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).img_2431
  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.”img_2432

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).

Mother’s Day Origami (Age 8)

The Activities

  1. Topics: Combinatorics, Combinations:  Anno’s Three Little Pigs by M. Anno.
  2. Topic: Origami:  To celebrate Mother’s Day, we made two different models: a double heart from Essential Origami and a rose from Origami Made Easy.IMG_1898

How Did It Go?

We had four kids this week.

Three Little Pigs

We spent some time looking at and understanding the pictures.  We’ve read this book before, and while I still don’t think they fully understood it, they understood a lot more than last time.

Origami

The double heart model was fairly tricky and they needed some help.  But it’s a pretty cool model!

How To Make a Square Number

The Activities

  1. Topic: Fractions: Book: Fraction Action by L. Leedy.
  2. Topic: Scale: This was a repeat from last week, where we had some line drawings on graph paper and the kids had to scale them up by a factor of 2. This week, the drawings were a bit harder, and for the kids who were better at this, I had them scale up by a factor of 3 instead.
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  3. Topics: Square Numbers, Sequences, Proofs:  I introduced and explored the fact that the sum of the first n odd numbers is n^2.
  4. Topic: Origami: I handed out origami instructions for an origami horse to each kid, with the goal that they would do it from the instructions without me showing them how to do it.

How Did It Go?

We had 4 kids this week.  This circle went pretty well, although our daughter did get frustrated and had to leave during the origami activity.

Fraction Action

The kids enjoyed this book quite a bit.  It wasn’t too complicated, but there was a slightly harder question at the end of each section that tested them a bit.

Upscaling Drawings

We did this in the previous circle with only two kids.  They liked it so much we decided to do it again.  The two kids who hadn’t done it before picked it up pretty easily.  Most of the figures weren’t too hard for them, but all the kids had an off-by-one error at least once.  The bow-and-arrow and the crown (not pictured) turned out to be the hardest.  I think the reason the bow-and-arrow was hard is that I didn’t draw a line for the “string”, which meant that in order to draw a nice curve, it was best to count out the top and bottom of the bow before you started drawing, which wasn’t what they normally did.  The crown was the only picture that had a slope that wasn’t 45 degrees, and this was pretty hard for them.  Again, the best solution is to find the two endpoints and connect them, but this wasn’t the way they had been doing the easier ones.

Sum of Odd Numbers

I started by having the kids make squares out of Base Ten Blocks for all side lengths from 1 to 10.  I kept track of the area (number of blocks) for each side length in a two-column table.  One of the kids has been practicing multiplication a lot, and already knew all the answers, but they were all still willing to arrange the blocks into squares.  All the kids liked the table and most of them made their own copy.  Next, I asked what they did when they were looking at the Fibonacci sequence, but no one remembered.  So I started writing down the successive differences, and they saw the pattern after the first few and completed it.  Next I made a table of 1, 1 + 3 + 5, 1 + 3 + 5 + 7, …, with the sum in the other column, to show that the squares were the sums of the first k odd numbers.  I was hoping they would be a bit excited by this “coincidence”, but not so much.  Next I made a 3 x 3 square of blue blocks, and asked how many I needed to add to make a 4 x 4 square.  I showed how you could add 3 on each of 2 sides, plus an additional 1, to complete the square.  Most of the kids were distracted by this point, but one of the kids learned the pattern and could answer questions like “How many do you need to add to an 11×11 square to make a 12×12 square?” without help.  Finally, I was hoping to teach them the trick that the sum of the odd numbers up to n was (n + 1) / 2 squared — but everyone just wanted to add up the numbers manually, they weren’t excited about a trick (and didn’t really understand it).  So in the end, there was some progress, but we didn’t succeed in proving anything.

Origami

The goal this time was to have them follow the instructions on their own, but they turned out mostly not to be ready.  One of the kids has been practicing quite a bit, and made a lot of progress on their own, but the rest needed lots of help.  Also, I accidentally picked one that required cutting, and the words on the instructions were very small — which was particularly a problem on the step that said “Repeat steps 4-7”.  Another challenge was knowing when to fold only one layer vs. several layers.  Yet another issue was that several steps showed multiple overlapping folds, and you needed to know that you were supposed to do the folds one at a time, unfolding after each.  We should try this again with a slightly easier model.

Mother’s Day Math

The Activities

1. Topic: Graphs: Book: The Great Graph Contest by Leedy. In this book, animals compete to make the coolest graphs. Everything by Leedy is awesome, including this book. The kids were very interested in the idea of a contest, and in the various graphs the animals made.

2. Topic: Origami: Each kid made a ‘Secret Heart Box’ origami model for their mother.

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3. Topic: Codes: Each kid wrote a short love letter to their mother, and then encoded it using a letter to number key I provided.  Then they folded up the notes and put them in the origami heart box.

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4. Topic: Graphs: The kids got a worksheet that list various points on the Cartesian Plane, e.g. (5, 3). When then plot the points, it draws a picture of a simple shape.

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How did it go?

I led the older kids circle this week. All five kids attended.

Origami

The kids all cheered when they heard we would do origami. My daughter complained because she did not want to give the heart box to me 😛

This model was a quite a bit harder than the ones the kids have done before, but they all did really well, staying patient and letting me help on the tricky steps. All 5 kids successfully completed the project.

Coded Letters

At first the kids were not sure what to write, but they soon each came up with their own mother’s day message. Some of the messages were quite a lot longer than others, so a couple kids had to finish their messages at the end of circle.  No one had any trouble translating their English message into coded numbers.

Coordinate Plane Graphing

None of the kids had graphed points before, but they very quickly caught on.  Some of the kids really zoomed through this.  They all enjoyed seeing what picture the coordinate drew.  The only question the kids had was whether they should start at 0 or start at the last point they had drawn.

Kitties, Cubs, and… Goatlings?

The Activities

  1. Topic: Patterns: Book: A-B-A-B-A- A Book of Pattern Play by B. Cleary.
  2. Topic: Patterns: We made rows of colored glass beads in different patterns.  I gave a starting pattern with 5-6 beads, which they copied and then continued the pattern.  They also each made up a pattern on their own.  Here’s the grid we put the stones in.IMG_1312
  3. Topic: Logic: We did three Halloween-themed logic puzzles.  For example, in the first one, there were 4 monsters and 4 candy buckets, and you have to figure out which one goes with which.  One of the clues is “The ghost likes orange”.  The full set of pictures and clues is available here (including 4 extra clue sets we didn’t do).
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  4. Topic: Verbal Discussions: I named a bunch of animals and asked the kids what the baby animals of that type were called.
  5. Topic: Counting: Book: Skip Counting with Meerkats by T. Steffora.
  6. Topic: Counting: We counted by 2’s and 3’s as a group taking turns going around the circle.
  7. Topics: Shapes, Charts: We have a large two-dimensional table with colors on the rows and shapes on the columns.  The task was to place the Attribute Blocks onto the chart.IMG_1314
  8. Topic: Origami: Each kid made the cat from Easy Origami by J. Montroll.

How Did It Go?

All the kids attended.  The kids were mostly attentive through out, although there was the usual “Is it time to go play?” starting about 40 minutes in.

A-B-A-B-A- A Book of Pattern Play

A decent introduction to simple patterns.

Glass Bead Patterns

All of the kids were able to complete all the patterns I made, (which went up to cycle length 4).  They weren’t fooled by an all-blue pattern.  One of the kids said they wanted to make their own pattern, so I gave them time to do that.  One kid used the columns of the grid for their own pattern instead of the rows.

Halloween Logic

The kids did well on this activity.  On several occasions, they remembered an important relevant clue from earlier in order to solve a puzzle.  However, they weren’t quite ready to solve the puzzles on their own — I still needed to ask leading questions several points, particularly on the 3rd (and hardest) puzzle.  The kids were pretty good at following the instructions, but there were a few times where someone said something like “I think the cat likes the hat because a cat can’t use a broom.”

Animal Babies

The kids knew a reasonable number, including kitty, puppy, calf, cub, and duckling.  They didn’t know foal/colt, piglet, tadpole, or gosling.  The most interesting guess was “goatling”.

Skip Counting with Meerkats

One of the kids knew the animals in the pictures were meerkats, from Happy Hollow (or rather, “Danny the Dragon place”).

Counting by 2’s and 3’s

The majority of the kids still aren’t very good at counting by 2’s or 3’s — two can do it reliable, three can’t.

Attribute Block Chart

I was impressed by the kids performance on this task, I remember Circle 1 having some issues, but all the kids got the idea and were able to do it.  Initially a number of the shapes were placed incorrectly because the kids didn’t want to put more than one shape in each box, but once I showed them stacking was ok, they placed all the rest of the shapes without problems.  The kids also put all the shapes back into the box afterwards.

Origami

The kids are decent at making folds, but the idea of making a fold from point A to point B, or so that an edge lines up with a given point, is tricky for them.

Micheltello, the Fifth Ninja Turtle

The Activities

  1. Topic: Story Problems: Book: How High Can a Dinosaur Count? …and Other Math Mysteries by Valorie Fisher.
  2. Topics: Codes, Combinations, Story Problems: We created a Teenage Mutant Ninja Turtles “adventure” with three stages.  All materials for this activity can be found here.  The story is that Shredder has discovered some ancient machines that, when activated, will cover the world in slime (I printed out some pictures of the characters for those less familiar with ninja turtles).  The turtles obviously want to stop him, and they find Beebop and Rocksteady, defeat them, and take from them a coded message from Shredder.  This was a number-letter substitution code, but unlike in the past, I did not give them the key.
    Deciphered message

    Deciphered message

    Then I said that Splinter remembered a legend of an ancient fifth ninja turtle, but he couldn’t quite remember his name.  However, he did remember that the name was a combination of two of the first four turtles’ names (e.g., Raphangelo).  If they can list out all the possible names, he will remember.

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    Now that they know the fifth turtle’s name (Micheltello), April O’Niel goes on TV and says that Micheltello should come help them.  He shows up, and says that that he knows where the slime machines are.  The turtles go to the slime machines, but each one has a story problem that the kids must solve in order to turn it off (see link to PowerPoint above for the story problems).  Once they solve the problems, the world is saved and the mayor gives them each a prize!

  3. Topic: Origami: We made the pinwheel from Easy Origami by John Montroll.

How Did It Go?

All six kids attended.  This circle went well, all the kids were engaged the whole time — having a theme definitely helps.

How High Can a Dinosaur Count?

This book has a bunch of story problems, mostly about addition but a few others, including a couple about money.  Most weren’t too hard for them, but they usually couldn’t solve them instantly.  I had the kids raise hands to answer, all the kids gave at least one answer.  The hardest one was to count up how much money two dimes, two nickels, and three pennies was.

Ninja Turtle Adventure

When I first announced the topic, one of the girls said “I’m not interested in boy things.”  Fortunately, once we started working on the problems, she jumped right in :).

The first breakthrough for the code was that one kid said that one of the three letter words could be “the”.  I pointed out there were several three letter words.  They noticed that two of the four three letter words were the same, and tried filling in “the” for that one (which was correct).  They made a bit more progress after this but then got stuck.  One kid kept asking “What are the two little dashes at the bottom?”  I just said “Part of the message.”  Eventually, that kid said “Maybe this says Shredder!”  They didn’t quite get the concept of making sure the pattern of the word worked (they certainly didn’t notice that certain words had double letters until after they filled them in).  Once they got Shredder and filled in the matching letters elsewhere, they guessed “turtle” and “ninja”, and got most of the rest.  They ended up having the whole message decoded except for “_i_th”.  One kid suggested “ninth”, but n was already used.  I suggested they try similar things, and they got “fifth” after a bit.  I had the kids take turns being the writers, since they all wanted to fill in the letters (4 kids writing at the same time is pretty tricky).

They were slower than I expected at generating names.  In similar activities in the past, there’s usually been a flood of suggestions, but I guess the concept of combining a prefix and suffix was trickier.  However, almost immediately, one of the kids suggested drawing a line between the prefix and suffix whenever they thought of a new name.  This was obviously extremely helpful for searching, although they still weren’t that good at that.  After a while, a different kid noticed there were a different number of lines coming from various suffixes, and found things to even it out.  However, they still didn’t realize there needed to be three coming from each one; after a while one of the kids suggested there should be four lines, but I pointed out that you wouldn’t have one for the actual real names, so only three.  All the kids came up with at least one name and wrote it on the chart.  This was the first time in a combinatorics activity that I think they had some concept of having found the all possibilities.

The story problems were somewhat difficult, in particular, understanding what the problem was asking was sometimes tricky.  Their reading is pretty good, but long complicated questions are still difficult.  I had to help almost all the kids understand their questions.  About half the kids finished their own questions fairly quickly, and then I had those kids help the others with their problems.  There some interesting incorrect answers.  For the problem “There are 17 pizzas with 10 slices each, Michelangelo eats one slice from each pizza, how many are left?” the initial answer was 7 (17 – 10).  When another kid came to help, they realized it should be 9 * 17.  I suggested that they might take the initial pieces (170) and subtract the eaten pieces (17), which they could do.  Another interesting problem was “Leo has 2 swords, Raph has 2 sais, Dona has 1 bo, and Michel has 2 nunchuks.  Also, each turtle has 3 ninja stars.  How many total weapons?”  The initial answer was 10, adding all the numbers in the problem.  So, there’s clearly room to improve in translating story problems into math expressions.

Origami

The pinwheel wasn’t too bad for them, we can probably do something harder.  But the final step where you pull out the points was tricky for some of them.  One of the kids was eager to tell me that she could make an origami dragon, but we didn’t have time for a demonstration.

Robot Dance Optimization

The Activities

  1. Topic: Charts: Book: The Great Graph Contest by Loreen Leedy.
  2. Topics: Multiplication, Proofs: Prove that multiplication is commutative, i.e., x * y = y * x.
  3. Topic: Programming: I did several robot dances.  The kids needed to write a program for that dance.  For example, R(ight turn)RRRJ(ump)F(orward)B(ackward)RRRRJ.  Then, they had to find the shortest program they could for that dance, using loops and functions.
  4. Topic: Origami: We all worked in parallel, each kid (and I) making two simple models, a rolling toy and an envelope.

How Did It Go?

All 6 kids made it this week again!

The Great Graph Contest

I asked the kids at the beginning what a graph was.  Kid 1 said they did graphs at school, and it was where they asked everyone a question and they said yes or no and then they saw who had more.  I asked for any other definitions, and Kid 2 said almost exactly the same thing.  I said those was one kind of graph but there were others.  The kids liked the book better than average I think, several kids said they liked it afterwards.

Commutivity of Multiplication

First, I asked which is bigger, 3 * 10 or 10 * 3.  Kids 1 and 2 both immediately said “the same!” although the other kids didn’t know.  I tried out some other numbers, and Kids 1 and 2 kept saying the same (going in, we thought that maybe none of the kids would know about this).  I asked them why it was true, Kid 1 had a circular answer and also said something like “because they’re both 30”.  Kid 1’s explanation for why 5 * 4 = 4 * 5 was that 5 * 4 is 5, 10, 15, 20, and then 4 * 5 is 4, 8, 12, 16, 20, so the same!  I asked them to prove it with blue blocks.  Several of the kids started making piles, Kid 1 started to make rectangles, and a couple kids did nothing.  After a while I asked them all to make 3 * 5, which Kid 3 explained as 3 groups of 5.  Then I asked them to make 5 * 3.  Most of them had 2 sets of piles at this point.  Kid 2 said “look, I can use one set of piles to make the other” but just did it adhoc without a pattern.  I pointed out Kid 1’s rectangles and asked if they were the same.  Kid 4 said something about them being the same squares (sic), and said something about the side lengths.  Kid 1, after a bit, said you could look at them from different ways and they would go back and forth.  I said “Kid 1 proved it!” and repeated his proof.  Then I showed them the proof where you take 1 thing from each pile to make a new pile, and do that until you’re out of blocks.  I don’t know if any of them understood it.

Writing Dance Programs

I performed several dances for them, and they had to write down the instructions for that dance.  I started with a simple dance, F(orward)B(ackward)BFJ(ump).  It was harder than expected.  I ended up doing it about 5 or 6 times.  By that time, all but Kid 1 had it correct (Kid 1 only had one B).

Next, I did R(ight)RRRJFBRRRJ.  Again, they had some issues keeping track of what I was doing.  After a while a several of the kids had correct programs.  Kids 1 and 2 were the fastest, followed by Kids 3 and 4, and then Kids 5 and 6.  About this time, Kid 5 said, unprompted, “Moopsy!”  I was very excited, because this was what I was looking for (Moopsy was a subprogram from last week), but Kid 5 didn’t follow up and use it.  After a little bit I hinted them towards Moopsy, and then several of them were able to quickly write the program in terms of Moopsy.

Finally, I did D(own)U(p)DUDUDUFBFBDUDUDUDU.  Kids 1 and 2 finished the whole thing, and Kids 3 and 4 made good progress.  Then I asked them who could do with the shortest program.  Kid 3 came up with “Do 4 times: DU FBFB Do 4 times: DU”.  It’s worth noting that most of the kids didn’t have any syntax; Kid 1 was the only one who wrote each instruction on a different line.  A couple kids mentioned Floopsy from the previous week (DUFB), which was usable but not super helpful.  Then I said “You can make up your own programs!”  Kid 1 did something interesting, Kid 1 erased the whole program and wrote “n” and then said that “n” was the whole program.  I said Kid 1 still needed to tell me what “n” meant, and Kid 1 got rather upset and said they didn’t know what I meant.  After a bit I was able to explain that I didn’t know how to do “n” so they needed to tell me.  But it turned out that wasn’t enough, and that the core problem was that the definition of the length of a program wasn’t clear.  So that’s something to be more careful about in the future.  Meanwhile, although most of the kids weren’t making progress (Kid 5 was busy writing the names “Poopsy” and “Peesie”), Kid 3 had written “t” and was doing DUDUDUDU.  I pointed this out to everyone and asked if they could write the program in terms of “t”, and several of them did.  I mentioned Kid 3 could use “do X times” inside “t”, and they did.  In the end, Kid 3 ended up defining “s” to be “FBFB”, so the final program was “tst”.

The kids are doing pretty well at programming, although there’s a noticeable range in ability among the kids.

After circle, our daughter kept working on her program with Corey and made a shorter program than we had come up with during circle.

Origami

We made two models, as a group, from the books “Origami: Fun with Paper Folding #5 and #8”.  The books don’t seem to be available online, but they’re pretty simple models, although a bit harder than the cat and dog we made last time.  One of the models was a cute rolling toy, and the other was an envelope.  There was lots and lots of “how do I do this step?” but everyone was doing pretty well.  The kids are good enough at making folds that I think we can try harder things in the near future, like the standard swan.  The kids all were able to handle the symmetry very well, once I showed them how to do one side they could do the other side without problems.  However, they did sometimes have problems when they were supposed to turn it and do the same thing on the other side.  A couple of the kids knew about “sandwich folds” and “hot dog folds” (for the two different initial ways of folding the paper in half).