www.inquirymaths.org © 2011 Andrew Blair
Visual proof n(n + 3)
Number line inquiry
(n + 1)(n + 2)
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Most classes, however, have suggested changing the prompt in one or more of four ways to see if the difference between the products is always two.
The teacher starts by asking for comments or questions, directing students (if necessary) towards the observation that, ‘the difference between the products is two.’ Is this always the case? Surely for larger numbers, the difference will be greater than two.
(1) One change is to the combinations. Typically, students will develop a conjecture from numerical results and then try to prove the general case.
At this point, students often select the regulatory card that requires the class to find more examples. They have gone on to verify that for consecutive numbers (however large), the difference is always 2. For example,
Combination 1 – 2, 3 – 4 n(n + 1) = n² + n (n + 2)(n + 3) = n² + 5n + 6 Difference between the products is 4n + 6
100 x 103 = 10 300 101 x 102 = 10 302 10 302 – 10 300 = 2
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At this point, the inquiry has gone one of two ways. A few classes have been enthusiastic to use algebra to prove the difference is two for all cases of consecutive numbers:
Combination 1 – 3, 2 – 4 n(n + 2) = n² + 2n (n + 1)(n + 3) = n² + 4n + 3
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Algebraic proof for four consecutive numbers Combination 1 – 4, 2 – 3
Difference between the products is 2n + 3
n(n + 3) = n² + 3n (n + 1)(n + 2) = n² + 3n + 2 x n 2
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n 1 n² n 2n 2
(2) A second change to the prompt that is regularly suggested relates to the sequence of numbers. Rather than use consecutive numbers, students will suggest ascending in different gaps. Thus, using the original 1 – 4, 2 – 3 combination, a difference of two between the
Difference between the products is: n² + 3n + 2 – (n² + 3n) = 2
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terms gives a difference of 8 between the products.
(3) A third change to the prompt is to use more than four numbers in the number line. For six numbers using the combination 1 – 6, 3 – 4, for example, the difference between the products is 6 n(n + 5) = n² + 5n (n + 2)(n + 3) = n² + 5n + 6 n² + 5n + 6 – (n² + 5n) = 6 Students have inquired into the relationship between three pairs of numbers (or more for longer number lines).
n(n + 6) = n² + 6n (n + 2)(n + 4) = n² + 6n + 8 Difference between the products is 8, i.e. 2 x 4
The general case for six numbers with a gap of a
A difference of three between the terms gives a difference of 18 between the products n(n + 9) = n² + 9n (n + 3)(n + 6) = n² + 9n + 18 Difference between the products is 18, i.e. 3 x 6
Combination 1 – 6, 3 – 4 n(n + 5a) = n² + 5an (n + 2a)(n + 3a) = n² + 5an + 6a² Difference between the products is 6a²
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General cases if the terms ascend with a gap of a (linear sequences)
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(4) A fourth change to the prompt involves using other types of sequences.
Combination 1 – 4, 2 – 3 n(n + 3a) = n² + 3an (n + a) (n + 2a) = n² + 3an + 2a² Difference between the products is n² + 3an + 2a² - (n² + 3an) = 2a²
Square numbers [n², (n + 1)², (n + 2)², (n + 3)²] give for the combination 1 – 4, 2 – 3: n²(n + 3)² = n4 + 6n³ + 9n² (n + 1)²(n + 2)² = (n² + 4n + 4)( n² + 2n + 1) = n4 + 6n³ + 13n² + 12n + 4 Difference between the products is 4n² + 12n + 4
If the terms in the number line have a difference of a, then the difference between the products is a x 2a = 2a². Combination 1 – 2, 3 – 4 n(n + a) = n² + an (n + 2a)(n + 3a) = n² + 5an + 6a² Difference between the products is 4an + 6a²
An exponential growth (2n, 2n+1, 2n+2, 2n+3) gives for the combination 1 – 4, 2 – 3: 2n (2n+3) = 22n+3 2n+1(2n+2) = 22n+3 Difference between the products is 0
Combination 1 – 3, 2 – 4 n(n + 2a) = n² + 2an (n + a)(n + 3a) = n² + 4an + 3a² Difference between the products is 2an + 3a²
Other sequences that have appeared in inquiries include: (1) Quadratic (2) Fibonacci (a, b, a + b, a + 2b)
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