I show my 6th graders this image, pointing out that this picture represents the two numbers 1 and 1 that I’d entered at the top.

I then ask them to give me two new numbers — any two positive integers [that are 10 or less, for now] — and the computer will draw a new picture. As each set of new numbers is entered and the corresponding picture is generated on the screen, I ask students to jot down their “I notice, I wonder” in Google Form and to draw a rough sketch of it in their journal. After a few sets of numbers, I ask students to imagine and/or draw a rough sketch of what they *think* the picture will look like **before** I hit the *update* button.

These are the pairs of numbers they’d asked for and their corresponding pictures, listed in the order that was asked.

I love that the kids are asking for…

- 6, 6 after the initial 1, 1
- 3, 8 after 3, 1 (keeping one number the same)
- 8, 3 after 3, 8 (reversing the numbers to see if anything changes)

But, when sets 9, 9, and 7, 1 are asked at the end there, I say to the class, “Hey, what if figuring out this puzzle — which is how the computer draws the picture given two numbers — gets you a million dollars. And you get to ask for sample sketches like you’ve been asking, except that each sample costs you some money! So, make each request worth it. Let it prove or disprove your conjecture. Ask carefully.”

I love the *OHHH*s and *AHHH*s after each picture is revealed. But no one is claiming that he/she had drawn the same diagram. I pause longer for them to write down their noticing and wondering.

I now say, “You may only ask for four more sets of numbers. Remember, make a request that would test your conjecture.”

I ask a normally quiet student. She says, “10, 3.”

Another student wants to know what “100, 5” looks like.

“What about 8, 5?” I reply, “Sure, but draw it in your journal first.” They are fully engaged. Then I say, “Now, share your drawing with a neighbor.”

I ask, “Did anyone sketch the same thing as their neighbor?” They’re shaking their heads, and I say, “That’s pretty crazy! Do you think yours is more ‘correct’ than your neighbor’s?”

I reveal 8, 5.

The last request is 23, 75.

What some of them have written [with minor edits from me]:

When we did the same two numbers the shape didn’t change but when we did different numbers it changed. Why does it divide into little parts within a square when we put 3,8? When we did 8,3 the number switched around. I wonder if the two numbers are dividing to make the shape. How can you figure out the number when it can’t divide easily. My drawing for 8,5 was one whole and 5 little squares. The 23,75 was a little confusing to me.

They’re different, they are the length and width, and when the two numbers are the same it’s just one cube. I notice that if it can be simplified, it is. Example: 6,2 = 3,1. I don’t understand 8,3, 5,9, 23,75 or 8,5. But I did notice that the smaller the parts of the shape are, the lighter shade of blue they are.

I observe that when the same numbers are entered it equals to a blue square. If the first number is bigger than 2 then it will add one more square. I wonder if you double the number for each number will it be the same shape. I wonder why for 3,8 it has one square with three parts. I observed that if you divided the first number by the second it will equal to the number of squares. For 8,5 I didn’t get the right sketch. The sketch was one square with half of a square cut in half, then in one half is has a strip that is cut in half. I wonder why it has half of a square. I think that my answer for how it figures it out is right, but I don’t know how it comes up with that picture for 8,5.

When you do 1 and 1 it doesn’t change because we tried 6 and 6 it didn’t change and if we put 3 and 1 it did change. I saw that when we did any number like 3 and 1 is 3 ones. So I think that all you have to do is divide something by something = the first number that you put in but if you can’t divide by 2 then I’m wrong. I’m not sure that I got this right but this is what I think.

For the first one 1 and 1 I thought it would be a small one by one cube. What threw my off was the 6 by 6 because the size did not change. For the 8 and 5 I drew a big block and and 5 little ones, but my image was wrong. I also wondered if the first number was the amount of shapes that would appear, but I was wrong again. I don’t understand yet. I tried looking for a pattern, but couldn’t find one.

When I tried 8,5, my answer was almost right. I had the one big square right, the half square right, but then I got the little squares wrong. I think that the way the computer does it is dividing the first number by the second number. I am confident that if you put the numbers 10 and 5 in, it will show 2 squares. When using the diagram, the second number will represent the vertical side.

What I’ve been noticing was that if you put the bigger # in the front and the small # last then it would be like a rectangle. I’ve also been noticing that if you put the same #’s it would like keep on drawing a square. So someone said what could (8,5) look like and Ms.Nguyen showed us the drawing and the I notice that nobody got it right. I was expecting something like smaller because the #’s were small they weren’t as big, but at least I tried to get it correct but I drew something a little bit smaller than that. I also wondered why when we put the same #’s together why do they all become a square that’s what I wonder.

For 5, 12 I notice that it is two big squares, two smaller squares, and two tiny squares, I thought it was going to show 1 big block and another big block but that one would be cut off at the bottom or not a whole block. I also notice that the pattern is the first number multiplied by what equals the second number and the number that is missing is the amount of blocks that is created. I thought I knew it but I don’t really get the ones with a bigger number first and the smaller one last. I thought I knew what 9, 23 was going to be but it the result was surprising. It didn’t look at all how I thought it was going to. The website is pretty cool, but one thing I didn’t understand was the placement of the small blocks and what they stand for. Like some were really tiny and some were small but I don’t know what they stand for. But I bet if someone explains it to me I probably will understand perfectly and feel dumb.

The only thing that I am sure that I know is that when the first number is larger than the second, the shape is wide and when the second number is larger, the shape is tall. Other than that I am very confused.

Then, together as a whole class, they agree on the following;

- When both numbers are the same, then the picture is one square.
- The computer simplifies the two numbers, such that a picture for 6, 3 is the same for 2, 1.
- When the second number is a 1, then the picture shows the first number of squares. For example, 7 and 1 would form a picture of 7 squares, and 100 and 1 would form 100 squares.
- The first number is the horizontal dimension, while the second is the vertical.
- They are all squares.

The dismissal bell is about to ring, and I want to teach forever.

Tomorrow, we’ll spend some time with one set of numbers, like 10, 3 or 8, 5. We’ll dissect the diagram. Play around with a few more. Practice sketching a few. We’ll write out the equations that go with each diagram. I’ll guide them into noticing the size of the smallest square in relation to the two numbers.

I found this investigation at underground mathematics. The site describes itself as having “rich resources for teaching A level mathematics.” From what I understand, “A level” means advanced level mathematics consisting of core modules ranging from quadratic, logarithms, geometric/arithmetic series, differentiation/integration, etc.

Perfect for my 6th graders who continue to torment me with their arithmetic atrocities, such as, 3² = 6 and 5 ÷ 10 = 2.

While the original task is scripted for older and more advanced students, I found in it what I needed to make it rich and appropriately complex for my 6th graders.

Hail, Euclid’s algorithm!

## 6 Comments

Thank you for sharing this – I plan to use it with my students as well (advanced fifth graders). I love that once you think you know the value of a side of each square, you can use that to prove your conjecture using the area formula.

This is gold! Thank you for your continued math mining and alchemy. Your findings and creations are making us richer teachers.

This is a great example of how math is an exploration. Thank you. I also like how you have the students write their “I notice, I wonder” in Google forms. Your classroom must be 1-1 then?

What’s interesting is that with the exception of [6,2], [100,5], and the identical pairs of [6,6] and [9,9], every set of values explored was relatively prime.

When we introduced Euclid’s Algorithm to our students, it came after the traditional factor tree method frustrated them. We explored the visual with well-known pairs like [18, 12]

and used the subtraction model of Euclid’s Algorithm before moving to relatively prime pairs because it allowed them to more easily recognize cases where the greatest common factor was 1.

Through this, Euclid’s Algorithm became a tool of empowerment for my students, as well as an early appreciation for the wisdom of the ancients.

Long story short: Euclid is my Homeboy.

Love this. Thank you for sharing their summary observations. I love how they put their new understanding into words.

– Elizabeth (@cheesemonkeysf)

I was so intrigued by this blog post and the visual models you showed to the students from Underground Mathematics. I would really like to try this with my sixth grade students and I anticipate that students will also be very excited to see what is revealed with each new set of numbers. I thought your set up of the activity as a puzzle that students were trying to figure out along with the drama of earning a million dollars to figure out the puzzle with as few clues as possible was brilliant.

The writing that you asked students to do in connection to this activity was also very powerful. Are you able to have students read eachother’s writing? I have tried several ways to allow students to read and critique eachother’s writing (Shared Google doc, Shared Google presentation, comments in Google Classroom). Have you figured out a way to do this in Google Forms?

Have you found any other resources from Underground Mathematics that you have been able to adapt for middle school students?

Thank you for sharing this resource and how you adapted it for your class.

Math to the 7th Power

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