Valentines Jeopardy! (Age 9)

The Activities

  1. Topic: Money. Book: The Story of Money by Maestro. This book traces the history of money from the earliest people to present day. We read until the Lydians invented the first coins. Both kids were really interested in this book, and didn’t want to stop reading. We had various interesting discussions, for example: what would happen if someone needed a blanket, but the blanket maker didn’t want any eggs.
  2. Topic: Story problems, coordinates, money, combinations. Valentines Jeopardy. We had 4 categories with 5 questions in each category. The questions were worth 100 – 500 points, with the higher point values being harder.  Our categories were “Broken Hearts”, “Time for Love”, “Map of My Heart”, and “Valentines Store”. Here are all the questions and answers.

Valentines Shop

Valentines Shop
Stickers…………12 for $2
Toys…………….5 for $3
Cards…………..25 for $4

Each Valentine is made of 1 card, 1 toy, and 1 sticker.

100: How much do 3 Valentines cost?
200: How much do 11 Valentines cost?
300: How much do 25 Valentines cost?
400: How much to 26 Valentines cost?
500: How much do 100 Valentines cost?

Time for Love

100: Katie sang a love song to Alex. She started singing at 5:22AM, and sang for 1 hour and 34 minutes. What time did she stop singing?
200: Fluffy bunny loved carrots so much she hopped around the garden with joy. Each hop was 2 feet long. She hopped 10 times per minute for 6 minutes. How far did she hop?
300: Luke has been waiting for Valentines day since December 8th. How many days did he have to wait?
400: Sam loves candy hearts. A pack contains 30 hearts, and it takes Sam 3 minutes to each one pack. How long does it take same to eat 5 hearts?
500: Corey loves numbers. She started at 5 and counted by fives for 30 minutes. She said one number every 2 seconds. What number did she end on?

Broken Hearts

100: You have 2 colors. How many ways can you color in a heart split into two sections?
200: You have 4 colors. How many ways can you color in a heart split into two sections?
300: You have 4 colors, and each heart has to use two different colors. How many ways can you color a heart split into two sections?
400: You have 2 colors. How many ways can you color in a heart split into 5 sections?
500: You have 3 colors. Each heart much use each color. How many ways can you color a heart split into 3 sections?

Map of My Heart

What word do the letters at the given coordinates spell? Starting at 300, the words are scrambled.img_20170212_181129

100: (7, 17) (11, 19) (3, 12) (16, 10)
200: (20, 1) (16, 10) (8, 5) (18, 3) (2, 20)
300: (2, 2) (8, 5) (8, 2) (9, 11) (6, 21)
400:
(16, 10) (4, 6) (18, 7) (7, 17) (9, 11) (13, 1) (15, 5) (8, 5) (2, 20) (16, 10)
500: (19, 13) (5, 10) (2, 2) (18, 7) (6, 21)

How did it go?

We only had two kids in the circle, which was unlucky, since competitive activities like jeopardy usually go better if you have teams. Otherwise there can be too much pressure on individual kids. My daughter had an especially hard time with the competition aspect, especially after she fell behind early. She started ripping up all the materials and crying in between questions, but refused my attempts to turn the activity into group problem solving instead of a competition. Here’s the room after the activity was done. Notice all the ripped up paper bits strewn around.

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Ultimately my daughter came back from a 1400 to 100 deficit, to win 2000 to 1900. The other kid was a great sport throughout the activity. She answered 7 questions correctly, compared to 6 from my daughter, but the point value was a bit lower.

The questions were just about the right difficulty. They had to work hard for the 500s.

Time for Love: they missed the 300 and the 500. They were close on the 300, but pretty far away from being able to solve the 500.

Valentines Shop: My daughter solved the 100 – 400, but could not compute the 500 (how many 12s make 100?). The other girl was uncomfortable with this category, even though I worked through each problem right afterward to show how it goes. I think she felt overwhelmed by having to compute how many packs you need to buy for each of 3 objects.

Map of my Heart: The other girl solved 100 – 400 very quickly. She was able to guess the Valentines words from just a couple coordinates. For the 100, she guessed the answer was “LOVE” after seeing the L and that the word was four letters long. The 300 was scrambled (CANDY), and it took both girls a while to figure it out. The 400 went quickly, guessed before all letters were searched.  My daughter got the 500 (CUPID), which was the trickiest word to unscramble.

Broken Hearts: I thought this wouldn’t be that hard, but neither girl knew how to compute color combinations through multiplication. They wanted to enumerate the colors. They only answered the 300 correctly. This was because I had enumerated the 16 options for the 200, and my daughter realized she just needed to remove the double color choices to get the 300. (12).

At the end of circle all the kids got a chocolate covered strawberry that me and my daughter made this afternoon.

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Happy girl, before tragical jeopardy.

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A Trick-or-Treat Circle (Age 8)

The Activities

  1. Topics: Proofs, Time, Logic:  I asked the kids to determine whether or not every year has at least one Friday the 13th.
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  2. Topics: Geometry, Time:  I made a map of a neighborhood for trick-or-treating.  The red house is your house.  Each inch of road is one block, and trick-or-treating along one block takes ten minutes.  First, I asked them how long it would take to trick-or-treat on every block (this requires repeating a few blocks).  Next, I asked what the most number of different blocks you could visit if you had 3 hours.  Finally, I asked how many different blocks you could visit if you had to return to your house to drop off candy every 5 blocks (i.e., after 40 houses).  A bonus question I didn’t get to was, if you wanted to minimize the time to visit every block, and you didn’t have to start at your house, where should you start?
  3. Topics: Combinations, Combinatorics, Logic:  I had a list of 10 possible trick-or-treaters:
    1. Evil Queen — Baddie, Girl
    2. Bride of Frankenstein — Baddie, Girl
    3. Vampire — Baddie, Boy
    4. Mummy — Baddie, Boy
    5. Princess — Goodie, Girl
    6. Fairy — Goodie, Girl
    7. Wizard — Goodie, Boy
    8. King — Goodie, Boy
    9. Alien — Neither, Neither
    10. Slime — Neither, Neither

    First, I asked how many ways there were to pick three trick-or-treaters.  Then I asked how many ways to pick three trick-or-treaters, with the requirement that there’s at least one Baddie, one Goodie, one Boy, and one Girl.  Note: Picking groups is much harder than picking ordered line-ups (where Evil Queen, Princess is different from Princess, Evil Queen).  If I were doing this again I would stick with ordered line-ups, it’s hard enough already.

How Did It Go?

We had all five kids this week.  This was a pretty hard circle; 2 of the kids were engaged through-out, with one saying how they liked the hard problems; the other 3 were distracted a lot of the time.

Friday the 13th

This is a pretty tricky problem, it’s not immediately obvious how to do it even for adults.  The kids made some good progress and had some interesting ideas.  First, one kid figured out that for there to be a Friday the 13th, the 1st had to be a Sunday.  Another kid wrote down the years starting with 2000 (she wanted to check “all the years”).  I used my phone to look up the calendars for each year, and we checked which months had a Friday the 13th each year.  One kid was really excited to try to find a year with no Friday the 13th, because then they’d be done.  But there is indeed a Friday the 13th each year, so we didn’t find one :).  At this point, I gave them a hint, which is to draw a pie chart like in the picture above.  The idea is to go through an entire year starting with January, assume that the 13th in January is, say, a Sunday, and then figure out what day of the week the 13th is in each month.  If you do this, you’ll find that every single piece of the pie is filled, which is what you need to prove that there’s always a Friday the 13th.  Unfortunately, the kids were not good at doing the calendar arithmetic to figure out what day of the week Feb 13 is given the day of the week for Jan 13.  So, we didn’t get that far, and since we had already spent 25 minutes I moved on to the next activity.  Most of the time, two of the kids were working on the problem while the others were drawing, etc.

Trick-or-Treat Optimization

The kids liked the theme of optimizing trick-or-treating.  Unfortunately, I made an error in how I set up the problem.  My intention was that they should concentrate on how many blocks you’d have to walk, but I drew the houses big enough that they focused on visiting houses instead of walking along blocks.  The map I included above I redid afterwards to make it clearer that it’s about blocks, not houses.  The problem with houses is that if you have houses on the corners of streets, it makes the counting a lot messier.  And counting houses is a bit more intuitive, so that’s what they defaulted to.  The result of this was that about half the kids thought I meant that it took 10 minutes to visit three houses, when I actually meant it took 10 minutes to walk one block.  All the kids paid attention during this activity.

The kids figured out that you’d have to backtrack or at least revisit some blocks.  They were all pretty comfortable with figuring out how long it would take to visit all the blocks, but the idea of the best route wasn’t as compelling.  They did understand the idea of visiting as many as possible in 3 hours.  The final problem, about returning home each time, isn’t actually that interesting with the map I had, but they still had to think about it some to figure out how to do it.

Picking Trick-or-Treaters

This problem turned out to be harder than I expected.  I just forgot that they weren’t that comfortable with combinations yet.  Even if I had done the ordered line version, they still didn’t immediately remember how to do the multiplication to figure out the answer to the unconstrained version.  They did figure out this part, and we moved on to the constrained version.

I actually gave them a four person version that required 2 baddies instead of 1 — it turns out to be a lot harder than the three person version.  Also, the non-ordered version is a lot harder to think about.  With the three person version, it’s not so bad to reason along the lines of “Let’s pick the baddy first, and the goody second.  For each of the possible combinations (there’s only 4 distinct ones), we can figure out what the third person can be.”  The four-person version gets a lot more complicated, so I switched to the three-person version — we made some progress but didn’t solve it.

Again, two of the kids worked hard, while the other three were distracted.

Leo the Rabbit (Age 8)

The Activities

  1. Topic: Logic. Book: Still More Stories to Solve by Shannon, Stories 11 – 14. The kids absolutely love this book of brain teaser stories, like what can you say to your two enemies to make them fight each other and leave you alone? Or how can a man get two wishes fulfilled when the genie only grants one wish? We spent about 25 minutes discussing the four stories we read. Most of them we could not solve on our own, but I would read the answer and give hints. Everyone understood the answers at the end.
  2. Topic: Logic, Combinations. We got this problem from the awesome site YouCubed.org. Leo the Rabbit is at the top of a staircase of ten steps. Leo can h0p down either one or two steps at a time. How many different ways can Leo hop down the stairs?
  3. Topic: Counting, Geometry. How many rhombuses are there in a heart made out of the YouCubed logo?

 

How did it go?

This was our first circle in a month, due to traveling. All five kids attended. This was a very high-energy circle, especially for my daughter who was having trouble staying on task. For each activity there were a couple kids complaining they were bored, but also at least a couple who stayed interested and learned something. I had a lot fun actually, because I intentionally didn’t solve the Leo the Rabbit problem ahead of time, and it was exciting to figure it out during circle.

Leo the Rabbit

First we started by drawing the rabbit at the top of a set of 10 stairs. We assigned a letter to each stair, and then each kid wrote a bunch of letter sequences representing the hops the rabbit makes. Kids came up with about 10 paths each before they started to want to find a faster way. My daughter suggested that you could first find all the paths that start with AB, and then all the paths that start AC.  I used this as a starting point. I asked the kids to consider just the last three steps in the staircase. If Leo is on step H, how many ways can he get to the bottom? Several kids were able to enumerate the 3 possibilities: HIJ, HI, or HJ.

Then I added step G. Now how many ways?  I pointed out that if the rabbit hops to step H, then his choices are now the same as the three ways we found for step H, namely: GHIJ, HI, GHJ. But Leo could skip step H, so we have to add in GIJ and GI as possibilities. Some kids understood this, but most did not. So I started even simpler.

What if there is only one step, step J? Then there is only one choice: J.

I ended up drawing a picture similiar to this:

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At least one kid really seemed to understand that to get the ways for Step N, you add together the ways down from N-1 and N-2 (since Leo could hop down to either of those). All the kids soon saw that to fill in the next step, you should add the numbers from the two steps below, but many of them probably did not fully understand why. We were all impressed to get 89 ways, and were glad we didn’t try to enumerate them all.

Everyone started out quite engaged during this activity, but people started dropping off and getting distracted. In the end, 3 of the kids were still paying attention and 2 were quite ready for the activity to end.

YouCubed Heart

I intentionally made this activity much easier. YouCubed has a number of interesting questions about the picture, but I just asked how many rhombuses there were, and then let them color the picture for the last five minutes of circle.

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Mother’s Day Origami (Age 8)

The Activities

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

How Did It Go?

We had four kids this week.

Three Little Pigs

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

Origami

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

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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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.

Spending a Million Dollars (Age 7)

The Activities

  1. Topics: Multiplication, Division: Perfectly Perilous Math, Challenge 3 — if you spent 50 cents every second, how many days does it take to spend 1 million dollars (to nearest day).IMG_1742
  2. Topic: Combinations: We continued the pumpkin activity from last week.  Last week, they had made a bunch of pumpkins but hadn’t checked for duplicates.  This week, we tried to come up with strategies for finding all the duplicates.  Also, I started with the simplest version of the problem (circular eyes, nose, and mouth), and then gradually added elements (non-symmetric parts, multiple choices), computing how many different combinations there were each time.
  3. Topics: Logic, Measurement: Perfectly Perilous Math, Challenge 4 — if you have one bucket that holds 3 quarts, and one that holds 5, how can you measure exactly 4 quarts using just those two buckets?  We worked as a group to solve this problem using various containers and beads (instead of water).

How Did It Go?

We had all five kids this week. All the activities this week were group activities, which turned out to be a problem. Most of the time, only two or three of the kids were actually working on solving the problem; the others were drawing, making paper airplanes, or otherwise not paying attention. Kid A grabbed and crumpled another Kid B’s paper, and later in circle (unrelatedly) Kid B threw a container of beads at Kid A, which we all had to help pick up. I think another issue was that the problems were all pretty hard; the kids working on them did a good job, but it discouraged the other kids.  One of the kids who drew during the Million Dollars activity said they couldn’t help because they didn’t know how to do multiplication, and then was very attentive for the remaining two problems.

Spend a Million Dollars

One of the kids is a lot better at large multiplication than the others, so that kid did most of the computations.  Another kid helped figure out what the right numbers to multiply were.  The kids figured out how many seconds were in a day, but then they were planning to multiply by 50 (cents) — they didn’t realize that 50 cents was half a dollar.  They also needed help figuring out how to divide 1 million by 43,200 — they know the method of repeated addition/multiplication, but I think their number sense suffers a lot past 1,000.  They got pretty close to getting the right answer, but I had to a help a lot for the final few steps.

Pumpkin Combinations

One of the kids suggested sorting the mouths by orientation, and then by type of mouth.  A different kid decided to implement this, but changed it to have 1 column for each of the combinations of 4 mouths.  Some of the other kids helped sort (there were about 35 pumpkins total), and eventually the pumpkins were all in columns.  There were still 8-9 pumpkins in each column, and they mostly stopped making progress at this point, besides finding a few duplicates ad hoc.  I suggested putting the circle eyes above the triangle eyes, but they didn’t take to this idea.

Then I switched gears a bit and started with a problem with circle eyes, nose, and oval mouth.  This meant there was only one possible pumpkin, which the kids figured out right away.  Then I made the nose triangular.  There were answers of both 2 and 4; the 4 answer was reasonable because they showed how you could put the straight edge in any of the 4 directions (although two of them make pretty weird noses).  I said we should only allow the two directions, and then added triangular eyes.  They got 4 right away.  Then I added a smiley/frowny mouth, and it got much harder.  I got answers of 4, 6, and 8.  6 is an interesting answer, because you could think of that if you considered varying each dimension one at a time.  I had the kids try to write out all the combinations — I really should have had them to all 4 for the previous problem first, because the key insight here is that if you group the faces by smiling vs. frowning, you have the same set of 4 faces from the round mouth case, once for each mouth.  In the end, one kid was able to figure out that there should be 32 different answers for the previous week’s problem, but wasn’t able to figure out which faces were still missing.

Tricky Buckets

The beads worked okay, although one of our containers was more like 2.6 than 3, so the measurements didn’t quite come out right.  Also, we ended up picking up a bunch of beads because of the bead-throwing incident.  One of the kids figured out a solution right away using a third bucket; and then they were able to come up with the 2-bucket solution with only a bit of help.  I asked a follow-up question about a 5 and 7 bucket (trying to make each of 1, 2, 3); the kid who had done the best on the first part was able to make progress and solved 2 and 3 but not 1.