Algebra

Algebra begins with a simple idea: a letter can stand for a number that is not fixed. The chapter starts with matchstick patterns. The letter L needs 2 matchsticks, so n Ls need 2n matchsticks, and a row of n joined squares needs 3n + 1. Such a letter, whose value can change, is called a variable. Variables let us write general rules briefly, such as the perimeter of a square 4l, the perimeter of a rectangle 2(l + b), and properties like a + b = b + a. You will then form expressions with variables, such as x + 10, 3y or 2m - 5, and turn word statements into expressions, for example "Sarita has 10 more marbles than Ameena" becomes x + 10. Finally, you learn that an equation is a statement of equality containing a variable, like x + 3 = 7, and find its solution by trial and error, checking which value makes both sides equal.

Detailed Worksheet: Algebra Section A - Definitions (10 marks) 1. What is a variable? Why is a letter used instead of a fixed number in a rule such as "number of matchsticks = 2n"? (2 marks) 2. What is an algebraic expression? Give two examples of expressions using the variable y. (2 marks) 3. Define an equation. Explain why 2x + 5 is not an equation but 2x + 5 = 11 is an equation. (2 marks) 4. What is meant by the solution of an equation? Check whether x = 4 is the solution of 3x - 2 = 10. (2 marks) 5. Write the general rule, using variables, for (i) the perimeter of a square of side l (ii) the commutative property of addition of two numbers. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find the rule for the number of matchsticks needed to make n copies of each pattern: (i) the letter T (ii) the letter Z (iii) the letter E. Use the rule to find the number of matchsticks needed for 10 copies of the letter E. (3 marks) 7. Write expressions for the following: (i) 7 added to p (ii) y multiplied by 5 and then 3 subtracted from the result (iii) one-fourth of z (iv) q divided by 4 and then 10 added (v) 11 subtracted from twice m (vi) x added to y. (3 marks) 8. Ravi's age is x years. Write expressions for: (i) his father's age if his father is 4 years more than three times Ravi's age (ii) his sister's age if she is 3 years younger than Ravi (iii) his age 5 years from now. If x = 12, find each age. (3 marks) 9. Complete the table of values for the equation m + 10 = 16 by substituting m = 4, 5, 6, 7, 8, state the left side value each time, and find the solution. Then solve 5p = 60 by trial and error. (3 marks) 10. The length of a rectangle is l cm and its breadth is b cm. (i) Write the rule for its perimeter. (ii) Find the perimeter when l = 12 cm and b = 7 cm. (iii) If the perimeter is 40 cm and l = 13 cm, write an equation for b and solve it. (3 marks) Section C - Diagrams (10 marks) 11. Draw the matchstick pattern of 1, 2 and 3 squares joined in a row. Count the matchsticks in each case, write the rule for n squares as an expression in n, and find the number of matchsticks needed for 15 squares. (4 marks) 12. Draw a rectangle and label its length l and breadth b. Shade a square inside it of side s and write expressions for the perimeter of the rectangle and the perimeter of the square. Find both perimeters if l = 9 cm, b = 5 cm and s = 4 cm. (3 marks) 13. Draw a row of n columns of dots with 3 dots in each column for n = 1, 2, 3 and 4. Write the rule for the total number of dots in terms of n, and use it to find the number of dots for n = 12. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Leela is Radha's younger sister. Leela is 4 years younger than Radha. (i) If Radha's age is x years, write Leela's age. (ii) Write an expression for the sum of their ages. (iii) If the sum of their ages is 30 years, form an equation and solve it by trial and error. (iv) Explain why Leela's age changes when x changes but the difference in their ages does not. (5 marks) 15. Sarita says that the expressions 2n + 3 and 2(n + 3) are the same. (i) Evaluate both for n = 1, 2 and 5. (ii) Is Sarita correct? (iii) Write each expression in words. (iv) Find an expression that equals 2(n + 3) for every value of n without brackets, using the distributive property. (5 marks) 16. A school is arranging chairs for a function. Each row has 12 chairs. (i) Write an expression for the number of chairs in r rows. (ii) If 20 more chairs are kept at the back, write an expression for the total number of chairs. (iii) If the total number of chairs is 140, form an equation and find r by trial and error. (iv) Explain why a variable is useful in planning such arrangements. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show the trial-and-error table wherever you solve an equation, and draw all patterns neatly in pencil.