Lesson Notes By Weeks and Term v3 - Primary 6
Order of operations
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Order of operations
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Subject: General Mathematics
Class: Primary 6
Term: 3rd Term
Week: 2
Theme: Basic Operations
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Use basics operations in the right or der. Solve quantitative aptitude problems in volving BODMAS
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations (such as addition, subtraction, multiplication, and division) should be performed in an expression to arrive at a unique and correct answer. Without a standard order, different individuals might obtain different results from the same expression. The universally accepted acronym for remembering the order of operations is BODMAS (or PEMDAS in some regions). In Nigeria, BODMAS is predominantly used.
BODMAS Breakdown: B - Brackets (Parentheses): Operations inside brackets must always be performed first. Expressions within brackets should be simplified completely before moving to operations outside them.
Examples: (3 + 5), (12 - 7), (2 × 4 + 1) O - Orders (Exponents/Indices/Powers and Square Roots): After brackets, any numbers raised to a power or any square roots should be calculated. At Primary 6, this usually involves simple powers (e.g., $2^2$, $3^3$) rather than complex roots.
Examples: $2^3$ (which means 2 × 2 × 2 = 8), $5^2$ (which means 5 × 5 = 25).
DM - Division and Multiplication: These two operations have equal precedence. They should be performed from left to right as they appear in the expression. It is incorrect to always do division before multiplication, or vice-versa. The order is simply from left to right.
Examples: If you have 10 ÷ 2 × 3, you first do 10 ÷ 2 = 5, then 5 × 3 =
1
5. If you have 10 × 2 ÷ 4, you first do 10 × 2 = 20, then 20 ÷ 4 =
5. AS - Addition and Subtraction: These two operations also have equal precedence. They should be performed from left to right as they appear in the expression. Similar to division and multiplication, it is incorrect to always do addition before subtraction, or vice-versa. The order is simply from left to right.
Examples: If you have 5 + 3 - 2, you first do 5 + 3 = 8, then 8 - 2 =
6. If you have 5 - 3 + 2, you first do 5 - 3 = 2, then 2 + 2 =
4. Step-by-Step Reasoning for Solving Expressions using BODMAS:
1. Identify all operations in the expression.
2. Look for Brackets. If present, solve the expression inside the brackets first. If there are nested brackets (brackets within brackets), solve the innermost bracket first.
3. Look for Orders. If any numbers are raised to a power, calculate them next.
4. Look for Division and Multiplication. Perform these operations from left to right as they appear in the expression.
5. Look for Addition and Subtraction. Perform these operations from left to right as they appear in the expression.
6. Simplify the expression at each step, writing down the intermediate result.
Worked Examples (Nigerian Context): Example 1: Basic Operations Calculate: $15 - 3 \times 4 + 2$ Step 1 (Multiplication first): According to BODMAS (DM before AS), perform multiplication. $15 - (3 \times 4) + 2$ $15 - 12 + 2$ Step 2 (Addition and Subtraction from left to right): $(15 - 12) + 2$ $3 + 2$ $5$ Therefore, $15 - 3 \times 4 + 2 = 5$.
Example 2: Including Brackets A Nigerian trader bought 5 bags of rice at N12,500 each and 2 cartons of oil at N8,000 each. She paid with a N100,000 note. How much change did she receive?
Expression: $100,000 - (5 \times 12,500 + 2 \times 8,000)$ Step 1 (Brackets first): Solve inside the brackets. Within the brackets, multiplication comes before addition.
Calculate cost of rice: $5 \times 12,500 = 62,500$ Calculate cost of oil: $2 \times 8,000 = 16,000$ Now, inside the bracket: $62,500 + 16,000 = 78,500$ The expression becomes: $100,000 - 78,500$ Step 2 (Subtraction): $100,000 - 78,500 = 21,500$ Therefore, the trader received N21,500 as change.
Example 3: Including Orders (Powers)
Calculate: $4 \times (2^3 + 7) - 10 \div 2$ * Step 1 (Brackets first): Solve inside the brackets.
Within the 1 (Brackets first): Solve inside the brackets. Within the brackets, multiplication comes before addition.
Calculate cost of rice: $5 \times 12,500 = 62,500$ Calculate cost of oil: $2 \times 8,000 = 16,000$ Now, inside the bracket: $62,500 + 16,000 = 78,500$ The expression becomes: $100,000 - 78,500$ Step 2 (Subtraction): $100,000 - 78,500 = 21,500$ Therefore, the trader received N21,500 as change.
Example 3: Including Orders (Powers)
Calculate: $4 \times (2^3 + 7) - 10 \div 2$ Step 1 (Brackets first): Solve inside the brackets. Within the bracket, 'Orders' ($2^3$) comes first. $2^3 = 2 \times 2 \times 2 = 8$ Now, inside the bracket: $(8 + 7) = 15$ The expression becomes: $4 \times 15 - 10 \div 2$ Step 2 (Multiplication and Division from left to right): Multiplication: $4 \times 15 = 60$ Division: $10 \div 2 = 5$ The expression becomes: $60 - 5$ Step 3 (Subtraction):** $60 - 5 = 55$ Therefore, $4 \times (2^3 + 7) - 10 \div 2 = 55$.* --- Teacher Activities: Introduction (5 minutes): The teacher presents a simple mathematical expression on the board, e.g., $6 + 4 \times 2$. The teacher asks students to calculate the answer individually and shares different possible answers (e.g., 20 if $6+4$ is done first, or 14 if $4 \times 2$ is done first). The teacher explains that to get a single, correct answer, mathematicians developed a specific order of operations, introducing BODMA
S. Explanation of BODMAS (15 minutes): The teacher writes BODMAS on the board and explains each letter with simple examples.
B (Brackets): Demonstrate with (5 + 3) ×
2. O (Orders): Demonstrate with $3^2 + 5$.
DM (Division/Multiplication): Emphasize left-to-right precedence with examples like $12 \div 4 \times 3$ and $12 \times 4 \div 3$.
AS (Addition/Subtraction): Emphasize left-to-right precedence with examples like $10 + 5 - 2$ and $10 - 5 + 2$. The teacher highlights common misconceptions, such as always doing division before multiplication or addition before subtraction. Worked Examples and Modelling (15 minutes): The teacher demonstrates solving 2-3 complex examples on the board, vocalising each step and referencing the BODMAS rule at every stage.
Example 1: $20 - (4 \times 3 + 6) \div 2$ Example 2: A market woman calculates her profit. If she buys 10 bags of oranges at N500 each, sells 8 bags at N750 each, and 2 bags at N600 each, what is her total profit? (This requires setting up the expression and then solving using BODMAS).
Guided Practice (10 minutes): The teacher provides 1-2 problems for students to attempt in small groups or pairs. The teacher walks around, observes students' work, provides immediate feedback, and clarifies any confusion. The teacher calls on a group to present their solution and explain their steps using BODMA
S. Quantitative Aptitude Problem Solving (10 minutes): The teacher presents a word problem that requires forming an expression first and then applying BODMA
S. Example: "A carpenter needs to buy 3 planks of wood costing N1,200 each and a box of nails for N
5
0
0. He initially had N5,
0
0
0. How much money does he have left after his purchases?" The teacher guides students in identifying the operations and setting up the correct expression before solving.
Conclusion (5 minutes): The teacher summarises the BODMAS rule and reiterates its importance for consistent and accurate calculations. The teacher assigns independent practice exercises.
Student Activities: Individual Calculation: Students individually calculate the initial problem presented by the teacher and share their answers.
Note-Taking: Students copy the BODMAS acronym and its explanation into their notebooks.
Active Participation: Students actively listen, ask questions during the explanation, and respond to the teacher's prompts.
Group Work/Pair Work: Students work in small groups or pairs to solve guided practice problems, discussing each step and applying BODMA
S. Problem Formulation: For word problems, students identify the numbers and operations involved, formulate the mathematical expression, and then solve it.
Presentation: Selected students or groups present their solutions on the board, explaining their step-by-step application of BODMAS. --- The teacher should guide students through these problems, encouraging them to articulate each step and the rule applied.
Question 1: Simplify: $18 + 6 \times 3 - 10$ Solution: Step 1 (Multiplication): According to BODMAS (DM before AS), perform multiplication first. $18 + (6 \times 3) - 10$ $18 + 18 - 10$ Step 2 (Addition and Subtraction from left to right): Perform addition, then subtraction. $(18 + 18) - 10$ $36 - 10$ $26$
Commentary: Emphasise that multiplication takes precedence over addition and subtraction. Once only addition and subtraction remain, work from left to right.
Question 2: Calculate: $36 \div (2^2 + 5) + 3$ Solution: Step 1 (Brackets): Solve the expression inside the brackets first. Within the bracket, 'Orders' ($2^2$) comes first. $2^2 = 2 \times 2 = 4$ Now, inside the bracket: $(4 + 5) = 9$ The expression becomes: $36 \div 9 + 3$ Step 2 (Division): Perform division next (DM before AS). $(36 \div 9) + 3$ $4 + 3$ Step 3 (Addition): Perform the final addition. $7$
Commentary: Highlight the nested application of BODMAS within brackets, then the main operations.
Question 3: Mrs. Ade sells garri. She had 250 kg of garri. She sold 5 bags of 20 kg each and gave 15 kg to her family. Then she bought another 100 kg. How much garri does she have now?
Expression: This is a quantitative aptitude problem. Students must first formulate the expression: Initial garri - (Sold garri + Given garri) + Bought garri $250 - (5 \times 20 + 15) + 100$ Solution: Step 1 (Brackets): Solve inside the brackets. Within the bracket, multiplication comes first. $5 \times 20 = 100$ Now, inside the bracket: $(100 + 15) = 115$ The expression becomes: $250 - 115 + 100$ Step 2 (Subtraction and Addition from left to right): $(250 - 115) + 100$ $135 + 100$ $235$
Commentary: Emphasise translating the word problem into a mathematical expression first, then strictly applying BODMAS. The answer is 235 kg of garri. ---
Market Calculations and Budgeting: In a typical Nigerian market, people often buy multiple items at varying unit prices. Applying BODMAS ensures accurate calculation of total expenditure or change received. For example, if a family has a budget of N10,000 for groceries and buys 2 kg of meat at N1,500/kg, 3 loaves of bread at N700/loaf, and a bag of tomatoes for N2,000, understanding BODMAS helps calculate the total cost $(2 \times 1500 + 3 \times 700 + 2000)$ and the remaining balance.
Resource Sharing in a Community: When sharing resources like bags of rice or gallons of palm oil among several households, especially when some families receive larger portions or additional items, BODMAS is implicitly used to ensure fair distribution and account for all items. For example, if 5 bags of rice are to be shared among 10 families, with 2 specific families receiving an extra 5kg each, calculations involve multiplication, addition, and division in a specific order.
Construction and Building Planning: In construction, calculating material costs, labour expenses, and total project budget often involves expressions with multiple operations. For instance, calculating the cost of cement, sand, and blocks for a building project with different unit prices and quantities, plus transport fees, requires careful application of the order of operations to arrive at the correct financial figures. ---