Distributivity and Algebra

IT'S PUZZLE TIME!

Coin Conjoin

Arrange 10 coins in a triangle as shown in the figure below on the left. The task is to turn the triangle upside down by moving one coin at a time. How many moves are needed? What is the minimum number of moves?

A triangle of 3 coins can be inverted (turned upside down) with a single move, and a triangle of 6 coins can be inverted by moving 2 coins.

The 10-coin triangle can be flipped with just 3 moves; did you figure out how? Find out the minimum possible moves needed to flip the next bigger triangle having 15 coins. Try the same for bigger triangular numbers.

Is there a simple way to calculate the minimum number of coin moves needed for any such triangular arrangement?

Observe the multiplication grid below. Each number inside the grid is formed by multiplying two numbers. If the middle number of a $3 \times 3$ frame is given by the expression $pq$, as shown in the figure, write the expressions for the other numbers in the grid.

Expand the following products.

(i) $(3 + u) (v - 3)$

(ii) $\frac{2}{3} (15 + 6a)$

(iii) $(10a + b) (10c + d)$

(iv) $(3 - x) (x - 6)$

(v) $(-5a + b) (c + d)$

(vi) $(5 + z) (y + 9)$

Find 3 examples where the product of two numbers remains unchanged when one of them is increased by 2 and the other is decreased by 4.

Expand:

(i) $(a + ab - 3b^2) (4 + b)$

(ii) $(4y + 7) (y + 11z - 3)$

Expand:

(i) $(a - b) (a + b)$

(ii) $(a - b) (a^2 + ab + b^2)$

(iii) $(a - b)(a^3 + a^2b + ab^2 + b^3)$

Do you see a pattern? What would be the next identity in the pattern that you see? Can you check it by expanding?

Which is greater: $(a - b)^2$ or $(b - a)^2$? Justify your answer.

Express 100 as the difference of two squares.

Find $406^2$, $72^2$, $145^2$, $1097^2$, and $124^2$ using the identities you have learnt so far.

Do Patterns 1 and 2 hold only for counting numbers? Do they hold for negative integers as well? What about fractions? Justify your answer.

Compute these products using the suggested identity.

(i) $46^2$ using Identity 1A for $(a + b)^2$

(ii) $397 \times 403$ using Identity 1C for $(a + b)(a - b)$

(iii) $91^2$ using Identity 1B for $(a - b)^2$

(iv) $43 \times 45$ using Identity 1C for $(a + b)(a - b)$

Use either a suitable identity or the distributive property to find each of the following products.

(i) $(p - 1)(p + 11)$

(ii) $(3a - 9b)(3a + 9b)$

(iii) $-(2y + 5)(3y + 4)$

(iv) $(6x + 5y)^2$

(v) $(2x - \frac{1}{2})^2$

(vi) $(7p) \times (3r) \times (p + 2)$

For each statement identify the appropriate algebraic expression(s).

(i) Two more than a square number.

$2 + s \qquad (s + 2)^2 \qquad s^2 + 2 \qquad s^2 + 4 \qquad 2s^2 \qquad 2^2s$

(ii) The sum of the squares of two consecutive numbers

$m^2 + n^2 \qquad (m + n)^2 \qquad m^2 + 1 \qquad m^2 + (m + 1)^2$ $m^2 + (m - 1)^2 \qquad (m + (m + 1))^2 \qquad (2m)^2 + (2m + 1)^2$

Consider any 2 by 2 square of numbers in a calendar, as shown in the figure.

Find products of numbers lying along each diagonal — $4 \times 12 = 48$, $5 \times 11 = 55$. Do this for the other 2 by 2 squares. What do you observe about the diagonal products? Explain why this happens.

Hint: Label the numbers in each 2 by 2 square as shown in the diagram.

Verify which of the following statements are true.

(i) $(k + 1) (k + 2) - (k + 3)$ is always 2.

(ii) $(2q + 1) (2q - 3)$ is a multiple of 4.

(iii) Squares of even numbers are multiples of 4, and squares of odd numbers are 1 more than multiples of 8.

(iv) $(6n + 2)^2 - (4n + 3)^2$ is 5 less than a square number.

A number leaves a remainder of 3 when divided by 7, and another number leaves a remainder of 5 when divided by 7. What is the remainder when their sum, difference, and product are divided by 7?

Choose three consecutive numbers, square the middle one, and subtract the product of the other two. Repeat the same with other sets of numbers. What pattern do you notice? How do we write this as an algebraic equation? Expand both sides of the equation to check that it is a true identity.

What is the algebraic expression describing the following steps—add any two numbers. Multiply this by half of the sum of the two numbers? Prove that this result will be half of the square of the sum of the two numbers.

Which is larger? Find out without fully computing the product.

(i) $14 \times 26$ or $16 \times 24$

(ii) $25 \times 75$ or $26 \times 74$

A tiny park is coming up in Dhauli. The plan is shown in the figure. The two square plots, each of area $g^2$ sq. ft., will have a green cover. All the remaining area is a walking path $w$ ft. wide that needs to be tiled. Write an expression for the area that needs to be tiled.

For each pattern shown below,

(i) Draw the next figure in the sequence. (ii) How many basic units are there in Step 10? (iii) Write an expression to describe the number of basic units in Step y.

Context: Consider the multiplication of two numbers, say, 23 × 27.

Q. By how much does the product increase if the first number (23) is increased by 1?

Context: Consider the multiplication of two numbers, say, 23 × 27.

Q. What if the second number (27) is increased by 1?

Context: Consider the multiplication of two numbers, say, 23 × 27.

Q. How about when both numbers are increased by 1?

Context: Consider the multiplication of two numbers, say, 23 × 27.

Q. Do you see a pattern that could help generalise our observations to the product of any two numbers?

What would we get if we had expanded $(a + 1) (b + 1)$ by first taking $(b + 1)$ as a single term? Try it!

Will the product always increase? Find 3 examples where the product decreases.

What happens when $a$ and $b$ are negative integers?

Check by substituting different values for $a$ and $b$ in each of the above cases. For example, $a = -5, b = 8; a = -4, b = -5$; etc.

Use Identity 1 to find how the product changes when

(i) one number is decreased by 2 and the other increased by 3;

(ii) both numbers are decreased, one by 3 and the other by 4.

Verify the answers by finding the products without converting the subtractions to additions.

Expand (i) $(a - u)(b + v)$, (ii) $(a - u)(b - v)$.

Use the following multiplications to find the product of a number with 11 in a single step.

(a) $3874 \times 11$ (b) $5678 \times 11$

Describe a general rule to multiply a number (of any number of digits) by 11 and write the product in one line.

Evaluate: (i) $94 \times 11$ (ii) $495 \times 11$ (iii) $3279 \times 11$ (iv) $4791256 \times 11$

Can we come up with a similar rule for multiplying a number by 101?

Multiply 3874 by 101.

Use this to multiply $3874 \times 101$ in one line.

What could be a general rule to multiply a number by 101 and write the product in one line? Extend this rule for multiplication by 1001, 10001, ...

Use this to find (i) $89 \times 101$, (ii) $949 \times 101$, (iii) $265831 \times 1001$, (iv) $1111 \times 1001$, (v) $9734 \times 99$ and (vi) $23478 \times 999$.

The area of a square of sidelength 60 units is 3600 sq. units ($60^2$) and that of a square of sidelength 5 units is 25 sq. units ($5^2$). Can we use this to find the area of a square of sidelength 65 units?

What if we write $65^2$ as $(30 + 35)^2$ or $(52 + 13)^2$? Draw the figures and check the area that you get.

If $a$ and $b$ are any two integers, is $(a + b)^2$ always greater than $a^2 + b^2$? If not, when is it greater?

Use Identity 1A to find the values of $104^2$, $37^2$. (Hint: Decompose 104 and 37 into sums or differences of numbers whose squares are easy to compute.)

Use Identity 1A to write the expressions for the following.

(i) $(m + 3)^2$

(ii) $(6 + p)^2$

Expand $(3j + 2k)^2$ using both the identity and by applying the distributive property.

Find the general expansion of $(a - b)^2$ using geometry, as we did for $55^2$.

Use the identity $(a - b)^2$ to find the values of (a) $99^2$ and (b) $58^2$.

Expand the following using both Identity 1B and by applying the distributive property

(i) $(b - 6)^2$ (ii) $(-2a + 3)^2$ (iii) $\left(7y - \frac{3}{4z}\right)^2$

Take a pair of natural numbers. Calculate the sum of their squares. Can you write twice this sum as a sum of two squares?

Try this with other pairs of numbers. Have you figured out a pattern?

Notice that $2 (5^2 + 6^2) = (6 + 5)^2 + (6 - 5)^2$.

Do the identities below help in explaining the observed pattern?

$(a + b)^2 = a^2 + 2ab + b^2$

$(a - b)^2 = a^2 - 2ab + b^2$

$(a + b)^2 + (a - b)^2 = (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)$

Adding the like terms $a^2 + a^2 = 2a^2$, $b^2 + b^2 = 2b^2$ and $2ab - 2ab = 0$, we get

$2(a^2 + b^2) = (a + b)^2 + (a - b)^2$

Pattern 2

Here is a related pattern. Try to describe the pattern using algebra to determine if the pattern always holds.

Use Identity 1C to calculate $98 \times 102$, and $45 \times 55$.

Show that $(a + b) \times (a - b) = a^2 - b^2$ geometrically.

Context: Sridharacharya (750 CE) gave an interesting method to quickly compute the squares of numbers using Identity 1C! Consider the following modified form of this identity — $a^2 = (a + b)(a - b) + b^2$

Q. Why is this identity true?

6.3 Mind the Mistake, Mend the Mistake

We have expanded and simplified some algebraic expressions below to their simplest forms.

(i) Check each of the simplifications and see if there is a mistake. (ii) If there is a mistake, try to explain what could have gone wrong. (iii) Then write the correct expression.

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

Observe the pattern in the figure below. Draw the next figure in the sequence. How many circles does it have? How many total circles are there in Step 10? Write an expression for the number of circles in Step k.

Context: The expression $k^2 + 2k$ gives the number of circles at Step $k$ of a pattern.

Q. Use this formula to find the number of circles in Step 15.

Consider the pattern made of square tiles in the picture below.

(i) How many square tiles are there in each figure? (ii) How many are there in Step 4 of the sequence? What about Step 10? (iii) Write an algebraic expression for the number of tiles in Step $n$. Share your methods with the class. Can you find more than one method to arrive at the answer?

Context: Consider the pattern made of square tiles in the picture below.

Q. How many square tiles are there in each figure?

Context: Consider the pattern made of square tiles in the picture below.

Q. How many are there in Step 4 of the sequence? What about Step 10?

Context: Consider the pattern made of square tiles in the picture below.

Q. Write an algebraic expression for the number of tiles in Step $n$. Share your methods with the class. Can you find more than one method to arrive at the answer?

By expanding both expressions, check that $(m + n)^2 - 4mn = (n - m)^2$.

By expanding the expressions, verify that all three expressions are equivalent. If $x = 8$ and $y = 3$, find the area of the shaded region.

Write an expression for the area of the dashed region in the figure below. Use more than one method to arrive at the answer. Substitute $p = 6$, $r = 3.5$, and $s = 9$, and calculate the area.