Lesson Notes By Weeks and Term v3 - Primary 6
Open sentences
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Open sentences
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Subject: General Mathematics
Class: Primary 6
Term: 2nd Term
Week: 4
Theme: Algebraic Processes
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Solve problems expressed as open sentence. In terpret words in open sentences and solve them. Solve related problems on quantitative aptitude.
Initially, she had 23 tubers of yam. 2.
6. Quantitative Aptitude Problems with Open Sentences These problems often involve multiple steps and combining different operations. The goal is to set up a correct open sentence (or a series of them) that represents the problem's conditions.
Example 6 (Nigerian context): "Musa saved N
5
0
0. He spent N150 on transport and then bought some groundnuts. If he has N200 left, how much did he spend on groundnuts?" Let the amount spent on groundnuts be
G. Initial savings: N500 Spent on transport: N150 Money remaining after transport: $500 - 150 = 350$ Then he bought groundnuts (G)
Money left after buying groundnuts: $350 - G = 200$ Solving: $350 - 200 = G$ $G = 150$ * Musa spent N150 on groundnuts. --- 2.
1. Definition of an Open Sentence An open sentence is a mathematical statement that contains one or more unknown values (variables) and becomes true or false once these unknown values are replaced by specific numbers. These unknown values are typically represented by letters of the alphabet (e.g., x, y, a, n) or symbols (e.g., ?, $\square$).
Examples: $x + 5 = 12$ $10 - y = 3$ $2 \times a = 14$ $\frac{n}{4} = 6$ $P + Q = 15$ 2.
2. Identifying the Unknown (Variable) The unknown quantity in an open sentence is called a variable. It is the value that needs to be determined to make the sentence true. In "$x + 5 = 12$", x is the variable. In "Three times a number is 21", "a number" is the unknown, which can be represented by a variable like m. 2.
3. Solving an Open Sentence Solving an open sentence means finding the value of the variable that makes the statement true. This often involves using inverse operations.
Inverse Operations: The inverse of addition is subtraction. The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication. 2.
4. Steps to Solve Simple Open Sentences: Addition: If a number is added to the variable, subtract that number from both sides of the equation.
Example 1: $x + 7 = 15$ To isolate x, subtract 7 from both sides: $x + 7 - 7 = 15 - 7$ $x = 8$ Verification: $8 + 7 = 15$ (True)
Subtraction: If a number is subtracted from the variable, add that number to both sides of the equation.
Example 2: $y - 9 = 11$ To isolate y, add 9 to both sides: $y - 9 + 9 = 11 + 9$ $y = 20$ Verification: $20 - 9 = 11$ (True)
Multiplication: If a variable is multiplied by a number, divide both sides of the equation by that number.
Example 3: $3 \times a = 24$ (can also be written as $3a = 24$) To isolate a, divide both sides by 3: $\frac{3a}{3} = \frac{24}{3}$ $a = 8$ Verification: $3 \times 8 = 24$ (True)
Division: If a variable is divided by a number, multiply both sides of the equation by that number.
Example 4: $\frac{m}{5} = 6$ To isolate m, multiply both sides by 5: $\frac{m}{5} \times 5 = 6 \times 5$ $m = 30$ Verification: $\frac{30}{5} = 6$ (True) 2.
5. Interpreting Word Problems into Open Sentences This involves translating verbal statements into mathematical expressions using variables and operators. Key words and their mathematical equivalents: "is equal to", "is", "gives", "was", "result is" $\rightarrow =$ "sum of", "added to", "more than", "increased by", "total" $\rightarrow +$ "difference between", "subtracted from", "less than", "decreased by", "take away" $\rightarrow -$ "product of", "times", "multiplied by", "of" (in fractions/percentages) $\rightarrow \times$ "quotient of", "divided by", "ratio of", "per" $\rightarrow \div$ "a number", "an unknown value", "what number" $\rightarrow$ variable (e.g., x)
Example 5 (Nigerian context): "A market woman sold 15 tubers of yam, and she has 8 tubers left. How many tubers did she have initially?" Let the initial number of tubers be
Y. She sold 15: Y - 15 She has 8 left: Y - 15 = 8 Solving: Y = 8 + 15 Y = 23 Initially, she had 23 tubers of yam. 2.
6. Quantitative Aptitude Problems with Open Sentences These problems often involve multiple steps and combining different operations. The goal is to set up a correct open sentence (or a series of them) that represents the problem's conditions.
Example 6 (Nigerian context): "Musa saved N
5
0
0. He spent N150 on transport and then bought some groundnuts. If he has N200 left, how much did he spend on groundnuts?" Let the amount spent on groundnuts be
G. Initial savings: N500 * Spent on transport: 3.
1. Introduction (10 minutes)
Teacher Activity: Begin by revising simple equations where a symbol is used for an unknown, e.g., "5 + ? = 12". Ask pupils how they find the missing number. Introduce the concept of using letters (variables) instead of symbols, explaining that x or y or a simply stand for "an unknown number".
Pose a simple real-life problem: "I have some oranges, and my friend gave me 3 more. Now I have 10 oranges. How many did I have initially?" Guide pupils to think about the unknown.
Student Activity: Pupils respond to revision questions, identifying inverse operations. Pupils participate in the discussion about the orange problem, verbally identifying the unknown. 3.
2. Presentation (30 minutes)
Teacher Activity: Step 1: Explaining Open Sentences: Clearly define open sentences and variables, writing examples on the board ($x+3=7$, $15-y=10$, $2a=16$, $m/4=5$). Explain that finding the variable's value makes the sentence true.
Step 2: Solving Simple Open Sentences: Demonstrate solving each type (addition, subtraction, multiplication, division) step-by-step using inverse operations. Emphasize balancing both sides of the equation. Use examples relevant to Nigerian currency (Naira), local goods, or common scenarios.
Example:* "If I bought 3 tins of milk for N600, how much did one tin cost?" (3x = 600)
Step 3: Interpreting Word Problems: Provide a list of common keywords and their mathematical operations (refer to Section 2.5). Write them on the board. Model the translation of a word problem into an open sentence, then solve it.
Example: "My age increased by 5 years is 18 years. How old am I?" (Let age be A. $A + 5 = 18$)
Step 4: Quantitative Aptitude: Present a quantitative aptitude problem that requires multiple steps or operations. Guide pupils through breaking down the problem into smaller, manageable parts and forming an open sentence for each part, or a single complex open sentence.
Example:* "A farmer harvested 50 tubers of yam. He sold 1/5 of them and gave 10 tubers to his neighbour. How many tubers are left?" (Guide pupils to find 1/5 of 50, then subtract that and 10 from
5
0. Let remaining be R: $R = 50 - (\frac{1}{5} \times 50) - 10$)
Student Activity: Pupils listen attentively, take notes, and ask clarifying questions. Pupils solve simple open sentences on their mini-whiteboards or in their notebooks as guided by the teacher. Pupils volunteer to translate word problem keywords and suggest variables. Pupils attempt to set up open sentences for word problems and quantitative aptitude problems in pairs or small groups, sharing their approaches. 3.
3. Application & Practice (15 minutes)
Teacher Activity: Provide 3-4 varied problems (simple open sentences, word problems, quantitative aptitude) for pupils to solve individually or in pairs. Circulate around the classroom, providing support, checking understanding, and correcting misconceptions. Select a few pupils to present their solutions on the board, explaining their steps.
Student Activity: Pupils work on the given problems. Pupils discuss their strategies with peers. Selected pupils present their solutions to the class. 3.
4. Conclusion (5 minutes)
Teacher Activity: Review the main concepts: what an open sentence is, how to identify variables, and how to solve different types of open sentences (including word problems and quantitative aptitude). Emphasize the importance of translating words carefully into mathematical symbols. Assign homework.
Student Activity: Pupils participate in the summary, highlighting key learning points. Pupils copy down homework assignment. --- Question 1 (Solving a simple open sentence): A trader at Onitsha Main Market bought a carton of biscuits. After selling 25 packs, he had 37 packs left. How many packs of biscuits were in the carton initially?
Target Objective: Solve problems expressed as open sentences.
Solution: Let the initial number of packs of biscuits be p. He sold 25 packs, so we subtract 25 from p: $p - 25$ He had 37 packs left: $p - 25 = 37$ To find p, add 25 to both sides: $p - 25 + 25 = 37 + 25$ $p = 62$ Answer: There were initially 62 packs of biscuits in the carton.
Commentary: This question tests the ability to set up and solve a simple open sentence involving subtraction, requiring the use of the inverse operation (addition). Question 2 (Interpreting a word problem and solving): Mrs. Ade needs to buy some bags of rice. If 4 bags of rice cost N48,000, what is the cost of one bag of rice?
Target Objective: Interpret words in open sentences and solve them.
Solution: Let the cost of one bag of rice be c. 4 bags of rice cost N48,000, which means 4 times the cost of one bag is N48,000: $4 \times c = 48000$ (or $4c = 48000$) To find c, divide both sides by 4: $\frac{4c}{4} = \frac{48000}{4}$ $c = 12000$ Answer: The cost of one bag of rice is N12,
0
0
0. Commentary: This problem requires translating a multiplication relationship into an open sentence and solving it using division. The Nigerian context (Naira, bags of rice) makes it relatable. Question 3 (Quantitative Aptitude with multiple operations): A farmer planted 120 seedlings of maize. One-quarter of the seedlings died due to drought. He then planted an additional 30 seedlings. How many seedlings does he have now?
Target Objective: Solve related problems on quantitative aptitude.
Solution: Let the initial number of seedlings be S =
1
2
0. One-quarter of the seedlings died: $\frac{1}{4} \times 120 = 30$ seedlings died. Number of seedlings remaining after drought: $120 - 30 = 90$ seedlings.
He planted an additional 30 seedlings: $90 + 30 = 120$ seedlings. Let N be the number of seedlings he has now.
The open sentence can be represented as: $N = 120 - (\frac{1}{4} \times 120) + 30$ $N = 120 - 30 + 30$ $N = 90 + 30$ $N = 120$ Answer: The farmer has 120 seedlings now.
Commentary: This problem involves three arithmetic operations (multiplication/division, subtraction, addition) and demonstrates how to handle multi-step problems by forming a single comprehensive open sentence. ---
Market Transactions and Budgeting (Economy): Open sentences are fundamental in daily market activities. A market seller uses them to calculate how much goods to buy to meet demand, how much profit to make, or how many items are left after sales. For instance, "I bought X bags of beans, sold 15, and have 5 left. How many did I buy?" ($X - 15 = 5$).
Families use open sentences for budgeting: "My salary is NS. After spending NF on food and NR on rent, I have NL left. What is NL?" ($S - F - R = L$). Resource Allocation and Community Projects (Community): In a community, deciding how to share resources often involves open sentences. For example, if a community receives 100 bags of rice to distribute among N households, such that each gets 2 bags, this can be expressed as $2 \times N = 100$. This helps in planning and ensuring equitable distribution for projects like boreholes, school supplies, or relief materials.
Time Management and Planning (Daily Life): Individuals and families use open sentences to manage time. For instance, if a pupil has H hours for homework, spends 1.5 hours on Maths, and wants to complete two other subjects in equal time, the problem can be set as $1.5 + 2T = H$, where T is the time for each other subject. This helps in planning daily schedules and meeting deadlines. ---