Topic
1.6 matrices
Sub-Topic
1.6.1 operation on matrices
General Competences
Critical Thinking, Problem Solving, Collaboration
Specific Competences
1.6.1.1 apply matrices to real life situations
Learning Activities
Carrying out operations on matrices in real life situations (addition, subtraction, multiplication; matrix by a scalar, matrix by a matrix up to 2×2)
Expected Standard
Matrices applied to real life situations correctly
References
Internet
Critical Thinking, Problem Solving, Collaboration
Lesson Goal
By the end of this lesson, learners should be able to perform addition, subtraction, scalar multiplication, and matrix-by-matrix multiplication (up to 2×2) on matrices and correctly apply these operations to solve real-life problems drawn from rural Zambian contexts such as market sales, farm produce tracking, and small-business record keeping.
Rationale
This lesson introduces Form 1 learners to the fundamental operations on matrices — addition, subtraction, scalar multiplication, and matrix-by-matrix multiplication — building directly on their prior knowledge of arithmetic operations, number patterns, and the arrangement of data in rows and columns from primary-level mathematics. Understanding matrix operations equips learners with a powerful mathematical tool for organising, comparing, and analysing quantitative data in everyday life, such as tracking farm produce sales across multiple markets or calculating group savings contributions, thereby promoting financial literacy and entrepreneurial thinking. The lesson employs a learner-centred, competence-based approach using the 5E instructional model, with group work, guided discovery, and real-life problem-solving activities that explicitly develop the competences of Critical Thinking, Problem Solving, and Collaboration.
Prior Knowledge / Prerequisite Knowledge
Form 1 learners already possess basic arithmetic skills — addition, subtraction, multiplication of whole numbers and decimals — from upper primary mathematics. They are also familiar with organising numerical data in rows and columns through work with simple tables, tally charts, and data handling in Grade 7. Additionally, learners have been introduced to the concept of a matrix as a rectangular array of numbers in the preceding sub-topic (1.6 matrices). This prior knowledge will be activated at the start of the lesson through a whole-class discussion in which the teacher displays a familiar two-week market sales table and asks learners to identify the monthly totals, thereby bridging the gap between ordinary tables and the formal idea of matrix operations.
Learning Environment
- Natural Environment: Being a rural setting, the teacher will draw examples directly from learners' daily surroundings — village markets, farm harvest records, community savings groups, and small roadside stalls. Learners will be encouraged to mention real crops (maize, groundnuts, sweet potatoes, beans) and livestock (chickens, goats) that they or their families trade, and the teacher will use these authentic data points to construct matrices during the lesson, making the content immediately relevant and tangible.
- Artificial Environment: The classroom will be arranged into six groups of 5–6 learners each, with desks pushed together to create shared workspaces. A large display board or wall chart at the front of the room will show a labelled matrix template with rows and columns clearly marked. Two flip-chart stands will be positioned at opposite corners of the room, each holding blank chart paper where groups will later display their findings. Each group will have a name card (e.g., "Group Maize", "Group Groundnuts") to foster ownership and identity.
- Technological Environment: The teacher will use a basic scientific calculator (or mobile phone calculator) to demonstrate one or two matrix calculations quickly, showing learners how technology can verify manual work. If a projector and laptop are available, a short pre-prepared slide showing images of a rural market with accompanying sales data in matrix form will be projected to enhance visual engagement. Learners will be reminded that calculators are tools for checking, not replacing, their own mathematical thinking.
Teaching and Learning Materials / Resources
- Chart paper (4 sheets) — pre-drawn with blank 2×2 and 2×3 matrices for group work
- Markers (four different colours) — one set per group for recording matrix operations
- Worksheet 1 (Explore phase) — contains pre-printed matrices with farm produce data for guided investigation
- Worksheet 2 (Elaborate phase) — contains a real-life village shop problem requiring matrix multiplication
- Set of 20 small counters or maize seeds per group — for physically arranging numbers in rows and columns during the Explore phase
- A3 laminated cards showing the definition and rules of matrix operations (one per group)
- Flip-chart stands (2) with blank paper for group presentations
- Masking tape for displaying group work on walls
- Timer or stopwatch for pacing each phase
Cross-Cutting Issues
The following cross-cutting issues are integrated into this lesson on "1.6.1 operation on matrices":
- Financial Education: All matrix operations in this lesson are contextualised using real financial data — weekly market sales of maize and groundnuts, group savings contributions, and small-shop inventory values. Learners calculate total revenue, differences in earnings between weeks, and projected sales using scalar multiplication, thereby developing practical financial computation skills essential for managing personal and household finances in rural Zambia.
- Entrepreneurship Education: Throughout the lesson, learners use matrices to simulate business decision-making — comparing sales across two markets, calculating the growth of a small poultry business, and determining the total stock needed for a village shop. These activities nurture an entrepreneurial mindset by demonstrating how mathematical tools can inform planning, budgeting, and resource allocation in micro-enterprises common in rural communities.
Lesson Progression (Model: 5E Model of Instruction)
| Phase |
Teacher Activities |
Learner Activities |
Assessment Criteria |
| INTRODUCTION |
ENGAGE
12 min |
- Hook: "Good morning, class. I want you to imagine that your family sells two things at Mpika Market — maize and groundnuts. Your elder sister keeps a record for two weeks. In Week 1, you sold 40 bags of maize and 25 bags of groundnuts. In Week 2, you sold 55 bags of maize and 20 bags of groundnuts. How could we write this information in a short, neat way that lets us quickly find the total for each crop over the two weeks?" Write the data as a simple table on the board as learners respond.
- Prior Knowledge Questions: Ask three targeted oral questions: (i) "What do we call a rectangular arrangement of numbers with rows and columns that you learned in the previous lesson?" (ii) "How do we label the size of a matrix — what two numbers do we state first?" (iii) "If I have two tables of the same size, what arithmetic operation could I use to combine them?"
- Lesson Goal: "Today we are going to learn how to add, subtract, and multiply matrices — both by a single number called a scalar and by another matrix — and then we will use these operations to solve real problems from your own village life." Write the goal on the top-right corner of the board.
- Note: Do NOT teach any operation yet. Keep the phase focused on curiosity and recall.
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- Learners listen to the hook scenario and suggest ways to organise the data, recalling their knowledge of tables and arrays. (Develops: Critical Thinking)
- Learners answer prior knowledge questions — they define a matrix, state that order is given as rows × columns, and identify addition as one way to combine two tables. (Develops: Communication)
- Learners listen to the lesson goal and one or two volunteers restate it in their own words. (Develops: Communication)
- Learners raise hands to share any experience they have with keeping records of sales or produce at home, building personal connection to the topic. (Develops: Collaboration)
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- Learners accurately recall the definition of a matrix as a rectangular array of numbers.
- Learners correctly state that matrix order is expressed as rows × columns.
- Learners confidently restate the lesson goal in their own words.
- Learners appropriately share a relevant personal experience related to record-keeping.
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| DEVELOPMENT |
EXPLORE
17 min |
- Learning Activities — Investigation: "In your groups, you will now carry out operations on matrices using real-life data. Each group will receive Worksheet 1 with two matrices showing sales of maize and groundnuts at two different markets — Kabwe Market and Ndola Market — for two weeks. Your task: find the total sales for each crop across both markets by adding the two matrices. Then find the difference between Week 1 and Week 2 sales by subtracting one matrix from another. Use your counters to physically arrange the numbers first, then write the results on the chart paper."
- Facilitation: Walk between groups, asking guiding questions: "Which numbers are in the same position in both matrices — why can we only add numbers that are in matching positions?" "If Kabwe Market sold 30 maize and 15 groundnuts in Week 1, and Ndola Market sold 20 maize and 10 groundnuts in Week 1, what is the combined total for maize?" Do NOT give answers; prompt learners to reason.
- Observation: Note which groups correctly align entries by position and which groups struggle with the concept of 'same order'. Provide an additional counter demonstration for struggling groups.
- Extension prompt for early finishers: "If both markets double their sales next month, how could we use a single operation to find the new matrix? Try it on your chart paper."
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- Learners read Worksheet 1 in their groups and arrange counters in rows and columns to model the two matrices physically. (Develops: Collaboration, Critical Thinking)
- Learners add corresponding entries of the two matrices and record the resulting matrix on chart paper, discussing why only matching positions can be added. (Develops: Problem Solving, Collaboration)
- Learners subtract one matrix from the other to find the difference in sales between weeks, writing their working step by step. (Develops: Critical Thinking)
- Learners present their group findings to the class briefly, explaining what the total and difference matrices mean in the context of the market data. (Develops: Communication, Collaboration)
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- Learners correctly identify corresponding entries in two matrices of the same order.
- Learners accurately add corresponding entries to produce a correct sum matrix.
- Learners correctly subtract corresponding entries to produce a correct difference matrix.
- Learners clearly explain the meaning of the resulting matrices in the real-life context of market sales.
|
EXPLAIN
17 min |
- Learner Sharing: Invite two groups to present their chart paper findings. Ask: "What did you discover about which numbers can be added or subtracted?" and "What did you notice about the size of the result compared to the original matrices?"
- Formalisation: Introduce precise terminology on the board: "Matrix addition and subtraction are defined ONLY for matrices of the same order. The result is a matrix of the same order where each element is the sum or difference of the corresponding elements." Write the notation: if A = [aᵢⱼ] and B = [bᵢⱼ] then A + B = [aᵢⱼ + bᵢⱼ]. Define 'scalar multiplication': if k is a scalar, then kA = [k × aᵢⱼ].
- Worked Example: On the board, show: A = [20 15; 10 25] (maize and groundnuts at Market 1) and B = [10 5; 15 10] (Market 2). Compute step by step: (i) A + B = [30 20; 25 35]; (ii) 3A = [60 45; 30 75]; (iii) A × B (2×2 by 2×2) explaining that the element in row i, column j is the sum of products of corresponding entries from row i of A and column j of B. Write each multiplication step clearly: e.g., row1×col1 = (20×10)+(15×15)=200+225=425.
- Guided Questions: Ask three mixed-type questions orally: (i) Fill-in-the-blank: "Two matrices can be added only if they have the same _______." (ii) Short answer: "If A is a 2×3 matrix and B is a 2×3 matrix, what is the order of A + B?" (iii) Problem-solving: "Given A = [5 8; 3 2] and B = [1 4; 6 7], find A − B." Learners write answers on mini whiteboards or in notebooks and show them simultaneously.
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- Learners present their group findings, explaining how they performed addition and subtraction and what patterns they noticed about matching positions. (Develops: Communication, Critical Thinking)
- Learners copy formal definitions and the notation for matrix addition, subtraction, and scalar multiplication into their exercise books. (Develops: Communication)
- Learners follow the worked example step by step, copying each calculation and asking clarification questions where needed. (Develops: Critical Thinking)
- Learners answer the three guided questions individually on mini whiteboards or in notebooks, then self-check when the teacher reveals correct answers. (Develops: Problem Solving)
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- Learners accurately explain the condition for matrix addition and subtraction in their own words.
- Learners correctly write the definition and notation for scalar multiplication.
- Learners correctly follow the worked example and complete each step without error.
- Learners accurately solve the three guided questions, demonstrating understanding of addition, subtraction, and order.
|
ELABORATE
22 min |
- New Task — Real-Life Application: "Each group will now receive Worksheet 2. A small village shop in Serenje sells three items: cooking oil, sugar, and salt. The shop has two branches — one in the village centre and one near the main road. Matrix S shows the stock (in units) at the start of the week, and matrix E shows the stock at the end of the week. Your group must: (a) Find the matrix that represents the items sold during the week (hint: which operation gives the difference?), (b) If each unit of cooking oil costs K25, sugar costs K15, and salt costs K8, use scalar multiplication to find the total value of stock sold at each branch, and (c) Challenge: If there are 5 such weeks in a month, find the total value of stock sold across all weeks at both branches combined."
- Instructions: "Work in your groups. Write ALL your steps on the chart paper. You have 15 minutes. I will choose two groups to present at the end." Do NOT re-teach; let learners apply what they have learned.
- Facilitation: Circulate and ask probing questions: "What operation tells you what was sold — addition or subtraction?" "How do you multiply a matrix by a scalar — do you multiply each entry or just some?" "If you need to find total value across all weeks, what operation combines the weekly matrices?"
- Extension for gifted learners: "Suppose the shop also sells soap at K12 per unit. The start matrix is now 2×4 instead of 2×3. How does this change your addition and subtraction? Write the new matrices and solve."
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- Learners read Worksheet 2 and discuss in their groups what each part of the problem is asking, clarifying the real-life context of the village shop. (Develops: Critical Thinking, Collaboration)
- Learners subtract the end-of-week stock from the start-of-week stock to find the number of units sold, recording the resulting matrix on chart paper. (Develops: Problem Solving)
- Learners multiply the 'units sold' matrix by the scalar price vector (or multiply each entry by the corresponding unit price if using a single price per item) to find the total value of stock sold at each branch. (Develops: Critical Thinking, Problem Solving)
- Learners present their chart paper to the class, explaining each step and how matrix operations helped solve a real business problem. (Develops: Communication, Collaboration)
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- Learners correctly identify that subtraction gives the matrix of items sold and perform the subtraction accurately.
- Learners accurately compute the matrix representing units sold and interpret it in context.
- Learners correctly apply scalar multiplication to find the total value of stock sold at each branch.
- Learners clearly explain their working and justify why each operation was chosen for the given business context.
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| CONCLUSION |
EVALUATE
12 min |
- Consolidation Questions: Ask three specific oral questions: (i) "What must be true about two matrices for us to add or subtract them?" (ii) "If matrix P = [4 7; 2 9] and the scalar k = 3, what is kP?" (iii) "In the village shop problem, why did we use subtraction to find the items sold, and not addition?"
- Learner-Led Summary: Invite 2–3 learners to state in their own words the most important thing they learned about matrix operations today. Affirm correct responses and gently clarify any persistent misconceptions.
- Link Forward: "Next lesson, we will learn how to use matrices to solve systems of equations — a very powerful application that farmers and businesses use to plan their production and budgets."
- Homework: "Complete the Class Exercise on the board. Write all five questions in your exercise book and show full working. Due tomorrow at the start of class."
- Closure: "Well done, everyone. Today you used matrices just like real business owners do. I am proud of your group work and your thinking. Pack up quietly."
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- Learners respond to consolidation questions individually, either chorally or by writing answers on mini whiteboards, demonstrating achievement of the lesson goal. (Develops: Critical Thinking)
- Learners volunteer to summarise key learning points — the order condition for addition/subtraction, the definition of scalar multiplication, and the real-life usefulness of matrix operations. (Develops: Communication)
- Learners copy the homework questions into their exercise books and check that they understand what each question requires. (Develops: Problem Solving)
- Learners share one thing they found interesting or challenging about the lesson, contributing to a positive closing discussion. (Develops: Collaboration, Communication)
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- Learners correctly state that matrices must have the same order to be added or subtracted.
- Learners accurately calculate a scalar multiplication and explain the process.
- Learners logically justify the use of subtraction in the shop problem context.
- Learners confidently articulate what they learned and what they found challenging.
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Class Exercise
Instructions to learners: Answer all five questions in your exercise book. Show all working where applicable. Time allowed: 10 minutes.
- (Fill-in-the-Blank) Two matrices can be added or subtracted only if they have the same ________. The resulting matrix will have the same ________ as the original matrices.
- (Short Answer) State the condition for multiplying two matrices, A (of order 2×3) and B (of order 3×2). What will be the order of the product matrix AB?
- (Problem-Solving) Given matrix P = [ 7 5 ; 3 8 ] and matrix Q = [ 2 4 ; 6 1 ], compute: (a) P + Q, (b) P − Q, (c) 4P.
- (Word Problem — Application) A farmer in Lundazi keeps records of his chicken and goat sales for two months. In January he sold 120 chickens and 45 goats. In February he sold 150 chickens and 30 goats. Write this information as two 1×2 matrices (one for January, one for February). Then find the matrix that shows the total chickens and goats sold over the two months, and the matrix that shows the increase or decrease from January to February.
- (Structured Problem — Analysis, Zambian Context) A women's cooperative in Mansa runs two small businesses — a vegetable stall and a bakery. Matrix V shows their weekly profit (in Kwacha) for three weeks: V = [Week1 Week2 Week3; Vegetables 450 520 480; Bakery 310 400 370]. (a) Write matrix V clearly showing rows and columns. (b) The cooperative plans to expand and expects profits to triple in the next quarter. Use scalar multiplication to find the projected weekly profit matrix. (c) If the cooperative's expenses matrix for the same three weeks is E = [280 310 300; 190 240 220], find the net profit matrix (profit − expenses) for each business for each week. (d) Which business had a higher total net profit over the three weeks? Show your working.
Answer Key (For Teacher Use Only)
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Question 1 — Fill-in-the-Blank
Answer: "Two matrices can be added or subtracted only if they have the same order. The resulting matrix will have the same order as the original matrices."
Marks: 2 marks (1 mark for each blank correctly filled).
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Question 2 — Short Answer
Answer: For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Since A is 2×3 and B is 3×2, the product AB is defined and will be of order 2×2.
Marks: 2 marks (1 mark for stating the condition correctly, 1 mark for giving the correct order 2×2).
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Question 3 — Problem-Solving
Answer: P = [7 5; 3 8] Q = [2 4; 6 1]
(a) P + Q = [7+2 5+4; 3+6 8+1] = [9 9; 9 9]
(b) P − Q = [7−2 5−4; 3−6 8−1] = [5 1; −3 7]
(c) 4P = [4×7 4×5; 4×3 4×8