What is 8?

I've had a chance to spend more time doing math with the kids again and am hoping to write up our activities more consistently.  Let's see how this works out!

Graham Fletcher created a set of  Progressions videos for various elementary school themes. J3 and I recently went back to his page and found he had a new(er than we knew) progression on early number and counting.  Even for this simple topic, the video highlights some points we hadn't considered explicitly, for example distinguishing producers (of a number) and counters. Also, the cardinality point that smaller natural numbers are nested within larger numbers wasn't something we had talked about, but we soon realized it was part of many examples in how we understand numbers.

With that as inspiration, J3 and I decided to search for a range of examples of a single number, we chose 8, in different forms.  There is at least one obvious version we're missing.

Add a comment (with picture, if you can) to show other forms of the number 8!

Marking 8 on the 100 board, an easy place to start:

8 beads on the abacus shows the relationships 3+5 = 8 and 10-2 = 8 (also 100- 92 = 8)

8 can hide in plain sight. Without labeling the three lengths, it would have been hard to recognize the longer one as 8 cm and, for you at home, impossible to know without reference to show the scale.

It happened that, within the precision of our scale, two chocolate wrapped chocolate bars were 8 oz (2x3.5 oz of chocolate + about half an ounce of wrapping for each):

8 cups of water ended up being a lot, so this version unintentionally revealed a relationship 4 + 2 + 2 = 8

Though I'm not sure I can articulate why or show supporting research, I feel it is very valuable to build experience with physical models of numbers to create familiarity and intuition about what they are/mean. In particular, I hope this helped J3 anchor the importance of units of measure and scale in the interpretation of numbers.

Finally, this construction has nothing to do with the number 8 (or does it???)

math recommendations for a 3 year old

I was recently asked for suggestions by a parent of a 3 year old.

There are a lot of different resources I could suggest, but they really depend on the child and the parents. The main question for customization is about the parents: what are their starting assumptions about math/math learning and how much do they want to engage on selecting/planning activities?

For example, if a parent doesn't really get the growth mindset, I would advise a heavy dose of Jo Boaler. If the parent wants open explorations and can build their own specific tasks, maybe the Vi Hart videos are good inspiration.

That aside, there are a few resources/products good enough that I’m willing to give blanket recommendations:

1. Lots of tools for measuring. Playing with measuring has so many benefits, I can’t list them all, but some of the highlights are (a) seeing math and numbers all around us, (b) tactile engagement, (c) inherent process of comparison, and (d) natural connection with language as the kids and parents talk about what they are measuring/why. The links I've provided just show examples, I am not necessarily recommending them over other versions.
1. Set of plastic measuring cups (imperial units and fractions)
2. Tape measure (we just used standard adult tape measures, but as a recommendation, you need to be careful about tape measures that have fast return springs for cutting or catching small fingers)
3. Balance scale and set of standard weights (this math balance is a good option and one we bought)
4. Timer (we liked this one)
5. For older kids, a step counter, GPS wrist-watch showing speed, thermometer, pH meter, electricity meter are all interesting additional measuring devices.
2. Talking Math with your Kids:
1. E-book
2. Blog. I recommend reading all the posts, I think they are a superset of the material in the e-book, so this is a better resource unless you want the “curated” highlights. This link goes directly to posts tagged 3 years old.
3. Tiling toys and shapes book in the TMWYK store. I particularly like Which on doesn’t belong? A better shapes book.
3. Denise Gaskin’s Playful Math books: these talk about general habits and methods in an intro section, then specific activities (mostly games) in the rest of the book.
4. I got a lot out of these storybooks (free to print) with my kids: CSMP Math Storybooks.
5. Standard gambling tools: playing cards and dice (I like pound-o-dice for the assorted colors, sizes, shapes)

There are some computer games/systems, a lot of board games, and mechanical puzzles, but the stuff above is where I think parents should start for young children.

What do you think of my recommendations? Any additions you think are worth adding to make a top 10?

Running, rates, rounding

My running session this morning gave me an idea for a kind of 3-act math discussion with J1 and J2. I will discuss this with them when they come back from camp and see what they think. I expect the last questions will be hard for them and I would like to see how much progress they can make working together.

First Act

Today, I went running and recorded some information on my GPS. For five laps, I ran moderately fast. Here is the data:

Time	Rate	Distance
3:00	12.7 kph	635 m
3:00	12.9 kph	647 m
3:00	12.6 kph	633 m
3:00	12.7 kph	637 m
3:00	12.8 kph	645 m

What do you notice?
What do you wonder?

Second Act

My target was actually to run 12 kph for each of these three minute segments. After the first lap, I knew that I could run more slowly and still hit my target. I wondered, how much less than 635m could I run and still hit my target?
If I compare two laps, both rates and distances, can I figure out the distance I get for each 0.1 kph? Is there another way to calculate the difference in distances for each 0.1 kph?

Third act

For some reason, this made me think about rounding that J1 had recently been studying. He is a bit disturbed about what to do with values that are halfway between the rounded levels, for example whether 15 should round up or down to the nearest ten. Since this investigation of running data involved calculations with measured values and rounding, I though it would be instructive to explore a couple of calculations:

• I have two distances, rounded to the nearest 10 cm of 20 cm and 10 cm. What is a reasonable range for the difference of those distances?
• My GPS measured a time of 3 minutes (3:00, rounded to the nearest second) and speed of 12 kph (12.0 kph rounded to the nearest tenth of a kilometer per hour). What distance did I run? What is a reasonable range for that distance?

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.

Equal? Ambiguous test questions

We have been looking at questions from a local exam contest. I thought two were worth discussing.

Climbing a mountain

Kaew had been in the mountains climbing for the last four days and recorded her distances on this graph (the first one below). The other graph shows the climbing distances from Ploy.


Below the graphs are 4 statements by other people and we're asked how many of the statements are true. The statements are:

1. Weera claims: "Each day, Kaew climbed more than Ploy."
2. Piti claims: "Kaew climbed more than 10,000 meters, in total."
3. Mana says, "In total, Ploy climbed more than Kaew."
4. Choojai says, "Each day, Ploy climbed 500 meters more than Kaew."

How many statements are true?

The math class answer

I'm only really interested in statement 2, did Kaew climb more than 10,000 meters?
In math class, we often ask students to enter a world of perfect accuracy and exact logic. Usually, the questions in this world only involve "nice" numbers.
Entering that framework,  we see that Kaew climbed 1500m+3000m+2000m+3500m = 10,000m
Because she climbed exactly 10,000 meters, she did not climb more than 10,000. Piti's statement is false.

Why is this unsatisfying?

We have two issues. First, how accurately did Kaew record her climbing distances? Second, how accurately are we reading the graph? Surely there is uncertainty here that prevents us from having any confidence in assessing Piti's comment.
My sense is that the data is accurate to the nearest 500 meters (so each day's distance is ±250m). Based on that, I feel confident that Kaew climbed between 9000m and 11,0000m over the four days.

Taking our medicine

In another question, we are told that a pack of pills comes in a 3x3 grid, pictured below. Each day, the patient has to take 3 pills. We are told that once a pill has been taken, the patient can't take a pill from the same row or column. How many ways are there to take the medicine?

Interpretation
How did you interpret the question? I guess you thought that each day the patient can take only one pill from each row and column, that we are asked how many ways the patient can take this medicine over three days. Right?

Looking carefully at the question, I came up with five interpretations:

• no two pills back to back from either the same row or column on the same day vs no two pills in sequence from the same row or column (possibly over two different days) vs no two pills on the same day from either the same row or column (choice 1)
• how many ways to take the medicine on the first day vs how many ways to take all 9 pills over three days (choice 2)

Putting it in perspective

I'm glad that we had ambiguous questions on the contest paper. It undermines the authority of the exam, the idea that there is only one right answer, and the importance of answering questions correctly. Instead, we are left with a collection of prompts that can lead us to interesting thoughts and interesting discussions.

Triangles and Angles: a proof for 5 year olds

who: J2 (also P and J0)
what did we use: polydrons, polydron protractor

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J2 and I were playing with polydrons and he got interested in measuring the various angles. Since there are three different triangles in the set, he got to test each of them and found that the angle sum was always 180 degrees. We went on to examine squares and pentagons to see what he could make of those.

In the evening, P and I were talking about about this investigation and I mentioned this proof that the interior angles of a polygon sum to $180 \times (n-2)$:

P was surprised and said that she likes this proof more:

In this version, you have to subtract the middle 360 degrees.

I was a bit surprised that I hadn't ever thought of this version or seen it elsewhere, but quickly realized we could repurpose it to finding the interior angles of a triangle itself.  Here's the picture:

Now, the argument:

1. Angles of the large triangle are equal to the sum of the angles of the smaller triangles - 360
2. Let x be the sum of interior angles of a triangle.
3. Based on 1, we have x = 3x - 360
4. Rearranging, we get x = 180.

Where we get stuck: a challenge for you

Here's the thing: the sum of interior angle measures of a triangle isn't 180 degrees. That is, it doesn't have to be if you are working in non-euclidean geometry. Here's a nice picture (not mine) of a triangle with angle sum of 270 degrees:

Comes from MathStackExchange

However, our proof doesn't seem to use any fancy postulates. Your challenge, why doesn't it work on a sphere? What extra postulate did we secretly use?

A side point about measurement

When my son was measuring the angles for an isosceles right triangle, he read 46 degrees off the protractor. I was too focused on getting to the punchline about all triangles having the same angle sum and I "corrected" the measurement to 45 degrees. He still said some things and we had a conversation about ideal mathematics and real measurement, but I felt it was a missed opportunity.