Open Sentences – Basic 6 Mathematics Lesson Note (Second Term)

What are Open Sentences?

An open sentence is a mathematical statement that contains one or more unknown values (variables). We use letters like x, y, or n to represent the unknown numbers.

Examples:

• x + 5 = 12
• 3n = 15
• y – 7 = 10

To solve an open sentence, we find the value of the unknown that makes the sentence true.

1. Addition and Subtraction

Solving Addition Open Sentences

Method: Subtract the known number from both sides

Example 1: x + 8 = 15 To find x: x = 15 – 8 = 7 Check: 7 + 8 = 15 ✓

Example 2: n + 12 = 25 To find n: n = 25 – 12 = 13 Check: 13 + 12 = 25 ✓

Solving Subtraction Open Sentences

Method: Add the subtracted number to both sides

Example 1: y – 6 = 14 To find y: y = 14 + 6 = 20 Check: 20 – 6 = 14 ✓

Example 2: m – 9 = 17 To find m: m = 17 + 9 = 26 Check: 26 – 9 = 17 ✓

Exercise A – Addition and Subtraction

Find the value of the unknown:

1. x + 7 = 18, so x = ______
2. n + 15 = 32, so n = ______
3. y – 8 = 12, so y = ______
4. a – 6 = 19, so a = ______
5. p + 24 = 45, so p = ______
6. m – 13 = 27, so m = ______
7. x + 56 = 89, so x = ______
8. b – 35 = 48, so b = ______

Check your answers by substituting back:

1. If x = 11 in x + 7 = 18, is this correct? ______
2. If y = 20 in y – 8 = 12, is this correct? ______

1. Multiplication and Division

Solving Multiplication Open Sentences

Method: Divide both sides by the multiplier

Example 1: 4x = 20 To find x: x = 20 ÷ 4 = 5 Check: 4 × 5 = 20 ✓

Example 2: 6n = 42 To find n: n = 42 ÷ 6 = 7 Check: 6 × 7 = 42 ✓

Solving Division Open Sentences

Method: Multiply both sides by the divisor

Example 1: y ÷ 3 = 8 To find y: y = 8 × 3 = 24 Check: 24 ÷ 3 = 8 ✓

Example 2: m ÷ 5 = 12 To find m: m = 12 × 5 = 60 Check: 60 ÷ 5 = 12 ✓

Exercise B – Multiplication and Division

Find the value of the unknown:

1. 5x = 35, so x = ______
2. 8n = 64, so n = ______
3. y ÷ 4 = 9, so y = ______
4. a ÷ 7 = 6, so a = ______
5. 12p = 84, so p = ______
6. m ÷ 8 = 15, so m = ______
7. 9x = 72, so x = ______
8. b ÷ 6 = 14, so b = ______

Mixed operations:

1. 3x + 6 = 21, so x = ______ (Hint: First subtract 6, then divide by 3)
2. 2y – 8 = 18, so y = ______ (Hint: First add 8, then divide by 2)

iii. Reciprocal of Numbers

What is a Reciprocal?

The reciprocal of a number is 1 divided by that number. When you multiply a number by its reciprocal, you get 1.

Examples:

• Reciprocal of 4 = 1/4 (because 4 × 1/4 = 1)
• Reciprocal of 1/3 = 3 (because 1/3 × 3 = 1)
• Reciprocal of 5/7 = 7/5 (flip the fraction)

Finding Reciprocals

For whole numbers: Put 1 over the number

• Reciprocal of 8 = 1/8

For fractions: Flip the numerator and denominator

• Reciprocal of 3/4 = 4/3

Exercise C – Reciprocals

Find the reciprocal of these numbers:

1. Reciprocal of 6 = ______
2. Reciprocal of 1/5 = ______
3. Reciprocal of 2/3 = ______
4. Reciprocal of 9 = ______
5. Reciprocal of 4/7 = ______
6. Reciprocal of 12 = ______
7. Reciprocal of 3/8 = ______
8. Reciprocal of 1/10 = ______

Using Reciprocals in Open Sentences

Example: x × 1/4 = 7 To solve: x = 7 × 4 = 28 (multiply by the reciprocal of 1/4)

Exercise D – Open Sentences with Reciprocals

Solve these equations:

1. x × 1/3 = 15, so x = ______
2. n × 1/5 = 8, so n = ______
3. y × 2/3 = 12, so y = ______
4. a × 3/4 = 21, so a = ______
5. p × 1/7 = 4, so p = ______

Real Life Problems on Open Sentences

Money Problems

Example: Kemi had some money. After spending ₦450, she had ₦1,200 left. How much did she have initially?

Open sentence: x – 450 = 1,200 Solution: x = 1,200 + 450 = ₦1,650

Exercise E – Money Problems

Write and solve open sentences:

1. Savings Problem: Tunde saved some money for 4 months. His total savings is ₦2,400. How much did he save each month? Open sentence: ______ Answer: ₦______
2. Shopping Problem: Mama bought items worth ₦3,500. After paying, she received ₦500 change. How much money did she give the shopkeeper? Open sentence: ______ Answer: ₦______
3. Pocket Money: After buying a book for ₦780, Chidi had ₦320 left. How much pocket money did he have initially? Open sentence: ______ Answer: ₦______

Exercise F – Age and Time Problems

Solve these real-life problems:

1. Age Problem: Father is 3 times as old as his son. If the son is 12 years old, how old is the father? Open sentence: ______ Answer: ______ years
2. Journey Time: A bus travels at 60 km/h. If it covers 240 km, how many hours does the journey take? Open sentence: ______ Answer: ______ hours
3. School Hours: Primary 6 students spend 8 hours in school daily. If they spend 280 hours in school per month, how many days are in the school month? Open sentence: ______ Answer: ______ days

Exercise G – Sharing and Distribution

Apply open sentences to sharing problems:

1. Equal Sharing: 240 books are shared equally among some classes. If each class gets 30 books, how many classes are there? Open sentence: ______ Answer: ______ classes
2. Food Distribution: A bag of rice feeds a family for 15 days. If the family wants the rice to last 20 days, by what fraction should they reduce their daily consumption? Open sentence: ______ Answer: ______
3. Sports Teams: 72 students are divided into teams of 8 players each. How many teams are formed? Open sentence: ______ Answer: ______ teams

Quantitative Reasoning

Pattern Recognition with Open Sentences

Example Pattern: 2, 5, 8, 11, __, 17 Each number increases by 3, so: n + 3 = next number Missing number: 11 + 3 = 14

Exercise H – Number Patterns

Find the missing numbers using open sentences:

1. Pattern: 7, 12, 17, 22, ____, 32 Rule: n + _____ = next number Missing number: ______
2. Pattern: 3, 6, 12, 24, ____, 96 Rule: n × _____ = next number Missing number: ______
3. Pattern: 100, 85, 70, 55, ____, 25 Rule: n – _____ = next number Missing number: ______

Exercise I – Logic Problems

Use open sentences to solve logic problems:

1. Number Puzzle: Think of a number. Add 15 to it. The result is 38. What was the original number? Open sentence: ______ Answer: ______
2. Age Logic: Mary is 4 years older than John. If Mary is 16 years old, how old is John? Open sentence: ______ Answer: ______ years
3. Multiplication Puzzle: A number multiplied by 7 gives 91. What is the number? Open sentence: ______ Answer: ______

Exercise J – Word Problem Analysis

Convert word problems to open sentences:

1. School Problem: “A number increased by 23 equals 56.” Open sentence: ______
2. Business Problem: “When 125 is subtracted from a number, the result is 78.” Open sentence: ______
3. Division Problem: “A number divided by 8 equals 15.” Open sentence: ______

Exercise K – Mental Problem Solving

Solve these quickly:

1. x + 50 = 100, x = ______
2. 10n = 150, n = ______
3. y – 25 = 75, y = ______
4. a ÷ 12 = 5, a = ______
5. 6p = 120, p = ______

Answer Key

Exercise A:

1. 11, 2. 17, 3. 20, 4. 25, 5. 21, 6. 40, 7. 33, 8. 83, 9. Yes, 10. Yes

Exercise B:

1. 7, 2. 8, 3. 36, 4. 42, 5. 7, 6. 120, 7. 8, 8. 84, 9. 5, 10. 13

Exercise C:

1. 1/6, 2. 5, 3. 3/2, 4. 1/9, 5. 7/4, 6. 1/12, 7. 8/3, 8. 10

Exercise D:

1. 45, 2. 40, 3. 18, 4. 28, 5. 28

Exercise E:

1. 4x = 2,400, ₦600
2. x – 500 = 3,500, ₦4,000
3. x – 780 = 320, ₦1,100

Exercise F:

1. 3 × 12 = x, 36 years
2. 60x = 240, 4 hours
3. 8x = 280, 35 days

Exercise G:

1. 30x = 240, 8 classes
2. 15x = 20, reduce by 1/4
3. 8x = 72, 9 teams

Exercise H:

1. 5, 27
2. 2, 48
3. 15, 40

Exercise I:

1. x + 15 = 38, 23
2. x + 4 = 16, 12 years
3. 7x = 91, 13

Exercise J:

1. x + 23 = 56
2. x – 125 = 78
3. x ÷ 8 = 15

Exercise K:

1. 50, 2. 15, 3. 100, 4. 60, 5. 20