January 2024
Welcome to EFM's January Newsletter!
It is essential that every family and caregiver in the world read books and do math with their young children!
EFM believes in every child’s mathematical right to equity, opportunity, and personal fulfillment.
News
Playing cards – Our first run of 2000 math puzzle playing card decks was such a hit that we have run out of them. We have begun manufacturing for K-3 and 2-5 decks in English and Spanish, as well as the first family deck. We hope to have those in a few months.
Animal Math and Ancient Numbers
A good story helps a child’s engagement with a math topic. Today’s story concerns how animals count when they have a different number of fingers than people do.
Animal math, less interestingly called “number bases,” is a fun way to make base 10 place value come to life. Where does that ten come from that keeps showing up in counting and arithmetic? Playing with animal math also provides lots of incidental practice with dividing and multiplying by single-digit numbers.
3-toed Sloth Counting
What is different when a 3-toed sloth, using only one paw, starts to count? It would count one, two, three, and then run out of toes. What do people do when they count up to ten on their fingers and run out of fingers?
It’s all about bundling. People bundle up groups of ten because that’s all the fingers they have. When we get ten of something, we bundle them up, and then all our fingers are available to start counting again. When we have ten bundles of ten, we bundle those up into a super bundle of 100, and keep going. That’s where our place value system comes from. We have 1’s, 10’s, 100’s, 1000’s, and so on – each place is ten times bigger, it is a bundle of ten of the previous place.
Sloths use bundles of three. They count 1, 2, and then they have a bundle of 3, which I’ll write <3> for now. Then four becomes a bundle of three plus 1 more, which is <3> + 1. The next number is <3> + 2. This is followed by <3> + <3> = 2<3>. Then 2<3> + 1 and 2<3> + 2. At nine we end up with three bundles of <3>, which is a super bundle <9>.
Writing <3>, <9>, <27>, and so on is rather clumsy. Instead we’ll use 3-toed sloth place value and write 103, 1003, and 10003. The columns, moving right to left, are the 1’s, 3’s, 9’s, and 27’s. Each column’s value is a bundle of three of the value of the previous column.
Sadly, SquareSpace does not support subscripts in blog postings, so all the carefully done subscripts in the rest of this post will appear inline. I’m so sorry!
Two quick things about these new numbers. The little 3 at the end of the number is important. If it’s left out, then 123 can be confused with 12. Also, 123 is said “one two base three” or simply “one two.” If you say “twelve,” you are using a name for a dozen things, whereas 123 (= 1 x 3 + 2) is a name for five things.
Counting aloud and writing these new forms is lots of fun, especially if you laugh together over the inevitable stumbles as your child fights the habit of counting in base ten. It may help to use piles of small objects to illustrate each quantity as you count.
1, 2, 103, 113, 123, 203, 213, 223, 1003, 1013, 1023, 1103, 1113, 1123, 1203, 1213, and so on.
Other Animals
How do things change if the sloth uses both paws (just as people use both hands)? Now the sloth will have bundles of size 6, and the counting will look like:
1, 2, 3, 4, 5, 106, 116, 126, 136, 146, 156, 206, 216, 226, 236, 246, 256, 306, and so on.
Note that the digits always start at 0 and go to one less then the bundle size. You can’t have a digit as big as the bundle size, because you would bundle those up.
Play with what happens with other animals. Sea stars (starfish) with five arms would use bundles of five. An octopus would use bundles of eight. A two-toed sloth would use bundles of two (for one paw) or bundles of four (for two paws).
Use different animals and spend lots of time counting up and down in this new way until it feels natural. There is nothing any more or less natural about counting using octopus math than with people math – we’re just accustomed to using base ten.
By the way, you can still see people bundling in groups of five when they use tally marks to count something, such as the number of people that have gone inside a room. Look at how much easier it is to recognize 23 tally marks when they have been bundled in groups of 5. With 4 groups of 5 with 3 left over, the sea star would write 23 as 435.
Counting to 31 on Just One Hand
Challenge your child to count up to 31 using just one hand. At first they may think you’re crazy. Show them how to count up to 3 with just their thumb and index finger – the thumb is the ones place and the index finger is the twos place. - you’re counting in binary. After that demonstration, see if your child can figure out how to count to 31 by using the remaining three fingers for the 4s, 8s, and 16s places.
How high can your child count if they use both hands?
Animal Math Magic Trick
Here’s a math magic trick that will surprise people who don’t know number bases. Choose a base to use. You will usually see this trick performed in base 2, so we’ll look at that first.
Make a list of the numbers from 1 to one less than a power of 2. We’ll just go from 1 to 7 (= 8 – 1) to keep this short (any power will work). It’s easier to see what’s going on if you include the leading 0s.
0012, 0102, 0112, 1002, 1012, 1102, 1112
Next, make some cards, one for each place value and possible nonzero number for that place. For this example, we’ll have three cards. On each card, record all the numbers that have a 1 in that place – the first card has numbers with 1s in the 1s place, the second has numbers with 1s in the 2s place, and the third has numbers with 1s in the 4s place.
0012, 0112, 1012, 1112 – write this as 1, 3, 5, 7
0102, 0112, 1102, 1112 – write this as 2, 3, 6, 7
1002, 1012, 1102, 1112 – write this as 4, 5, 6, 7
Now for the trick. Hand the three cards to someone. Have them think of a number from 1 to 7, and then have them hand you back only the cards that contain their number. For example, if they think of 5, they will hand you the first and third cards. You can tell them their number by adding up the smallest numbers on the cards they give you – in this case, that would be 1 + 4 = 5.
This works for 3-toed sloths too. List the numbers from 1 to 8 (= 9 – 1). Again, put in leading 0’s.
013, 023, 103, 113, 123, 203, 213, 223
This time, you have cards for 1s and 2s for each of the places. Here are the numbers on the cards.
013, 113, 213 – written as 1, 4, 7
023, 123, 223 – written as 2, 5, 8
103, 113, 123 – written as 3, 4, 5
203, 213, 223 – written as 6, 7, 8
The trick is done exactly as before, and it works for the same reason.
Animal Math – Animals Doing Addition and Subtraction
Addition and subtraction work the same for other animals as it does for people. The only difference is the size of the groups for regrouping.
Let’s watch a sea star add 1345 + 4335.
Starting with the 1s column, 3 + 4 = 7 = 125, so write the 2 and carry the 1 to the 5s column. In the 5s column, 1 + 3 + 3 = 7 = 125, so once again write the 2 and carry the 1 to the 25s column. Finally, in the 25s column, 1 + 1 + 4 = 6 = 115, so write the 1 and carry the 1 to the 125s column. The final answer is 11225. In each of those calculations, a group of 5 in a column became a 1 in the next column.
Animals Talking with Other Animals
Suppose you are a sea star and your child is an octopus, and one of you wants to understand a number from the other. For example, suppose the sea star is looking at the answer from the last section. What would 11225 be to an octopus.
The way this is most commonly done is to turn the number into a base 10 number and then translate it again to an octopus number.
Use place value to convert another animal’s number to base 10. 11225 is
(1 x 125) + (1 x 25) + (2 x 5) + (2 x 1) = 125 + 25 + 10 + 2 = 142.
To convert a base 10 number to an octopus number, start subtracting off the highest powers of 8 that you can, and keep track of them.
142 – 64 = 78; 78 – 64 = 14; 14 – 8 = 6 → so, 142 = 64 + 64 + 8 + 6 = 2168.
Another technique people like to use for converting from base 10 is keeping track of the remainders as you do repeated division by the base. 142 / 8 = 17 with a remainder of 6. 17/ 8 = 2 with a remainder of 1. Writing the remainders in order gives the same result as before: 2168.
Letter Substitution Puzzles for Other Animals
Letter substitution puzzles take on an extra bit of spice when done in a different base.
The rules for these puzzles is the same as usual: 1) each letter represents a number from 0 to one less than the number of the base, 2) in a given puzzle, different letters must have different values, 3) no number can have 0 as its leftmost digit. Here are some animal math puzzles of varying degrees of difficulty (the last is pretty hard).
1) Solve this base 3 puzzle: KL + KK = LKM.
2) Solve this base 8 puzzle: CARE + CERF = ARRFE.
3) Solve this base 8 multi-digit multiplication puzzle:
JKL
x DL
------
FJMK
+ GGFK
-------
GJMKK
Computer Math
Computers use the same math as two-toed sloths – binary! When computer scientists look at numbers being used by computers, they often look at the binary digits of 0s and 1s in groups of four. This leads them to work in base 16, which is called hexadecimal. Computer memory is made up of bytes, which are groups of 8 binary digits. It is much easier to read and think about 2 hexadecimal digits than 8 binary ones.
Base 16 uses numbers from 0 to 15, which leads to the question – what do we use for a single symbol for each of the numbers above 9? I’ll leave the answer to that for you and your child to discover should you choose to learn more about bases larger than 10.
Ancient Number Systems
Ancient number systems used many combinations of strategies for place value. The Babylonians used base 60 (imagine their multiplication flash cards!), Chinese used base 10, and Mayans used base 20 (with a little bit of base 5 thrown in). Here are some examples of Mayan numerals:
Other civilizations, such as the Egyptians and Romans, used numbers built out of multiples of 5 and/or 10, but did not use place value. Have fun with your child learning the various systems and interesting symbols they used – the Egyptian hieroglyphs are particular enjoyable.
Wrap Up
Over the years, number bases have received an unfortunate reputation early math education. I hope you see them now as a fun topic that leads to math explorations and insights.
If you have any questions or comments, please send them our way! We would enjoy the opportunity to chat with you. Also, if you are interested in collaborating with us or supporting us in any fashion, we would love to talk with you about ways we can work together!
January 18, 2024
Chris Wright
Chris@EarlyFamilyMath.org
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