Table Talk

You don’t teach students the problem-solving strategy of Organize the Information: Make a Table by having them “complete the table.”

The activity “That’s Sum Challenge!” from AIMS asks “What sums from one to 25 can by obtained by adding two, three, four, five, or six consecutive numbers?”

One of the student pages looks like this:


I’ve designed this type of thing before. Fortunately, there’s a quick fix: ask the question, allow students time to work on the problem, ask the groups–or regroup and ask the class–”How can we organize this information?”

Likely, students’ tables won’t match the one above. Some students will probably make a table for two consecutive numbers, then three, and so on. To highlight the impossible sums, the helpful folks at AIMS have done the work of merging these tables into one. In their defence, kinda, the teacher pages has this under “Management”:

If you have a class that functions well with open-ended problems, you can explain the problem to them and have them solve it without using the student pages.

Subtracting the table engages more students at more levels. From “two consecutive numbers are always even and odd (or odd and even) and that gives us all the odd sums” to “the sums made by adding three consecutive numbers are all multiples of three” to “powers of two cannot be obtained because…,” each student can contribute to answering the key question “What sums can be obtained by adding consecutive numbers?” (The ellipsis is there because the reason isn’t immediately obvious to me.)

In the past, I had it back-asswards. Take the “How many different possible meal combinations are there on the kids¹ menu?” problem. I’d give ’em tables and tree diagrams up front. A problem became practice. Once I “turned the tables” and allowed students time to get started, I could later ask groups to share their tables or I could step in at just the right time with tree diagrams to help make sense of spaghetti nightmares.

¹Kid’s? Kids’? This is why I’m not a prolific blogger.

Related: The more sides you have, the smarter you are.

Recommended: “You Can Always Add. You Can’t Subtract.” Ctd. by Dan Meyer

The role of Ponyboy Curtis will by played by the Mathematics Department Head.

“English teachers differentiate all the time. Why don’t math teachers?”

I’ve heard this more than once. It irks me for a couple of reasons.

First, I’m not convinced that most English teachers do differentiate. After all, students still read The Outsiders in English 8. I read The Outsiders in English 8. The year was 1987. Do the math. Twenty. Five. Years. Are we to believe that this same group of educators have been too busy in the last quarter of a century meeting the diverse needs of all of their learners to find time to pick a different novel? During this time, Tom Cruise, who starred in the movie adaptation, found time to get married and divorced– three times! Google outsiders essay. Three million five hundred ninety thousand results. I’m  just sayin’.

Second, if differentiated instruction is more common in English class than it is in math class, it may be because it is easier. Some teachers of English 8 may simply assign an alternate book to read based on reading level. What can teachers of Math 8 simply do?

I’ve seen samples of those Outsiders essays. I’m no English teacher, but some of them wouldn’t look out of place in a Grade 3 classroom. Others could easily have been written by a first-year university student. In fact, Google search results suggest that maybe they were.

The Greasers’ reaction to “No WiFi”

In English Language Arts, from Grade 1 to 12, students brainstorm, draft, revise, edit, and publish. In short, they write. What is the equivalent in Mathematics?¹ Complete this sentence: In Mathematics, from Kindergarten to Calculus, students…

It’s not so easy, is it? Two-thirds of the Three Rs can be verbs. English gets to read and to write. Math gets a noun. Differentiating narrow nouns–numbers to 10 000 in Grade 4, integers in Grade 8, logarithms in Grade 12–is difficult. What verb could math teachers have?

The answer, I think, is to problem-solve. The BC mathematics curriculum document supports this: “Learning through problem-solving should be the focus of mathematics at all grade levels.” However, school mathematics is often taught in such a way that students do not encounter problem-solving on a regular basis. Sadly, to practice might be more accurate of students’ math classroom experiences. This is not mathematics.

Yeah, but it’s a nice bus.

Regardless of how or if English teachers differentiate, one size fits all math instruction is not acceptable. I am in no way letting my fellow math teachers off the hook. I am, however, suggesting that questions like “Why don’t math teachers differentiate like English teachers?” are not accurate or helpful. We’re not so different after all.

Pushback, as always, is welcome but must be expressed in the form of a five-paragraph essay.

Stay gold.

¹ I’m having a “Scrambled Eggs” moment. If you believe I have plagiarized this part, won’t you please, please help me?

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.

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.

Turn it Around

In More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction, Dr. Marian Small discusses the turn-around strategy to create open questions.

Instead of asking “The legs of a right triangle are 3 cm and 6 cm long. What is the hypotenuse?” the teacher can ask “The answer is √45. What could the question be?”

There are many possible questions. For example,

Determine the length of the hypotenuse.

Determine the length of x.

A square has an area of 45 cm². What is the side length?

What is an example of a square root that has a value between 6 and 7?

Which number is the greatest: √37, 6, 6½, √45?

Students will come up with a variety of questions. However, at first, I imagine the response to open questions such as “The answer is √45. What could the question be?” will be silence. Students are used to being asked questions where there is one correct answer. In math, you either get it or you don’t. It’s not just questions that need turning around. This black and white view of mathematics also needs turning around. With time and practice, class discussions about open questions can help change this attitude.