Kitchen Tiles

I start class today by letting students know that they will take a look at more tile problem to expand their thinking about patterns. In this lesson, their task will be to explain why an algebraic expression works, rather than come up with their own expressions like they did in the previous tile lessons. My students may be used to writing algebraic expressions, but they are unfamiliar with starting with an expression and explaining why it works.  I let students know this is a good opportunity for them to practice SMP 3: Construct viable arguments and critique the reasoning of others. In particular, they will be critiquing someone else's argument, which may be new to them.  

As we begin I show students the context for the problem and the visuals that accompany it. I do not show students the algebraic expression or the accompanying questions yet.  With the pattern on the board, ask students to respond to two prompts:

1. What do you notice about the diagrams?  
2. What do you wonder about the diagrams? 

I record their responses.

Teacher's Note: This lesson is adapted from a task created by Illustrative Mathematics. Please see the citations section for more information.

Now I hand students the full problem, Kitchen Floor Tiles. Once they have begun to notice and to wonder about the problem, they are ready to investigate it.  

In the task, students are first asked to explain why Fred's algebraic expression is correct. Students may try to take the easy way out here. For example, a student may say something like, "Fred is correct because his expression gives the correct number of border tiles."  I make sure to explain to students that I am looking for much more mathematical detail than this!  I say something like, "Right, we can see that the expression works because when we plug in the diagram number for b, we get out the correct number of border tiles, but WHY does that happen?"  I push students to investigate the why behind the expression. 

Throughout this segment of the class, I circulate around the room to take a look at student writing.  I continually push students to be explicit in their writing about how they know Fred's expression is correct.

If students have trouble getting started or need scaffolding to access this problem:

• Some students may need to physically generate the pattern themselves in order to understand it. I suggest they use graph paper as a tool or even a set of manipulatives. I think any opportunity for students to interact with the pattern will be help them make connections to the algebraic expression.

• I may further scaffold the task by asking students to mark the original 10 shaded tiles from Border 1.  If students mark the orginal 10, and can identify the originals in Border 2, they may start to see that 4 new tiles are added in Border 2 and then 8 new tiles are added in Border 3.  

• I ask students to create a table that shows the relationship between the In/The Border Number and the Out/The Number of Border tiles. (If these headers are too confusing, you might refer to the In as the Diagram Number and the Out as The Number of Shaded Tiles Needed.)  

• If students are still having trouble connecting the pattern to the expression, I will ask them to show those orginal 10 tiles separately in the Out column.  So for Border 2, intead of writing just 14 in the Out column, students could write 10 + 4. This may help them again, to see that 4 tiles need to be added to each consecutive diagram.

• Some students my struggle to explain the (b-1) part of the expression.  Again,I have them break down the results of the Out column. Once they do so, they will see that Border 3 is 4(2) + 10.  You then want to ask them "How does the 2 relate to the border number?  If Border 1 is 4(1) + 10, how does the 1 relate to the Border number?  Students may start to see that it is always one less than the order number.  You can ask them why this is so, and also how can they write that algebraically.

If students struggle with Question 2, I ask them to draw out the situation.  I might also ask them to consider how their Border 1 is different for Emma than it was for Fred. I want them to relate that difference back to the constant number in the expression.

Question 3 may be more of an extension question for this lesson in my class. What is tricky about Question 3 is that students will need to work with two variables instead of one:

Students are asked to generalize the first pattern where there is a tile above and below each non-shaded tile and two tiles on the end.  So, if the starting diagram had 10 tiles, you would need 10 shaded tiles on the top, 10 on the bottom and 1 on each end.  

One approach is to use this line of thinking to help students to generalize that the original number of tiles will always be 2n + 2.  From there, I ask my students where that should go in the expression and how can they account for the added tiles as the diagrams grow and change.  I think that my students should see that the 4(b-1) will remain the same regardless of the number of starting tiles.

Much of the discussion today will happen one-on-one, or in small groups with students as they work, but I leave time at the end of class to debrief and reflect on the activity.  I plan to have students share out their responses to all three parts of the question.  I pay special attention to students who have different ways of relating the algebraic expression to the original problem.  

I emphasize with students the importance of being able to discuss and critique the solutions of others (MP 3).  This problem is a great opportunity, because Emma (in the problem) has reasoned incorrectly about what the 4 represents in the expression and by default the 10 as well. I have students practice their responses to Emma (even though she is not a real student in the room!). This kind of role playing can help students to eventually critique each other's work.  Of course, it may be necessary to encourage students to focus on the math when they explain why Emma is incorrect, rather than doing any kind of character assassination!

For an Exit Ticket, I ask them to reflect on the prompt:

Write about one way you were able to see part of the pattern in part of an algebraic expression.  Was seeing that relationship challenging for you? Why or why not?