Where does the number begin and end?

Look at your hands and your fingers, 1 to 5 on one hand and 6 to 10 on another. Doesn’t it look like 5 might be closer to 0? It wasn’t anything I contemplated recently until my students brought it up during Open Middle/Clothesline math session. Make a number that is closer to 200 than 300. Where is 250?

“I think, it is closer to 200, because you count 250 first.”

“Because a teacher told if it’s 50, you round down.”

This made me pause, “What about numbers from 0 to 10, Is 5 closer to 0 or to 10?”

“Closer to zero.”

“Raise your hand, if you think it’s closer to 0.” – More than half of the hands go up.

I did not expect this at all.

“I am not sure. Is 5.1 still 5? Where does the number begin and end?”

Developing the Question and the Claims

In the next few days I’ve been obsessed with this question and with the possible ways to approach it, and I think this question made it into most of the twitter chats. Kent Haines wrote about the rounding conversation in his blog.

In another conversation about natural numbers,  David Butler pushed my thinking in an unexpected direction when we started talking about calendar.

David Calendar Super Important

I wondered if that might have been the reason for my students’ uncertainty. Ordinal and Cardinal number duality causing a lot of confusion!

I was trying to figure out how to approach it when Malke Rosenfeld’s book, Math on the Move, finally arrived in the mail. The next day, I made four large number lines on the floor of my classroom.

Students explored the line in groups to test their ideas and to discuss them.

S: “You start here (points at 0), you count 1, 2, 3, 4, 5, 6. But when I go to 10 if I count from 5, it’s 1, 2, 3, 4, 5.”

Me: ”Why does it change?”

S: “You don’t count 5 because you already counted it.”

Me: “Do you count your starting point?”

S: “Yea.”

Me: “So what happens if you count from 10?”

S: “It will still be 5!”

Me: “Try it.”

S: “1, 2, 3, 4, 5.. (stops)…. 6? It changes. When you walk up, 5 is closer to 10. When you walk down, 5 is closer to 0.”

We were thoroughly confused.

This time, Malke Rosenfeld, Sarah Caban, Simon Gregg and Kristin Gray joined the conversation and shared some ideas on using kid-sized math tools. The calendar also reappeared.

Simon Calendar 2Simon CalendarSimon Calendar 3

I didn’t want to stop just yet. The discussions were great, and I was looking for some structures to prompt students to verbalize their conjectures and to listen to each others’ arguments. After the recent online session with Kristin Gray at Mount Holyoke College, I’ve decided to start with the Talking Point and this time limit it to one statement to focus our conversation. Talking points helped students to find like-minded peers to do more proving together.

Developing Arguments and Revisiting Claims

We ended up making four teams as there was always one more popular claim. Each team got their own large number line and any other manipulatives or measuring tools that they could find in the classroom. Our most popular choices were meter sticks, snap cubes and balances. I also handed each group a set of cards with numbers 0 to 10 and some tape to arrange their numbers on the number line when and as needed. Some added fractions. Some took rulers to be precise.

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Groups presented to each other and asked questions. Many students changed their opinions in the process.

reflect change idea

Counting Argument: Five is in the middle.

argument, one not zero

S: “You go 1, 2, 3, 4, 5 then 1, 2, 3, 4, 5.”

Me: “Do you count zero?”

S: “No.”

Me: “Why no? Shouldn’t you?”

S: “You don’t count zero because it’s a number that you don’t use in counting.

Semiotic Argument: Five is closer to zero.

argument 1
“Zero is one digit and ten is two digits. And five is one digit. So it would be closer to zero than to two-digit number.”

Calculations argument: Five is in the middle.

Change argument: It depends.

argument, it changes
“It shifts when you count from different sides.”

Snap cubes argument: Five is in the middle.

“The number of blocks between zero and 5 is 4, and the number of blocks between 5 and 10 is also 4. The number of blocks stays the same so it is the same distance.”

Hotel argument: Five is in the middle.

hotel argument

“Say you wanted to stay for 5 nights and then go to a different hotel for 5 more nights. And then you leave. Days are spaces and nights are numbers.”

In my final move, I asked students to take the numbers away, then find the middle and stand there. Then together we put the numbers back. Five was landing on the middle spot. We proceeded to our journal reflection.

reflect1

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argument, number line

reflect6 hard to pick

reflection uncertain
Some students are still uncertain.
Reflect not convinced
And some are unconvinced.

Closing Thoughts

This question appeared to be much trickier than it sounded at first and it brought forward more questions. What is the middle? Are numbers “spaces or points?” Are fractions numbers? How do you count? Do you count zero even if it is nothing?

Actual Answer

I thought about a difference between the models of number line and a hundreds chart. I never considered that essential difference before. I wondered if developmentally ordinal understanding of numbers comes  first and it is a an abstraction leap to the cardinal numbers. I also wondered about historical development of number concepts and at which point numbers stopped being counters. My twitter and classroom conversations made me think about our zero-less calendar. We count years, days and months as they begin. Our whole calendar system is based on the intuition of ordinality. Zero is hidden in the moment of midnight, too dark and fleeting to count it.

And a large number line is a keeper.

large number line

I Noticed a Pattern in My Pattern

Introduction

“What is a pattern?” – “It is when something is repeating like ABAB.” I found at the beginning of grade 3 that my students could build repeating patterns. We’ve been working throughout the year on identifying rules in increasing and decreasing numerical and visual patterns, and I wanted to find a way for all this scattered practice to come together.

The credit for the idea goes, yet again, to Simon Gregg  who shared how he used Desmos with his students to graph the visual patterns they created. He shared his blog post with the details of the lesson and students’ work. I wanted to adapt it for my classroom and I hoped that my students would notice the connections between the model, the symbolic numerical representation and the diagram/graph.

Questions and Thoughts: Before

I was anxious about the logistics of the lesson. Thank you, Sarah, for moral support!

twitter

 

Will everyone be able to build an increasing pattern?

Will all students identify which step is which when translating the pattern into the table?

If we work in math journals, will someone spend an hour creating a table to fill out?

Will the students be able to handle the technology part independently?

Preparing for the Lesson: Getting Ready to Make a Pattern

I used one of the three-act math prompts from Graham Fletcher’s website to get students thinking about the growing patterns. After watching the video, I asked my students to build steps 4 and 5 of the pattern, label the steps and the number of tiles in each step and write down the pattern rule. Then students had to figure out how many tiles would be in step 10 without building or drawing all intermediate steps. This part has gone smoothly enough to proceed to the next lesson.

Preparing for the Lesson: Getting Ready to Make a Table and a Graph

Many of my students used the table to organize the numerical information about the pattern to find the number of tiles for step 10. We created and extended the table together with the class. I decided to use teacher.desmos.com to utilize the dashboard feature that would allow me to keep track of the students work and to bring it on the interactive whiteboard for future discussions. Here is the activity screen I used to for my lesson. We graphed the pattern together and discussed the behaviour of the graph.

The Creation

The lesson went much smoother than it did in my nightmares. I made a table template to simplify the logistics. Most of the students were successful in creating an increasing pattern that follows a specific rule. Some students created patterns that were increasing, but they needed support to identify that their rule was not consistent.

table-inconsistent
There was often a “break” in the rule between step 1 and the rest of the pattern.
pat16
Some patterns had obvious visual rules but were tricky to represent in a table.

While most of the students worked with addition, some students tried to triple or double the number of cubes they use in the pattern.

Pattern 1: A Pyramid

Pattern 2: A Fire

graph1

Pattern 3: Plus Four

Pattern 4: “It’s a bit up and down but still increasing.”

pat15graph4

Students had some great notices and wonders.

noticewonder1

noticewonder2

noticewonder3

noticewonder4

 

The really exciting things started happening when students crowded around the interactive whiteboard and started analyzing their own and each others’ graphs. Someone noticed that if there is an unusual change in behaviour in a graph, there might be a mistake somewhere in the calculations. One student said, “If the number of cubes is the same on every step, I think the graph will be just the straight line, not tilted.” Someone else asked, “How can you make a circle if there will be more than one Y for one X?”

I had a moment of insight.

Making Predictions

During my lunch break I threw together my template for the afternoon reflection which included three questions and three spaces for the sketches.

-Without numbers, sketch how the graph of your increasing pattern looked like.

-Sketch how you think a graph of a decreasing pattern might look like.

-Sketch how you think a graph of a repeating pattern might look like.

I asked students to try and explain their intuition.

sketch3

There were a few interesting ideas considering the behaviour of the graph of a decreasing pattern.

decrease6.PNG

Many students tried to look at the graph differently rather than switching the direction of the slope. As if they re-located their point of view rather than changing the orientation of the graph to represent the decrease. I wonder why and I currently have no explanation.

decrease5

This reminded me of a graph of an absolute value function, and it seems to me that the student’s logic was also describing something similar.

As expected, repeating patterns had the largest variety of predictions. Most of the students went with the equally spaced bumps. A few sketched a graph of a constant.

repeat2

There was an attempt to express a repeating pattern as a periodic event.

repeat6

And I need to ask some questions about this graph: How far do you think these lines can go up and down?

repeat1

Questions and Thoughts: After

My students need more Desmos in their lives. The interface appeared to be 8 year old friendly. I did not need to explain anything about the coordinate grid. I think, if we try graphing decreasing patterns, there will be more clarity about the structure. I loved the teacher dashboard with all the functionality of tracking my students’ work and the ability to pause the lesson and to control the slides. I would love to have the function to print students’ work from the dashboard implemented because printing 50 clipped screenshots was somewhat time consuming.

I wonder if showing all three steps of the process before engaging with it pushed many of my students to work numerically rather than visually. Many of the students were just building the “rods” with the cubes instead of going for more creative shapes. Would the outcome have been different if I asked them to build the patterns first, took the pictures, and then introduced the next two steps? If anyone tries, please let me know how it goes.

I also wonder what my next step(s) should be. I am considering giving kids their graphs and pictures of their patterns and finishing their notices and wonders first. Some of the students wanted to test their predictions for decreasing and repeating patterns. Some started experimenting with creating pictures with their lines. I am also thinking about reversing the process; giving my students a graph and asking to create a table and to build a pattern. We will start with wonders and then I will make my decision.