Kaktovik Numerals

 In August 2021, I stumbled across the wonderful Katovik numeral system.

Kaktovik is a city in northern Alaska. In the 1990s, the students at a school in Kaktovik realised that there were problems matching Hindu-Arabic numerals to the patterns of their Iñupiaq language. As part of their math enrichment class, they set out to create new numerals that matched their language.

The Iñupiaq language of Alaska uses a Base-20 for its numerals with subdivisions for groups of 5. There are some rule breakers (looking at you, 6) and numbers before a multiple of 5 have a subtractive element '-utaiḷaq' included so 14 is a bit like saying 15 – 1.

The students devised a brilliant numeral system that matched their language (revealed below). I created and shared a task that teachers could use to get their students to learn about these wonderful numerals. This was originally meant to be a task for Grade 9s but today I tried it with two Grade 8 classes for the first time.

First, I showed them that they are already familiar withe different number systems that we use:

I told the students the story of how the students in Kaktovik wanted to create numerals that matched their language (as shown above) and then (after splitting the students into visibly random groups) I gave them the following clues and challenged them to find all twenty Kaktovik numerals:

Immediately, all groups got stuck into the task. The first two clues were quickly deciphered to give the symbols for 1 and 2. These allow us to make sense of the third clue to find the symbol for 3 which can then be used in the fourth clue to find the symbol for 5. The next clue (which represents 4 + 4) helps us to get the symbol for 8:

At this point, I did remind some groups that there was a pattern behind the symbols that matched the Iñupiaq language. Once they understood that the symbols were not random, they began to make sense of the pattern to complete the numerals to ten. The sixth and seventh clues helped them fill in some other values for 14 and 15:

The penultimate clue was quickly deciphered by most groups to get the symbol for 0 which then allowed them to make sense of the last clue: 10 times by 2 equals 20. Using Kaktovik numerals, this is written as one group of twenty and zero ones. There was a lot of excitement in the room as students completed their work:

I then showed them how the place value columns in Base 20 differ to those in Base 10 i.e. to the left of the ones column, we have the twenties column and to the left of that we have the 20 squared (or four hundreds) column. I then challenged them to rewrite this Kaktovik numerals: 

This was definitely a challenge for some groups but I was pleased to see a lot of success with this.

Finally, I put up some 'Mild, Medium and Spicy' questions for them to try: 

This worked really well as students self-differentiated and all felt that they rose to the challenge:

There is now a Kaktovik numerals calculator app and I was able to show this to the students to check any given calculation. They really enjoyed seeing this!

I finished the lesson by making three important points:

• There are pros and cons to any numeral system.
• Don't assume that what you are familiar with is always the best way of doing things.
• Anyone can create math that can have a positive impact on their community.

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More information about the development of Kaktovik numerals and how they can be used to facilitate arithmetic operations can be found in this Scientific American article from April 2023.

UPDATE:

Some colleagues tried this in another Grade 8 class and saw this on one group's work!

A Visual Approach to Solving Linear Systems by Elimination

I've been lucky enough to work with a teacher of a Grade 10 Applied class and we have been trying to adopt a more visual approach to help students make sense of solving linear systems. We began by challenging students with this puzzle:

The students worked in groups of three on vertical whiteboards and they did a pretty good job of solving this by mainly using a guess-and-check approach. It was interesting that most focussed on the middle two equations first. It allowed me to tell them that we would now learn some cool techniques to solve this pair of equations (i.e. b – c = 5 and b + c = 19). As there were some magnetic algebra tiles handy I then created some zero pairs and asked them what the value of each was:

Now it might seem like an obvious question, but in particular I needed to be sure that they were good with the idea that when they see 3x and -3x (six things), they combine to give 0.

I then used Polypad to show them this linear system:

I asked them to tell me what the two equations were (i.e. x + y = 14 and x – y = 4) and then showed them how we could combine these two equations and, with the power of the zero pair, end up with a simplified equation to solve.

When students saw the y and the -y bars actually be eliminated in front of their eyes, it really did seem to help them make sense of the algebra. What I like about Polypad is that I can show something visually and then alongside it write what is happening symbolically.

The students were then given other linear systems to solve in groups at the whiteboards and we were happy with what they did. Having the students work on the vertical whiteboards allowed us to move from group to group giving feedback when necessary.

This dealt with elimination when one pair of the variables are opposite quantities (e.g. 3y and -3y). But what happens when they are the same quantities (e.g. 4x and 4x)? Well, Polypad can help us again:

Finally, what happens if the linear systems do not have variables in equal or opposite amounts (e.g. 3x + 2y = 16 and x + y = 4)? Well, Polypad allows us to visualise manipulating one of these equations to create a 'match':

When the students reviewed solving linear systems by elimination, it was pleasing to see how well they did:

My thanks go to Ms. K. Perri and her wonderful class of students.

Here are the links to the various Polypads that I have created to help teach this:

Linear Systems 1

Linear Systems 2

Linear Systems 3
Linear Systems 4

Linear Systems 5

Linear Systems 6

Linear Systems 7

Algebraic Expressions and Polypad

I tried this activity in Heather Lyon's MTH1W class using Polypad on Mathigon to help illustrate algebraic expressions. First Q: if this is x + 2 what would 3(x+2) look like? Students go to the VNPS to work in small groups:

Some puzzled looks at first but once they began sketching what it might look like, the students saw what was happening:

I could then easily use Polypad to confirm this:

I followed this up with 5(x+4) and we could now talk of more efficient ways of drawing our thinking. All the while I'm nudging them to the array model:

Now, I want to chunk the ones so give this question. Students' array models are now becoming more efficient:

We do this for a few more example. As they are feeling confident, I want to push them so ask them to sketch out what x(2x+3) would look like. Though they have never seen this sort of question before, they connect it to the model that they have just been using:

I can quickly confirm this on Polypad:

Some students choose to use the concrete algebra tiles but I can then show them how this connects to the diagrammatic model. One of these students says 'Now I see it... and that will save time!'

A few more examples follow and while I help a couple of groups, I give an extension to others: the answer is 18x^2-27 x... what is the question? They factor this successfully without any help from me:

They have been working hard so we take a little break and I perform a little mathmagic: think of a number... add 5, double it... add 8... half it... take away the number you first thought of. By using a little chicanery with a pack of cars, I show them the nine of hearts... the number they were all left with!! I then show how I used algebra to 'rig the system'!

With a little bit of time left, we decide to see if they can use this new knowledge to solve equations. I scroll to the top of my polypad to create an equation from the first two expressions we created:

And with no fuss whatsoever, they successfully solve it!

I was thinking how this approach differed to the one I would have used when I began teaching. Then, I would have given ten examples, no visuals, all symbolic and got the students to do more exercises. Now the students (thanks to Mathigon!) are doing the math!

The link to the polypad I used is here.

A Nice Algebra Puzzle

Last week I went into a Grade 9 class that had just begun to learn about simplifying polynomials by collecting like terms. I had an idea for a task that I thought would help them with this so began by showing them this pyramid.

I explained that the numbers in two adjacent squares add to give the number in the square directly above them. With this information, I split them into visibly random groups of 3 and had them work at whiteboards to find four numbers that go in the bottom row that would give 54 in the top square.

After completing this, I then gave them this pyramid:

I wanted to see how comfortable they were with collecting like terms before giving them something more thought-provoking. Some were able to complete this symbolically and others were happy to use algebra tiles to help their thinking:

I then gave them this task:

By now, I could hear how adept the students were at collecting like terms and was impressed at the different ways they went about solving the task:

Next, I gave them this task with the restriction that all the terms on the bottom row had to be different:

Again, I was really pleased how they went about solving this (some symbolically, some using algebra tiles again) and by listening to the students talk, I could tell that they were really understanding this. I even overheard a few groups say how much fun the task was!

As this had taken less time than I expected, I had to then think quick and come up with an extension. I asked them that if the value of the top brick was 57, what would the value of the four bottom bricks be?

This required a bit of clarification for two or three groups, but once they understood it, it allowed them to demonstrate their algebraic skills in solving equations and substitution:

It was great to see all the different solutions as well as see that in some cases, the different terms actually resulted in the same value once the substitution was made.

I finished by asking them to figure out:

a) the side lengths of an isosceles triangle, given that the perimeter is 120 cm and that the two equal sides are double the size of the other other side

b) the dimensions of a rectangle given that the perimeter is 1000 cm and that the length is triple the width.

For the first problem, they did this by trial and improvement, so I walked them through how to set this up algebraically. I was pleased to see them al use this approach for the rectangle problem.

I was really happy with the way they remained engaged throughout these tasks, especially as it was a Friday afternoon and would definitely use these again.