Patterns

Patterns

Patterns are everywhere in mathematics. From the humble pattern of the squares: 1, 4, 9, 16, 25, 36, \ldots to the intricate pattern of Fibonacci: 1, 1, 2, 3, 5, 8, 13, \ldots One thing I realized recently is that these patterns, and the human mind’s ability to recognize them, can be a powerful teaching technique.

Take for example the age-old question: why is a negative times a negative a positive?

We first start with the fact that (3) \times (-4) = -12 which I don’t think is too controversial. If you don’t believe the result, imagine 3 groups of 4 negative signs floating around. There should be a total of 12 negative signs in your mind.

Now look at the following equations. Can you spot the pattern?

(3) \times (-4) = -12

(2) \times (-4) = -8

(1) \times (-4) = -4

(0) \times (-4) = 0

(-1) \times (-4) = ?

 

 

 

 

Each time the answer increases by 4. This pattern leads to the logical conclusion that (-1) \times (-4) = 4.

Another example I find my students struggle to comprehend is 3^0. What does raising a number to the power of zero mean? Since exponentiation makes ‘copies’ of the base, 3^4 = 3 \times 3 \times 3 \times 3 and 3^1 = 3. But for 3^0 how can  you make no copies of the base and still get an answer? Again, a pattern can help us to determine the answer:

3^4 = 81

3^3 = 27

3^2 = 9

3^1 = 3

3^0 = ?

 

 

 

 

In this case, the pattern is division. Each time the answer is divided by 3. This pattern leads to the logical conclusion that 3^0 = 3 \div 3 = 1.

As one final example to illustrate the power of this technique, consider 0! This concept is especially troublesome for students since factorial is conceptualized as ‘the number of ways of arranging objects.’ As such, a concrete example can be constructed to demonstrate that 3!=6, the number of ways to arrange 3 objects. But how is the world would someone count the ‘number of ways to arrange no objects? What if we made a pattern?

4! = 24

3! = 6

2! = 2

1! = 1

0! = ?

 

 

 

 

As in the previous case, the pattern is division. However, in this case the number that you divide by decreases by 1: 24 \div  4 = 6, \, \, \, 6 \div 3 = 2. This pattern leads to the logical conclusion that 0!=1.

I am curious if you have ever used a pattern like the ones above to help explain a tricky concept. Let me know in the comments below!

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2 Comments

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2 responses to “Patterns

  1. Janet Nguyen

    Hi Jehu!

    I’ve been teaching secondary maths for six weeks and I absolutely love your reading your blog! This post was really interesting because I didn’t think of patterns as a teaching technique before! Right now, I’m teaching my year 7s integers and my year 8s indices so I’ll definitely refer to the examples in this post! Keep up with the blogging! All the best!

    Graduate teacher from a “land down under”

    • Thanks Janet, I am glad you liked it! One pattern I didn’t include in the post but would be relevant for grade 7 would be demonstrating that 5 – (-1) = 6 (Minus minus makes a plus)
      5 – (3) = 2
      5 – (2) = 3
      5 – (1) = 4
      5 – (0) = 5
      5 – (-1) = ?

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