Lesson Notes By Weeks and Term v3 - Primary 5
Open sentences
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Open sentences
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Subject: General Mathematics
Class: Primary 5
Term: 2nd Term
Week: 3
Theme: Algebraic Processes
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pupils should be ableto: find the missingnumber in opensentences. use letters to represent boxesin open sentences find the missingnumbers that the letters represents. in terpret each boxin a mathematicalstatementrepresent a letterthat could befound use letters to represent the missing numbersin quantitativeaptitude problemsand find the irvalues.
Division (÷): The method to find the missing number depends on its position.
Case 1: Missing number is at the beginning (dividend).
Example 5: A group of children shared some oranges equally among 4 friends. Each friend received 5 oranges. How many oranges were there in total?
Mathematical statement: ☐ ÷ 4 = 5 Step 1: Understand the problem. We need to find the total number of oranges.
Step 2: To find the missing number, multiply the number of groups by the amount in each group. ☐ = 5 × 4 Step 3: Calculate the result. ☐ = 20 Check: 20 ÷ 4 =
5. The statement is true.
Interpretation: There were 20 oranges in total.
Case 2: Missing number is in the middle (divisor).
Example 6: Mama Kemi shared 24 groundnuts equally among some children. Each child received 6 groundnuts. How many children shared the groundnuts?
Mathematical statement: 24 ÷ ☐ = 6 Step 1: Understand the problem. We need to find the number of children.
Step 2: To find the missing number, divide the total by the amount each received. ☐ = 24 ÷ 6 Step 3: Calculate the result. ☐ = 4 Check: 24 ÷ 4 =
6. The statement is true.
Interpretation: 4 children shared the groundnuts. Using Letters to Represent Missing Numbers (Variables): Instead of boxes, letters are commonly used to represent unknown quantities. This is an introduction to algebraic concepts.
Example 7: Represent the missing number with a letter and solve: 12 + p = 20 Step 1: The letter p represents the missing number.
Step 2: To find p, subtract 12 from 20. p = 20 - 12 Step 3: Calculate the result. p = 8 Check: 12 + 8 =
2
0. The statement is true.
Example 8: Represent the missing number with a letter and solve: 7 × m = 35 Step 1: The letter m represents the missing number.
Step 2: To find m, divide 35 by 7. m = 35 ÷ 7 Step 3: Calculate the result. m = 5 Check: 7 × 5 =
3
5. The statement is true. Interpreting Boxes/Letters in Quantitative Aptitude Problems: Quantitative aptitude problems often present scenarios where a symbol (box, letter, or even an abstract shape) represents a consistent missing value across different parts of a problem.
Example 9: If ☐ + 3 = 9 and 7 - ☐ = y, find the value of y.
Step 1: Find the value of ☐ from the first open sentence. ☐ + 3 = 9 ☐ = 9 - 3 ☐ = 6 Step 2: Substitute the value of ☐ into the second open sentence to find y.** 7 - ☐ = y 7 - 6 = y y = 1 Interpretation: In this problem, the box represents the number 6, and y represents the number
1. Definition of Open Sentence: An open sentence is a mathematical statement that contains an unknown quantity or a variable. It is a statement that is neither true nor false until the unknown quantity is replaced by a specific value. The unknown quantity is often represented by a box (☐) or a letter (e.g., n, x, y).
Examples: ☐ + 5 = 12 (The box represents the missing number) 10 - x = 4 (The letter x represents the missing number) 3 × y = 18 (The letter y represents the missing number)
The Missing Number (Unknown/Variable): The missing number is the value that makes the open sentence true. In algebra, this missing number is called a variable.
Steps to Find the Missing Number/Value:
A. Open Sentences Involving Addition (+): When the missing number is part of an addition problem, subtraction is used to find it.
Example 1: A farmer harvested some yams and added 15 more yams from his neighbour. He now has 30 yams. How many yams did he harvest initially?
Mathematical statement: ☐ + 15 = 30 Step 1: Understand the problem. We need to find the initial quantity of yams.
Step 2: To find the missing number, subtract the known number from the total. ☐ = 30 - 15 Step 3: Calculate the result. ☐ = 15 Check: 15 + 15 =
3
0. The statement is true.
Interpretation: The farmer harvested 15 yams initially.
B. Open Sentences Involving Subtraction (-): The method to find the missing number depends on its position.
Case 1: Missing number is at the beginning (minuend).
Example 2: A trader had some bags of rice. She sold 8 bags, and 12 bags were left. How many bags of rice did she have initially?
Mathematical statement: ☐ - 8 = 12 Step 1: Understand the problem. We need to find the initial quantity of rice.
Step 2: To find the missing number, add the number subtracted to the remainder. ☐ = 12 + 8 Step 3: Calculate the result. ☐ = 20 Check: 20 - 8 =
1
2. The statement is true.
Interpretation: The trader initially had 20 bags of rice.
Case 2: Missing number is in the middle (subtrahend).
Example 3: Adamu had 25 kola nuts. He gave some to his friends and now has 10 kola nuts left. How many kola nuts did he give away?
Mathematical statement: 25 - ☐ = 10 Step 1: Understand the problem. We need to find the number of kola nuts given away.
Step 2: To find the missing number, subtract the remainder from the initial amount. ☐ = 25 - 10 Step 3: Calculate the result. ☐ = 15 Check: 25 - 15 =
1
0. The statement is true.
Interpretation: Adamu gave away 15 kola nuts.
C. Open Sentences Involving Multiplication (×): When the missing number is part of a multiplication problem, division is used to find it.
Example 4: A tailor needs 3 metres of ankara fabric to make one dress. She has made 18 metres of ankara fabric into dresses. How many dresses did she make?
Mathematical statement: ☐ × 3 = 18 Step 1: Understand the problem. We need to find the number of dresses made.
Step 2: To find the missing number, divide the total by the known number. ☐ = 18 ÷ 3 Step 3: Calculate the result. ☐ = 6 Check: 6 × 3 =
1
8. The statement is true.
Interpretation: The tailor made 6 dresses.
D. Open Sentences Involving Division (÷): The method to find the missing number depends on its position.
Case 1: Missing number is at the beginning (dividend).
Example 5: A group of children shared some oranges equally among 4 friends. Each friend received 5 oranges. How many oranges were there in total?
Mathematical statement: ☐ ÷ 4 = 5 Step 1: Understand the problem. We need to find the total number of oranges. * Step 2: To find the missing number, multiply the number of groups by the amount in each group. ☐ = 5 × Materials: Real-life objects (beans, stones, bottle caps), flashcards with open sentences, chalk/marker, whiteboard/blackboard, exercise books.
A. Introduction (10 minutes)
Teacher Activity: Begin by posing a simple riddle or real-life scenario to students. "I have a certain number of oranges. If I add 5 more, I will have 12 oranges. How many oranges did I have at first?" Student Activity: Students discuss in pairs or individually and share their answers. The teacher elicits the answer (7).
Teacher Activity: Introduce the concept of a "missing number" and write the problem as a number sentence with a box: ☐ + 5 =
1
2. Explain that this is called an "open sentence." Teacher Activity: Briefly review inverse operations (addition/subtraction, multiplication/division) as these are key to solving open sentences.
B. Development (30 minutes)
Activity 1: Finding Missing Numbers in Open Sentences (Using Boxes)
Teacher Activity: Write various open sentences on the board using boxes for the missing numbers (covering all four basic operations).
Example: 8 + ☐ = 15; ☐ - 6 = 10; 4 × ☐ = 24; ☐ ÷ 5 = 7 Student Activity: Students work in small groups using concrete materials (e.g., beans, bottle caps) to model and solve the problems. For 8 + ☐ = 15, they might put 8 beans, then add more until they count 15, then count the added beans.
Teacher Activity: Guide groups, observe their strategies, and correct misconceptions. Demonstrate the inverse operation method for each example on the board, showing step-by-step calculations. For 8 + ☐ = 15, demonstrate ☐ = 15 - 8 =
7. For ☐ - 6 = 10, demonstrate ☐ = 10 + 6 =
1
6. Activity 2: Using Letters to Represent Missing Numbers Teacher Activity: Explain that instead of boxes, letters (like a, b, x, y) are commonly used to represent unknown numbers. Emphasize that any letter can be used.
Teacher Activity: Rewrite some of the previous examples using letters.
Example: 8 + x = 15; y - 6 = 10; 4 × m = 24; n ÷ 5 = 7 Student Activity: Students solve these new open sentences with letters in their notebooks, applying the inverse operations they just learned.
Teacher Activity: Call on students to share their solutions and explain their steps. Reinforce the concept that the letter stands for the specific numerical value.
Activity 3: Interpreting Letters in Quantitative Aptitude Problems Teacher Activity: Present quantitative aptitude problems that involve letters or symbols representing missing numbers.
Example: If 5 + a = 12, what is the value of a?
Example: If p - 7 = 11, then what is p + 3?
Student Activity: Students work individually or in pairs to solve these problems, first finding the value of the letter, then using it in the subsequent part of the problem.
Teacher Activity: Facilitate a class discussion, allowing students to explain their reasoning for each step. Emphasize the systematic approach.
C. Conclusion (5 minutes)
Teacher Activity: Summarize the key concepts: what an open sentence is, how to find missing numbers using inverse operations, and how letters can represent these missing numbers.
Student Activity: Students share one new thing they learned or one challenging aspect.
Teacher Activity: Assign homework.
Objective Alignment: These questions directly target finding missing numbers, using letters, and interpreting letters in simple problems.
Question 1: A bus departed from Lagos with a certain number of passengers. At Ibadan, 7 more passengers boarded, bringing the total to 25 passengers. How many passengers were on the bus when it left Lagos? Represent the missing number with a box and solve.
Solution 1: Representation: ☐ + 7 = 25 Calculation: ☐ = 25 - 7 ☐ = 18
Commentary: The missing number is found by using the inverse operation of addition, which is subtraction. This helps students identify the initial quantity before an increase. There were 18 passengers on the bus when it left Lagos.
Question 2: The Local Government Primary School in Kano received x textbooks. After distributing 15 textbooks to Primary 5 students, there were 35 textbooks left. Find the value of x.
Solution 2: Representation: x - 15 = 35 Calculation: x = 35 + 15 x = 50
Commentary: Here, x represents the initial number of textbooks. Since 15 were removed, we add the remaining amount back to find the original total. The value of x is
5
0. Question 3: If 6 × y = 42, find the value of y. Also, if y is used in the expression y + 10, what would be the result?
Solution 3: Part 1: Find y 6 × y = 42 y = 42 ÷ 6 y = 7 Part 2: Calculate y + 10 y + 10 = 7 + 10 y + 10 = 17
Commentary: This problem first requires students to solve a multiplication open sentence using division. Then, they must use the determined value of y in a subsequent calculation, demonstrating interpretation of the letter's value. The value of y is 7, and y + 10 is 17.
Market Transactions and Budgeting: Students can apply open sentences when calculating change after buying items, determining how many items can be bought with a certain amount of money, or figuring out how much more money is needed for a purchase. For example, "A customer gave N1000 for yams costing N
6
5
0. How much change should they receive?" (1000 - ☐ = 650 or 650 + ☐ = 1000). Or, "If one sachet of water costs N20, how many sachets can I buy with N500?" (N500 ÷ N20 = ☐).
Resource Allocation in the Community: In local communities, open sentences can be used to divide resources fairly. For instance, if 6 bags of rice are to be shared equally among a certain number of families, and each family receives 2 bags, how many families received rice? (6 ÷ ☐ = 2). This teaches problem-solving in a practical community context.
Simple Event Planning: If a community event requires a certain number of chairs, and there are only a few available, students can use open sentences to figure out how many more chairs are needed. "We need 50 chairs for the meeting, but we only have
3
5. How many more chairs do we need?" (35 + ☐ = 50). This helps them understand practical planning and logistics.