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).

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 Pride of Fish? (Age 6)

The Activities

  1. Topic: Comparisons: Book: Too Tall Tina by D. Merritt.
  2. Topic: Measurement: I gave each kid a 12″ ruler and asked them to look around the first floor for something 3″ long.  Then I asked 6″, 11″, and 1″.IMG_2065
  3. Topic: Verbal Discussions: I asked the kids a bunch of questions about what you call groups of things: cows (herd); sheep, birds (flock); wolves (pack); flowers (bouquet, bunch, garden); fish (school); geese (gaggle); cats (?); ants (colony); bees (hive, swarm); lions (pride); people (crowd); whales (pod); witches (coven); rabbits (warren); thieves, robbers, musicians (band); soccer players (team); dancers (troupe, company); soldiers (troop, army, legion); girl/boy scouts (troop); kittens, puppies (litter); math students (circle); cards (deck, pack); grapes, bananas (bunch); books (shelf, stack, library); wheat (field); hay (bale); knives (rack); ships (fleet); stars (galaxy, cluster, universe); planets (solar system); sailors (crew); actors (cast).
  4. Topics: Logic, Numbers: I did an activity from  Math Logic & Word Problems, Gr. 1-2, Guess Benny’s Number and Guess Jenny’s Number.  Each had a series of clues that narrowed down to a single number.  We used a 100 Number Board to keep track of which numbers were eliminated.The first puzzle was
    1. The number has two digits.
    2. Both digits are greater than or equal to 5.
    3. The tens digit is greater than the ones digit.
    4. The sum of the digits is 12.
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    The second puzzle was

    1. The number has two digits.
    2. Both digits are less than 8.
    3. The ones digit is greater than the tens digit.
    4. The sum of the digits is 10.
    5. The number is even.
  5. Topics: Counting, Games:  Using the 100 board again, we played the following game.  Each turn, a kid rolled a six-sided die.  They could then advance that number of spaces up to 10 times (so if they rolled a 5, they could advance 0, 5, 10, …, 45, 50).  The goal was to get to 100.  The first time they started at 0, but the second time I had them start at 1 since it’s more interesting.

How Did It Go?

We had three kids this week.

Too Tall Tina

Not much math in this book, but loosely ties into the next activity.

Finding Objects

Some of the kids needed some help measuring at first.  One of the kids spent a lot of time measuring different parts of their mom’s body.  They were pretty excited when they found matching things.

Groups

The kids weren’t able to think of many of them — e.g., for cows, they only knew herd once I told them.  One of the few that they did get was bees, where our son got both “hive” and “swarm” right away — which is pretty funny, because he’s rather afraid of bees.  They also got band of musicians, circle of math students, pack and deck of cards, and solar system of planets.  One kid guessed “pride” for fish, and then when we got to lions realized that it actually went with lions.  For stars, with some help one of the kids thought of pictures in the sky, but couldn’t remember the word constellation.

Guess Jenny’s Number

This activity was kind of hard for them.  First, they weren’t that familiar with the concept of ones and tens digit.  Second, it’s pretty tricky that you need to cover all the squares that DON’T match.  They kept trying, though, and with some help, they were able to do it.  One neat thing is you get some nice patterns along the way.  Our 8-year-old daughter worked on one of them after circle, and it wasn’t trivial for her either.

Skip Counting

This was a good exercise for skip counting — the game made it a bit more interesting, but mostly it was about practicing skip counting.  Switching to starting at 1 made for a much more interesting game — the first time, two players finished in 3 rolls.  One of the kids realized that once you were on 96, if you rolled a 3, you should stay on 96 because there are more ways to win, which is the most interesting part of this activity.

 

Eggs and Boxes (Age 6)

Age 6

The Activities

  1. Topics: Numbers, Codes, Algebra:  The Cat in Numberland, Chapter 3, by I. Ekeland.  In this chapter the letters come to visit the numbers, and we learn about letter/number ciphers and letters standing in for numbers.
  2. Topic: Algebra:  I made problems of the form “X + 3 = 5” using unit cubes from Base Ten Blocks and a small cardboard box.  I.e., I would secretly put 2 blocks into the box and close it, put 3 blocks next to it, and then say “There are 5 blocks total, how many are in the box?”
  3. Topic: Primes:  I introduced the idea of primes using Base Ten Blocks: a number N is a prime if the only rectangle you can make using N blocks is 1 X N.  I gave different numbers to each kid and had them figure out whether it was prime or not.
  4. Topics: Combinations, Combinatorics:  I printed a bunch of “Easter eggs” with a top and bottom section.  Using five different colors of crayons, I asked the kids to make as many different eggs as they could, coloring each section in solid colors (not stripes/dots/etc.).  I taped each one to the wall (stacking repeats).IMG_1886

How Did It Go?

We had four kids this week.

Cat In Numberland

The algebra in this chapter is tricky because it includes addition, subtraction, multiplication, and division; most of the kids don’t know multiplication or division yet.

Box Algebra

This worked pretty well.  The kids understood what was going on right away, and they were always excited when I opened the box and dumped out the blocks inside to see if their guess was correct.  At the end they made a problem for me, which was something like “X + 3 = 39” (of course, they used as many blocks as they could).

Rectangle Primes

We did up to about 14.  I kept track of each result.  The only odd composite number <= 14 is 9, so for the most part they just needed to check a 2 row rectangle.  Proving something is prime is tricky, of course, and whenever a kid said that something was prime, I always asked them “did you check 3-wide”?  Whoever had 9 didn’t initially check 3×3.

Easter Eggs

The kids were really into this activity and worked very hard to get all the combinations.  They got all 10 two-color combinations pretty quickly and without help (first two rows in picture above) — but there was no pattern to which color was on top vs. bottom.  Then one of the kids realized that you could flip the colors.  They quickly got 6 more, but the next 3 took them a lot longer to find, and I had to help them find the last one.  This got them to 20, but they didn’t think of having the same color on top and bottom.  I suggested it to them and they quickly made the last 5.  Then I rearranged them so that there were same color tops along the rows and same color bottoms along the columns.  I realized afterwards that I should have made this chart before I gave them the hint about same color top/bottom, because then there would have been gaps and I could ask them what went in the gaps.

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My Friend’s Mom Is Always Right (Age 6)

The Activities

  1. Topics: Numbers, Large Numbers: The Cat in Numberland, Chapter 1, by I. Ekeland.
  2. Topic: Large Numbers: First, I asked the kids to come up with numbers with increasing sizes (first 1 digit, then 2 digits, …).  After they stopped being able to come up with things, I wrote powers of 10 up to a quintillion.  I also wrote a googol and had them help me count the zeroes.  Next, each kid wrote down the biggest number they could on a sheet of paper.  I asked them which kid’s number was largest, and then I asked if it was the largest possible number.  Then I guided them through a proof by contradiction that there is no largest number.
  3. Topic: Programming: We revisited the Dance Programming activity.  First I called out commands (Down, Up, Jump, Spin) and the kids did them.  Then I taught them a few programs I had made that used the four commands plus “Do X times {… }” loops and function calls.  Next I had them each say a 3-command function for us all to do.  Finally, I asked each kid to write down a named function, with the rule that if you used a “Do X” loop X was at most 5.  Then I made a program that called all their functions and a couple of the kids did it.
    IMG_1864
  4. Topic: Programming: We revisited the Cargo Bots activity.  I made a new version of the board, download here.  First, we moved a single block from one end to the other; then two blocks; and finally, they needed to move two blocks of different colors from one end to other so that the final ordering was the same (i.e., if red is on top initially, it should be on top at the end as well).
    IMG_1865

How Did It Go?

We had four kids this week.  This circle went pretty well, we had some good discussions and some good problem solving.  The kids sometimes thank us after circle(usually prompted by parents) , this time one of them said “Thanks for giving circle to me.”

The Cat in Numberland

This was a popular book with the older circle.  There was a part about how not all numbers can play division together — two of the four kids got the idea.  When it got to the part about infinity, several of the kids already knew something about infinity.  One kid said “My friend says that her mom is always right, and her mom says infinity is a number.”

Large Numbers

Most of the kids had trouble coming up with a number above 1000.  Even the kids that knew larger numbers like 1,000,000 couldn’t make arbitrary seven digit numbers.  The kids enjoyed counting out 100 zeroes when I was writing a googol.  The kids varied in their approaches to writing large numbers.  Some did 1 followed by lots of zeroes, others wrote somewhat random sequences.  I wrote a large number as well — I started with 9’s for obvious reasons, but I discovered that 9 is pretty slow to write, and so I switched to 1’s which are really fast to write.

Next, I asked them which number was biggest; I’m pretty sure it was mine, but one of the kids had filled up their page with larger numbers, so they picked that one.  I asked whether it was the biggest number.  3 of the kids said no, one said yes.  I asked whether they could make a bigger number.  They didn’t really come up with adding 1, but they did suggest adding more digits, including adding on someone else’s sheet of numbers.  Then I said the idea of proof by contradiction (assume the opposite, find a problem), and said “Suppose you took all paper in the world and filled it with numbers.  Could you make a bigger number?”  After a bit, one of the kids said “You could cut down more trees and make more paper so you could add more numbers.”  So I think some of them got the idea.  Since they had suggested taping together sheets, I taped together all the sheets that we had made and laid it on the floor.  For the rest of circle, when a kid finished their work and was waiting on someone else, they asked to go over and add more numbers to their sheets.

Dance Programming

For an activity involving jumping up and down, the kids payed pretty good attention.  They all understood sequences of instructions, and I think they understood loops and functions.  When it came time to write a program, only one used a loop; the loop they wrote was a copy of one of my functions with a different number.  It’s a good thing I limited X to 5, because they immediately said “I wanted to do 100.”  No one figured out the weakness in my problem specification — nested loops.

Cargo Bots

One of the kids missed the first time we did this but caught up fairly quickly.  All the kids were able to solve the first two problems, at different speeds.  A common mistake on the two-block problem was forgetting to move the crane back to the beginning for the second block.  The two-color two-block problem was way harder.  One of the kids came up with some interesting rule-breaking solutions, such as using their other hand.  After a while I gave them a hint by making a program that flipped the stack onto the middle square of the track.  From this, one of the kids (same one who came up with the alternate solutions) realized if they repeated this again, they would solve the problem.  They had some bugs along the way, which I demonstrated by tracing through their program, and they were able to keep fixing things until they got a correct solution (with a few unneeded instructions).  Another kid understood the solution and tried to copy what the first kid had done and got pretty close.  The other kids were still far off and hadn’t gotten the idea of double flipping.

Mount Everest and Submarines (Age 6)

The Activities

  1. Topic: Addition: Book: Mission, Addition by L. Leedy.
  2. Topics: Tangrams, Geometry: We did six different easy tangram puzzles, download here.
    IMG_1837
  3. Topics: Numbers, Negative Numbers: I made a vertical number line from -10 to 10.  We played the number guessing game (higher/lower) with numbers in this range.  I had the secret number the first time, and then each kid took turns having a secret number.  Then I described several different places numbers are used — money, elevators, traveling on the Earth, and age.  For each one, I described what the positive numbers were, and then I asked what negative numbers would mean (for example, for money, it means you owe someone else money).
    IMG_1838

How Did It Go?

We had all five kids this week.

Mission, Addition

This book is slightly on the long side but we finished it.  The kids loved the last story, where the teacher gave them all bills for the food they ate but did the addition wrong (sometimes by a lot).

Tangrams

Everyone worked hard on the tangrams.  Some of them are pretty easy, and no one really had problems on them.  A few are harder, particularly the scarab beetle. Three of the kids finished all six pictures.  One of the kids (one of the two who didn’t finish all of them) liked to say how easy the problems were after they solved them but then asked for help more quickly than the other kids.

Negative Numbers

The guessing game went surprisingly well.  There were only a few times where someone went the wrong way for a negative number.  Having the vertical chart for them to look at helped a lot.

We had a pretty good discussion about the different scenarios.  We started with money, which they had discussed last week.  Of the things we discussed, I think they understood money the least well — except for our son, who has gone negative on his allowance before.  They figured out that elevators would go underground right away.  For traveling on the Earth, I described sea level and said how we were at about 50 feet above sea level and the nearby hills were a couple thousand feet above sea level.  One of the kids mentioned Mount Everest and I said it was 27K feet (I was wrong, it’s actually 29K).  Then I asked what negative was, and they said going underwater.  I asked how they would get there, they said submarine, and two of the kids had seen a movie about going deep underwater.   I asked how they would go negative where there wasn’t water, and they said dig.  For age, I asked them how old they were, and they said 5-6 years old.  Then I asked what -3 years old would mean.  Our son, who we’ve discussed this with, said you’d be an egg in your mommy.  I asked how old our son was when I was born, they enjoyed that.  I asked how old they were when the Universe was created.  A few kids got somewhat distracted by the end of the conversation, but overall they were very attentive and engaged.

Drawing Castles and Trees (Age 6)

The Activities

  1. Topic: Addition: A Fair Bear Share by S. Murphy.
  2. Topics: Numbers, Addition, Base Ten Blocks:  We did several addition problems using Base Ten Blocks.  We practiced making numbers, combining the piles, and interpreting the sum.
  3. Topic: Logic: I drew some pictures of paths leading to a castle with labeled river crossings.  Each picture corresponds to a particular logical formula.  For example, for a path crossing two rivers before reaching the castle, you need a bridge at A AND B in order to reach the castle.  If there is a path that forks and crosses the same river in two places, then you need A OR B.  We interpreted a number of pictures, and then I gave them a new formula and asked them to draw a picture of it.  You can download my pictures here.
  4. Topic: Decision Trees, Attributes, Attribute Blocks:  I drew some decision trees using color and shape as attributes to split on.  First we practiced sorting shapes according to the decision tree.  Next, I sorted the shapes onto A and B in a simple way (e.g., triangles on A and circles on B) and asked the kids to draw a tree that would give you that sorting.

How Did It Go?

We had all five kids this week.

A Fair Bear Share

This book went well, the kids were interested in the baby bear not doing her fair share.

Base Ten Addition

The kids are getting better at making numbers using blue blocks, and I think most of them understand how to use blue blocks to add two digit numbers.

Castle Logic

The kids did a pretty good job understanding the pictures.  They were very good at telling whether a particular configuration of bridges allowed you to get to the castle (I had little squares of paper that you could place on the drawing to show whether there was a bridge).  I used ^ and V for AND and OR, but I didn’t ask them to use these symbols later so I don’t know if they understood.

Drawing a picture was much harder.  By far the hardest part is parentheses, which is important for any mildly complicated situation.  I gave them a problem with parentheses to start, involving A, B, C, and D; one of the kids made a picture of Y ^ Z.  Then I gave them a much simpler problem and some of them got a correct picture.

Decision Trees

Again, the kids were pretty good at tracing a tree.  I think all of them understood this pretty well by the end.  However, drawing a tree was extremely hard for them.  Only one kid got even close (see 3rd picture above).  A big difficulty for them was that the original depth-1 tree I gave them split on colors, so when I gave them that as an example of how to draw a tree, they just copied it, including the colors, even though they needed to split on shapes.