# 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.