Year 10 Extended · Unit 2 Generalizing, coordinate geometry and linear programming · Lesson 2 of 6

Does the pattern survive when the grid changes?

Choose a box on a number grid, multiply opposite corners and subtract. This lesson asks what the answer depends on, and how letters can answer that for every box at once.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: write “any even number” using a letter.
    Answer2n, where n is a whole number.
  2. From Year 9: expand (n − 3)².
    Answern² − 6n + 9.
  3. From lesson 1: what must a counterexample do?
    AnswerMeet every condition of the claim and still break its conclusion.
Words used in this lesson
consecutive
following one after another with no gap. For example: n, n + 1 and n + 2.
expression
numbers, letters and operations written together. For example: 4n(n + 1).
identity
a statement that is true for every value of its letters. For example: (n + 3)² − n² = 6n + 9.
general case
a case with letters, standing for every case at once. For example: the box with top-left corner a.
multiple
a number you get by multiplying by a whole number. For example: the multiples of 9 are 9, 18, 27, and on.
factor
a number that divides exactly into another. For example: 5 is a factor of 40.

One box, then every box

A teacher in Kampala gives each student a number grid with rows of 10 (1 to 10, then 11 to 20, and so on). Each student draws a 2 by 2 box anywhere on the grid, multiplies the top-right and bottom-left numbers, multiplies the top-left and bottom-right numbers, and subtracts the smaller product from the larger one. The students compare answers.

Predict first: will students whose boxes are in different places get different answers? What might the answer be? Write a guess and a reason, then try boxes in the figure. It opens on a larger box than the teacher's.

3
4
3
24top-left number
40difference of the two products

A 3 by 3 box with 24 in its top-left corner: the corners are 24, 26, 44 and 46. 26 × 44 = 1144 and 24 × 46 = 1104, so the difference is 40.

Show me the steps
  1. A 2 by 2 box at the top-left of the grid: the corners are 1, 2, 11 and 12. 2 × 11 = 22 and 1 × 12 = 12, so the difference is 10.
  2. A 2 by 2 box with 46 in its corner: 47 × 56 = 2632 and 46 × 57 = 2622, so the difference is 10 again.
  3. A 3 by 3 box with 24 in its corner: 26 × 44 = 1144 and 24 × 46 = 1104, so the difference is 40.
  4. A 4 by 4 box with 13 in its corner: 16 × 43 = 688 and 13 × 46 = 598, so the difference is 90.

One letter for every free choice. To prove a statement about any whole number, give it a letter: n is any whole number, 2n is even, 2n + 1 is odd, and n, n + 1, n + 2 are consecutive. Two numbers that may differ need two letters, a and b, never the same letter twice.

Then expand, collect like terms, and look for the factor that shows the property. The proof ends with a sentence that says what has been shown: “this is 4 times a whole number, so it is a multiple of 4 for every n.” Examples can check a result; only the letters prove it.

8
3
96area of the ring
964 × n × d

n = 8 and d = 3: the squares have sides 11 and 5, areas 121 and 25. The ring is 121 − 25 = 96, and 4 × 8 × 3 = 96.

Show me the steps
  1. n = 8, d = 3: the sides are 11 and 5, areas 121 and 25, ring 96 = 4 × 8 × 3.
  2. n = 6, d = 2: the sides are 8 and 4, areas 64 and 16, ring 48 = 4 × 6 × 2.
  3. n = 10, d = 5: the sides are 15 and 5, areas 225 and 25, ring 200 = 4 × 10 × 5.
  4. n = 12, d = 4: the sides are 16 and 8, areas 256 and 64, ring 192 = 4 × 12 × 4.

Two common mistakes. Using one letter for two numbers that can differ: 2n + 1 and 2n + 1 are always the same number, so they cannot stand for two different odd numbers (use 2a + 1 and 2b + 1). And stopping at a number: “it gives 40 for these boxes” is a test; the proof is the line in which the box's position has cancelled out.

Worked examples

1. Prove that the product of two consecutive even numbers is a multiple of 8.

  1. Call them 2n and 2n + 2. Their product is 2n(2n + 2) = 4n(n + 1).
  2. Of any two consecutive numbers n and n + 1 one is even, so n(n + 1) = 2k for some whole number k. Then 4n(n + 1) = 4 × 2k = 8k, a multiple of 8 for every n.

2. Prove that for a 3 by 3 box on a grid with rows of w, the two diagonal products differ by 4w.

  1. Let the top-left number be a. The corners are a, a + 2, a + 2w and a + 2w + 2.
  2. (a + 2)(a + 2w) − a(a + 2w + 2) = (a² + 2aw + 2a + 4w) − (a² + 2aw + 2a) = 4w. The a has gone, so the box can sit anywhere.

3. Prove that the squares of three consecutive whole numbers always add up to a number that is 2 more than a multiple of 3.

  1. n² + (n + 1)² + (n + 2)² = n² + (n² + 2n + 1) + (n² + 4n + 4) = 3n² + 6n + 5.
  2. 3n² + 6n + 5 = 3(n² + 2n + 1) + 2, which is 2 more than a multiple of 3 for every n.

Your go. The first step is given. Prove that the sum of three consecutive even numbers is a multiple of 6. Let the numbers be 2n, 2n + 2 and 2n + 4.

  1. The sum is 2n + (2n + 2) + (2n + 4) = 6n + 6.
  2. Now write 6n + 6 as 6 times a whole number, and finish with a sentence.
Check the answer

6n + 6 = 6(n + 1). Since n + 1 is a whole number, the sum is 6 times a whole number, so it is a multiple of 6 for every whole number n.

Check before you go on

1. Which list shows three consecutive odd numbers, written with a letter n that stands for any whole number?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. (n + 1)² − (n − 1)² simplifies to ?n. What number goes in front of n?

3. On a number grid with rows of 12, a 2 by 2 box has corners a, a + 1, a + 12 and a + 13. The product of the top-right and bottom-left corners minus the product of the top-left and bottom-right corners comes to a single number. What is it?

Level 2 · Building

4. (2n + 1)² − (2n − 1)² simplifies to ?n. What number goes in front of n?

5. On a number grid with rows of 7, a 4 by 4 box is drawn anywhere. The top-right corner times the bottom-left corner, minus the top-left corner times the bottom-right corner, always comes to the same number. What is it?

Level 3 · Stretch

6. On a number grid with rows of w, the same difference is worked out for a 5 by 5 box and it comes to 96. What is w?

7. Write two consecutive odd numbers as 2n − 1 and 2n + 1. Their product is 483. What is the smaller of the two numbers?

Something to think about

On a grid with rows of 10, an s by s box gives 10(s − 1)². What if the box is a rectangle, p columns wide and r rows tall? Predict the difference for a box 3 columns wide and 5 rows tall, then prove a rule with letters.

One way to think about it

With top-left number a, the corners are a, a + (p − 1), a + 10(r − 1) and a + (p − 1) + 10(r − 1). Expanding, every a term cancels and the difference is 10(p − 1)(r − 1). For 3 columns and 5 rows that is 10 × 2 × 4 = 80. On rows of w it is w(p − 1)(r − 1), and a square box is the case p = r = s.

Exit ticket

8. Which of these is a proof that the sum of any two consecutive odd numbers is a multiple of 4?