Kid #1 color the first row on the wall chart…She colored in the 1st circle. Then I asked Kid #3 if she wanted to do the next, but she wasn’t sure what to do yet, so I asked Kids #4 and #5, but they also didn’t want to answer. At that point, Kid #2 volunteered so I had her go. She colored in the 2nd circle.

I checked Kid #6 paper, and she had a new way, so she colored in the 3rd circle.  Then Kid #3 and #5 caught on and added a row with just the 4th circle colored, and another with 5th circle colored.  Then the kids said that was all.  I asked how they were sure, and they explained that they had colored in each circle, so there was no other way.  I said, ok, and wrote 1 (for the number of circles) at the top, and 5 (for the number of ways we found) at the bottom, on the wall chart.

Next was much harder: color in 2 out of 5 circles. The kids started doing this on their own paper, and if they found a new way, they would stand in the line to add it to the wallchart (after I checked it).  At this point most of the kids seemed to understand the activity, and were excited to contribute. For the quieter kids, I tried to explicitly check if their paper had a new coloring, since they were less likely to yell for me to check.

After awhile we had 10 ways (which is all of them).  I asked the kids if they thought we were done. They weren’t sure.  I asked if there was any way to check…did they see any patterns here, the way they did with one circle colored in?  Someone said that you could check if the 1st and 3rd were colored, and the 1st and 5th.  I said it was a good idea and structured it a bit differently so we checked if we had 1st and 2nd, 1st and 3rd, 1st and 4th, 1st and 5th.  Three of the kids definitely understood this.

Next I said we had checked all the ways that had the 1st box colored, so we should check the ways that have the 2nd box colored.  The kids then suggested checking 2nd and 3rd, 2nd and 4th, 2nd and 5th.  We continued on with this pattern until we verified that we had all possible colorings. So I wrote 2 at the top of the chart, and 10 at the bottom.

Next we colored in 3 circles at a time.  By this point, the kids really wanted to add to the wallchart, so they would show me as soon as they had a new coloring.  After we got 10 different 3s, I suggested we should check if we had all ways.  We tried to do this, but it was tricky to describe having the 1st 2nd 3rd colored in.  After a bit of trying, I wrote 3 at the top, and 10 at the bottom.

For coloring in 4 circles, the kids did not immediately see that there were 5 ways to do it. However, they quickly came up with all 5 ways.  I asked how we could tell if we were done, and someone said that in each row only one circle was empty, and we had all 5 choices for the empty circle.  I wrote 4 at the top, and 5 at the bottom.

For 5 circles, everyone knew there was only one way.

Now I asked if anyone saw any patterns.  Kid #1 was really excited to see 1 5 10 10 5 1 for the number of ways to color.  I asked the kids why there was 1 way to color 0 and one way to color 5.  One kid explained that either you must color all or none.  I said that was a good point, it was like 0 and 5 were opposites of each other.

Next I asked about 1 and 4.  Are they opposites of each other?  Kid #1 and #2 were very excited about this, and pointed out that with 1 circle, there was always one that was colored, while for 4 circles there was always one that was not colored.  Then I spent some time asking different kids if they could find the corresponding rows between 1 colored circle and 4 colored circles.  All the kids were eventually able to find the “opposite” row in each chart.  I saw several nice “A-ha!” moments where a kid really understood how it was working.

Next I said we should check if 2 & 3 circles also had opposites.  We found that they are opposites. Every kid wanted a chance for me to point to a row in the 3 circle chart, and them to find the corresponding opposite in the 2 circle chart.

Keva Towers

I started by telling the kids that everyone would get a treasure at the end of this activity. There was tons of screaming and yelling and excitement.  Then I said they would pick the treasure in the order they finished in the activity.  Each kid gets 20 Keva blocks. Whoever builds the tallest tower gets to pick their treasure first.

I counted out the blocks and put each kid in a separate spot on the floor.  The kids then built their first tower.  3 of the kids built a tower that had 2 blocks on each side, turned up on their edges. They all ended up 9.5” towers.  The other 3 kids built a straight stack of blocks on their flat side. This was a 7” tower.

As soon as I measured, I encouraged the kids to try to build a new, taller tower.  After about 15 minutes of building, the winning tower ended up being 29” tall, 2nd place was 24.5”, last place was 7”.  It was quite fun making the rounds measuring each kid’s tower.  Some kids came up with lots of novel ways to build, and other kids mostly learned strategies by watching.

Then all the kids put back their blocks.  I congratulated them on their hard work, and their good sportsmanship (much improved in the last 2 years!), and told them what place each kid got.  They really were surprisingly good sports about this activity!

Everyone was ecstatic to get to pick a treasure.  They definitely took their time evaluating all the treasure options. (Monetary value ~25 cents 🙂 ).

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Don’t Let a 6-year-old Do Your Taxes

The Activities

  1. Topic: Proofs:The Boy Who Cried Ninja” by Alex Latimer.
  2. Topics: Multiplication, Teamwork, Counting: Working as a group, evaluate 29 times 73 using Base Ten Blocks.
  3. Topic: Programming: More programming problems like the previous week.  This week, we had a new chart for keeping track of the program variables and output and introduced multiple variables and assigning one variable to another variable or itself.
    Program 1:
    ———-
    Print “Joe_has_”
    Box_A = 3
    Box_B = Box_A + 1
    Print Box_B
    Print “_dogs.”
    Program 2:
    ———-
    Box_A = 5
    Print Box_A
    Box_A = Box_A – 1
    Print Box_A
    Box_A = Box_A – 1
    Print Box_A
    Box_A = Box_A – 1
    Box_A = Box_A – 1
    Print Box_A
    Program 3:
    ———-
    Box_A = 10
    Box_A = 10 * Box_A
    Box_A = 10 * Box_A
    Print “Sabrina_ate_”
    Print Box_A
    Print “_cookies.”
    Program 4:
    ———-
    Box_A = “Shannon”
    Box_B = “Lucy”
    Box_A = “Sophia”
    Print Box_A
    Print “_ate_”
    Print Box_B
    Print “‘s_lunch.”
    Program 5:
    ———-
    Box_A = 3
    Box_B = 5
    Box_B = Box_A
    Box_A = Box_B
    Box_B = Box_A
    Print Box_B
    Program 6:
    ———-
    Box_A = “John”
    Box_B = “Monkey”
    Print Box_A
    Print “_eats_bananas.”

    Program2Program5

Preparation

The programming worksheet is available here, and the programming problems are available here.  We used the kids’ names in the problems (anonymized in the link) — they loved this!

How did it go?

We had all 6 kids this week.

The Boy Who Cried Ninja

This is a light retelling of “The Boy Who Cried Wolf”, with a twist.  It’s vaguely related to the idea of proofs.

Large Multiplication

First I asked them what multiplication is.  Some knew, although they mostly only knew the word “times”.  The best definition was adding the first number the second number of times (3 x 4 = 3 + 3 + 3 + 3).

Then, I gave them the problem 29 x 73.  I initially gave them 10 minutes.  I put the whole box of blue blocks on the table.  One of the kids said they should do 29 73 times.  I suggested they might want to do 73 29 times.  I don’t think they understood why, but they took my suggestion (which is good, because it would have taken forever the other way).

They started by each separately making a few piles of 73.  After each kid had made about 3, they got kind of stuck; many of the kids stopped making piles.  Kid #1 started changing their tens into hundreds.  I suggested they could put their groups on the island in the kitchen.  Mostly they didn’t listen to me, but kid #2 said a little later they should put their groups on the ground.  They didn’t listen to kid #2 either, but I encouraged kid #2 to start doing it by themselves.  Once the other kids saw what kid #2 was doing, they began to join in.  They weren’t very careful about blocks that weren’t part of groups though, there were some random 100s that kid #1 had moved which didn’t end up being used, had I not cleared them away they probably would have included them accidentally.  10 minutes was up by this point, so I added 10 more, and then later, 10 more (they ended up finishing almost exactly at the 30 minute mark).

When they were at about 20 piles, kid #1 started to trade in some of the existing piles.  Kid #2 said not to do that, and I agreed.  Kid #2 started counting the piles over and over seeing if they had gotten to 29.  They actually overshot by 3 or 4 piles, because kids #3 and #4 were still busily making piles.  I suggested they might want to come up with a better way to count, but kid #2 wasn’t interested.  Kid #1 suggested grouping into fives, which I said was a good idea, but by that point kid #2 and others had decided that they had 32 (I don’t know if they were right).

Then they started trading in, beginning with 1’s.  I think? most of the kids understood they were supposed to be trading, but they were pretty bad at it.  Almost all the kids picked up a pile of 1’s, took it over to the table, and counted it.  Of course, they didn’t have multiples of 10, and so who knows what happened to the extras.  Also, they were really worried about the uneven 1’s — kid #5 got the big bag of extra 1’s and brought it over to start counting out new 1’s until she got to 10!  Kid #1 said “No! No!” so that didn’t end up happening.  Once they got to 10’s, kid #1 went over and grabbed a stack of 7 100-plates, and then took them off the pile one at a time, measuring against the bars (rather than counting out 10).  Only after (physically) measuring out the bars did kid #1 take them to the table, which of course is a procedure with a high risk of error, plus the extra 100’s appear to be part of the total.  Some of the kids did the trades correctly, picking up 10 bars, taking them to the table, and bringing back a 100.  But I’m pretty sure kid #4 was just taking stacks of 10 bars to the table (I caught kid #4 doing this at least once and made sure they took a 100).

Kid #1 was able to read the number correctly (they’ve had trouble with this in the past).  Their final answer was 1757 (correct answer: 2117).  Not at all surprising.  I think that if we set up one person to be the “trader”, they could get much closer to the right answer, but not sure if we should do that.  Another interesting note is that had they realized that 10 * 73 is 730, they could have skipped a lot of work.

Programming

This was the same as the previous week, except that I had a new sheet with a clearer section for the variable boxes and for the output.  There were two new concepts: multiple boxes (which our daughter had suggested), and setting one box equal to something based on either itself or another box (e.g., Box_A = Box_B + 1 or Box_A = Box_A – 1).  I accidentally used * for multiplication, but once I said it was times they were able to do it.

This time, I think all the kids got the idea.  Kid #1 (different numbering from previous problem) really liked this problem.  We did the first couple programs in sync, and then people started working at different pace.  Everyone but kid #2 and #3 were able to consistently make independent progress.  Kid #2 was able to do it, but would sometimes pause until I encouraged them to do the next step.  Kid #3 got distracted by kid #1 (who was able to guess most of the programs just by thinking in their head).  I think kid #3 also had a bit of trouble determining what the next statement was.

As before, the kids loved having their names in the problems.  I did the problems out of order, because problems 2 and 5 were trickier.  We were out of time after the 3rd program, but kid #4 was sad because we hadn’t done the one with their name yet, so I brought that one out.  Several of the kids finished the 4th quickly, and did the 5th as well; they wanted to do the 6th problem too, so I sent it home with them.  On the problem with B = A; A = B; B = A; they thought it was very funny that you kept crossing off 3 and writing 3.  Kid #5 had an “optimization” in the cookies problem where instead of crossing off 10 and writing 100, they added a 0.

Computers Love Love Love Math

The Activities

  1. Topic: Proofs: Book: Cat Secrets by Czekaj.  This is a very funny book about a bunch of cats trying to prove whether the reader is cat or not.  Afterward we talked a bit about proofs and how they would prove this.
  2. Topic: Proofs, Pigeon Hole Principle: Prove that if your first grade class has 15 kids in it, then at least 2 of them must share the same birthday month.
  3. Topic: Spatial Reasoning:  I made some shapes (some of them very similar to each other), out of Pattern Play Blocks, and then I took pictures of various sides of the shapes.  We printed out a worksheet with the pictures, and the kids had to say which shapes was in the picture, and which side the camera was on.

    One of the worksheets.

  4. Topic: Programming:  Teach the kids a simple computer language that has ‘print’ ‘loops’ and ‘variables’.  I told the kids that they were all computers, and that they had to follow some programs I had written, just like real computers have to follow programs.
Program 1
———————-
Print “i”
Print “love”
Print “math”
Program 2
———————–
Print “i”
Do 3 times:
___Print “love”
Print “math”
Program 3
———————-
Print “L”
Print “i”
Do 2 times:
___Print “t”
Print “L”
Print “E”
Program 4
———————–
Box_A = 1
Print Box_A
Box_A = 2
Box_A = 3
Print Box_A
Program 5
———————-
Box_A = 100
Do 3 times:
___Print Box_A
___Box_A = 5
Program 6
——————–
Box_A = “Elsa”
Print “Anna’s sister is”
Print Box_A

One student’s solution to Program 5 and 6.

Preparation

We decided on the commands we wanted to teach, and made short programs demonstrating them.  I built the rainbow block structures, took pictures, and David made the worksheets.  Here are the two worksheets: One, Two.

How did it go?

All 6 kids were at the circle, which led to a high-energy environment, but overall it went pretty well. The spatial reasoning was easier than expected, and the programming was much harder.

Birthday Proof

I started this by asking the kids how many people are in their 1st grade class.  The answers varied from 15 to 24.  Then I asked them if they thought two of the kids in their class had the same birthday month.  They started giving examples of kids in their class who have the same birthday.  So I said that I claimed that in any class of 15 kids, at least two would have the same birthday month.

Unfortunately, some of the kids got stuck on the idea of their own classes, so they kept wanting to work on 22 or 24 kids instead of 15. Eventually one kid said that it must be true because 15 is higher than 12.  I asked her why that was important? and she had a bit of trouble explaining.  I got a paper, and wrote the 12 months on it.  Then I got out 15 base ten cubes to be the kids.

They immediately started assigning one block to each month.  At the end there were 3 blocks left, so they said they’d proved it.  I said, but what if you just put all the leftovers on September? But the kids said “No, then September would have more than 2 kids”.  What if I just take these 3 cubes?  “Then you wouldn’t have 15!”.  So I was pretty convinced.

Spatial Reasoning

I expected this to be a bit difficult, because drawing the shapes was so tough last week.  But this turned out to much easier than expected.  Almost all the kids did well on this.  As the kids finished the first sheet of photos, I passed out the second one.  Eventually, one of the kids noticed that one of the photos could be from either of two shapes.  I said to just put down both answers.  Several kids later independently also noticed this.

Programming

The kids loved the idea of pretending to be a computer and following the programs.

The first 3 programs were quite easy…everyone understood the idea of the “print” statement, and even loops were no problem.  The kids were able to guess what that meant.

Program 4 is the first one that has variables.  This concept turned out to be way harder than I expected, though I think I’ll be able to a better job explaining it next time.

I told the kids that “Box_A” is called a variable, and that we should write “Box_A” on our paper, and underneath, put the value: 1.  They all did this.  The next command says Print Box_A.  I asked what they thought that meant. One kid said we should write “Box_A”, and several other kids agreed.  I then explained that it means we should print whatever is the value of Box_A.

The next line: Box_A = 2.  The kids figured out we should cross off the 1 and put a 2 in Box_A.  I agreed, and then all the kids started to do that…but they had different ideas about whether to cross off the old value, or keep it there, or erase it. Also, a couple of the kids crossed off the ‘1’ that they had printed, and put a ‘2’ there too.  I tried to explain that what we print can never change but what we keep in the box can.

One kid didn’t want to put a 2 in Box_A because she saw that the next command was Box_A = 3, so she’d just have to cross off the 2.  Several kids were concerned that we didn’t print the 2, but I said that they are computers, and they have to do what the program says.

By this time, 4 of the 6 kids were pretty confused, so we restarted Program 4 with clean sheets of paper.  This time I asked the kids to draw a line that would be our “Printing Area”.  Then we put “Box A” off to the side so it wasn’t in the way.  This time it went a bit more smoothly and the kids seemed less confused.  They were concerned at the end that the answer was ‘13’ which didn’t make sense.

After this, we worked slowly through Program 5 and 6…the kids seemed closer to understanding, but next time I’ll make sure to differentiate the ‘Printing area’ from the variables more clearly.

Prove it!

The Activities

  1. Topic: Numbers: Book: Missing Math: A Number Mystery by Loreen Leedy.
  2. Topics: Spatial Reasoning and Drawing: I made a fairly simple 3-dimensional sculpture out of Pattern Play blocks.  The kids each had a piece of paper that they divided into four quadrants.  In each quadrant, they drew the shape from each of 4 directions (we rotated the sculpture 90 degrees each time).  Finally, they drew a top view on the back of their sheet.  We did this for 2 different sculptures.

    Sculpture A — simple rectangle, but note the triangular holes

    4 kids’ drawings of sculpture  A

    Sculpture B

    Drawings of sculpture B

  3. Topics: Proofs, Primes, Even and Odd: First, I asked the kids to prove that seven was prime.  Next, I asked them to prove that an odd number plus and odd number is an even number.  We got the idea for the latter problem from Building Better Teachers, a review in The Atlantic of Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone) by Elizabeth Green.
  4. Topics: Puzzles: I read the book How Many Feet? How Many Tails by Marilyn Burns interactively with the kids.

Preparation

The only special preparation for this circle was building the sculptures.

How did it go?

We had all six kids in circle this week.

Missing Math: A Number Mystery

This book was about someone stealing all the numbers, so no one could use money, measure anything, etc.  It was a hit.  Two of the girls wanted to look at it after circle, and our daughter later said it was her favorite part of circle (she rarely chooses the book as her favorite), and took it to her room at bedtime to read.

Drawing from Different Angles

The proof activity is best suited for a smaller group, so I asked one of the parents to lead the drawing activity.  The kids liked this activity quite a bit.  They got the idea of drawing directly from the side, but it was a challenge for them.  Only two of the kids were very close on Sculpture A (with the triangular cutouts on the side), although they all got the top view correct (which had no cutouts).  The second sculpture was quite a bit more challenging, and none of the drawings were very close.

At the end, we circled back and looked at the pictures each had drawn, but there wasn’t really any discussion.

Proofs

I led the proof activity, with 3 kids at a time.  First, I asked them what a prime number was.  No one remembered right away, but when I reminded them they picked it up again right away.  I gave 4, 6, and 8 blue blocks to different kids and asked them to prove they weren’t prime, they all did it easily.  Then, I gave them each 7 blocks and asked them to prove that 7 was prime.  They all started arranging in different patterns.  After a bit they decided they couldn’t make a rectangle, and I asked them to prove it.  They kept trying things for a while, and I think I gave a minor hint, like “What about a rectangle with width 1?”  At this point, one kid said something like “Oh” and started trying 2, 3, 4, 5, …  So she got it.  One of the other kids in her group was able to repeat her proof.  In the other group, one of the kids got the proof once I said “What about width 2?”  The other three kids didn’t quite get the idea of systematic search, and kept trying things without making progress.

Then I asked them to prove that odd + odd = even.  I started by asking them to define odd and even.  Only half of the kids had a clean definition.  Two of the kids defined even = “when you split in two piles, has the same number”, odd = “when you split in two piles, different numbers.”  This is interesting because it isn’t quite right — it should be “there exists an equal split” and “there exists no equal split”.  The other definition was even = “numbers ending in 2, 4, 6, 8, 0” and odd = “numbers ending in 1, 3, 5, 7, 9”.  This definition is interesting because it shows knowledge, but it’s a consequence of the real definition — it could just as well have been “numbers ending in 1, 3, 6, 7, 8”.

Odd + odd = even is a bit tricky with their definition of even/odd — it’s considerably easier with a definition based on arranging the number into pairs, with either 0 or 1 left over.  In the first group, after a while I changed to “prove that if you add 1 to an odd, you get an even”.  They were able to get this, seeing that it either added a left-over or matched with it.  One of the kids knew that odd/even alternated, and so we added one a number of times, using the blue blocks to see odd/even.  Then I asked “prove that even + 2 = even”.  Some of the kids were able to get this eventually, after I strongly stressed the definition of even = two equal piles.  I made two equal piles out of blue blocks and added 2 new blocks, and a couple of the kids saw that you could add one of the new blocks to each pile, and it was still two equal piles.

Doing proofs for the first time revealed that the kids have a weakness around definitions — they tend to think intuitively about the problem, rather than strictly sticking to the given definitions.  The solution to the even + 2 problem is simply “Divide into two equal piles.  Add one block to each pile.  It’s even!”  But you need to use the definition of even twice, once before and once after adding, which was difficult for them.

How Many Feet? How Many Tails?

This book was a pretty good level for them, they usually didn’t get the riddles right away, even with the picture, but at least a couple of kids figured out each one.

 

We started a blog!

Corey and I have been leading a math circle of 6 kids for over a year and a half.  Lots of people have been interested in the problems and activities we have been making, so we decided to publish a blog.  We’re currently in the process of backfilling to earlier circles, and we plan to stay up to date moving forward.