Tag: three-part

Posted on

Quadratic Patterns

Having students write an equation that describes a pattern involving toothpicks, pattern blocks, or colour tiles is nothing new. However, students (teachers?) often focus on patterns in the table of values rather than properties of the pattern itself. Visualizing the pattern can help students write the equation. For some, this approach may be new.

For example, consider the following pattern:

In each figure, students may see a rectangle with two squares attached, one above and one below. That rectangle has a width of n and a length of n + 2. The expression is n(n + 2) + 2.

Some students may see the pattern in a different way. But what about the students who don’t see anything? For them, some scaffolding is necessary. Note the scaffolding in the pattern below.

Students may see one red square, two green rectangles, and two blue tiles in each figure. That is, they see n^2 + 2n + 2. The use of colour is intended to be helpful. Of course, some students may ignore this hint. I’m cool with that. They may see a large square with one tile attached, or (n + 1)^2 + 1.

Again, look for the scaffolding in the pattern below.

Students may see a rectangle with a number of tiles being removed, as suggested by the dotted lines. That rectangle has a width of n + 1 and a length of n + 2. The number of tiles being removed is equal to the figure number. Alternatively, students may visualize  2(n + 1) + n^2.

Did you notice that each of the expressions above are equivalent? They must be. Each of the three patterns begin with 5, 10, and 17 tiles. Each pattern/expression tells the same story, but in a different way.

My goal was to design three parallel tasks. Have students choose one of the three representations… just don’t tell them they’re the same.

My three-part lesson plan:

Marc and I created two more sets of patterns. All three:

For more, please see Fawn Nguyen’s Pattern Posters.

Posted on

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.

Posted on

The more sides you have, the smarter you are.

“How does shape affect your place in society?”
“The more sides you have, the greater your angles. So, the smarter you are.”

Two years ago, I created a lesson on Angles in a Polygon. The ‘hook’ was the opening minutes of the animated film Flatland: The Movie. In the story, Arthur Square asks his curious granddaughter if she has memorized her ‘laws of inheritance’.

Hex replies “Isosceles triangles have baby equilateral triangles. Equilateral triangles have baby squares. Squares have pentagons. Pentagons have hexagons, like me! And each new generation gets one new side until they get so many sides they look like a circle and become a priest.”

This film interestingly addresses many mathematical concepts, such as points, lines, and shapes in zero, one, and two dimensions as well as larger themes such as critical thinking.

Here it is:

I think it’s a pretty good lesson, but I decided to tinker with it. Here’s the new and improved version:

Yep. That’s it. Blank space.

I learned that from Sandra Ball when planning together for elementary school demonstration or team-teaching lessons. Just one of the many things I have learned from Sandra since joining the team a year ago.

The first activity is overly scaffolded. In the second version of the activity, the scaffolding is removed. Students will ask “How can I solve the problem?” versus “How does Mr. Hunter want me to solve the problem?”. Some students may need scaffolding, but I can better support these students by listening to and observing them. In the first assignment, I assumed all students would need scaffolding. And, really, if my students can’t think of using a table to organize information, what does that say about how numeracy is taught in my classroom?

Here are the documents as well as the three-part lesson plan:

Flatland Assignment
Flatland Assignment 2.0
Flatland Three-Part Lesson Plan

Posted on

Math Manipulative of the Month – Pattern Blocks

MMM September 2011 Pattern Blocks (colour printer, double-sided)

Last year, a group of Surrey teachers suggested having a “Math Manipulative of the Month” at their school. Instantly, I thought this was a great idea. After this conversation, I created the brochure above. My hope is that this series of brochures can be used to generate conversations between teachers (and students, of course!).

Before trying the problems, I would ask teachers to get to know each MMM and list all they know about them. For example,

  1. “Two reds cover 1 yellow”, “Three triangles make 1 trapezoid”, etc.
  2. “All sides are the same length, except the base of the red trapezoid. It’s twice as long.”
  3. “The orange square and tan rhombus do not cover the other tiles.”

The symmetry problem ended up on the cutting room floor. Here it is: Pattern Blocks Symmetry.

Also, please see how the question “How many ways can you make 360 degrees?” becomes a problem-based lesson in Grade 6. Here’s the three-part lesson plan: Angles (format from Van de Walle).

I attempted to have a balance of primary and intermediate problems. How can each problem be adapted for the grade level that you teach?

Next month… Base Ten Blocks.