6.5 This Way or That Way, All Ways Lead to the Bay

A growing dot pattern can often be seen in several ways. Remarkably, every correct way leads to the same algebraic expression, that is the surprise, and the proof.

Consider a sequence of figures where Step k looks like a (k+1)\times(k+1) square of circles with one corner missing. Four students describe the count differently:

Method Expression for Step k Simplified
1, full square minus 1 (k+1)^2 - 1 k^2 + 2k
2, square plus two arms k^2 + 2k k^2 + 2k
3, rectangle plus column k(k+1) + k k^2 + 2k
4, wide rectangle k(k+2) k^2 + 2k

Every route simplifies to k^2 + 2k. So Step k has k^2 + 2k circles; Step 10 has 100 + 20 = 120, and Step 15 has 225 + 30 = 255.

One Pattern, Many Views In mathematics there are often many ways to look at the same pattern and many routes to the same answer. Expressions that look different can be proved equal by expanding them. Finding fresh ways takes creativity, and seeing them collapse into one identity is deeply satisfying.

Areas in disguise

The same idea explains why students’ different area formulas agree.

A shaded interior. A square of side (m+n) has four identical rectangles tucked into its corners. Meera computes the leftover as the whole minus the four rectangles, (m+n)^2 - 4mn. Kabir instead spots the leftover as a single square of side (n-m), with area (n-m)^2. Both are correct, because (m+n)^2 - 4mn = m^2 + 2mn + n^2 - 4mn = m^2 - 2mn + n^2 = (n-m)^2.

Math Talk Three students measure the same slanted region in three ways: Riya gets x^2 - xy, Sneha gets x(x+2y) - 3xy, Dev gets x(x-y). Expand them all: x(x+2y) - 3xy = x^2 + 2xy - 3xy = x^2 - xy and x(x-y) = x^2 - xy. Every route lands on x^2 - xy! For x=9,\ y=4: area = 81 - 36 = 45 sq. units.

Figure it Out: Many Methods

Practice

  1. Compute using the suggested identity: (i) 46^2 via 1A   (ii) 397\times403 via 1C   (iii) 91^2 via 1B   (iv) 43\times45 via 1C.
  2. Find each product: (i) (p-1)(p+11)   (ii) (3a-9b)(3a+9b)   (iii) -(2y+5)(3y+4)   (iv) (6x+5y)^2   (v) \left(2x-\tfrac12\right)^2   (vi) (7p)(3r)(p+2).
  3. Pick the expression(s) that fit each phrase: (i) A square number with 2 added on: choose from 2+s,\ (s+2)^2,\ s^2+2,\ s^2+4,\ 2s^2,\ 2^2 s. (ii) The two squares of a number and the one just after it, added together.
  4. In any 2\times2 block of a calendar, compare the two diagonal products. What do you notice, and why?
  5. True or false? (i) (k+1)(k+2)-(k+3) always equals 2. (ii) (2q+1)(2q-3) is a multiple of 4. (iii) An even number squared is divisible by 4, while an odd number squared sits 1 above a multiple of 8. (iv) (6n+2)^2 - (4n+3)^2 falls 5 short of a perfect square.
  6. One number leaves remainder 3 on division by 7, another leaves remainder 5. What are the remainders of their sum, difference, and product?
  7. Square the middle of three consecutive numbers and subtract the product of the outer two. Find the pattern and prove it.
  8. Add two numbers, then multiply by half their sum. Show the result is half the square of the sum.
  9. Which is larger (without full multiplication)? (i) 14\times26 or 16\times24   (ii) 25\times75 or 26\times74.
  10. A park has two square plots of area g^2 each, with a path of width w around and between them. Write the area to be tiled.
  11. For a tiling pattern with Step y, write the count of basic units (two given patterns).
    1. 46^2 = (40+6)^2 = 1600 + 480 + 36 = \mathbf{2116}.   (ii) 397\times403 = (400-3)(400+3) = 160000 - 9 = \mathbf{159991}.   (iii) 91^2 = (100-9)^2 = 10000 - 1800 + 81 = \mathbf{8281}.   (iv) 43\times45 = (44-1)(44+1) = 1936 - 1 = \mathbf{1935}.
    1. p^2 + 10p - 11   (ii) 9a^2 - 81b^2   (iii) -6y^2 - 23y - 20   (iv) 36x^2 + 60xy + 25y^2   (v) 4x^2 - 2x + \tfrac14   (vi) 21p^2 r + 42pr.
    1. \mathbf{s^2 + 2}.   (ii) \mathbf{m^2 + (m+1)^2} (the squares of m and the next number m+1).
  4. The diagonal products differ by 7 (one minus the other). Labelling the block a,\ a+1,\ a+7,\ a+8, the difference is (a+1)(a+7) - a(a+8) = (a^2+8a+7) - (a^2+8a) = 7, independent of a.
    1. False, (k+1)(k+2) - (k+3) = k^2 + 3k + 2 - k - 3 = k^2 + 2k - 1, which varies. (ii) False, e.g. q=1 gives 3\times(-1) = -3, not a multiple of 4. (iii) True, (2k)^2 = 4k^2; and (2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1)+1, where k(k+1) is even, so it is 8m+1. (iv) False, (6n+2)^2 - (4n+3)^2 = 36n^2 + 24n + 4 - 16n^2 - 24n - 9 = 20n^2 - 5, which is not generally 5 less than a perfect square.
  6. Write the numbers as 7a+3 and 7b+5. Sum \to 3+5 = 8 \equiv 1. Difference \to 5 - 3 = 2. Product \to 3\times5 = 15 \equiv 1. Remainders: 1,\ 2,\ 1.
  7. With numbers n-1,\ n,\ n+1: \;n^2 - (n-1)(n+1) = n^2 - (n^2 - 1) = 1. The answer is always 1.
  8. (a+b)\times \tfrac12(a+b) = \tfrac12(a+b)^2, half the square of the sum.
    1. 14\times26 = (16-2)(24+2) = 16\times24 + 32 - 48 - 4 = (16\times24) - 20, so \mathbf{16\times24} is larger. (ii) 25\times75 = (26-1)(74+1) = 26\times74 + 26 - 74 - 1 = (26\times74) - 49, so \mathbf{26\times74} is larger.
  10. The whole rectangle is (2g+4w) by (g+2w), area 2g^2 + 8wg + 8w^2. Subtract the two green squares 2g^2: tiled area = \mathbf{8w(w+g)} sq. ft.
    1. Step y has \mathbf{(y+2)^2} units (Step 10: 144).   (b) Step y has \mathbf{(y+1)^2 + y} units (Step 10: 131).