Translating Word Statements to Algebraic Expressions

1. Write an algebraic expression for: "8 more than a number $x$."

2. Write an algebraic expression for: "5 less than 4 times a number $y$."

3. Write an algebraic expression for: "12 less than 2 times a number $z$."

4. Describe a real-life situation that could correspond to the expression $10x + 5y$.

5. Describe a real-life situation that could correspond to the expression $50 - 3m$.

6. Write an algebraic expression for: "The sum of a number $p$ and 7, multiplied by 3."

7. Translate the following into a mathematical expression: "Half of a number $k$, decreased by 4."

Mind the Mistake, Mend the Mistake

For each of the following, identify the mathematical mistake, explain what went wrong, and solve it correctly.

1. Statement: "If $x = -5$, then $12 - x = 7$."

2. Statement: "If $y = 6$, then $4y = 46$."

3. Statement: "If $p = -3$, then $2(p + 4) = 14$."

4. Statement: "If $m = 4$ and $n = 2$, then $3m - 2n = 2$."

5. Statement: "If $a = 7$, then $2a + 3 = 14$."

6. Statement: "If $k = -2$, then $10 + k = 12$."

7. Statement: "If $g = 3$ and $h = 1$, then $5g - 5h = 5$."

Evaluating and Simplifying Algebraic Expressions

1. Charu's score in a quiz is calculated using the expression $8p - 4q$, where $p$ represents the points for correct answers and $q$ represents the penalty for incorrect answers. Evaluate her score if $p = 5$ and $q = 2$.

2. Simplify the following expression by grouping the like terms: $5x - 2y + 3x + 6y$.

3. Simplify Charu's total score across two rounds if her round one score is $6p - 2q$ and her round two score is $4p - 3q$.

4. Find the value of the expression $15y - 4$ when $y = 3$.

5. Now, evaluate the expression $15(y - 4)$ when $y = 3$, and compare this result to your answer in Question 4. Are they the same?

6. Simplify the expression $12a + 4b - (3a + 2b)$. (Hint: Remember that when we remove brackets preceded by a negative sign, the terms within the bracket change their sign)

7. Simplify the algebraic expression $10m - 3m + 5n - 2n$.

8. Evaluate the simplified expression you found in Question 7 when $m = 2$ and $n = 5$.

Formula Detective — Finding the Rule

Find the formula for each machine below (in terms of a and b), then find the missing output.

Machine A

Machine B

Machine C

Machine D

Machine E

Machine F

Machine G (this one gives a negative output somewhere)

Machine H

Patterns and Position Formulas

1. A pattern repeats every 4 shapes: Square, Circle, Triangle, Star, then repeats. Circle first appears at position 2, then 6, then 10. Write the formula for the position of the _n_th Circle.

• Using the same pattern, write the formula for the _n_th occurrence of: (a) Square (b) Triangle (c) Star.
• Which shape appears at position 45?
• Which shape appears at position 100?

2. A pattern of colours Red, Blue, Yellow repeats. Yellow appears at positions 3, 6, 9, … Write the formula for the _n_th occurrence of Yellow.

• In the same pattern, Blue appears one position before Yellow each time. Write the formula for the _n_th occurrence of Blue.
• In the same pattern, Red appears first each cycle (positions 1, 4, 7, …). At what position will Red appear for the 15th time?

3. A tile pattern of A, B, C, D, E repeats every 5 tiles. Design C appears at positions 3, 8, 13, … Write the formula for the _n_th occurrence of C.

• Using the same 5-tile pattern (A first, then B, C, D, E), which design appears at position 61?

4. In a 6-design repeating border (A, B, C, D, E, F), Design F appears at positions 6, 12, 18, … and Design E appears one position before F each time.

• (a) Write the formula for the _n_th occurrence of E.
• (b) Does Design E appear at position 150? Explain your reasoning.

5. A sequence of numbers grows by a multiple of 5: 5, 10, 15, 20… What is the algebraic expression for the $n^{\text{th}}$ term?

• A mathematical sequence is defined by the expression $4n - 1$. What are the first three terms of this sequence?

Translating Word Statements to Algebraic Expressions

1. Write an algebraic expression for: "8 more than a number x."

• Solution: x + 8

2. Write an algebraic expression for: "5 less than 4 times a number y."

• Solution: 4y - 5

3. Write an algebraic expression for: "12 less than 2 times a number z."

• Solution: 2z - 12

4. Describe a real-life situation that could correspond to the expression 10x + 5y.

• Solution: A shop sells large notebooks for ₹10 each and pens for ₹5 each. The total cost of buying x notebooks and y pens is represented by 10x + 5y.

5. Describe a real-life situation that could correspond to the expression 50 - 3m.

• Solution: You have a jar of 50 candies. Every day, you eat 3 candies. The number of candies left in the jar after m days is 50 - 3m.

6. Write an algebraic expression for: "The sum of a number p and 7, multiplied by 3."

• Solution: 3(p + 7)

7. Translate the following into a mathematical expression: "Half of a number k, decreased by 4."

• Solution: $\frac{k}{2}$ - 4

Mind the Mistake, Mend the Mistake

1. Statement: "If x = -5 then 12 - x = 7."

• The Mistake: The negative sign of the value was ignored. The person calculated 12 - 5 instead of substituting -5 correctly.
• Correction: 12 - (-5) = 12 + 5 = 17.

2. Statement: "If y = 6, then 4y = 46."

• The Mistake: The numbers 4 and 6 were just placed next to each other (concatenation) rather than being multiplied. 4y means 4 $\times$ y.
• Correction: 4(6) = 24.

3. Statement: "If p = -3 then 2(p + 4) = 14."

• The Mistake: There was an error in calculating the value inside the bracket or distributing the 2. The person likely added 3 + 4 = 7 and multiplied by 2 to get 14, ignoring the negative sign.
• Correction: 2(-3 + 4) = 2(1) = 2.

4. Statement: "If m = 4 and n = 2, then 3m - 2n = 2."

• The Mistake: Arithmetic error. The person likely calculated 3(4) = 12 and perhaps subtracted 10 instead of 2(2).
• Correction: 3(4) - 2(2) = 12 - 4 = 8.

5. Statement: "If a = 7, then 2a + 3 = 14."

• The Mistake: The + 3 part of the expression was completely ignored after multiplying 2 \times 7.
• Correction: 2(7) + 3 = 14 + 3 = 17.

6. Statement: "If k = -2, then 10 + k = 12."

• The Mistake: Added 2 instead of taking the negative integer into account (subtraction).
• Correction: 10 + (-2) = 10 - 2 = 8.

7. Statement: "If g = 3 and h = 1, then 5g - 5h = 5."

• The Mistake: Calculation error. They might have done 5(3) = 15, but then mistakenly evaluated 15 - 10 to get 5 instead of 5(1) = 5.
• Correction: 5(3) - 5(1) = 15 - 5 = 10.

Evaluating and Simplifying Algebraic Expressions

1. Evaluate her score if p = 5 and q = 2 for the expression 8p - 4q.

• Step 1: Substitute the values: 8(5) - 4(2)
• Step 2: Multiply: 40 - 8
• Answer: 32

2. Simplify: 5x - 2y + 3x + 6y

• Step 1: Group like terms together: (5x + 3x) + (-2y + 6y)
• Answer: 8x + 4y

3. Simplify Charu's total score across two rounds if her round one score is 6p - 2q and her round two score is 4p - 3q.

• Step 1: Add the two expressions: (6p - 2q) + (4p - 3q)
• Step 2: Group like terms: (6p + 4p) + (-2q - 3q)
• Answer: 10p - 5q

4. Find the value of 15y - 4 when y = 3.

• Step 1: Substitute the value: 15(3) - 4
• Step 2: Multiply and subtract: 45 - 4
• Answer: 41

5. Evaluate 15(y - 4) when y = 3, and compare this result to your answer in Question 4. Are they the same?

• Step 1: Substitute the value: 15(3 - 4)
• Step 2: Solve brackets first: 15(-1)
• Answer: -15. No, they are not the same (41 \neq -15). Brackets change the order of operations entirely.

6. Simplify: 12a + 4b - (3a + 2b)

• Step 1: Remove brackets. The negative sign changes the signs inside: 12a + 4b - 3a - 2b
• Step 2: Group and simplify: (12a - 3a) + (4b - 2b)
• Answer: 9a + 2b

7. Simplify: 10m - 3m + 5n - 2n

• Step 1: Combine like terms directly: (10m - 3m) + (5n - 2n)
• Answer: 7m + 3n

8. Evaluate the simplified expression you found in Question 7 when m = 2 and n = 5.

• Step 1: Substitute into 7m + 3n: 7(2) + 3(5)
• Step 2: Multiply and add: 14 + 15
• Answer: 29

Formula Detective

Machine A

• Formula: Notice 3 + 5 = 8, 7 + 2 = 9. The operation is addition. Rule: a + b
• Missing Output: 6 + 8 = $\mathbf{14}$

Machine B

• Formula: Notice 9 - 4 = 5, 10 - 2 = 8. The operation is subtraction. Rule: a - b
• Missing Output: 11 - 5 = $\mathbf{6}$

Machine C

• Formula: Look closely: 2(2) + 3 = 7; 2(4) + 1 = 9; 2(0) + 6 = 6. Rule: 2a + b
• Missing Output: 2(3) + 4 = $\mathbf{10}$

Machine D

• Formula: Look closely: 3 + 2(2) = 7; 5 + 2(4) = 13; 0 + 2(5) = 10. Rule: a + 2b
• Missing Output: 4 + 2(3) = $\mathbf{10}$

Machine E

• Formula: Look closely: 3(2) - 1 = 5; 3(4) - 3 = 9; 3(1) - 0 = 3. Rule: 3a - b
• Missing Output: 3(3) - 2 = $\mathbf{7}$

Machine F

• Formula: Look closely: 2(5) - 2(3) = 4; 2(8) - 2(2) = 12; 2(10) - 2(10) = 0. Rule: 2a - 2b (or 2(a - b))
• Missing Output: 2(7) - 2(4) = 14 - 8 = $\mathbf{6}$

Machine G

• Formula: Look closely: 10 - 2(3) = 4; 8 - 2(1) = 6; 6 - 2(6) = -6. Rule: a - 2b
• Missing Output: 5 - 2(2) = 5 - 4 = $\mathbf{1}$

Machine H

• Formula: Look closely: 4(2) - 3(1) = 5; 4(5) - 3(3) = 11; 4(4) - 3(4) = 4. Rule: 4a - 3b
• Missing Output: 4(6) - 3(5) = 24 - 15 = $\mathbf{9}$

Patterns and Position Formulas

1. 4-Shape Pattern (Square, Circle, Triangle, Star)

• Formula for the $n^{\text{th}}$ Circle: It appears at 2, 6, 10. The difference is 4. Rule: $\mathbf{4n - 2}.$
• (a) Square Formula: Appears at 1, 5, 9. Rule: $\mathbf{4n - 3}.$
• (b) Triangle Formula: Appears at 3, 7, 11. Rule: $\mathbf{4n - 1}.$
• (c) Star Formula: Appears at 4, 8, 12 (multiples of 4). Rule: $\mathbf{4n}.$
• Shape at position 45: Divide 45 by 4. 45 = 11 $\times$ 4 + 1. Because there is a remainder of 1, it matches the $1^{\text{st}}$ shape in the sequence. Shape: Square.
• Shape at position 100: Divide 100 by 4. 100 = 25 $\times$ 4. There is no remainder (it is a multiple of 4), meaning it falls on the last shape of the sequence. Shape: Star.

2. Colour Pattern (Red, Blue, Yellow)

• Formula for Yellow: Appears at multiples of 3 (3, 6, 9). Rule: $\mathbf{3n}.$
• Formula for Blue: It is one position before Yellow (3n). Rule: $\mathbf{3n - 1}.$
• Red 15th time: Red appears first each cycle (1, 4, 7). Its formula is 3n - 2. For the $15^{\text{th}}$ time, substitute n = 15: 3(15) - 2 = 45 - 2 = $\mathbf{43}.$

3. Tile Patterns

• Formula for C (5-tile pattern A..E): Appears at 3, 8, 13 (difference is 5). Rule: $\mathbf{5n - 2}.$
• Design at position 61: Divide 61 by 5. 61 = 12 $\times$ 5 + 1. The remainder is 1, so it is the $1^{\text{st}}$ design in the pattern. Design: A.

4. 6-Design repeating border A..F: - (a) Formula for E: Design F is at multiples of 6 (6n). E is one before F. Rule: $\mathbf{6n - 1}.$ - (b) Does E appear at position 150? Divide 150 by 6. 150 $\div$ 6 = 25 with 0 remainder. Because 150 is a perfect multiple of 6, position 150 holds Design F, not E. Therefore, No, E does not appear at 150; it appears at 149.

5. Number Sequences

• Algebraic expression for 5, 10, 15, 20: These are multiples of 5. Expression: $\mathbf{5n}.$ - First three terms of 4n - 1:
  • $1^{\text{st}} term (n = 1): 4(1) - 1 = \mathbf{3}$
  • $2^{\text{nd}} term (n = 2): 4(2) - 1 = \mathbf{7}$
  • $3^{\text{rd}} term (n = 3): 4(3) - 1 = \mathbf{11}$