How many legs does a starfish have?

The Activities

  1. Topic: Place Values, Exponents: Book: The Number Devil, by Enzensberger.
  2. Topic: Functions, Charts: We made input/output charts for various functions, and then made bar charts out of some of them
    1. Baby(x): cat, dog, person, frog, cow, bear, tiger, table
    2. Opposite(x): up, top, bottom, big, long, far, open

      A student’s chart for Baby(x) and Opposite(x).

    3. Legs(x):  bird, cat, crab, human, spider, starfish, worm
    4. F(x) = x + 1

      My daughter’s graph of Legs(x)

  3. Topic: Venn Diagrams, Logic:
    1. There are 10 students. The students had pizza for lunch. 3 students had only pepperoni on their pizza. 3 students had only mushroom on their pizza. 3 students had no toppings. How many students had both pepperoni and mushroom on their pizza?
    2. Everyone is having ice cream. 6 students have bubblegum, 7 students have vanilla. How many students have both kinds?

      Diagram solving the ice cream problem.

    3. The students brought their parents to school. 4 students brought their mom, 5 students brought their dad, 3 students did not bring a parent. How many student brought both parents? How many students brought one parent? How many brought at least one parent?

How did it go?

We had 3 kids for most of the circle, but a 4th kids came when there were 10 minutes left.

The Number Devil

The number devil. We read the second half of Chapter 2 this week, about place values and exponents. The kids were interested in 5^3 notation to mean 5 x 5 x 5.  I’m not sure how much they really understood the book, but they laughed when the devil yelled at Robert.

The Function Game

At first the kids protested that they didn’t know what functions were, but soon they were really into this.

First we wrote a chart with x, and Baby(x). The girls quickly caught on.  I started with cat -> kitten.  The kids then added:

  • owl -> owlet
  • horse -> foal
  • lion -> cub
  • person -> kid
  • cheetah -> cub
  • manatee -> calf
  • dolphin -> calf

I asked what about table? Kid A suggested that chair was the baby of a table.  I said that didn’t sound quite right, so I just crossed out the Baby(chair) column. Later Kid A suggested that Baby(boulder) = pebble.  I said that was interesting, but didn’t seem quite right.

Next we did x, Opposite(x).  This was harder for the kids. First Kid A proposed blue and red as opposites.  I suggested top, and Kid B said opposite(top) = bottom.  Then we added black -> white, big -> small.  Kid B wanted to add cat -> dog, but we decided it wasn’t quite clear.  The girls did decide to add arms -> legs.  Then Kid A came up with back -> front.  Kid C said she wanted to instead write front -> back.  Then I asked them if we could switch the order of the other rows too, and they agreed we could.

Next I asked Kid A if we could swap the order for the Baby(x) function.  She said no because a cat is not the baby of a kitten.  Kid B said we’d have to change the function name to Grownup(x), so Grownup(kitten) = cat.

Next we did Legs(x).  For example, Legs(bird) = 2.  The kids answered all my inputs easily, except Kid C said Legs(starfish) = 0, not 5.  After circle, Kid C’s dad was looking at her chart and said “A starfish has 5 legs?? It should have 0 legs!”.  I said they could change her chart at home 🙂  We later looked this up on wikipedia, and apparently the starfish appendages are actually called ‘arms’, and each arm has a bunch of ‘tube feet’ on the underside.

After everyone filled in their function table, I handed out graph paper that had the x-axis labelled with the animals from their chart.  The girls then easily made a bar chart out of the Legs function.  It was really no problem for them. 

As the girls finished their Legs chart, I explained the x+1 function to them, and they filled in their table too.  Next we made bar charts out of f(x) = x + 1.  Kid B caught on pretty quickly, but A and C were surprisingly puzzled by what to do with two numbers.  Interestingly, Kid B insisted on labelling the x-axis as ‘x’, which is a good a idea…maybe it would have clarified it for the other kids too.

As the kids finished their charts for x = 0 .. 5, I then helped them add in negative values of x.  At first they wrote f(-1) = -2, but after I asked them clearly what is -1 + 1? they knew it was 0.  I also showed them a number line to demonstrate adding to negatives.  After we had the chart, it was fairly easy for them to fill in the graph with the negative values of x. One kid figured out that negative would mean filling in the boxes below the x-axis. The other kids were able to do it too.

Venn Diagrams

At this point, Kid D arrived at circle. He joined in the next activity.

First I started by giving the kids 10 unit blocks each.  Then I told them we had a tough logic problem about 10 students.  The kids all said that the 10 blocks would represent the students.

 Then I read the first problem:  “There are 10 students. The students had pizza for lunch. 3 students had only pepperoni on their pizza. 3 students had only mushroom on their pizza. 3 students had no toppings. How many students had both pepperoni and mushroom on their pizza?”

Everyone was initially confused, but then I read the problem again, and some of the kids started making groups of students. At that point Kid B said, “But that only makes 9 students!”.  We all checked and saw that 1 student was unaccounted for.  We all agreed that student must have had both pepperoni and mushroom.

At this point, I introduced Venn diagrams. Kid A immediately knew what they were called, and all the kids seemed pretty comfortable with them.  I drew two circles for pepperoni and mushroom, and I asked the kids what the overlap meant.  They all said it meant both pepperoni and mushroom.

Then I read the problem again, and the kids put the blocks in the circle as I went, and saw that we could put the last block in the ‘both’ section.  Most kids added  a third non-overlapping circle to mean ‘plain’ or ‘no toppings’.

I told everyone to turn over their papers for a new problem, and read, “Everyone is having ice cream. 6 students have bubblegum, 7 students have vanilla. How many students have both kinds?”

All the kids immediately realized we should draw a vanilla circle and a bubblegum circle.  I asked the kids which flavor they would pick.  Kid C and D said Vanilla. Kid A said Bubblegum, and Kid B said she’d be one of the students who got both.

Next I read the problem again, and kids started trying stuff, like putting 6 blocks in the bubblegum circle, but then they complained there were not enough to put in the vanilla circle.  Kid B then suddenly figured it out by putting 3 blocks in the ‘both’ section. I moved her paper to the center and we went through the cases, and saw that it worked, and the answer was 3 kids had both kinds of ice cream.

Then I handed out new papers, and read:  “The students brought their parents to school. 4 students brought their mom, 5 students brought their dad, 3 students did not bring a parent. How many student brought both parents?”

The kids started telling me about when they had brought their parents to school.  Then I read the problem again, and the kids started trying different things.  After trying a few different solutions, Kid B came up with the correct answer.  Unfortunately, Kid B was so excited about solving the problem first, that she started interrupting me loudly when I tried to talk to the other students.

Finally, we used Kid B’s diagram to answer other questions: How many kids brought only their mom? How many brought exactly one parent? How many brought at least one parent? This was surprisingly hard for some of the kids.  The difference between “How many brought only their mom?” and “How many brought their mom?” was subtle.  So there’s lots more to be done on this topic.

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Table Full of Cows

The Activities

  1. Topic: Numbers: Book: The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, Ch. 1.
  2. Topic: Logic: Castle bridges, part 2.  This time, I added NOT to the set of operations.

    (¬A ^ B ^ C) v D

    ¬A ^ B ^ (C v D)

  3. Topic: Geometry: Fencing animals.  This was a continuation of last week’s activity with the Keva blocks.  This time, they had animals which they had to build pastures for, according to several rules.  The rules, which I added gradually, were: 1) Only 2 animals per 1×1 square covered (so a 3×1 enclosure could hold 6 animals), 2) Each pig needed to be next to an enclosure with donkeys, and vice-versa, and 3) The donkeys didn’t want to be on the edge.

How Did It Go?

Four kids attended this week.

The Number Devil

We read about 2/3rds of the first chapter, stopping right after the discussion of very small numbers.  The kids were very interested in the story and complained when I stopped.  The first mathy part, about infinity, the kids already knew, and sort of remembered (when I asked if they remembered the proof we did, they said yes, at least).  When we got to the small numbers, we had a great sequence of kids saying ½, ¼, ⅛, 1/16, 1/18, 1/20, 1/30, 1/100, 1/100, 1/1000000000.  After that they had to think a bit, and then Kid 1 said 1/googleplex.  I said that if we divided a piece of gum (which is what the book talks about) into that many pieces, it would be really really small; the kids said “smaller than an ant?”  “smaller than a germ?” and Kid 2 said “smaller than an atom?”  So this was a very successful discussion spurred by the book.

Castle Logic

First I reviewed the pictures from last week, they were able to quickly remember what each picture meant.  Then I gave then D v (A ^ B ^ C).  They did well on this, although they had a bit of trouble drawing the river; Kid 1 drew it joined together from 3 rivers into 1, but Kids 2 and 3 both had 3 rivers that crossed both paths, but then didn’t add letters to all the places on the D path.  Kid 4 drew an abstract drawing, where she drew two lines instead of a forked path, and drew perpendicular segments for the crossings and labeled them.  So she had more of a diagram than a drawing.

Next, I showed a picture with a path going off into nowhere, with a B crossing; at first they said A v B, but then Kid 2 said A, which is correct.  Then I added the wolf on the path.  They said A at first for that as well, but then, without prompting, Kid 2 said “No B”.  Then I introduced NOT, showed them how to write the ¬ sign.  I also mentioned that you could use !, which they were intrigued by since they recognized it.  I then had a series of increasingly complicated ones using NOT.  Not surprisingly, the difference between ¬A ^ ¬B vs ¬A v ¬B was less obvious than the difference between A ^ B vs A v B.  They did fairly well.  The final one involved parentheses, which they didn’t use.  I mentioned this, and gave them the same formula but with different parentheses.  They had a bit of trouble but all got the right drawings.

Fences and Animals

I started them with 2 sheep and 2 cows and just rule 1 (2 animals per square), and then gradually gave them more animals and more rules.  As we added things, they needed more blocks, which I gave out gradually as well.  The kids were all pretty good at rearranging their fences to meet the requirements.  One of the trickiest parts was going from 2 sheep and 4 cows to 4 sheep and 4 cows, because only Kid 1 had made an L shape for the previous part, the others all had the 2 enclosures end to end; and so, not surprisingly, Kid 1 was the first to solve the problem with 4 of each (because the correct solution is two 2×1 rectangles sharing one of the long edges).  The final enclosure for this part ended up being a 4-pointed star with the donkeys in the middle; the trickiest part was you needed to break the cows into two groups.

Just like last time, I decided to give them a group activity, so I had them join all their animals and then add some more.  At first, they made a lot of little squares, but then gradually started making bigger and bigger enclosures.  Their final solution was pretty good but had a few unnecessary fences, such as two 3×1 rectangles sharing the long edge, and both rectangles having cows inside.  I helped them get rid of the extras, and then they had a pretty nice looking large farm.  I asked if they thought it was optimal, and Kid 2 said yes, but I’m fairly sure it wasn’t because there were still 2 separate pastures for some of the animal types.

After circle our daughter decided to sort all the pieces from Caverna (which is the board game that the animals came from) into groups and put them inside pastures, although she didn’t follow the rules about only 2 animals/vegetables/minerals per square.

Boolean Bridges

The Activities

  1. Topic: Division: Book: One Hungry Cat by Joanne Rocklin.
  2. Topic: Logic: I made several pictures of paths and rivers, where you needed bridges in one or more spots in order to be able to make it to the castle.  Each picture corresponded to a Boolean logic formula.  The kids both had to write down a formula given a picture, and draw a picture given a formula.  All the pictures can be found here, as well as the PowerPoint if you want to make your own.

    A v B

    A ^ (B v C)

    A v B v C v D

    (A v B) ^ (C v D)

  3. Topics: Addition, Subtraction, Multiplication:  I did several “math magic” problems where each kid starts with a different number between 1 and 9, I give them instructions, and then at the end they get the same number.  The three “tricks” were “add 6, subtract 8, add 2”, “add 2, add 2, add 2, subtract 6”, and “multiply by 2, add 3, multiply by 5, subtract 15, cross off the trailing 0”.
  4. Topic: Geometry:  Build some number of squares using as few fences (Keva blocks) as possible.  Later, build some number of triangles.

How Did It Go?

One Hungry Cat

Kid 1 said she had heard the book before, but she didn’t remember what happened.  The kids thought it was pretty funny when the cat kept eating all the food.  Kid 2 pointed out that when there were 8 cookies left, the cat could have eaten only 2, but ate all 8 instead.

 Castle Logic

 I started with one bridge, and then showed them the pictures for “A and B” and “A or B”.  I gave them ^ and v as the “and” and “or” symbols; Kid 1 mentioned that & could stand for “and”.  Then I showed them pictures for A v B v C and A ^ B ^ C, and had them write down the formula.  They didn’t have trouble with this, except that Kid 2 left out the v on the first one, and Kids 2 and 3 both wrote v instead of ^ for the second one.  Then we did A ^ (B v C).  We ended up getting two correct answers: Kids 4 and 5 wrote A ^ B v A ^ C, while the rest wrote A ^ B v C.  This is nice, because it’s the distributive low for Boolean formulas.  Of course, no one used parentheses, but before discussing that I showed them the next one, which was (A ^ B) v C.  Again, a bunch of good answers, including both the distributed and undistributed versions.  As it happened, no one wrote down the same thing both times (they randomly varied the order, probably accidentally although it’s possible Kid 6 did it on purpose to signify which came first).  So, I wrote down A ^ B v C for both of them, and explained how the parentheses tell you the precedence.  They may or may not have understood.  Kid 6 finished these quickly and started drawing a picture of a bridge with a funny troll, and then showed it to other kids.

After this, I gave them A v B v C v D and asked them to draw a picture. Four of the kids drew the right thing; Kid 1’s was the nicest, while two of them didn’t initially connect the roads at the bottom (each road went off the paper at a different place).  Kid 6 finished early and started adding to the troll again, so I gave Kid 6 another problem, (A v B) ^ (C v D).  Kid 6 didn’t want to (and didn’t) work on it, but Kid 4 drew a correct picture for it!  I thought Kid 4 had been working on A v B v C v D, and I’m still not absolutely sure she didn’t just do that one incorrectly, but it’s quite a coincidence that she got a correct diagram for the 2nd one; and when I asked, she said that they had been working on the 2nd one.  Kid 2 had the most trouble throughout the activity, and for the final one drew A ^ B ^ C ^ D instead of A v B v C v D.

 Math Magic

The first one was “add 6, subtract 8, add 2”.  Subtracting 8 is still not that easy for many of them.  They were suitably amazed when they got both the original number, and didn’t know how I did it.  I explained how it all added up to 0.  I also showed the chain formula, and explained how I could group together the terms so I got 0.  Not sure exactly how much they understood; Kid 1 was confused because she had started with 2, had gotten down to 0 after subtracting 8, and was getting confused between having 0 as a partial result and the fact that the operations they performed summed to 0.  Next, I did “add 2, add 2, add 2, subtract 6”.  This time, Kid 2 explained how 2 + 2 + 2 = 6, so it canceled.  Finally, I did a hard one: “multiply by 2, add 3, multiply by 5, subtract 15, cross off the trailing 0”.  Multiply by 5 was, not too surprisingly, pretty hard for them.  Initially, they tried to add their number 5 times together, but since most of the numbers were above 10, they had problems.  When I suggested counting by 5’s, that worked better.  Kid 3 is really fast at counting by 5’s, although when I helped her do 15 * 5, I’m not sure she understood what she was doing.  What was most surprising was how difficult it was for them to subtract 15 from 75/85/95.  I’m not sure a single kid got it right on the first try — usually they would say that 75 – 15 = 65.  Similar to some times in the past, Kid 2 was more stubborn about wanting to do the multiplication her way, so it took her longer than anyone else — eventually she took my suggestion to count by 5’s and then did the counting correctly on her own.  I asked the kids if I should explain how it worked but they said it should stay magic for now.  Kid 4 wrote down the numbers from 5 – 225 counting by 5’s while I was helping Kid 2.

Fences 

Interestingly, the kids immediately, without thinking about it, started using shared fences when building multiple squares (so to build 2 squares, you only need 7 fences).  When we got to 3 squares, some kids had done 3 in a row, but they quickly changed to a 2 x 2 grid once we got to 4.  The first time that we got a less efficient answer was 9 squares, where Kids 1 and 2 had an irregular shape that used 26 blocks instead of 24 (it had a part that was only one square thick with thicker parts on either side).  I showed them how you could move squares in order to save some fences.  Next, I asked them to build 1, 2, … triangles.  Kid 3 quickly went to a “gridded” hexagon for 6 triangles.  We were almost out of time and they were having fun, so I said they should cover the whole table with triangles, and they did.  They didn’t work from a single place, so the interfaces between the sections weren’t great, but they did a pretty good job anyway.  Kid 3 mentioned at some point they should all work from one place, but it didn’t happen.  At the end, the kids were happy to get a little wild and knock over all the blocks and not-that-carefully slide them into the box.

Table full of triangles!

71 x 25 = 1230?

The Activities

  1. Topic: Multiplication: Book: 2 X 2 = Boo by Leedy.
  2. Topic: Teamwork, Multiplication: As a group, figure out 71 x 25.  If you get it within 100, you get a treasure.

    My daughter makes stacks of 71.

  3. Topic: Programming:  Do the following programs.
    The functions:
    Cry {
    __Do 2 times {
    ____Print “Wah!”
    __}
    }
    Eat (Box_W, Box_Y) {
    __Print Box_W
    __Print “ate”
    __Print Box_Y
    }
    SOS (Box_Y) {
    __Do Box_Y times {
    ____Print “Help!”
    __}
    }

    The programs:

    Print “A ghost cried”
    Cry
    Cry
    Eat(“Avery”, “candy”)
    Eat(“Arun”, “broccoli”) Print “Joe shouted”
    SOS(3)
    Eat(“Laura”, “Poison”)
    Print “Shelly yelled”
    SOS(2)
    Print “Laura was saved!”
    Eat (“Monsters”, “Lucy”)
    Print “Lucy said”
    Cry

How did it go?

2×2 = Boo!

This book was really fun and clever. The kids really enjoyed it, and laughed throughout it.

Big Multiplication

This is a hard problem of teamwork. I decided to step in a bit more than usual, to help amplify good ideas that kids came up with.

Kid A immediately said that everyone should make 25 stacks of 71.  She started doing this, and other kids joined in.  However, the stacks were all over the floor.  After a few minutes of this, Kid A wanted to start trading in 10s for 100s, but Kid B and Kid C both said that wouldn’t work because they wouldn’t know how many stacks they had made.  Kid A then got very distracted and started wandering around, saying she was bored.

Kid D really wanted to count the stacks, but several kids did NOT want her to count until after they had arranged the stacks in rows.  Kid C mentioned that the stacks should be organized, and she started making a line.  Kid B and Kid A meanwhile started making a rows of stacks.  Kid B and Kid C both told everyone it was important that the stacks should not touch.

Eventually they got 25 stacks each with about 71 cubes in them.  Then chaos started.  Kids started complaining that it was too hard to count by 71s. Kid B suggested that they should trade in for hundreds.  Some very haphazard trading ensued, with 10 bars getting mixed in from the extras, and kids taking away 10 ten bars without bringing back a 100 square.

Kid B got a bit frustrated and wanted to quit, but I said we should figure out their answer.  They answered 1230, when the answer was actually 1775.  So they did not earn a treasure.

I told them they’d get one more chance.  Before they started I suggested that they should again make rows of stacks.  And I suggested that all the 100 squares stay on the table. When someone collected 10 ten bars, they should leave the 10 bars on the table and take a 100 square back to the pile.  All 6 kids said they wanted to try again, so they could earn a treasure.

The new problem was 62 * 23.  They immediately started making 23 piles of 62 blocks.  I had to keep reminding various kids what we were doing, and asking if they had any more piles of 62.  This time they assembled the piles and immediately added them to the final rows (keeping the stacks separate).  Soon they had 23 piles. Everyone verified this, and then I reiterated that each person who collected 10 ten bars should trade in for a 100 from the table.

They started to do this, in quite an organized way, except that Kid C was using a 100 square to measure the ten bars (instead of counting out ten of them).  I was pretty sure that she accidentally left behind her measuring square AND added another square to the pile.  I pointed this out, but she thought it was ok.

Kid B took over the singles, and traded them in for 10 bars.

Then they started counting the 100s.  They got to 1200, 1300, 1400…and there was one more…but Kid A suddenly said “No, this one doesn’t belong, because it was Kid C’s measuring square!” and took it out.  The other kids let her, and their final answer was 1436. The correct answer was 1426, so they celebrated their victory, and then helped pick up.

I asked Kid A how she knew to take out that 100 square, and she came up with a variety of somewhat strange reasons, like “I knew it should be smaller than our last one, so I took one out.”  I suspect she either saw the answer somehow, or saw my face when I realized they had one too many hundred square.

Programming

We had new functions and new programs this week.  Kid A missed last week, so she didn’t yet know about functions.  Three of the kids were able to independently do almost all the programs, and enjoyed reading the silly sentences.  My daughter was again super-focused and zoomed through.  She was often waiting for me to hand her the next program. The only thing that tripped up her was the SOS function.  She didn’t understand how to “Do Box_X times”

Kid D was doing ok on the programs.  However, she lagged behind quite a bit and was distracted, drawing on her sheet.  Perhaps the time-change had put her in a weird mood.

I helped Kid A quite a bit.  She remembered what “Print” meant, but she didn’t initially understand the functions (since she had missed that circle).  After some help from me, she solved 3 of the problems…but I’m not completely sure if she was just pattern matching or actually following the commands.

Kid E got very frustrated on the first program.  The other 4 kids finished before her, and then she started to doubt herself.  I came over to help, but she started to cry, so I asked if she wanted a break. Her dad noticed and came over to help her trace through.  After a bit more frustration, she started to make progress again, and solved 4 of the problems.  She’s just not quite comfortable with the functions yet (she wanted to write “Cry” instead of execute the commands in the Cry function, which was the same problem she’d had last week).  It was also extra noisy this week which didn’t help.

Hopefully we can get all 6 kids to the point where they are comfortable with functions and loops.  Then we can try having the kids write their own programs.  There is lots of good logic in programming.