In language arts class it is not unusual to have students freewrite or journal periodically, sometimes even daily. The idea is to practice fluency in writing, generating ideas for stories, creating characters or settings, and much more. The activity only takes a few minutes and usually gets students focused and ready for the main lesson of the class.
I propose a similar activity for math classes where students play around with numbers and patterns – something known under the general heading of “recreational mathematics.” The purposes would be similar to journaling, but of course, geared toward mathematics and numeracy: get students focused, generate ideas, make connections between numbers, create and/or extrapolate patterns, and much more.
What would this look like? Well, of course, it would depend on the lesson plans, specific goals, and time available, but this “thought experiment” time could be as short as a two minute doodle or as long as half a period of working with things like geoboards, manipulatives, or just a scrap of paper and a pencil.
Here are some ideas to get things started:
1. Turn a 6 upside-down and you get a 9. Turn an 8 sideways and you get the infinity symbol. But what other patterns or ideas are there when you rotate or morph various digits and numbers? Have students doodle in a notebook to see what they can discover. How many numbers have symmetry?
2. Square numbers actually form squares: 1, 4, 9, 16… Triangular numbers form triangles: 1, 1+2, 1+2+3, 1+2+3+4 and so on. Have students draw different figurate numbers to visually see the patterns inherent in numbers. Many of them could probably create their own named patterns! If we don’t represent four with the numeral “4,” how would you represent it? Four dots? Four tallies? Four coins? Fours toothpicks? Once we have four toothpicks, how many different ways can they be arranged? Which shapes seem more natural to four-ness? What symbols do you recognize? What happens when you add a fifth toothpick?
3. The number 13 is associated with bad luck. 666 is supposed to be “the number of the beast.” 7 is often considered good luck. But what about other numbers? What are their associations and potential meanings? How about a number like 40? Have the entire class pick a number and compile as many examples of how that number shows up in history, nature, mythology, school, etc. Here are some twelves: 12 months in a year, 12 signs of the zodiac, 12 Olympian gods in Greek mythology, 12 Tribes of Israel, 12 Apostles of Jesus, 12 Days of Christmas, 12 peers of Charlemagne, 12 knights of the round table, 12 ounces in a troy pound, 12 hours in the AM and 12 hours in the PM…
4. Have the students play a game and analyze and extrapolate the various math concepts. I have seen plenty of teachers use decks of cards, backgammon, checkers and more. How about a game of Connect Four? Check out Numberphile’s video. Similarly, try mathematizing a read-aloud.
5. There are even and odd numbers, composite and prime numbers, happy and unhappy numbers. Happy numbers are those that when you continuously add the squares of the individual digits, eventually yields the number 1. Unhappy numbers are those that do not cycle down to the number 1 and instead fall into a repeating pattern: 4, 16, 37, 58, 89, 145, 42, 20, 4… For example, 313 is a happy number and 61 is unhappy. What are some of the other types of numbers and can your students create their own type? All they really need to do is find a unique pattern! Here are some other types to give you some ideas: Keith Numbers, Brown Numbers, and Perfect Numbers.
6. Personalize numbers. What if the number 7 came up and introduced itself to you? What would it be like? How would it behave? What about the number 4? How would it be similar and different? What are the various personalities of numbers? For example, what makes odd numbers odd? The ancient Greeks believed that odd numbers were masculine and even numbers were feminine. Why is that?
7. Look for patterns like “Highly Composite Numbers.” A positive integer with more divisors than any smaller number. Examples include: 1, 2, 4, 6, 12, 24, 36, 48, 60 … 5040 and 10080. Notice the twelves. This may be why the idea of a dozen developed instead of base 10. Plato may have known about these when he suggested that the ideal number of citizens in a city should be 5040.
8. There are many types of numbers beyond the basics that most students learn. For example, notice that continuously dividing numbers by 2 will demonstrate different types of even numbers: 12 ÷ 2 = 6 ÷ 2 = 3 (oddly-even). 14 ÷ 2 = 7 (evenly-odd). And 16 ÷ 2 = 8 ÷ 2 = 4 ÷ 2 = 2 (evenly-even).
Image by Gerd Altmann