Distributivity and Algebra | FIO

Question 20

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 $y$ .

Solution

Understand the Question

To determine the next figure and find general formulas for the patterns, we analyze how the basic units grow at each step $n$ .
Yellow Pattern: Deconstructed into three distinct sections:
- Bottom vertical column: starts at $3$ and increases by $2$ each step ( $2n + 1$ ).
- Middle horizontal row: starts at $2$ and increases by $2$ each step ( $2n$ ).
- Top vertical column: grows quadratically as $2n^2 - 2n + 2$ .
- Combined total units: $S_n = 2n^2 + 2n + 3$ .
Blue Pattern: Formed by an $(n+1) \times (n+2)$ $(n + 1) \times (n + 2)$ grid of squares with $1$ $1$ corner square removed:
- Total units: $S_n = (n + 1)(n + 2) - 1 = n^2 + 3n + 1$ .

(i) Draw the next figure in the sequence.

Step 1 · Analyze and Draw Step 4 for Yellow Pattern

Analyzing each component for Step $n$ :

Bottom vertical column: $3, 5, 7, \dots \implies 2n + 1$
Middle horizontal row: $2, 4, 6, \dots \implies 2n$
Top vertical column: $2, 6, 14, \dots \implies 2n^2 - 2n + 2$

Total units for Step $n$ :

\begin{aligned} S_n &= (2n + 1) + 2n + (2n^2 - 2n + 2) \\ &= 2n^2 + 2n + 3 \end{aligned}

For Step $4$ ( $n = 4$ ):

Bottom vertical column $= 2(4) + 1 = 9$ units
Middle horizontal row $= 2(4) = 8$ units
Top vertical column $= 2(4)^2 - 2(4) + 2 = 32 - 8 + 2 = 26$ units
Total units $= 9 + 8 + 26 = 43$ units

Step 2 · Analyze and Draw Step 4 for Blue Pattern

For Step $n$ , the figure is an $(n + 1) \times (n + 2)$ grid with $1$ square missing from the bottom-right corner:

\begin{aligned} S_n &= (n + 1)(n + 2) - 1 \\ &= (n^2 + 3n + 2) - 1 \\ &= n^2 + 3n + 1 \end{aligned}

For Step $4$ ( $n = 4$ ):

Grid dimensions $= (4 + 1) \times (4 + 2) = 5 \times 6$
Total units $= (5 \times 6) - 1 = 30 - 1 = 29$ units

Answer

(i) Yellow pattern: Step 4 has a bottom column of $9$ units, a middle row of $8$ units, and a top column of $26$ units (total $43$ units).

Blue pattern: Step 4 is a $5 \times 6$ grid with $1$ corner unit removed (total $29$ units).

(ii) How many basic units are there in Step 10?

Step 1 · Calculate Units in Step 10

For the Yellow pattern, substitute $n = 10$ into $S_n = 2n^2 + 2n + 3$ :

\begin{aligned} S_{10} &= 2(10)^2 + 2(10) + 3 \\ &= 2(100) + 20 + 3 \\ &= 200 + 20 + 3 \\ &= 223 \text{ units} \end{aligned}

For the Blue pattern, substitute $n = 10$ into $S_n = n^2 + 3n + 1$ :

\begin{aligned} S_{10} &= 10^2 + 3(10) + 1 \\ &= 100 + 30 + 1 \\ &= 131 \text{ units} \end{aligned}

Answer

(ii) Yellow pattern: $223 \text{ units}$ ; Blue pattern: $131 \text{ units}$

(iii) Write an expression to describe the number of basic units in Step $y$ .

Step 1 · Substitute $y$ for $n$ in the General Formulas

Replace $n$ with $y$ in the general formulas:

Yellow pattern: $S_y = 2y^2 + 2y + 3$
Blue pattern: $S_y = y^2 + 3y + 1$

Answer

(iii) Yellow pattern: $2y^2 + 2y + 3$ ; Blue pattern: $y^2 + 3y + 1$

Common Mistakes

Assuming Linear Growth: Mistaking non-linear sequences for simple arithmetic progressions instead of checking second differences for quadratic growth.
Grid Dimension Offsets: For the blue pattern, using $n \times (n+1)$ instead of $(n+1) \times (n+2)$ for the dimensions of the rectangle at Step $n$ .
Forgetting the Missing Corner: Forgetting to subtract $1$ from the total area $(n+1)(n+2)$ for the missing square in the blue pattern.

More questions in FIO

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

(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.

Q10

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

Q11

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) $\left(2x - \dfrac{1}{2}\right)^2$

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

Q12

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

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

Q13

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.

Q14

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.

Q15

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?

Q16

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.

Q17

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.

Q18

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

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

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

Q19

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

Q20

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 $y$ .