Number Magic (Age 8)

The Activities

This whole circle is built from activities described in the book Games for Math by Peggy Kaye.

  1. Topic: Reducing Fractions.  Book: Fractions in Disguise by Einhorn. A millionaire collects fractions for fun, but then a villain steals a rare fraction and tries to disguise it. Only reducing the fractions to their true values can find the lost fraction.
  2. Topic: Addition, Subtraction, Number Properties. I performed a math magic trick. Each kid picked a three digit number where no two digits could match, e.g. 581. Then I turned all their numbers into 1089 by:
    1. Reverse the kid’s number.
    2. Subtract the smaller number from the larger. (e.g. 581 – 185 = 396).
    3. Reverse the result and add it to the result (e.g. 396 + 693 = 1089).IMG_20160814_175721
  3. Topic: Logic, Strategy.  I taught the kids “Tapatan” a tic-tac-toe like game. Each person takes turns placing one of their three stones on the board. After all six stones are placed, you take turns sliding the pieces from point to point along the board lines. You cannot jump over another piece or land on top it. The first person to get their three stones in a line (vertical, horizontal, or diagonal) wins.IMG_20160814_175308
  4. Topic: Logic, Addition. I gave the kids a series of ‘number bubble’ puzzles. Place the given digits in the bubbles to make each row add up to the required sum.IMG_20160814_175551
    IMG_20160814_175344

    A solved puzzle, placing 1,2,3,4,5,6 so that each side adds to 12.

     

How did it go?

There were only two kids this week, so it was a good, focused circle. My daughter had a few angry moments, but settled down after some warnings.

Fractions in Disguise

Both girls really enjoyed this book. The mystery of the stolen fraction was quite compelling, but they were each a bit reluctant to spend energy trying to reduce the fractions in the book.

Number Magic

The kids were quite impressed by this trick. Right away they started trying to figure out how it worked. One girl noticed that the middle digit is always 9 after the initial subtraction. Both kids wanted to try again several times.

I told them there is one class of numbers that the trick does not work for. Eventually, my daughter stumbled upon it. If the first and last digits are consecutive, then the final answer will be 198 instead of 1089. For example: 231 – 132 = 99, 99 + 99 = 198. We noticed how the subtraction always results in 99 in this case.

Tapatan

This game proved to be pretty fun. The girls quickly started thinking a move or two ahead to make sure they didn’t let their opponent win. My daughter was quite a poor sport whenever she lost, crumpling up the board, or throwing the pieces. The other girl was very calm during these tantrums. There is a lot more to this game than to tic tac toe. We added one extra rule: you cannot undo a move on your next turn, i.e. you can’t move a stone back to the same place it had been the previous turn. This helps prevent stalemates.

Bubble Logic

Both girls quickly got the idea of these problems, and had some good insights. On the first class of problems, with the 4 bubbles in a cross shape, my daughter quickly noticed that you should always put the middle two numbers together, and the largest and smallest together.

Later, when we switched from the L-shaped 5-bubble puzzle to the cross-shaped 5-bubble puzzle, both girls independently realized that they could reuse their answer from the L-shape. This was a great insight.

We stayed late at circle a few minutes, because both girls wanted to finish all the bubble puzzles.

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Star Wars Battles and Bent Legs

The Activities

  1. Topics: Addition, Graphs, Time:  Book: Get Up and Go! by S. Murphy.
  2. Topics: Logic, Venn Diagrams:  First, we did our standard Venn diagrams activity using fairy tale characters.  The two problems we did were “Animals” and “Magic”; and “Scary Things” and “Girls”.  Then I shuffled up the cards and flipped over sets of 4 (later 6) cards, and the kids had to come up with as many different ways to group the cards into two groups, with explanations.

    IMG_2031

    Just Animals vs. Not Just Animals

  3. Topics: Simulation, Charts: I introduced “Star Wars battles”.  The idea is that you have two characters, each with a certain amount of Attack and Health.  Each simultaneously deals damage to the other, and when you get to zero health you are knocked out.  The battles end in a tie if both are simultaneously knocked out.  They also can have armor, which reduces the damage received by one each time.  I designed 6 characters (we happened to have figures for all of them): Kylo Ren (4 attack, 4 health), Rey (2 attack, 8 health), BB-8 (1 attack, 12 health), Flame Trooper (6 attack, 1 health), Finn (6 attack, 6 health), and Phasma (1 attack, 9 health, 1 armor).  We used glass beads to keep track of health, and the kids took turns setting up and running the battles.  We started with the first 4 characters and played all pairs; I kept track of the results on a chart (see picture).  Then we added Finn, did all the pairs with him; and then Phasma.  Then we figured out the win-tie-loss records for each character and compared them.  Finally, I asked them whether they could make a character that tied with Finn.

How Did It Go?

We only had two kids this week; as usual things were easier with such a small group.

Get Up And Go!

A straight-forward book about getting ready in the morning, adding up the time for each individual activity in order to get the total time to get ready.  I gave each of the kids a worksheet to take home and fill out for their own routine.

Venn Diagrams

It’s been a while since we did Venn diagrams, one of the kids remembered them pretty well but the other was rusty.  The fairy tale Venn diagrams is always fun because the kids have to decide what’s an animal, what’s magical, what’s scary, etc.  This time, the gingerbread man wasn’t magical, for Pinocchio: Child: “Is this magical?” Me: “It’s a living puppet.”  Child: “Ok, no.  Wait, yes?”, ogres and trolls are animals.

Grouping the cards is also fun.  I stumped them once by grouping a cat, bear, and wolf together vs. a dragon, goose, and frog.  They often went for very small bits of color when grouping cards.  My favorite was “bent legs”, when “legs” would have accomplished the same split.

Star Wars Battles

Our son has been doing something similar outside of circle on his own, so naturally he loved it.  The other kid also liked it quite a bit.  It took a bit of time for the other kid to catch on, but by the end both kids could run the battles smoothly using the glass beads.  With the stats I picked, it’s pretty interesting because it’s quite non-transitive: Finn has the best record (4 Wins, 1 Tie) while the Flame Trooper has the worst (2 Ties, 3 Losses), yet Finn and the Flame Trooper tie.  For the final question about tying Finn, our son was able to figure out that he would tie with a character with 1 attack and 36 health, because 6 * 6 = 36 damage from Finn.  Pretty nice!

Reading the chart was somewhat tricky, so only one of the kids followed the second part about calculating the records for each character and comparing them.

91 Is Not Prime (Age 6)

The Activities

  1. Topic: Addition: Book: Mall Mania by S. Murphy.
  2. Topic: Primes: As a followup to last week, I made bags of 65, 91, and 95 unit cubes, gave one bag to each of the three kids, and asked them to prove that those numbers weren’t prime by making them into a rectangle.
  3. Topic: Logic: We did about ten puzzles from Logic Links, numbers 1-8, 50, and 51.  To make a set of pieces for each kid I used Unifix cubes and printed-out boards.  These puzzles have clues like “There is a blue cube directly to the left of the orange cube.” and you have to figure out the position of all the cubes.  IMG_1892

How Did It Go?

We had three kids this week.  Two of them were somewhat out of sorts, so the group was less attentive than usual.

Mall Mania

This book has a bunch of different interesting adding strategies, so it would be a good lead-in to an addition activity.

Large Composites

Kid 65 recalled that you could get to 65 counting by 5’s, and tried that out right away successfully.  Kid 91 tried a very long (and spread out) 2-wide rectangle.  Kid 95 decided to use the hundred plate as a guide and tried out a 10-wide rectangle.  When 2-wide didn’t work, kid 91 didn’t want to try anything else.  I mentioned that you could check things quickly by skip counting and seeing whether you got your number.  We skip counted 3, 5, and finally 7, but kid 91 wasn’t interested in checking whether you could make a 7-wide rectangle.  Kid 95 worked slowly and eventually found that 10-wide didn’t work.  Kid 91 had noticed that you got 95 counting by 5’s, but kid 95 didn’t see how that helped.  I showed them that 95 did work with a 5-wide rectangle.  Kid 95 had suggested doing the prime rectangles activity with larger numbers the previous week, so it was surprising that the kids weren’t a bit more interested in this activity — might have just been a one-off problem, but it does take quite a while for them to make rectangles with this many cubes.

Logic Links

Even with the L and R printed on the boards, understanding what “The blue block is directly to the left of the red block” means was challenging.  In particular, you really have to pay attention to the order the blocks are mentioned.  Besides that, the kids were good at following the directions individually, and decent at combining all the clues to get the final answer.  However, they definitely aren’t good at abstracting what the clues imply — for example, one of the clues was “There are 3 red cubes.  One of the red cubes only touches red cubes” which means that there must be an L shape of red cubes in the corner.  One of the kids got tired of the puzzles and said they were bored and wanted to stop.  Again, might be a one-off, we’ll have to see how it goes next time.

Who Gets to Jump the Most? (Age 6)

The Activities

  1. Topic: Infinity, Even and Odd. Book: The Cat in Numberland ,Chapter 4, by Ekelund. In this chapter, Mr Hildebrand wants exactly one half of the numbers to go visit the letters. But how do we take one half of infinity? Later, the hotel is half empty because all the odds have gone. How can the evens move down so there are no lonely, empty rooms? We acted this part out with a diagram.
  2. Topic: Addition. The kids took turns throwing bean bags at a number chart on the floor. Then they used based-10 blocks to add up the three numbers. Once all the kids had their sum, we all jumped together, and each kid stopped jumping when we got to their sum. This means the kid with the highest sum gets to jump the longest.Which, believe me, was considered quite a high reward 🙂IMG_20160424_174531
  3. Topic: Primes and Composites. We worked to prove that various numbers between 1 and 100 were prime or composite. To prove something is composite, you should make a rectangle out of that many cubes. Proving something is prime is much harder, since you have to convince me that it is not possible to make a rectangle out of the blocks.

 

How did it go?

We had four kids this week. Everyone *loved* jumping, but my son had to sit out a couple activities because he was bored/wild/frustrated. Everyone else was pretty excited, especially about the beanbag addition.

Hotel Infinity

There were several interesting discussions this week. The hotel owners, the Hildebrands, decide that half of the numbers should go to visit their friends for one week. How do we make sure exactly half of all numbers go? One kid had a great suggestion: send the negative numbers away, and keep the positives. I was very impressed by this idea, but in this book, there are no negative numbers. Someone also suggested sending the numbers away, 50 at a time. I pointed out that this would definitely not be half of the number. And, we had yet another discussion about whether there is a biggest number. I think I (again) convinced them that we can always make a higher number, so there is no biggest.

The next part of the book is about the even numbers being lonely because the odd rooms are empty. Zero figures out that each even number can divide itself in two, and move into that room. We made a diagram of the hotel, and worked to divide the evens in half to find their new room number.IMG_20160424_174658

Three of the kids figured out the new room number by making two equal rows of base 10 blocks. For example, above we see that ’14’ will go to Room 7. Then the kid would write the number in the correct number, and cross it off the list. I gave out numbers in a random order, so it would not be obvious which room each belonged in.

My son got pretty impatient with this, because he could divide the numbers in his head, and just wanted to write them all into the hotel. After 2 or 3 numbers for each kid, we filled in the rest of the hotel by counting by twos and filling in the empty.

Bean Bag Addition

Everyone was excited to throw the bean bags at the chart. All four kids were able to add three numbers between 1 and 30 together, using base ten blocks. There were a few adding mistakes, but these were easily corrected.

After each kid added up their number we discussed who had the highest and lowest numbers, and then we all started jumping. When we got to each kid’s number, they had to stop jumping. At the end, it was just me and the kid with the highest number who were jumping.  Everyone absolutely loved the jumping part, and desperately wanted to get the highest number. The highest number of the day was 64, and all the kids knew exactly who had gotten it.

This caused some problems for my son, who kept wanting to redo his throws to try to get higher numbers. Also, the adding was too easy for him. Initially I told him to multiply his three numbers, but his first set was 5 * 12 * 12. He knew 12*12 = 144, but 144 * 5 was pretty hard for him. This frustrated him, and eventually he decided he wanted to add his numbers like everyone else was doing.

We played 3 rounds of this game, and the last round we used 5 bean bags instead of 3. The kids all cheered when I asked if they wanted 5.

IMG_20160424_174520

Primes and Composites

Last week David and the kids tested which number 1 – 14 were prime. This week one girl was able to explain that a number is prime if you cannot make a rectangle out of that number cubes (unless one side is just one block wide).

This week, we explored some of the numbers above 14. We tracked our progress on the 100 board, using a blue square to indicate prime numbers and red to indicate composites. I just told them which were prime this time, but made them prove the composites by making a rectangle.

IMG_20160424_174539

The kids started to ask to do very large numbers like 99. Unfortunately it takes forever to count out 99 cubes. In a future week I might do it ahead of time, and let each kid try one big number.

Corey’s Pretend Party IV (Age 6)

The Activities

  1. Topic: Graphs. Book: The Great Graph Contest by Leedy. In this book, a frog and a salamander compete to see who makes the best graphs. They make Venn Diagrams, Bar Graphs, Pie Charts, and others, covering a variety of subjects. Leedy is a great author, and the kids always love her book. This is no exception.
  2. Topic: Voting. Each year around the time of my birthday, I have the kids plan my pretend birthday party. This is the fourth year in a row we have done this activity (twice with the older circle, twice with the younger). This time, I planned to have each kid own one question, and have them collect votes. Then I wanted to make a fancy graph out of each, similar to the graphs in the The Great Graph Contest.
  3. Topic: Adding and Subtracting, Word Problems. The kids used Base Ten Blocks to solve problems about my age (36).
    1. How old will I be in 36 more years?
    2. How many years older am I than you?
    3. How much is my age plus yours?
  4. Topic: Advanced Birthday Math. My son is extremely good at calculation, so I knew he would cause problems during birthday math. I gave him a much harder problem to work on separately: How old will I be in the year 3022?

How did it go?

We had four kids this week. The circle was a bit scattered, partly due to me being less organized and partly due to the advanced birthday math not working out.

Happy Birthday to Me!

The kids all enjoyed coming up the questions for my party. We came up with:

  1. Where should the party be? Pump It Up, Park, Restaurant, Arcade.
  2. What treat should we have? Crackers, Oreos, Cupcackes, Tacos.
  3. What should be in the gift bag? Bouncy ball, Legos, Light Saber, Balloon.
  4. What drink should I server? Water, Apple Juice, Pink Lemonade, Grape Juice

Each kid owned one sheet for voting. Everyone voted on each other’s sheets. The kids made tally marks to record the votes. Two kids remembered that you can cross the fifth tally mark. We had an interesting discussion about why you should cross the fifth one. Some kids said it was because they had seen parents do it that way. I made an example on paper of 15 tallymarks, vs 3 groups of 5 tallymarks. Then one girl said that crossing the fifth meant you could count by fives. We then tried counting the two rows of marks, and found that counting by 5s was much faster.

I had planned to have the kids make graphs out of the votes, but they were getting a bit distracted by the end, and I hadn’t planned it out that well, so I decided to move on to birthday math instead.

Birthday Math

First I gave my son his problem (How old will I be in the year 3022?), then started working with the other three. First I asked each kid to count out 36 Base Ten Blocks. All three of them decided to use 10 bars, so they picked out 3 10-bars and 6 units.

Next I asked how old I would be in 36 more years? The kids had lots of silly guesses, like 1000 years old, but finally one girl suggested we should start at 36 and count up 36 more. I suggested counting out 36 more Base Ten Blocks, and then counting them to see how many there were in total.  The three kids eventually did this, though there was some messing around, and also some mistakes in counting (I got answers of 71, 72, and 73).

Meanwhile, my son had decided the advanced problem was too hard. He was trying to subtract 3022 – 2016 in his head, and had come up with 1008, which I said was close but not correct. This frustrated him, so he crumpled up the paper and threw his pencil. He then rejoined us at the table for a bit until he got too wild and had to sit on the couch for the rest of circle.

Next we each collected 36 blocks again. I asked, how old was I when each of you was born? No one knew how to figure this out. Some people said my age was negative, some people said I was 1000.  I suggested we could go back in time by taking away one block at a time. They enjoyed watching me take one block away saying: Now you’re 4, now 3, now 2, now 1, now 0. Then we counted how many blocks were left and found I was 31 years old when this student was born.

My final question what is your age plus my age? By this time, the kids were quite restless, some were saying it was too boring. But we persevered and figured it out, and we even added all 4 of our ages together. Everyone thought it was funny to think about being 53 years old (the sum of our ages).

 

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.

Pokemon on Parade (Age 8)

The Activities

  1. Topics: History of Math, Mathematicians:  Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians (Volume One) by L. Reimer and W. Reimer, Chapter 4 (Hypatia).  Note: I skipped the ending of this story, only briefly summarizing (see notes below).
  2. Topic: Decision Trees, Addition:  First I reviewed the Pokemon decision tree from last week (a couple kids weren’t there).  Next, each kid “drafted” a team of Pokemon cards.  I split the cards into 5 piles, gave one pile to each kid; they each picked one and passed the pile to the right, and we repeated until each kid had 5 Pokemon.  Then, I gave out prizes based on the attributes of the Pokemon; some were based on the single Pokemon (e.g., most HP), and others were based on the entire team (largest total weight).  You can download the prizes here (the Pikachus are to give out when there are ties).   After all the prizes were awarded, the kids worked together to built a decision tree which identified whose team each Pokemon was on (that is, the labels at the leaves were the kids’ names).  I did the initial few splits until I had 5 different leaves, numbered them 1-5, and then had each kid work on one of the pages.
    IMG_1832

How Did It Go?

We had all five kids this week.

Hypatia

I definitely wanted to read this one because it’s one of the few stories about a female mathematician.  Unfortunately, she was killed by a religious mob.  So I stopped about 2 pages from the end of the story and just summarized as “some people didn’t like that she asked questions about the things they believed in, and so they killed her”.  The kids really wanted to know how she died, but I said “No one knows for sure” (which is true, although it was almost certainly quite unpleasant).  Out of the four mathematician stories, three have ended with the murder of the mathematician — so if you decide to read this book to kids, be aware of that.  I checked out the Wikipedia page for Hypatia, and it’s consistent with the book, so they’re not making things worse than they actually were.

Pokemon Round-up

For the review, I had the kids who were there last week show the other 2 how to use the big decision tree from last week.  They had seen decision trees before, so they caught on very quickly.

The drafting went fine, although some kids were confused about the mechanics of the draft (only take 1 from each pile, don’t mix the ones you’ve already picked into the current pile, etc.)  Most kids picked carefully, but one picked randomly.  Overall, most of the kids were pretty excited by the theme.

The prize giving went well, everyone was motivated even when they had to do somewhat complicated sums.  The trickiest part was computing the total height, because it involved feet and inches.  Everyone got some prizes.  One of the kids really wanted to win the prize with the picture of Heracross (a bug with lots of attack).

I asked the kids whether they wanted to work alone or in pairs, 3 said pairs and 2 alone.  I worked with one of the 3.  They all had trouble getting started, but after some help they made pretty good progress on their own.  One kid noted that they could use a split to identify a single Pokemon that wouldn’t have worked earlier but because of an earlier split was now unique.  Some of the kids were drawing Pokemon by the end, but they did manage to finish the entire tree of 25 Pokemon.  They used a few interesting splits that were different from our big tree, including “Is bird?” and “Name starts with ‘s'”.