# 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.

## 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.

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 🙂
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.

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.

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.

#### 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.

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.

# 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).

## 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.

# I Want To Go Last! (Age 7)

## The Activities

1. Topics: Division, Primes: Book: The Number Devil by H. Enzensberger, first half of third chapter.
2. Topics: Primes, Multiplication: Following the chapter from the Number Devil, each kid did a sieve of Eratosthenes up to 70.
3. Topics: Games, Probability: Using percentile dice (two 10-sided dice which together roll a number from 0 to 99), we played this game: going around the circle in turn, each kid picks a number.  I roll the dice, and whoever is closest gets a point (if there’s a tie, each kid gets half a point).  After doing this a few times, we did the same thing except that instead of rolling the dice, we computed how many numbers would make each person win, and they got that many points.  E.g., if the numbers were A: 10, B: 45, and C: 85, then A wins from 0-27 for 28 points, B wins 28-65 (tie on 65) for 37.5, and C wins 65-99 (tie on 65) for 34.5 points.

## How Did It Go?

We had 4 kids this week.

#### The Number Devil

This chapter talks about the connection between multiplication and division, and about prime numbers.  It introduces the sieve of Eratosthenes.  One interesting thing that came up is one of the kids, who knows division already, first said that they hadn’t done division this way before, but then later said that they probably knew this way of doing it because they knew how to do division.

#### Sieve of Eratosthenes

We’ve tried this before, and this time the kids were definitely better.  But some of the kids still made multiple mistakes, particularly when counting by threes.  I tried to explain why it makes sense to cross out every third number, but I’m not sure they fully understand that counting by 3’s gives you multiples of 3.

#### Dice Guessing

The number picks were pretty random for a while; one of the kids guessed lucky numbers, and most of them liked to pick larger numbers.  They did all realize they should pick between 0-99.  After a bit, one kid realized that guessing right next to another guess might be a good idea — but it then backfired on them when the next person did the same thing.  They soon decided that they all wanted to go last — with good guessing, it’s not an advantage to go last, but with the way they were guessing, it definitely was.  I had initially planned to use the dice the whole time, but quickly realized that the variance was too high — one of the kids was winning by a sizable margin despite not having made the best picks.  So I switched to giving points based on number of ways to win (I did all the calculations, it would have been hard for them).  Some of the kids understood this pretty well, but some of them were pretty confused and didn’t know what I was doing.  For one thing, they hadn’t seen notation like 45-58 before, and the idea of writing down all the numbers that would win for them wasn’t obvious.

# Monsters, Bears, and Division

## The Activities

1. Topic: Prime Numbers. Book: You Can Count On Monsters by Schwartz. This is one of our family’s favorite books. Each page shows a number, starting with 1, and then if the number is prime, there is a picture of a new ‘monster’ representing that number. If the number is composite, the picture shows the factor monsters squished together.

A.  Read the first 15 or so pages of the book.  Count the dots, and look at the picture to see if the page is for a new prime monster, or which factor monsters are in the picture.

B. Give the kids print outs of the 1, 2, 3, 5, and 7 monsters. They can glue the numbers onto the paper to make their own composite number. When they are done, multiply all the factors together and tell the kid the number the made.

My son gluing monsters.

My son’s finished picture.

2. Topic: Numberline, Number Recognition.  Number guessing. The kids get seven guesses to get my number, which is between 1 and 40.  For example, they may guess 25, and I would say, “No, it’s lower than 25.”.   We had a number line on the table so the kids could cross off the numbers that are not possible.

The theme of this game was a bear who wants to steal our picnic food. If you don’t get the number in time, he steals one snack.

The bear is racing toward the snacks!

3. Topic: Symmetry, Division. Book: Rabbit and Hare Divide an Apple by Ziefert.  This is a funny book where two bunnies try to divide food evenly, while a ‘helpful’ raccoon eats it up.

4. Topic: Symmetry, Division. Divide a geometric shape into equal pieces. For example, divide a square in half, or in 4 pieces.

## How did it go?

We had just 3 kids this week…Everyone behaved pretty well, though my son had to sit out briefly for telling another kid to ‘shut up’.

#### You Can Count on Monsters

The kids all enjoyed looking at the pictures and seeing which monsters were on each page.  They also liked counting the dots on the page to make sure it matched the number.  After about 15 pages, they started to get restless, so we skipped to the end of the book, since one kid wanted to know what the biggest number in the book was.

The kids also liked making the monster picture. Each kid had a different style. My son just glued every monster he could reach onto the page. Another kid glued on 12 three monsters, and the last one folded the paper carefully and glued on just 4 monsters.

As the kids finished, I used Google to multiply together all their factors. The numbers ranged from 175 to 1.1 billion.

#### Number Guessing to Save the Picnic

The kids like the excitement of the bear sneaking up to get the snacks.  This activity was really good number recognition and numberline practice for them. For example, if I say the number is higher than 15, then what numbers do you cross off the numberline?

The kids did not have any strategy for how to quickly find my number. The numbers ranged from 1 – 40.  One round started with one kid guessing 40, and the next kid guessing 1.  None of them noticed that those questions weren’t very helpful.  I did later point out when a guess allowed us to cross off lots of numbers. Eventually they’ll see strategy, but for now they’re just getting used to numbers.

#### Symmetry and Division

The kids enjoyed this activity, and did some interesting things like splitting a triangle non-symmetrically.

# Kakuros are Surprisingly Hard!

I led the big kids’ circle this week. 5 kids attended.

## The Activities

1. Topic: Primes: Book: The Number Devil, by Enzenberger.

2. Topic: Measurement: Measuring the kids bodies with ribbons. We measured the kids’ ears, wrists, hands, feet and height with ribbons, and glued the ribbons onto their charts. We discussed measurement error and compared different kids’ measurements.

My daughter’s measurement chart.

3. Topic: Patterns, Fibonacci Numbers, Graphing: I showed the kids the start of the Fibonacci pattern: 1, 1, 2, 3, 5, 8, 13, 21, and asked them to figure out what came next. Then we used graph paper to draw the Fibonacci numbers as squares of each size.

One of the kids’ Fibonacci drawing

4. Topic: Logic Puzzles, Kakuro: I gave the kids a sheet of Kakuro puzzles and we worked together to solve them.  Kakuro is similar to Sudoku.  You fill in the boxes with numbers that sum to a given value (like 6), and you can’t use the same number more than one in a single column or row.

## How did it go?

#### Book: The Number Devil.

This week we read about large prime numbers, and the fact that between any number and it’s double is at least one prime.  This was really over the kids’ heads and they got confused and distracted. I think it may be time to take a break from this book and come back in a few months or so.

#### Body Measurement

Two parents and I used ribbons to measure the various body parts of the kids.  This went fine except for general loudness and high-spirits from the kids (which is no problem).  We noticed that several kids ears and wrists seem to have shrunk.  The kids said they did not think they had really shrunk. They mentioned that perhaps the parent that was doing the measuring had done it differently this time.  We discussed the idea of measurement error, and I said maybe this meant they hadn’t grown very much since the last time.

While their hands, wrists and feet were not much bigger, all the kids had grown noticeably since last July. Their growths varied from 2cm to 4 cm.

#### Fibonacci Numbers

I wrote the first few Fibonacci numbers on a sheet of paper: 1, 1, 2, 3, 5, 8 and asked the kids what should come next. First they suggested it should start back over at 1, but I said it was not a repeating pattern.  Then they guessed various numbers like 9, 10, etc with no particular reason.  Someone suggested maybe it was counting by 3, but another kid pointed out that the gap between the numbers was not always the same.

Above the original pattern, I helped the kids write the gap between numbers. We saw the first gap was 1 then 1 then 2 then 3 then 5.  The kids then caught on that the gap followed the same pattern.  We extended the pattern: 13, 21, 34, 55.  I expected these sums to be easy for the kids at this point, but all of them had some trouble with these.  Looks like it would be good to bring back the Base 10 blocks and practice some more.

After this, I got out some graph paper, and we drew a graphical version of the Fibonacci numbers. We drew a 1×1 sqaure, 1×1, 2×2, 3×3, in a spiralling pattern.  The kids did pretty well drawing the squares and figuring out where the next one should go. However, I wasn’t paying enough attention and we ended up not making the spiral.  Therefore I cancelled the second half of the activity, which was to use compasses to draw the spiral on top of the squares, and moved on.  A few kids complained because they saw the compass kits but we didn’t use them.

#### Kakuro Puzzles

I handed out a sheet of Kakuro puzzles that focused on adding numbers up to 10.  I had never done a Kakuro, but expected to be able to quickly learn it, since these were aimed at kids.  Actually it was much harder than I expected to come up with strategies.  I sat with a few of the kids and we wrote down some possibilities: this row adds to 3, so it can be 1,2 or 2,1.  Some of the kids followed along with me, some tried out their own guesses, and some were just confused.

After I had figured out 3 of the numbers in the puzzle (which had ~10 numbers), I filled them in in all the kids sheets. At that point one kid finished solving the puzzle, and that encouraged several others to have another try.  Finally they all ended up with the right answer.

We moved on and did one more puzzle. Some of the kids understood the puzzle by the end, but a couple others didn’t quite get the idea.  I think I’d be able to teach it better next time.

#### Paying Attention

2 or 3 of the kids were pretty rowdy this time, not listening to me, or not giving back the materials when it was time to move on.  I stopped circle in the middle and discussed this with the kids. They agreed that the purpose of circle is to learn Math, and we should not distract other kids. They suggested that kids who were being disruptive should get a warning, and then sit out of circle for a few minutes to calm down.  I’m planning to implement this next circle.

# A Circle Circle

Happy 2015! This was our daughter’s first circle of the year, and our son’s first circle ever.  We now do 2 circles each week, at the same time.  We alternate leading each circle.  To keep the kids focused, we moved the big kids circle upstairs. Oddly enough, the only room with enough space is our master bathroom.  We bought a folding picnic table and benches, and it worked out really well, despite being slightly strange.

The Bathroom Picnic Table

## The Activities

1. Topics: Numbers, Primes: Book:  The Number Devil by Hans Magnus Enzensberger.  We read most of chapter three.

2. Topic: Venn Diagrams:  First I made a Venn diagram with necklaces as the objects, some with emeralds, some with rubies, some with both, some with neither.  I asked them questions such as “How many necklaces do not have rubies?”  Next, I had them draw an abstract version with one number per region (i.e., a region is labeled “4” if it contains 4 necklaces).  Finally, I gave them an abstract diagram for a new problem: superheroes who could fly, had super strength, both, or neither.  I gave each kid a sheet with a bunch of small Venn diagrams, and then when I asked a question, they had to fill in the appropriate regions which contribute to that answer.  For example, “How many heroes are super strong?” means filling in the two regions inside the “Strong” circle.  Materials available here.

3. Topics: Geometry, Geometrical Drawing, Angles:  We supplied one geometry kit per kid, which contains a small ruler, 90-45-45 triangle, 90-60-30 triangle, protractor, and compass.  I explained that you could use a ruler for measuring distances; drawing lines of a certain length; and connected two points.  Each kid made a constellation by connecting random points.  Next, we traced the 90-45-45 triangle and measured the angles using the protractor.  Finally, we practiced drawing circles with the compass.

## How Did It Go?

We had four kids this week.

#### The Number Devil

This chapter was a bit harder than the previous ones, because it talked about division and only one of the kids was able to readily do simple divisions (she knew 17 / 2 = 8 ½).  There was an interesting discussion of why you can’t divide by 0, but it was kind of tricky — I think the kids might have been able to understand it if we spent a whole activity on it, but they didn’t get it in passing.  Then there was the prime number sieve.  One of the kids pointed out you could cross out all the evens for 2.  I think a couple of the kids were a bit bored during the division part but they were all paying attention for the sieve (which also had some nice charts).  Everyone wanted to finish the chapter but we needed to move on.

#### Venn Diagrams

After I showed them the necklaces picture, I asked what kind of chart it was, no one remembered, but when I said “Venn diagram” someone said “Oh yeah”.  I asked them a bunch of questions such as “How many necklaces have emeralds?”  “How many have emeralds or rubies?”  They did quite well with only a couple issues.  They even were able to do “How many have either both rubies or emeralds or neither?”  The kids got the idea of coloring the small diagrams pretty quickly, and we did a bunch, going both ways (I say something, they figure out the regions; I color some regions, they tell me what it means).  One of the kids got behind because she was coloring so carefully (she colored the picture above).

#### Geometry Kits

Using a ruler to connect two points was new to them, so we did the constellation activity to practice.  At first, they were inclined to connect the dots freehand.  When I showed them the 90-45-45 triangle, I asked if anyone knew what a right angle was, and one kid said yes and made a big L shape with her arms (one straight up, one out to the left).  Measuring with a protractor was kind of hard to grasp, I think a couple of kids kind of got it but I’m not sure about the others.  As expected, it was fairly hard for them to draw perfect circles using the compass, but they were very impressed by my circles.  One kid really liked the compass and didn’t want to stop.