Block blobs redux

Last December, we played Block Blobs (notes here). This week, we are trying a slightly modified version for two digit multiplication.

The Game

Materials

• 4d6. Two dice are re-labelled 0, 1, 1, 2, 2, 3 (see notes below)
• Graph paper (we are using paper that is roughly 20 cm x 28 cm, lines about 0.5 cm apart)
• colored pencils

Taking a turn

Roll all four dice. Form two 2-digit numbers using the standard dice as ones digits. Then, use your color to outline and shade a rectangle in the grid so that:

1. the side lengths are the 2-digit numbers you formed with the dice
2. At least one unit of the rectangle's border is on the border of your block blob
3. Note: the first player on their first turn must have a corner of their rectangle on the vertex at the center of the grid. The second player has a free play on their first turn.
4. Write down the area of your rectangle

Ending the game

The game ends when one player can't place a rectangle of the required dimensions legally.

When that happens, add up the area of your block blob. Higher value wins.

Notes

"Counting" sides
The side lengths of the rectangles are long enough that counting on the graph paper will be irritating. Instead of counting directly, they can measure the side lengths. For our graph paper, the link between the measurement and the count is nice, since the paper is very nearly 5mm ruled, so they just double the measure. I think this is a really nice measurement and doubling practice, too.

Dice labels

Other labels could be used on the special dice. We chose this arrangement because of the size of the graph paper. Rectangles with sides longer than 40 often won't fit and we think we will even need some single digit rectangles to allow a fun game length. An alternative we are considering is 0, 0, 1, 1, 2, 2.

As an alternative to labeling the dice with new numbers, you could label them with colored dots and give out a mapping table. For example:

blue corresponds to 3
red corresponds to 2
green corresponds to 2
yellow corresponds to 1
black corresponds to 1
white corresponds to 0

This would keep the tens digit dice distinct from the ones digit dice and allow rapid modification if you want to change the allowed tens digits (just tell everyone a new mapping). Alternatively, you could create a mapping using "raw" dice and even allow more strategic flexibility from the players. I have a feeling that this would be confusing to most kids, though.

Reinforcing the distributive law

To facilitate calculating the area and reinforce the distributive law, you might have the students split their rectangles into two (or four or more) pieces and calculate the partial products. You can further decide whether to ask them to split the sides in particular ways or encourage them to find the easiest way to split to help them calculate.

Bedtime math extensions

Who: J1 and J2
Where: all around the house
When: 6 am

It is summer vacation for J2 and J3, so we have been doing a lot of activities. Unfortunately, there is almost no free time to blog about it.  For now, let me share some of the conversations J1 and I have had about recent Bedtime Math posts.

Doubling Volcanoes

Today, we talked about this post: Instant Island. In particular, we focused on the last part, the "big kids bonus" question: if an island is ¹/₂ mile wide and the width doubles every month, how wide will it be in 4 months?

Okay, forget about the answer for a moment, does the question make sense? First, we asked: if this were true, how long would it take the volcano to stretch around the circumference of the Earth? Using some familiar powers of 2, we figured out it would be slightly less than 16 months. It seemed pretty clear that this didn't make physical sense, since we clearly don't see small islands cover the Earth like that.

We talked about this doubling growth process for a while. What things do we know that work like this? J1 listed:

• bacteria
• people
• computer viruses
• plants

We talked about what is happening: basically, the new "material" is able to reproduce itself, so the more you have, the more productive potential you have.

Does the growth keep going forever? No, otherwise everything would be covered, actually become, the thing that is doubling. At some point, these all run into a limiting factor, food, water, space, etc.

Back to the island: J1 realized that the growing island would hit other landmasses before it went around the Earth. If you consider connecting with Asia to count as "growth" for the island, then there could be moments of extremely fast growth. At the same time, we know that the Eurasian landmass isn't growing with an exponential process, so this connection won't contribute to the further growth of the volcanic island.

Stuffed with lead?

We were scratching our heads about the weights in this Stuffed Animal post. The average weight per stuffed animal assumed in this post is 7 pounds. We had two conversations about this: how much is that in grams and how much do our stuffed animals actually weigh?

First attempt, a scant 27 grams

Hefty Panda is only 281 grams

Massive Diplodocus is about 100 grams lighter

One of our heavy-weights: still under 400 grams

Not a stuffed animal. This is the trickier we had to use to break 1 kg

Suffice it to say, we had no stuffed animals that exceeded 1 kg.

Heavy Pencils

After having an experience with unreasonable weight measurements, Pointy Gorilla helped launch a similar conversation. Actually, it came from misreading! We connected the following comments:

1. Gorilla weighs 300 pounds (we read this as the pencil gorilla)
2. The big kids question implies a certain number of pencils used (under 600)

So, how heavy are those pencils? Given what we know of real pencils, how much would that gorilla actually weigh?

the measure is 27

Who: J3 (also something for 13+)
Where: at home on the reception floor
When: just after breakfast

---

A quick picture to show some standard activities in our home.

When in doubt (i.e., too tired to think creatively), I reach for the trio blocks and polydrons and start putting them together. Inevitably, the children will join and take over the activity. We had our tape measure lying around, so J3 started measuring our creations and Ms Rabbit.  After putting the tape measure up to something and looking carefully at the numbers, she would proclaim: "27." She did this several times, each time announcing the same length: 27.

I guess this is similar to her lack of 1-1 correspondence when she's counting: just a developmental step she hasn't yet taken.

A further exploration
Did you notice the star-shaped polydron construction?  It is a cube with the faces replaced with square pyramids. Though it is pretty obvious, I was delighted when we realized that square faces in our constructions could be replaced with 4 triangles arranged as a square pyramid and equilateral triangles could be replaced by 3 sides of a tetrahedron.  Here's an NRICH exploration I found when trying to determine the name of our construction (the cube with pyramids instead of faces).

Mathematical distraction (argh!)

who: J0
when: J3's naptime
what material: scale, bottle, water, formula

---

I looked down at my hands and saw a partially prepared bottle of baby formula.  Hmm, did I add the right amount of each ingredient?!

Stepping back, how I had I gotten into this (minor) predicament? By day-dreaming about some other math (which I hope to post in the near future), I had managed to mess up this simple recipe:
1. add 4 ounces of water to the bottle
2. add 2 scoops of formula powder to the bottle
3. close and shake the bottle
4. dispense to the consumer

My solution: use a little more (very simple) math to figure out what I had done.

I got out my trusty scale and made the following measurements:
(1) empty bottle: 60 grams
(2) 2 scoops of formula: 20 grams (pictured below is one scoop, I had tared out the lid and the scooper)
(3) 4 ounces of water is approximately 120 mL/approximately 120 grams
(4) my partially made bottle: 155 grams

Hmm, 155 grams? My target mass was supposed to be 200 grams. If I had forgotten one scoop of powder, it should have been 190 grams. So...

Looking more carefully, I had only put in about 3 ounces of water (mass about 90 grams).  I concluded that I needed an extra 30-40 gr of water and another scoop of powder.

What lesson did I learn?
Maybe I need to be cautious thinking about engaging math problems when I'm driving?

Baking math

Who: J3 (2 year old)
When: late morning, after J1 and J2 had gone to school
What did we use: water, yeast, flour, sugar, salt, oil, measuring cup, measuring spoon, scale, bread machine
Where: kitchen

See how little of the finished product was left before I remembered to take a picture:

---

I'd posted before (here!) about some of the (mathematical) reasons to bake with kids.  Here is the math we did today while making bread:

1. counting
2. measuring volume and mass
3. estimating
4. next steps: ideas I forgot, but you can include them!

Counting
Maybe I go overboard, but my habit is to count everything around the smaller children.  Based on the recipe alone, there wasn't much scope for counting (2 eggs, 2 tsp salt, 2 tsp yeast) but we also counted the number of measuring spoons in our set and how many scoops of flour we needed to get our target mass. Along the way, we probably counted fingers and maybe even toes, too.

Measuring volume and mass
This is the obvious place to develop number sense while cooking.  We end up doing a lot of extra measuring during the process, playing with the different tools, and talking about comparisons between them.  For example, flour (21 ounces, about 600 grams) was the only thing we needed to weigh. What we actually weighed: flour for the bread, flour not used for the bread, the mixing bowl, the measuring cups (full and empty), a water bottle and box of crackers near us on the counter, the measuring spoons, and the sugar jar.
That allowed us to explore a range of weights from about 20 grams up to 1.5 kg.

We talked about metric and imperial units and looked at all the quantities in each. In particular, I use the imperial units as a good place to talk about fractions.  For today, I just read out the fractions close to what J3 measured, and included those in our chatter about comparisons.  My goal is to head off any (distant) future anxiety that comes from seeing fractions as a different "type" of number.

Estimating
I try to encourage them to estimate measures before they see the result.  This is easiest for mass, I just say "I think that will be 100 grams (or whatever I guess)" before they put the object on the scale. For the older kids, they will almost always respond "I think it will be less/more/[x grams]." J3 will usually just copy me, but she makes a delightful sound when we read the measurement together and comment on how close our estimate was.

Next steps
In truth, I didn't really go through every step with J3.  We measured the ingredients into a bread machine pan and then let the machine do the mixing and kneading.  Of course that saves a lot of time and effort, but we didn't see the exciting phase transitions where the dough goes from  separate liquid+dry flour to lumpy/sticky to smooth/elastic.  For older kids, it is interesting to ask them why they think the dough changes. Also, I did the shaping myself and that could be an opening to talk about braids/twists/knots.

As I wrote up these notes, however, I realized that I missed an opportunity with J3 to talk about temperature and time.  I think time was the big one: time for mixing/kneading the dough, time for a rise (actually there are 3 rises in this recipe) and time to bake.  With that in mind, I'll leave you with a picture of our kid-friendly timer: