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.

Fool me once, shame on… shame on you. Fool me… you can’t get fooled again.

Lately I’ve been enjoying Veritasium’s videos on misconceptions about science. From the Veritasium YouTube channel:

If you hold views that are consistent with the majority of the population, does that make you stupid? I don’t think so. Science has uncovered a lot of counterintuitive things about the universe, so it’s unsurprising that non-scientists hold beliefs inconsistent with science. But when we teach, we must take into account what the learners know, including their incorrect knowledge. That is the reason a lot of Veritasium videos start with the misconceptions.

I’ve been thinking about students’ misconceptions about mathematics. What math concepts are counterintuitive? How might starting with the misconception play out in the math classroom? Probability probably provides the most potential, from a pedagogical point of view. (Do robot graders give high marks for alliteration?) The classic Monty Hall problem or birthday problem are just two examples of this. Exponential growth can also be counterintuitive – see Chris Lusto’s alternative to the doubling penny problem.

One common misconception students have is that (a + b)^2 is equal to a^2 + b^2. In my classroom, I’d start with this misconception then have students substitute values before exploring this with algebra tiles. Not exactly Why does the Earth spin? type stuff. Still, addressing this misconception right off the bat provided us with a problem to solve – if (a + b)^2 is not equal to a^2 + b^2, then what is it equal to and why?

Recently, I was fascinated by Dan Meyer’s Coke v. Sprite question because my gut reaction was wrong. Twice. Please watch Dan’s act one video now. I’ll wait.

What fraction must you drink to balance the Coke can on edge?

My guess was that there was more Sprite in the Sprite glass than there was Coke in the Coke glass. After all, I reasoned, the Coke that was added to the Sprite also contained a small amount of Sprite.

When I did the calculations, I was surprised to learn that the amount of Sprite in the Sprite glass and the amount of Coke in the Coke glass were the same:

  • Assume the original amount of each is 100 mL.
  • Assume 10 mL of Sprite is transferred to the Coke.
  • 10 mL of pop is transferred back to the Sprite. Stirring means 10/110, or 1/11, of this is Sprite. 100/110, or 10/11, of this is Coke.
  • The amount of Sprite in the Sprite glass is now 90 mL + (1/11)*10 mL = 90 10/11 mL.
  • The amount of Coke in the Coke glass is now 100 mL – (10/11)*10 mL = 90 10/11 mL.

Before watching Dan’s act 3 video, my colleague Shelagh Lim and I modelled this with colour tiles:

  • Start with 12 green tiles on the left and 12 red tiles on the right.
  • Move 4 green tiles to the right. Now, 4/16, or 1/4, of the tiles on the left are green. 12/16, or 3/4, are red.
  • 4 tiles are moved back to the left. To simulate the effect of stirring, 1 of these 4 are green. 3 of these 4 are red.
  • The number of green tiles on the left is now 8 + 1 = 9.
  • The number of red tiles on the right is now 12 – 3 = 9.

Shelagh asked, “What if you don’t move back 1 green and 3 red? What if you close your eyes and take out 4 random tiles?” In other words, does stirring matter? I argued it did. “Something something proportions,” I said.

Mind. Blown.

I want students to experience this feeling of enjoyment at being led astray by their intuition. But, more importantly, students must also experience the feeling of enjoyment that comes from following their intuition and being correct. The former is not possible without the latter; to be amused by failure, there needs to be an expectation of success.

Linear Functions – Concretely, Pictorially, Symbolically

Welcome to my blog!

I really enjoyed Marc’s Patterning the Blues activity (taken from Marian Small’s Big Ideas book that department heads received).

Teachers often talk about how manipulatives can help the struggling learner. I’m suggesting that having students solve problems concretely can assist all learners.

When I experienced this problem using the blue and yellow tiles, I gained a deeper understanding of the problem. The equation y = 3x + 2 now had meaning. I was able to find the pattern in the table to determine the number 3. By modelling the problem using tiles, I was able to see this as adding an extra 3 blue tiles every time the figure grew.

In the past, I had difficulty explaining to students where the 2 came from. I could convince them that it had to be there. For example, take the point (2, 8). Multiplying the 2 by 3 gives  6, so we need to add 2 more. Looking at this concretely & pictorially, the 2 now has meaning. For me, it is how many blue tiles there were before we start adding yellow & blue tiles. (See the photo below.)

Your students who used to get it symbolically will still get it if they approach it concretely. However, what it means to “get it” in your classroom will start to change.

Patterning the Blues
Patterning the Blues Concretely

I’d appreciate your comments. Maybe you have some thoughts on how this activity addresses the 7 processes?