Distributivity and Algebra | IT

Question 33

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.

Solution

Understand the Question

To check and correct algebraic simplifications, apply the core rules of algebra:

Distributive Property: $a(b + c) = ab + ac$ . Every term inside the parentheses must be multiplied by the factor outside.
Binomial Square Identity: $(a + b)^2 = a^2 + 2ab + b^2$ . Do not omit the middle term $2ab$ .
Like vs. Unlike Terms: Only terms with identical variable parts and powers can be combined by addition or subtraction.
Associativity of Multiplication: In $a(b \times c)$ , the operation inside is multiplication, so $a(bc) = abc$ (distributivity does not apply over multiplication).

Step 1 · Check Expression 1

Question diagram

Given expression: $-3p(-5p + 2q)$

Mistake: The distributive property was not applied correctly to all terms inside the parentheses.
Correction: Multiply $-3p$ by each term inside the parentheses:

\begin{aligned} -3p(-5p + 2q) &= (-3p)(-5p) + (-3p)(2q) \\[0.6em] &= 15p^2 - 6pq \end{aligned}

Step 2 · Check Expression 2

Given expression: $2(x-1) + 3(x+4)$

Mistake: In the first term, $2$ was not multiplied with $-1$ .
Correction: Distribute and combine like terms:

\begin{aligned} 2(x-1) + 3(x+4) &= 2x - 2 + 3x + 12 \\[0.6em] &= (2x + 3x) + (-2 + 12) \\[0.6em] &= 5x + 10 \end{aligned}

Step 3 · Check Expression 3

Given expression: $y + 2(y+2)$

Mistake: Incorrectly assumed to be the square of a binomial, $(y+2)^2$ .
Correction: Distribute $2$ and combine like terms:

\begin{aligned} y + 2(y+2) &= y + 2y + 4 \\[0.6em] &= 3y + 4 \end{aligned}

Step 4 · Check Expression 4

Given expression: $(5m + 6n)^2$

Mistake: Missed the middle term $2ab$ when squaring the binomial.
Correction: Using $(a+b)^2 = a^2 + 2ab + b^2$ :

\begin{aligned} (5m + 6n)^2 &= (5m)^2 + 2(5m)(6n) + (6n)^2 \\[0.6em] &= 25m^2 + 60mn + 36n^2 \end{aligned}

Step 5 · Check Expression 5

Given expression: $(-q + 2)^2$

Status: Correct.

\begin{aligned} (-q + 2)^2 &= (-q)^2 + 2(-q)(2) + (2)^2 \\[0.6em] &= q^2 - 4q + 4 \end{aligned}

Step 6 · Check Expression 6

Given expression: $3a(2b \times 3c)$

Mistake: Distributivity was incorrectly applied over a product of terms instead of multiplying directly.
Correction: Multiply the terms:

\begin{aligned} 3a(2b \times 3c) &= 3a(6bc) \\[0.6em] &= 18abc \end{aligned}

Step 7 · Check Expression 7

Given expression: $\dfrac{1}{2}(10s - 6) + 3$

Status: Correct.

\begin{aligned} \dfrac{1}{2}(10s - 6) + 3 &= \dfrac{1}{2}(10s) - \dfrac{1}{2}(6) + 3 \\[0.6em] &= 5s - 3 + 3 \\[0.6em] &= 5s \end{aligned}

Step 8 · Check Expression 8

Given expression: $5w^2 + 6w$

Mistake: Unlike terms ( $w^2$ and $w$ ) were incorrectly added together.
Correction: Terms with different powers cannot be combined. $5w^2 + 6w$

Step 9 · Check Expression 9

Given expression: $2a^2 + 3a^3 + 6a^2b + 6ab^2$

Mistake: Unlike terms were incorrectly combined.
Correction: Since all terms have distinct variable powers or combinations, no further addition is possible. $2a^2 + 3a^3 + 6a^2b + 6ab^2$

Step 10 · Check Expression 10

Given expression: $(x+2)(x+5)$

Status: Correct.

\begin{aligned} (x+2)(x+5) &= x(x+5) + 2(x+5) \\[0.6em] &= x^2 + 5x + 2x + 10 \\[0.6em] &= x^2 + 7x + 10 \end{aligned}

Step 11 · Check Expression 11

Given expression: $(a+2)(b+4)$

Mistake: Only the first and last terms were multiplied, missing the inner and outer terms.
Correction: Expand fully using the distributive property:

\begin{aligned} (a+2)(b+4) &= a(b+4) + 2(b+4) \\[0.6em] &= ab + 4a + 2b + 8 \end{aligned}

Step 12 · Check Expression 12

Given expression: $ab^2 + a^2b + a^2b^2$

Status: Correct.

\begin{aligned} ab^2 + a^2b + a^2b^2 &= ab \cdot b + ab \cdot a + ab \cdot ab \\[0.6em] &= ab(b + a + ab) \\[0.6em] &= ab(a + b + ab) \end{aligned}

Answer

(i) Expressions with mistakes: 1, 2, 3, 4, 6, 8, 9, and 11.

(ii) Explanation of errors:

Exp 1: Distributive property not applied to all terms inside brackets.
Exp 2: Constant $-1$ was not multiplied by $2$ .
Exp 3: Incorrectly treated as $(y+2)^2$ .
Exp 4: Omitted the middle term $2ab$ in $(a+b)^2$ .
Exp 6: Distributivity wrongly applied over a product.
Exp 8 & 9: Unlike terms added together.
Exp 11: Omitted inner and outer products during binomial expansion.

(iii) Corrected expressions:

Expression 1: $15p^2 - 6pq$
Expression 2: $5x + 10$
Expression 3: $3y + 4$
Expression 4: $25m^2 + 60mn + 36n^2$
Expression 6: $18abc$
Expression 8: $5w^2 + 6w$
Expression 9: $2a^2 + 3a^3 + 6a^2b + 6ab^2$
Expression 11: $ab + 4a + 2b + 8$

Common Mistakes

Freshman's Dream Error: Writing $(a + b)^2 = a^2 + b^2$ instead of $a^2 + 2ab + b^2$ .
Incomplete Distribution: Multiplying the outer factor only with the first term inside parentheses, e.g., $2(x - 1) \neq 2x - 1$ .
Adding Unlike Terms: Adding terms with different powers, such as writing $5w^2 + 6w = 11w^3$ .
Distributing over Multiplication: Incorrectly writing $a(b \times c) = ab \times ac$ instead of $abc$ .

More questions in IT

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

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, $23 \times 27$ .

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

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

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

Context: Consider the multiplication of two numbers, say, $23 \times 27$ .

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.

Q10

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

Q11

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

(a) $3874 \times 11$

(b) $5678 \times 11$

Q12

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) $4791256 \times 11$

Q13

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

Q14

Multiply 3874 by 101.

Q15

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

Q16

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 $1001$ , $10001$ , $\dots$

Q17

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) $23478 \times 999$ .

Q18

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?

Q19

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

Q20

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?

Q21

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.)

Q22

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

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

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

Q23

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

Q24

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

Q25

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

Q26

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

(i) $(b - 6)^2$

(ii) $(-2a + 3)^2$

(iii) $\left(7y - \dfrac{3}{4}z\right)^2$

Q27

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

Q28

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

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

Q29

Pattern 2

\begin{aligned} 9 \times 9 - 1 \times 1 &= 10 \times 8 \\ 8 \times 8 - 6 \times 6 &= 14 \times 2 \\ 7 \times 7 - 2 \times 2 &= 9 \times 5 \\ 10 \times 10 - 4 \times 4 &= 14 \times 6 \end{aligned}

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

Q30

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

Q31

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

Q32

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?

Q33

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.

Q34

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 $k$ .

Q35

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.

Q36

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?

Q37

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

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

Q38

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?

Q39

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?

Q40

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

Q41

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

Q42

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.