Friday, May 18, 2012

Connecting Area Models to Linear Models to 100

First let me start off with a big HELLO! It's been months since I've posted. The content of this blog will change from a technology component to that of a math focus. It was always my intent to use this blog as a space to open up the conversation about math thinking in a primary classroom, namely my own and that of the students, with a focus on grade 2 curriculum. It's a work in progress and I've got lots to share so please return frequently and I look forward to the conversations to come.

Our grade 2 curriculum in Alberta says, students need to represent and describe numbers to 100, concretely, pictorially, and symbolically. As a class we've been working on this all year but we're revisiting it again to help pull and develop math strategies.

One of the strategies AB Learning recommends is using addition o subtract. The goal in the images you'll see below is to help build the connections between the area model and move it into the linear model of a number line. My goal is to show how the two are connected as well as work on placement of numbers in the area model and linear model. 

Students built 25 and 100 and the task was to compare them to each other. Many students were very good at pulling the chunk of 75 from the 25 and then using strategies like making tens, transforming the chunk of 25 and doubling it, some counted 10, 20, 30, 70 and then added 5.
You'll see from the model, the students could easily count the tens and fives. This picture does not lend itself to easily seeing the adding on to get to 30 from 25 or adding a 10 to get to 35 but it's the model the student built and there were others who built it as two tens and the five ones. I just didn't get a picture :-).

From here we talked about the decade numbers (tens) and where they would go on the grid. Lots of good discussion was had around where it should go and why. In the end, students placed their numbers in various places on the grid. Some up in the left hand corner others bottom left. It depended on what they were thinking and where they placed their 25.
You'll see here that this student is building it backwards. It's not a concern of mine because they are understanding their own sequence. I must add that writing the numbers on the blank grid was a challenge for some of them. I didn't leave them hanging because they then stretched it out into a number line to see the connections between the numbers. Now this student I watched because of where the numbers were being placed on his rod.
This student is going to have difficulty taking the area model out into a number line because they are just writing out the pattern of numbers and placing all the stickers on the one rod. Many ran into this problem but eventually worked it out with help from another student or listening to what others were saying.

I didn't want all of the numbers written on the grid because it's not about seeing all the numbers it's about making connections to the relationships between the numbers and where they fall on the open number line. Remember, my goal is to make connections and eventually build the understanding of adding to subtract. Before I can get them to this strategy they need to understand several different ideas here.

  1. Where are the numbers in relationship to each other on the area model and the linear model.
  2. Comparing the two models and being able to use them the same way. 
  3. It's possible to move from 25 to 100 or from 100 to 25 without moving the large chunk. HUGE IDEA!!!
  4. The two models the same but just organized in a different way. 
  5. The number 25 is embedded in the number 100. Another HUGE IDEA!!
After placing the numbers on their paper copy and their manipulative copy, the students took the area model apart and stretched it out. A lot of misunderstandings were somewhat cleared up and others realized they'd made some mistakes along the way. It's okay because we'll come back and build more and more of these models. Unfortunately, I didn't get pictures of the linear model but I will next week because all of them didn't build it.

There were some interesting connections being made once they moved to the linear model. Some said, "Ms. Brown, did you know there are 4 - 25's in the 100?" Great connections. "There are 5 - 5's in 25?" Others were asking, "Why didn't the paper model match with the unifix model?" They kept trying to place the numbers side by side on both models but the paper model was longer. Sorry, no explanation for that one other than the paper was actually a bit bigger and wider than the blocks. Hopefully, I can get a picture of that next day.

When they built both models I wanted them to compare the two. I think it's also important to realize that students don't always have to put the 1 in the same place. You'll see various images where the 1 is opposite. It's all about their understanding and how they build it. It's our job to take their thinking and help them make sense of it. Not to put them into our way of doing it.

Have they finished making connections and do they know everything they should? Absolutely not!!! We've only just begun. More building and discussion needs to take place. We haven't even gotten to the idea of using addition to subtract but I know someone will lead us into this strategy without me having to tell them. 

1 comment:

  1. I'm so excited to read this blog. I am always striving to "up" my math teaching. I look forward to reading lots more!

    2nd Grade Pad