A Linear Functions Lesson Across the Grades

How many people can sit at 100 (or n) triangular tables? Square tables? Hexagonal tables? What if you join the tables so that one side of the next table touches one side of the previous table?

I appreciate this problem for a few reasons:

  1. I can present it in grades 4 through 10. In grade 4, students write a recursive relationship (e.g., for joined hexagonal tables, start at 2 and add 4 each time). In grade 6, students write a functional relationship (e.g., 4n + 2). In grade 8, students graph a linear relation (e.g., y = 4x + 2). In grade 10, students interpret the slope and y-intercept (e.g., each added table provides 4 additional seats, there are 2 additional seats at the ends of the table). When I teach and discuss this lesson at different grade levels within a school, I think a common activity helps teachers connect the big ideas across the grades.
  2. I can easily adapt and extend the task. When I have taught this lesson in grade 6 (see three-part lesson plan), most students can write an expression for joined square or hexagonal tables. Some students may choose to solve a simpler problem and write an expression for joined triangular tables. Other students can be challenged to write an expression for tables with any number of sides. All students can participate in the class discussion.
  3. The use of pattern blocks can help students gain a deeper understanding. Most students were able to make sense of the 4 in 4n + 2. Each time a table is added to an end, 4 seats are added. (Two seats are lost when tables are joined.) When one student showed how he added tables to the middle rather than an end, this helped his classmates make sense of the 2 in 4n + 2. There are two more tables at the ends. Pattern blocks allow students to make sense of the expression beyond “add 2 to make the numbers in the table of values work”.

This problem appears in several resources including The Super Source.

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3 thoughts on “A Linear Functions Lesson Across the Grades

  1. I have used these problems teaching sixth grade. This year I will be teaching eighth grade (Math 8 and also Algebra) so I will have some of the students I had two years ago. I love the idea of bringing it back and moving it to the next level with slope and intercept.

    The most amazing part of the activity in my classes was the variety of expressions students came up with for the same situation. For the hexagon chain, aside from the 4n + 2, I would have students see it as:
    2n + 2n + 2 (two chairs for each table along one side, two chairs for each table along the other side, and one on each end.)
    10 + 4(n – 2) (five chairs on each of the tables on the end is ten, and then four on all of the “middle tables” but there were only n – 2 tables in the middle since you used the ones on the end.)
    I am sure there were more that I don’t remember. This was followed by the opportunity to discuss WHY the different expressions were equivalent!

  2. Cindy,

    Please don’t read anything into the fact that it’s taking me a month to thank you for stopping by and commenting. I decided that I would stay off my blog during summer BREAK. I was able to do this but had a much tougher time with lurking. Love your blog, btw.

    It is amazing to see the variety of expressions that students come up with and the excitement this brings. In my current role, I get to see this with teachers too.

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