Exercise practice 2: We Distribute, Yet Things Multiply

4.1 Figure it Out: identities and products

This page answers the third Figure it Out set, which runs across textbook pages 154 to 157. Every question, sub-part and verification is solved below with the key steps.

4.1.1 FIO 3, Q1: four products with suggested identities

(i) 46² using Identity 1A: 46 = 40 + 6, so 46² = 1600 + 2 × 40 × 6 + 36 = 1600 + 480 + 36 = 2116. (ii) 397 × 403 using Identity 1C: 397 = 400 − 3 and 403 = 400 + 3, so the product is 400² − 3² = 160000 − 9 = 159991. (iii) 91² using Identity 1B: 91 = 100 − 9, so 91² = 10000 − 2 × 100 × 9 + 81 = 10000 − 1800 + 81 = 8281. (iv) 43 × 45 using Identity 1C: write both around 44, so 43 × 45 = (44 − 1)(44 + 1) = 44² − 1 = 1936 − 1 = 1935.

4.1.2 FIO 3, Q2: six products by identity or distributivity

(i) (p − 1)(p + 11) = p² + 11p − p − 11 = p² + 10p − 11. (ii) (3a − 9b)(3a + 9b) = (3a)² − (9b)² = 9a² − 81b². (iii) −(2y + 5)(3y + 4) = −(6y² + 8y + 15y + 20) = −6y² − 23y − 20. (iv) (6x + 5y)² = 36x² + 60xy + 25y². (v) (2x − 1/2)² = 4x² − 2x + 1/4. (vi) (7p)(3r)(p + 2) = 21pr(p + 2) = 21p²r + 42pr.

In (iii) the minus sign in front of the bracket applies to every term of the expansion, which is why all three signs change. In (v) the middle term is 2 × 2x × 1/2 = 2x and the square of 1/2 is 1/4.

4.1.3 FIO 3, Q3: matching statements to expressions

(i) "Two more than a square number" means take an unknown number s, square it, then add 2. Among the printed choices, s² + 2 is the match. 2 + s is not a square plus 2, (s + 2)² squares the sum, and s² + 4 adds 4 instead of 2. (ii) The expression asked for is the total of the squares of two whole numbers that sit next to each other, which means n² + (n + 1)². Among the choices, m² + (m + 1)² matches; the option (m + (m + 1))² squares the sum instead of adding the squares. The expression m² + (m − 1)² also describes the same idea with the pair (m − 1, m), so the question has two acceptable forms; the printed answer uses m² + (m + 1)².

4.1.4 FIO 3, Q4: the calendar 2 x 2 pattern

Statement: in any 2 x 2 block of a calendar, compare the products along the two diagonals and explain what you notice.

Key step: let the top-left number of a 2 x 2 block be n. The block then holds n, n + 1, n + 7 and n + 8, and the two diagonal products are n(n + 8) and (n + 1)(n + 7). Their difference is always 7, whichever block is chosen.

Example from the textbook calendar: the block containing 4, 5, 11 and 12 has diagonal products 48 and 55, which differ by 7.

Answer: the diagonal products always differ by 7. The reason is that the two diagonals differ by exactly one move across and one move down, and that difference is 7 in the calendar's layout.

4.1.5 FIO 3, Q5: verify four statements

(i) (k + 1)(k + 2) − (k + 3) = k² + 3k + 2 − k − 3 = k² + 2k − 1. This is not always 2, so the statement is false. (ii) (2q + 1)(2q − 3) = 4q² − 4q − 3 = 4(q² − q − 1) + 1. It is one more than a multiple of 4, never a multiple of 4, so the statement is false. (iii) Even numbers are 2m, so their squares are 4m², which are multiples of 4. Odd numbers are 2m + 1, so their squares are 4m² + 4m + 1 = 4m(m + 1) + 1. Since m(m + 1) is even, 4m(m + 1) is a multiple of 8, and the square is 1 more than a multiple of 8. The statement is true. (iv) (6n + 2)² − (4n + 3)² = (36n² + 24n + 4) − (16n² + 24n + 9) = 20n² − 5. For n = 1 this is 15, which is not 5 less than a square number, so the statement is false.

4.1.6 FIO 3, Q6: remainders of a sum, difference and product

Statement: one number leaves remainder 3 and another leaves remainder 5 when divided by 7. Find the remainders of their sum, difference and product on division by 7.

Key step: write the numbers as 7a + 3 and 7b + 5. The sum is 7(a + b) + 8 = 7(a + b + 1) + 1, so the remainder is 1. The difference is 7(a − b) − 2 = 7(a − b − 1) + 5, so the remainder is 5 when the second is subtracted from the first; taking the absolute difference gives remainder 2. The product is 49ab + 35a + 21b + 15 = 7(7ab + 5a + 3b + 2) + 1, so the remainder is 1.

Answer: sum 1, difference 2 (or 5, depending on order), product 1.

4.1.7 FIO 3, Q7: square the middle of three consecutive numbers

Statement: take three whole numbers in a row, square the middle number, then subtract the product of the other two. What pattern appears?

Key step: let the numbers be n − 1, n and n + 1. Then n² − (n − 1)(n + 1) = n² − (n² − 1) = 1.

Answer: the result is always 1. The identity n² − (n − 1)(n + 1) = 1 is true for every integer n, including negative values.

4.1.8 FIO 3, Q8: an algebraic expression for a word rule

Statement: add any two numbers, multiply the sum by half of the sum, and prove the result is half of the square of the sum.

Key step: let the numbers be a and b. The sum is a + b, half of the sum is (a + b)/2, and the product is (a + b) × (a + b)/2 = (a + b)²/2.

Answer: the expression is (a + b)²/2, which is exactly half of the square of the sum.

4.1.9 FIO 3, Q9: which product is larger?

Key step: write each product as a rectangle around the middle value of its two factors, then compare the two expansions shown in the math blocks below. In (i) the first product is smaller by 20; in (ii) it is smaller by 49.

Answer: in (i) the product 16 × 24 is larger; in (ii) the product 26 × 74 is larger. In both pairs the factors have the same sum, and the product is larger when the two factors are closer together.

4.1.10 FIO 3, Q10: the park in Dhauli

Statement: two square plots of area g² square feet each have green cover. The rest of the park is a path of width w feet, and the outer boundary is a rectangle. Find the tiled area.

Key step: from the plan, the outer rectangle measures (2g + 4w) feet by (g + 2w) feet. Its area is (2g + 4w)(g + 2w) = 2g² + 8gw + 8w². The two square plots take 2g², so the path area is 2g² + 8gw + 8w² − 2g² = 8w(g + w).

Answer: the tiled area is 8w(g + w) square feet. Check with g = 10 and w = 2: 8 × 2 × 12 = 192 square feet, and the rectangle (28 × 14) minus 200 gives 392 − 200 = 192.

4.1.11 FIO 3, Q11: two patterns of basic units

Statement: for each of the two patterns, draw the next figure, find the number of basic units in Step 10, and write an expression for Step y.

Pattern (a): Step 1 has 9 units, Step 2 has 16, Step 3 has 25. The counts are (y + 2)², so Step 4 has 36 and Step 10 has 144. Pattern (b): Step 1 has 5 units, Step 2 has 11, Step 3 has 19. The counts are (y + 1)² + y, so Step 4 has 25 + 4 = 29 and Step 10 has 121 + 10 = 131.

Answer: pattern (a) gives (y + 2)² units; pattern (b) gives (y + 1)² + y units. Both expressions can be expanded: (y + 2)² = y² + 4y + 4 and (y + 1)² + y = y² + 3y + 1. The next figure in pattern (a) is a square frame with 36 units, and in pattern (b) a ring-plus-tail shape with 29 units.