Protocol and Ideas for Number System 1.4 '17-'18

I had some trouble doing this activity when I did it with my group during the PD. My initial misconception was what I had to do to start it. After I worked on it for a little bit, I realized we were just creating all of the permutations that could be created with a circle, square, and triangle. Since I haven’t worked with permutations in a while, I had trouble coming up with all of the patterns. My initial thought was to just create all of the patterns right away, so I started to do that. It was at this time when a math teacher said that we could figure out a pattern to come up with all of the permutations.

I foresee that my students will want to create all of the permutations one by one like I did. I intend to have the students work through it a while with the hope that they figure out that they could come up with an algorithm to create all of the permutations rather easily. I hope at least one of the groups will be able to figure this out, and teach the other students how they did it.

I had the same difficulty with this lesson during the PD as I suspect my students will have. I had issues articulating the algorithm for generating all of the patterns. Some of the other teachers ended up representing the circle, triangle, and square as yet other symbols (talk about abstraction!) and writing out the patterns using the replacement symbols. For example, the instructions would say, “Replace * with circle. Replace - with triangle. Replace ! with square.”




In the end, the replacement strategy worked because you only had to do write it out once and instruct the reader to repeat the pattern, but now the * means triangle, the - means square, and the ! means circle. Finally, do it one last time with the * = square, the - = circle, and the ! = triangle.


The beauty of this lesson is that student groups will come up with different ways to solve the problem of generating all the patterns AND that the ‘ordering’ of those patterns are arbitrary. By listening or seeing the solutions of other groups, it may spark something. I have students who will get stuck on trying to get all the patterns and may not be able to get to the second point. One thought I had was to have sets of all the patterns available for use. After spending time with the individual pieces, groups could be given the full set. Then, the focus would be to organize them in an order and writing the instructions for another group to recreate their order may help. Some groups may not need them.

Hope that helps,

The protocol my partner and I came up with was very straight forward. We immediately recognized that each shape could be placed in each position; first, second, and third. We would have three sets of patterns; Circle in the first position, Square in the first position, and Triangle in the first position. The same procedure for the second and third positions. To identify all possible patterns, we started with a shape such as a circle. Then place a circle in the next two positions: c-1, c-2, c-3, then change out position 3 with another shape: c-1, c-2, s-3 and c-1, c-2, t-3. Next return to the starting pattern but change out the shape in the second position: c-1, s-2, c-3; c-1, s-2, s-3; c-1, s-2, t-3, etc. Continue this process until all possible combinations for a circle in position 1 have been created.
The protocol, aka algorithm, my partner and I completed was the easy part. The difficulty I experienced was that I wanted to make the idea of a number system much too complicated. I was thinking how do I create a number system of shapes circles, squares, and squares in terms of addition, subtraction, etc. Although, I doubt I’ll I have many students who think in terms of closed number systems, sets, and groups, I do think some students may over- think the problem to be solved. Suggestions on how to gently guide students to the simpler concept would be greatly appreciated.

I would suggest starting with one place and focusing on ways of arranging the three shapes in one place. Then move on arranging the three shapes in 2 places and then 3 places.

The students could start out random as like a brainstorming session then look for patterns. Then come up with a systematic way to do it. I’m not taking credit for what my group did, but the math guys came up with using a tree so all the different combinations could be found and used. Also, make sure that you use the video resources, they’ve been helping me get through all of this.

@thornhillr3 thanks for sharing this strategy and I totally agree, the teaching tips and tricks videos can be really useful as you’re prepping for the class! Not every lesson has them, but a lot of the lessons that are more conceptual do.

I think @randle.moore makes a good point that some students will over-think the task which might be ok - it is kinda nice when a student who knows the answer of “27” right away and a student who needs to draw it out come together - it forces the “27” student to articulate their thinking a bit more clearly. I agree that without a partner, it could get much to complicated for some students. The way I mediate that is to monitor pairs throughout the room - once they have a pattern written down, I make them write directions so a different group could follow their system. If I tell students they are going to share out, I notice they practice a bit more to clarify their thinking before they get up under the document camera to present. I do have students over think it, but I think partners and monitoring helps control that in my room.

Having students start with a brainstorming session to look for patterns first is important. Then, I would have the students write out all the possible patterns they have found. Then we do a gallery walk and students can see what other groups were able to come up with before coming up with a systematic way or organizing the patterns. Once the students have done the gallery walk, I will give them a few minutes to discuss what they have seen. Students may notice some systematic patterns when when going through the walk. They may also discuss if they have seen fewer or more patterns then their group came up with.

I really like how @bhatnagars suggests starting with one place and focusing on the ways of arranging the 3 shapes in one place and then moving on to arrange the 3 shapes in the 2 place and 3 place.

One misconception that students might have is that the same shape can not be in all 3 places.

When doing this activity during TeacherCon, my partner and I quickly understood that we needed to see how many different combinations we could make with the 3 shapes provided. We also quickly determined that there would be 27 possible combinations 3 to the third power). What we didn’t immediately see was the pattern in which we could begin building our combinations that would have set the ground work for verbalizing our protocol in an easy to understand manner.

I foresee that my students will stuggle with identifying that the symbols we use to represent numbers are not, in fact numbers, but just symbols and that anything could be used. In addition, I think some of my students will have the same issue I had in identifying a pattern in the combinations. I would definately prompt students to look for a systematic way to build the combinations and then use the pattern to simplify their instuctions. One thing I do when I teach this lesson is to have groups share their instructions with another group (normally across the room so there is less of a possibility of the groups to have heard each other as they worked through the combinations) and to see if the groups can follow each other’s instructions (protocol) to build the combinations (pattern).

The first time I did this lesson, I found students were using rotation as a way to show something ‘different’. This year I used pasta that came in three different shapes (much easier to manipulate and no cutting :slight_smile: ). Before the students started I held up a pasta piece and said ‘this is one symbol, no matter what direction I hold it in.’ They took the assignment and ran from there.

We had lots of good ideas, most groups took a few minutes to figure out what was going on, and then started laying down the sequences. Many picked up on patterns. I am thinking about adding a prediction (in the journal) for students to give a minute of thought to the problem before working with their team to solve it.


I thought this lesson was a bit confusing too when I revisited it after teacher con. I looked at the presentation where it says previously on CSP, and watching that helped me better understand the lesson and its goal

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That is what I should have done. During our PD day on this lesson I struggled at first with starting but once one of our group members got started I started to see the pattern. My students did struggle with getting starts (most of them). I had a hard time keeping myself from getting them started. I wanted them to think through it. I think I write down this tip on my lesson for next year.

My protocol was to use the three symbols to create a ternary system much like the binary system. The circle was the first “digit” since it had the fewest number of sides, the triangle was the second “digit”, and the square was the third “digit”. (I don’t think my students saw the geometric connection but that was fine.) Once I decided the “value” of each digit, it became much easier to create the sequences needed.

My students that struggled with this lesson were caught up on the values they wanted to assign to each symbol. A couple of groups tried to assign money values to the three shapes (1 cent, 5 cents, 10 cents). After letting them struggle for a bit I would ask them how to represent 4 cents. When they realized they couldn’t do it with just 3 digits, they began again. This time they were much more successful.

My protocol started by knowing there should be 27 possible outcomes 3 to the third power 3 x 3 x 3 = 27
Also I knew that each shape should be in each position 9 times
Started by picking a shape and drawing it vertically nine times in the first place,
Then knew that there will be 3 of each shape in the second place
The third place will go through a single rotation of each shape three times as you go vertically down

Repeat the above sequence twice.

My students struggled with the writing down of their actions. They could do it, but writing down a clear protocol was difficult for them.

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Thank you for sharing your protocol. Students often find it difficult to articulate the steps clearly.

I did not attend the PD, so this feedback is from my classroom experience.

I found that students caught on to the idea of generating all of the patterns rather quickly, which was reassuring since most of my class is on the younger side (9th graders). A few groups were able to come up with a more ordered way of generating all of the patterns (a protocol), but no group was able to identify all the way down to a variation of 3 patterns with layers (ie, adding other shapes) added on.

They were very engaged watching the lecture afterwards because of this activity!! Well done, superstars :slight_smile:

I did cut out all of the patterns beforehand, and handed each group of 2 a paperclipped set of 3 of each shape. This saved a lot of time, and it helped keep both group members engaged. Everybody had something to do with their hands!

I taught this with mixed age levels (9th-12th grade). Some were ready to jump into college level number theory, while some where struggling with their fears of numbers entirely. So the concrete nature of this activity and discussion enabled all students to participate meaningfully. The patterns they generated ranged from random to cyclical to group-based to numerically representations.

My frustration at the time was not feeling able to help students to comminate effectively enough for their (differently-abled) listener to understand. In retrospect, I see that that’s beyond the intended scope of the lesson.

In the future, I’d like to focus on just finding ways of organizing the patterns, realizing that there are many ways to do it, and that any further use of a system among many people will require agreeing on a system. This will segue into students realizing they already have a working understanding of the decimal number system which solves the problem, allowing the group students to feel relief about knowing the number system, rather than the dread that some math-phobic students may have had otherwise.