Amanda Hathaway William J. Ostiguy, Boston, MA
Algebra I : Unit #2 - Multiple Representations: Situations, Tables, Graphs, and Equations : Lesson #1

Border Tiles: Seeing Structure in Algebraic Expressions

Objective: SWBAT use algebraic language and symbols to represent geometric situations. SWBAT understand and use equivalent expressions. SWBAT describe underlying structures in algebraic expressions.
Standards: HSA-SSE.A.1 HSG-MG.A.3 MP1 MP2 MP3 MP4 MP7
Subject(s): Math
60 minutes
1 Overview - 0 minutes

The focus of the three tile lessons in this series is for students to practice Mathematical Practice 7: See and make use of structure along with HSA-SSE.1.A: Interpret parts of an expression, such as terms, factors, and coefficients.  

The population in my school is untraditional, small and variable. This school year I am working with a group of struggling students, all of whom have significant educational interruptions and most of whom have IEPs. They often need scaffolding and differentiation in order to access content. This lesson offers an opportunity for students to revisit some 7th algebra standards in service of accessing the standards at grade level.  In particular, this lesson builds on the 7th grade strands: Use properties of operations to generate equivalent expressions and solve real-life and mathematical problems using numerical and algebraic expressions and equations.

The task I use to address these Common Core Standards (HSA-SSE.A.1 & SMP #7) is from the IMP Year 1 textbook (2010, p. 43).  We start with this task and then build to a  similar task from the Mathematics Visions Project.  It is called Checkerboard Borders ‐ A Develop Understanding Task.

I find this series of lessons to be rich and engaging  for all students and I like the way it can be scaffolded to meet different student needs.  I teach this lesson is two class periods, but if your students are at grade level, you can probably cover it all in one lesson.  This lesson also builds so students are ready to work on the third lesson, Kitchen Tiles, which continues work on HSA-SEE.A.1 and brings in HSA-SSE.B.3.

Feel free to tweak this lesson so it fits your students' needs!


2 Opening - 20 minutes

The task I use to address these Common Core Standards (HSA-SSE.A.1 & SMP #7) is from the IMP Year 1 textbook (2010, p. 43).  I begin class by posting a picture of the garden with the tiles around the edge.  I ask students to find out how many tiles are needed WITHOUT counting each tile individually.  

My students will want to just count up the tiles - I don't let them! Instead, I encourage them to think of creative ways that they could figure out how many tiles there are.  You may need to point out to students that the garden is 10 feet by 10 feet INCLUDING the tiles themselves.

Some methods you might see:

  • Some students may multiply the length of each side by 4 to get 40, and then subtract the 4 corner tiles to get 36 tiles.
  • Some students may realize that the garden itself is 8 x 8 so they will need 8 tiles on each side to get 32 and then 4 for the corners to make 36.
  • Some students may count the top and bottom of the garden with the corner tiles as 10 x 2 to get 20 and then add the 8 tiles on each side to add 16 more.

When students have found one method for getting the number of tiles without counting individually, encourage them to keep working to find as many methods as possible.

Once all students have found at least one method, have them share out their responses on the Smartboard or white board.  I typically ask students to mark up my diagram (or their own) whenever possible to show their method.

To bring this opening to a close, I ask students how the number of tiles needed would be different if the garden was a different size. Then, I let students suggest examples that are different than 10 x 10 and ask them to apply one of the methods they came up with for the original problem.


Differentiating the Lesson Opening
Diverse Entry Points

This year I am working with many special education students in my Algebra 1 class.  In addition, all of my students have had significant interruptions to their education.  Several have been out of school for over one year.  When I introduced today's mathematical task, some students struggled to get started. I had to tell them again and again NOT to count the tiles!

Other students found ways to find the number of tiles, but were at a loss for how to describe (or diagram) how they found them.  What I did with those students was to ask them (one-on-one) how they found the number of tiles and I scribed what they said. From there, we tried to translate that writing into a diagram or a mark up of the original paper.  Students were using really interesting and different ways to figure out the number of tiles, they just weren't always comfortable describing those ways.

3 Investigation - 30 minutes

This activity segues nicely to the IMP activity named Border Varieties on page 44 of the Year 1 IMP textbook.  Border Varieties uses diagrams and arithmetic for an s sided garden.  See an example in the Border Varieties PowerPoint in the resources section.  Students are asked to examine five different methods that include diagrams and calculations and write an expression using s as the length of one side of the square.  In the example in the PowerPoint slide, if the length of a side was s instead of 10, the algebraic expression would be 4(s-2) + 4.  

Once students have come up with all of the expressions, you can lead a discussion about how all the rules are equivalent.  This should lead to some discussion about the distributive property and combining like terms.  You can also have students check their answers with different sized garden plots.  

In my class, students were able to just complete Border Varieties by the time it was time for a reflection question.  In the next lesson, I begin class by having different pairs of students present the connection between the diagram and the arithmetic to the algebraic expression.  If your class works more quickly, you may be able to fit the entire discussion in here.

Looking at Student Work
Complex Tasks

This piece of student work shows a misconception on the part of the student about how to incorporate a variable into the problem. At first I thought this student was disengaged and not trying, but when I looked closer at his work, I realized that he was trying to put an x in that would make the expression come out to 36. With the expression 10x - x, if x was equal to 4, then the expression would equal 36 (which is the correct number of tiles around the border).  A conversation around this misconception is a good way to talk about the value of even using a variable in a problem like this. In fact, a student in class was sort of hinting at this idea when he asked why we had to bring algebra into the problem when we could just count the tiles around the border.  We were able to talk about a different sized garden plot that would be much larger, where algebra would help us to find the number of tiles.  You could also use this problem to talk about the value of being a "translator" of algebra - in a sense, learning a new language.  

Many students in this class had trouble incorporating the variable. One way I addressed it was to keep saying, well if one side of the garden has a length of 10, and we're calling that side x, how we can describe this side that has two tiles taken out, in relationship to x.  This seemed to help.  Students got as far as writing the expressions, but we will have to discuss how they are all related in the next lesson. We were not able to cover all the material in a one hour block.

4 Closing - 10 minutes

This lesson has many different phases. First, students find a method for finding the number of tiles without counting them individually.  Next, they examine those different methods and generalize to an s x s sized garden.  Along they way, they translate diagrams and arithmetic into algebraic expressions.  They then examine and explain why those expressions are equivalent which leads to discussion about the distributive property and combining like terms.  That's a lot of math!  Give them some time to let it all sink in.  You might decide which part of the lesson you want them to "take away" or you could have different groups reflect on different parts of the lesson.  

I like to ask students to complete an exit ticket at the end of class.  For example:

Think about the first task you worked on, finding the number of tiles without counting them individually and answer the following prompt: What did you find most challenging about this task? Why?

I plan to give each group 5 minutes to come up with their responses and then have each group share out.  This reflection should help students make sense of the work they did. An alternative closing activity is contained in the Garden Reflection resource.

What was challenging about not counting?
Student Self-Assessment

Instead of having different groups answer different reflection questions for this lesson, I asked all students to answer the prompt:

  • Think about the first task you worked on, finding the number of tiles without counting them individually and answer the following prompt: What did you find most challenging about this task? Why?

I got some interesting responses.  Here are some reflections:

  • "Not being able to count them was challenging because then I actually had to sit down and step by step do the problem."
  • "Not taking the simplest answer to the problem, and having to do it differently."
  • "I found it hard because you had to just use equations to get the answer."
  • "The thing that made it most challenging is that I couldn't count the blocks and I didn't know any other way to find the answer."

These reflections remind me about how different this task is.  We are asking students to take a fairly simple situation and go about finding an answer in a different, and possibly counterintuitive way. I really like the richness of this problem. It was accessible to all students but there was plenty of challenge built in.

5 Citations - 0 minutes

This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.