Opening questions…

We Distribute, Yet Things Multiply

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Page 136: By how much does the product increase if the first number (23) is increased by 1?
Page 136: What if the second number (27) is increased by 1?
Page 136: How about when both numbers are increased by 1?
Page 136: Do you see a pattern that could help generalise our observations to the product of any two numbers?
Page 138: How do we expand this?
Page 138: What would we get if we had expanded (a + 1) (b + 1) by first taking (b + 1) as a single term? Try it?
Page 138: What happens when one of the numbers in a product is increased by 1 and the other is decreased by 1? Will there be…
Page 138: Will the product always increase? Find 3 examples where the product decreases.
Page 138: What happens when a and b are negative integers? Check by substituting different values for a and b in each of the…
Page 139: By how much will the product of two numbers change if one of the numbers is increased by m and the other by n?
Page 140: This identity can be used to find how products change when the numbers being multiplied are increased or decreased…
Page 140: Use Identity 1 to find how the product changes when (i) one number is decreased by 2 and the other increased by 3;…
Page 140: Verify the answers by finding the products without converting the subtractions to additions.
Page 140: Expand (i) (a – u) (b + v), (ii) (a – u) (b – v).
Page 140: Example 1: Expand 3a/2(a – b + 1/5).
Page 141: Can any two terms be added to get a single term? For example, can 3/2a² and 3/10a be added to get a single term?
Page 141: Example 2: Expand (a + b) (a + b).
Page 141: Example 3: Expand (a + b) (a² + 2ab + b²).
Page 142: Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the…
Page 143: Expand the following products. (i) (3 + u) (v – 3) (ii) 2/3(15 + 6a) (iii) (10a + b) (10c + d) (iv) (3 – x) (x – 6)…
Page 143: Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other…
Page 143: Expand (i) (a + ab – 3b²) (4 + b), and (ii) (4y + 7) (y + 11z – 3).
Page 143: Expand (i) (a – b) (a + b), (ii) (a – b) (a² + ab + b²) and (iii) (a – b)(a³ + a²b + ab² + b³), Do you see a…
Page 143: Use the following multiplications to find the product of a number with 11 in a single step. (a) 3874 × 11 (b) 5678 × 11
Page 144: Describe a general rule to multiply a number (of any number of digits) by 11 and write the product in one line.…
Page 144: Can we come up with a similar rule for multiplying a number by 101?
Page 144: Multiply 3874 by 101.
Page 144: Use this to multiply 3874 × 101 in one line.
Page 144: What could be a general rule to multiply a number by 101 and write the product in one line? Extend this rule for…
Page 144: Use this to find (i) 89 × 101, (ii) 949 × 101, (iii) 265831 × 1001, (iv) 1111 × 1001, (v) 9734 × 99 and (vi) 23478 ×…
Page 145: The area of a square of sidelength 60 units is 3600 sq. units (60²) and that of a square of sidelength 5 units is 25…
Page 145: Can you find the areas of the four parts in the figure above?
Page 145: What if we write 65² as (30 + 35)² or (52 + 13)²? Draw the figures and check the area that you get.
Page 145: If a and b are any two integers, is (a + b)² always greater than a² + b²? If not, when is it greater?
Page 145: Use Identity 1A to find the values of 104², 37². (Hint: Decompose 104 and 37 into sums or differences of numbers…
Page 146: Use Identity 1A to write the expressions for the following. (i) (m + 3)² (ii) (6 + p)²
Page 146: Expand (6x + 5)².
Page 146: Expand (3j + 2k)² using both the identity and by applying the distributive property.
Page 146: Can we use 60² (=3600) and 5² (=25) to find the value of (60 – 5)² or 55²? Let us approach this through geometry by…
Page 146: What can we do with this expression to get the actual area?
Page 147: We have seen what (a + b)² gives when expanded. What is the expansion of (a – b)²?
Page 147: We can also use the expansion of (a + b)² to find the expansion of (a – b)². Think how. Hint: (a – b)² = (a + (–b))².
Page 147: Find the general expansion of (a – b)² using geometry, as we did for 55².
Page 147: Use the identity (a – b)² to find the values of (a) 99² and (b) 58².
Page 147: Expand the following using both Identity 1B and by applying the distributive property (i) (b – 6)² (ii) (–2a + 3)²…
Page 148: Take a pair of natural numbers. Calculate the sum of their squares. Can you write twice this sum as a sum of two…
Page 148: Do the identities below help in explaining the observed pattern?
Page 148: Here is a related pattern. Try to describe the pattern using algebra to determine if the pattern always holds.
Page 148: Use Identity 1C to calculate 98 × 102, and 45 × 55.
Page 148: Show that (a + b) × (a – b) = a² – b² geometrically. Hint:
Page 149: Why is this identity true? a² = (a + b) (a – b) + b²
Page 149: Which is greater: (a – b)² or (b – a)²? Justify your answer.
Page 149: Express 100 as the difference of two squares.
Page 149: Find 406², 72², 145², 1097², and 124² using the identities you have learnt so far.
Page 149: Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about…
Page 150: –3p (–5p + 2q)
Page 150: 2(x – 1) + 3(x + 4)
Page 150: y + 2 (y + 2)
Page 150: (5m + 6n)²
Page 150: (–q + 2)²
Page 150: 3a (2b × 3c)
Page 150: 1/2 (10s – 6) + 3
Page 150: 5w² + 6w
Page 150: 2a³ + 3a³ + 6a²b
Page 150: (x + 2)(x + 5)
Page 150: (a + 2) (b + 4)
Page 150: ab² + a²b + a²b²
Page 150: Observe the pattern in the figure below. Draw the next figure in the sequence. How many circles does it have? How…
Page 152: Does your method match any of these, or is it different? Each expression that we have identified appears different,…
Page 152: Use this formula to find the number of circles in Step 15.
Page 153: How many square tiles are there in each figure?
Page 153: How many are there in Step 4 of the sequence? What about Step 10?
Page 153: Write an algebraic expression for the number of tiles in Step n. Share your methods with the class. Can you find…
Page 153: Find the area of the (interior) shaded region in the figure below. All four rectangles have the same dimensions.
Page 153: By expanding both expressions, check that (m + n)² – 4mn = (n – m)².
Page 153: Find out the area of the region with slanting lines in the figure. All three rectangles have the same dimensions…
Page 154: By expanding the expressions, verify that all three expressions are equivalent. If x = 8 and y = 3, find the area of…
Page 154: Write an expression for the area of the dashed region in the figure below. Use more than one method to arrive at the…
Page 154: Compute these products using the suggested identity. (i) 46² using Identity 1A for (a + b)² (ii) 397 × 403 using…
Page 154: Use either a suitable identity or the distributive property to find each of the following products. (i) (p – 1) (p +…
Page 155: For each statement identify the appropriate algebraic expression(s). (i) Two more than a square number.2 + s(s +…
Page 155: Consider any 2 by 2 square of numbers in a calendar, as shown in the figure. Find products of numbers lying along…
Page 155: Verify which of the following statements are true. (i) (k + 1) (k + 2) – (k + 3) is always 2. (ii) (2q + 1) (2q – 3)…
Page 155: A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7.…
Page 155: Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same…
Page 156: What is the algebraic expression describing the following steps—add any two numbers. Multiply this by half of the…
Page 156: Which is larger? Find out without fully computing the product. (i) 14 × 26 or 16 × 24 (ii) 25 × 75 or 26 × 74
Page 156: A tiny park is coming up in Dhauli. The plan is shown in the figure. The two square plots, each of area g² sq. ft.,…
Page 156: For each pattern shown below, (i) Draw the next figure in the sequence. (ii) How many basic units are there in Step…
Page 158: Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside…

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