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.

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9 thoughts on “Quadratic Patterns

  1. This is awesome. I do lots of this with linear patterns, but for some reason, it never occurred to do this with quadratic patterns too. But of course, they can build the equation for each pattern in the same way – so great! Do you find it more helpful to color it for the students? I’ve always asked them to color it themselves (“color the tiles that are new in each subsequent figure the same color… “) after drawing in Figure 0, Figure 4, and describing Figure 100. Then, kids can show the way they colored the pattern to the class and explain how the colors show the different components of the pattern. But I wonder if that would be too hard for them to do on their own with the quadratic patterns.

  2. I really liked this activity as a professional development activity as well. I plan on sharing it with my colleagues (if we can ever find some time to meet as a department), so thank you for sharing it with us here.

  3. You know these patterns rock my world, Chris. Scaffolding for them is key. You just gave me an idea to have the kids do the coloring to show how they see the pattern and thus is reflected in the equation they write. They can write the terms of the equation using the same colors too. Thank you for the mention!

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  5. Anna, I do find it more helpful to present the pattern to students using different colours. For example, in 2A, some students determine the expressions for red (1, 4, 9), green (2, 4, 6), and blue (2, 2, 2) as n^2, 2n, and 2. If this pattern were one colour, finding an expression for the total number of tiles (5, 10, 17) would prove to be much more difficult. Of course, I’m hoping that after students have seen this strategy, they would be able to apply it themselves when presented with patterns in one colour.

    It’s interesting that you ask students to colour the new tiles in each subsequent figure a new colour. In my L-shape pattern (see lesson plan), this means n groups of 2, whereas the blue tiles in the second picture illustrate 2 groups of n. I had always thought of the pattern in one way– the way that you suggested– until I sat down with colleagues and listened to the different strategies that we all had. So, David, I’d agree that this has potential to be a great pro-d activity.

    As Fawn points out (I think), even if students visualize a pattern as (n + 1)(n + 2) – n, after they simplify it to n^2 + 2n + 2, they can be asked to visualize/illustrate this representation as well.

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  7. Genius. I’ve been trying to find the quadratics by finding patterns in the table of values…I like using the figures much better. Some of my kids will really get this!

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