Egg, Head, …

Take a moment to think about the following image:

#whatrepeats? #patternchat

What comes next? What comes before? How do you know?

You might have sensed (the start of) a repeating pattern. Whether you considered the materials that make up the egg cups (glass, porcelain, …) or the position of the eggs (down, up, …), it’s a simple AB pattern. Or rather, like 🍀💎🍀💎🍀💎…, two synchronous AB patterns. If you were to extend the pattern, you’d get this:

AB; AB

Not so fast. Check out the video in the following tweet:

An AB pattern is maintained in the materials: still glass-porcelain. But the video hints at a new possible pattern–an ABAA pattern–with respect to the elliptical “dome”: egg-head-egg-egg.

AB; ABAA

Again, not so fast. These first four elements may not be what repeats; they may not be the pattern core. What if the pattern core were instead egg-head-egg-egg-egg-head (all the while still maintaining glass-porcelain)?

AB; ABAAAB

Patterns repeat. Repetition is what makes a pattern a pattern. Sometimes items repeat, sometimes a rule (e.g., add 3 each time) repeats. How would you describe what repeats in the following pattern?

All of these possibilities illustrate that without knowing what repeats, you can’t know for certain what comes next. For example, consider the following open question: Extend the pattern 5, 10, … in as many ways as you can. Common classroom responses include: 5, 10, 5, 10, 5, 10, …; 5, 10, 25, 5, 10, 25, …; 5, 10, 15, 20, 25, …; 5, 10, 15, 25, 40, …; 5, 10, 20, 40, 80, …; etc. (Variation: Extend the pattern ▲◾️… in as many ways as you can.)

The two attributes in the egghead examples–container and “contents”–made the task more interesting. In the classroom, this plays out by looking at repeating patterns with multiple attributes (i.e., colour, shape, size, orientation). Consider the pattern below:

What’s missing?

What’s missing? If you focus on colour, it’s an ABC pattern; it must be teal. If you focus on shape, it’s an AABB pattern; it must be a triangle. If you focus on orientation, it’s an ABBA pattern; it must “sit” on a vertex. If you hold all three asynchronous patterns in your mind, it must be a teal square resting on a vertex (a/k/a “diamond”). But I’m not looking for one right answer. In the classroom, I’d happily accept a teal triangle (or circle) from a student who sees a teal-orange-green pattern; an orange (or purple) square from a student who spots a triangle-triangle-square-square pattern; etc. If the claim is true, the answer is correct.

Pattern Fix-Its present another opportunity for students to examine patterns involving multiple attributes. Here, a pattern is messed with by adding or removing an element, changing one or more attributes of an element, or swapping the order of two adjacent elements. The math picture book Beep Beep, Vroom Vroom by Stuart J. Murphy provides a context: Molly plays with her big brother’s toy cars and must put them back in the right order before he returns. Using this context, I swapped the last two cars in a big-small and yellow-blue-green pattern:

Can you fix it?

Press Here by Hervé Tullet also includes some mixed-up pattern pages. That probably inspired my shaking effect here:

Like Which One Doesn’t Belong?, these questions allow all students to confidently contribute to and benefit from the discussion, whether they notice one or many patterns, whether they attend to simple (colour and shape) or more challenging (orientation) attributes, or whether they examine single or multiple attributes at a time.

* * ** *** ***** ********

I’d be remiss not to include Marc’s tweet somewhere in this post:

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