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.

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The Parallelogram and the Pendulum (Age 8)

The Activities

  1. Topics: Logic:  Still More Stories to Solve, by G. Shannon.  We read and discussed the first two stories.
  2. Topics: Spatial Reasoning, Tangrams:  We did the same set of tangrams from a few weeks ago (letters, numbers, and things from Cinderella).
  3. Topics: Physics, Experiments:  Inspired by the Galileo chapter of Mathematicians are People Too from a few weeks ago, I hung a makeshift pendulum from the ceiling — a roll of tape suspended from an 8′ thread, hanging from a sticky hook attached to the ceiling.  I had pre-marked the 2′, 4′, 6′, and 8′ points away from the center of the roll of tape.  We released the pendulum twice at each length, varying the height that we released it at, and timed how long it took to go 20 swings.IMG_1888
  4. Topics: Sorting, Patterns:  We have a card game called Blink — basically a racing version of Uno.  Each card has some number of symbols 1-5, one of six colors, and one of six shapes.  There are 180 possible combinations, but only 60 cards in the deck.  After we figured out there should be 180, I asked the kids to find out which ones are missing.

How Did It Go?

We sat on the floor this week to make room for the pendulum; this tends to make them a little crazier since they can easily roll around on the floor.

Still More Stories to Solve

I wasn’t crazy about the first puzzle, but the second one, about two brothers having a contest to see whose horse would get somewhere LAST, was nice.  The kids figured it out with some hints.

Tangrams

Corey and I discovered that I’m better at Tangrams than she is :), so unlike last time, where Corey AND the kids were stuck I was able to help them solve the puzzles.  The main thing I tried to teach them was to figure out where the big triangles go first; it’ll be interesting to see if next time we do Tangrams they remember this.

Timing a Pendulum

As you can see from the chart above, we had really reproducible results.  I believe we were actually only counting 19 swings (we started on 1 as we let go and then stopped when we said 20, when we should have let it swing again).  Anyway, I had incorrectly remembered from physics long ago that the time was linearly proportional to the length of the pendulum, so I was initially worried about the timings we were getting — but once they were all in, it become obvious (to me, not the kids) that the time is proportional to the square root of the length.  I asked how long the pendulum should be to get 15 seconds; and also, how long would a 32′ pendulum take.  They were comfortable assuming a linear relationship, but when I pointed out that 8′ was four times 2′ while 60 s was only two times 30 s, they couldn’t really use that information — one kid did guess 1/2 foot for the 15 seconds question, but they didn’t stick to their answer so I think it was just a guess.

One thing that worked out well is that the pieces of tape served as resting spots for the thread so that it would stay at the right length (I didn’t cut the string, we had looped it over the hook like a pulley).  If I hadn’t had the tape sticking out, it would have been hard to maintain a constant length.

Blink

I only had time to explain the problem and figure out how many cards there should be before circle ended.  We’ll probably do the main activity next week.

Is it a keychain? Is it a pony? (Age 8)

The Activities

  1. Topic: Logic. Book: True Lies, by Shannon. We read four more chapters from this book, and the kids were begging for more.
  2. Topic: Spatial Reasoning, Tangrams. Each kid had a set of tangrams and tried to recreate shapes from a tangram book.IMG_20160403_175820
  3. Topic: Logic, Decision Trees. I made several sets of objects, and the kids had to write a decision tree that would classify each object into my groupings.IMG_20160403_175210IMG_20160403_175546

    How did it go?

We had four kids this week. The circle was great overall. The Tangram activity was really hard for everyone. The decision trees were easy for some kids, and hard for others.

True Lies

This time the kids had lots of ideas for how to solve the riddles. In fact, the kids solved 2 out of 4 all by themselves, and had good ideas for the others. The best one today was:

A man said “I have a hog so tall that a fellow can’t touch its back if he stretches his hand as high as he can.” His friends went to go see the hog, and it was just a normal size. Was he lying?

One kid said you can’t pat a normal sized hog if you stretch your hand up high, because your hand will be above its back. This turned out to be the correct answer.

Tangrams

We did Tangrams two ways. First, we had some tangram pictures printed out, where the picture was the full tangram size. You could do the tangram by arranging the pieces on top of the picture. This was easy for everyone.

Second, we had printed out a few pages from a Tangram book, where each picture was much smaller than the Tangrams. Only one kid matched any picture from that book. I also failed…it was surprisingly hard.  The kids did give it a good try, but after a few minutes there were lots of comments about how this was too hard, or impossible. After a few more minutes, we stopped, and my daughter said she was so exhausted she could hardly open her eyes.

Decision Trees

I had several stations set up with two or three groups of objects at each station. The task was to make a decision that would classify the objects into the groups I had made. You could ask any yes/no question, as long as you didn’t use the word ‘and’ or ‘or’.

First we did one tree as a group, and then kids cycled around to the different stations, moving one whenever they finished one. Some of the stations were easy, and some were tricky.

The easiest was a station that had one pile with yellow stones in it, and another pile with a mix of blue and green stones.  3 of the 4 kids started their tree with “Is it yellow?” if yes, then pile A, if no, then pile B.  The fourth kid started with “Is it green?”, but then didn’t know how to continue. I suggested looking at the other colors, and the kid switched to “Is it yellow?”

IMG_20160403_171449

My daughter was very interested in making sure her decision trees used different questions from everyone else’s. At the end of circle we compared trees, and saw the different ways people had solved the same problem.

IMG_20160403_175423I watched one girl working on this set of objects. First she said “Oh, is it a keychain?” but then she noticed that there is a keychain in 2A.  She then changed the top question to “Is it a pony?”, and then followed up with “Is it a keychain?” in the “no” branch.

IMG_20160403_173918

One kid’s decision trees.

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.

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.IMG_1759
  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.

Captain Invincible and the Magformers (Age 5)

The Activities

  1. Topic: Geometry:  Book: Captain Invincible and the Space Shapes by S. Murphy.
  2. Topic: Geometry:  Each kid made a cube, rectangular prism, pyramid, and tetrahedron using Magformers.
    IMG_1498
  3. Topics: Numbers, Games:  We played the higher/lower guessing game from 1-100, with each kid taking turns having the secret number.  Rules summary: each of the other kids, in turn, guessed a number and the kid with the secret number said whether the secret number was higher or lower.  There was no guess limit, but we didn’t use any visual aides to keep track of the information so far so it wasn’t trivial to win.
  4. Topics: Geometry, Tangrams:  I had 6 not-too-hard to-scale tangram diagrams.  The kids each solved as many individually as they could.
    IMG_1497
  5. Topic: Numbers:  We practiced making moderate size numbers (e.g. 63) using Base Ten Blocks (each kid made their own).  I also made a number and asked the kids what it was.  Finally, each kid made a challenge number for me to count, using as many blocks as they wanted.

How Did It Go?

We had three kids this week.  This circle went very well, all the kids paid attention the whole time.

Captain Invincible

Not very complicated mathematically but with interesting pictures.

Magformer Solids

Except for our son who has played with Magformers quite a bit, it took them a bit of time to get the hang of it.  But once they started, all the kids were able to make each of the shapes.

Higher/Lower

The kids have been getting much better at number sense; I had them tell me the secret number each time so I could check their answers, but I hardly ever had to correct the answer of the secret holder.  On the other hand, their guessing sophistication varied quite a bit.  One kid could easily solve 1-100 puzzles on their own, keeping track of the bounds so far.  Another kid often, but not always, made good guesses.  The final kid frequently guessed numbers outside of the current range, often far outside.  For 1-100, these guesses didn’t distract the other kids too much.  After we had done two full rounds of secret numbers (6 games total), I did a secret number between 1-1000.  One of the kids was capable of winning had they been playing on their own; but this time, the non-useful guesses kept “resetting” the current progress, and they weren’t ever able to win.  One of the kids kept guess numbers between 100 and 200, even after they knew the answer was between 700 and 800.

Tangrams

I was afraid this was going to be terribly hard and the kids would give up; but actually, they were pretty good at it!  I often had to give a small hint, but generally only one hint per puzzle.  They each finished at least 4 of the 6 puzzles.  They also enjoyed this quite a bit, so we’ll probably do it again soon.

Base Ten Numbers

This also was better than previous times, although it’s still a bit tricky switching from counting by 10’s on the rods to counting by 1’s on the cubes.  The challenge problems for me ranged from 69 to 382.

Math Jeopardy

The Activities

1. Topic: Multiplication: Book: Mulitplying Menace: The Revenge of Rumplestiltskin by Calvert.

2. Topic: Many. Math Jeopardy.  Here are all the questions and pictures you need to play this.

We divided the kids into two teams and played a variation of Jeopardy. The categories were Multiplication, Estimation, Patterns, Tangrams, and Algebra.  The first team to write down their answer and raise their hand got to guess. If the first team was wrong the second team got 2.5 minutes to answer.  If they were wrong, then the first team got one last chance to guess. This way the teams are never just waiting for someone to answer.

My daughter working on a Tangram question.

My daughter working on a Tangram question.

The Jeopardy Board

The Jeopardy Board

How did it go?

We had 4 kids this week. The younger circle was cancelled because many of the kids were out of town, so my son was the score keeper for the big kids circle.

Rumplestiltskin

This book has a lot of story, and little bit of multiplication mixed in.  All the kids were really into it, and my daughter asked if she could have it in her room at bedtime.

Jeopardy

We divided the kids into two teams of two, and explained the rules.  I think none of the kids had ever played trivia games before, so they didn’t know some basic strategy: for example, if the first team guesses wrong, the second team should take plenty of time before answering, to be sure to get it right.

Team 1 started out by getting pretty far ahead. This is mainly because one kid was really fast on all the multiplication problems, answering all of them except the 500 point one. No one got that one…it was 101 * 37.  Team 1 tried to do it by writing down 101 thirty-seven times, but they ran out of time. Team 2 tried to do it using base 10 blocks, making 37 piles of 101 each.  I really thought Team 2 might realized that 37 one hundred squares makes 3700, but they didn’t.

Both teams did very well on Algebra, with several kids being very close when the right answer was given.  They didn’t get to the 500 question.

Estimating was hard for the kids. The teams solved the first two by counting each object.  From the 300 onward they tried to estimate, but were never close enough to score points.

The 100 Tangram was pretty hard because the kids assumed our Tangram pictures would be to-scale. We had to give a couple hints for that one.  The kids did much better on the 200, 300, and 400, but ultimately the one kid from Team 2 solved all the Tangrams.

Patterns was a very close category, with multiple kids figuring out what the pattern was, but Team 1 was faster at writing down the answer. We only did the 100, 200, and 300.  The 300 was the hardest: Nov, Oct, Sept, Aug.  Team 2 guessed that the next three would be Sept, Oct, Nov.  Team 1 realized that it should be the months backward, but ended up guessing July, April, March.

Ultimately Team 1 by a score of 1900 to 1600.  Everyone was a good sport, though my daughter had started to get upset when Team 1 was pretty far ahead at the start (because they did all the multiplication problems early).

Overall this was very fun and motivating for the kids, and we’ll have to do it again!