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- 1Learners will recall and identify all possible outcomes when drawing one object from a bag containing different coloured items. This objective matters because understanding single outcomes builds the foundation for identifying all possible results (sample space) when performing two independent events in probability
- 2Quick Recall Activity — Show learners a brown paper bag containing 3 red bottle tops, 2 green bottle tops, and 1 pink bottle top (items visible on the table before they go in the bag). Ask: 'When Ama puts her hand in this bag without looking and pulls out one bottle top, what colours COULD she get?' Write learners' answers on the board (Red, Green, Pink). Ask a volunteer to stand and repeat all three possible colours aloud. This activates prior knowledge of outcomes from a single draw
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- UNDERSTANDING SAMPLE SPACE AND INDEPENDENT EVENTS WITH REPLACEMENT
- 1Main Activity — Introduce Sample Space Concept Using Physical Demonstration. Place the bag of 3 red, 2 green, and 1 pink bottle tops on a table visible to all 35 learners. Say: 'A sample space is a list of EVERY possible result we can get when we do an experiment. Today, we do the experiment TWICE — we draw a bottle top, we write down its colour, we PUT IT BACK, then we draw again.' Demonstrate this physically: draw one bottle top (say it is red), show the class, write 'R' on the board, return the bottle top to the bag, shake the bag, draw again (say it is green), write 'G' on the board. Explain: 'This pair of draws — Red then Green — is ONE outcome. The sample space is ALL the pairs we could possibly get.' Ask: 'If the first draw could be Red, Green, or Pink, and the second draw could also be Red, Green, or Pink, how many total pairs do we think are possible?' Let learners call out estimates. Write the number 9 on the board and say: 'Actually, there are 9 possible pairs. Let us list them all together.'
- 2Sub-Activity 1 — Guided Co-Construction of the Sample Space. Organise the 35 learners into 5 mixed-ability groups of 7 each. Give each group a manila card and a marker. Say: 'When the first draw is RED, the second draw can be Red, Green, or Pink. Write these three pairs on your card: (R,R), (R,G), (R,P).' Circulate and check each group's card. Once all groups have written their three pairs, ask the group at the front to read theirs aloud chorally with the class: '(R,R), (R,G), (R,P).' Write these on the main board. Repeat: 'When the first draw is GREEN, the second can be Red, Green, or Pink. Groups 2 and 3, write: (G,R), (G,G), (G,P).' Continue until all 9 pairs are listed: (R,R), (R,G), (R,P), (G,R), (G,G), (G,P), (P,R), (P,G), (P,P). Point to the complete list and say: 'This complete list of 9 pairs is the sample space.'
- 3Sub-Activity 2 — Learners Record Sample Space in Exercise Books. Ask all learners to open their exercise books and copy the complete sample space from the board exactly as shown: a list of all 9 outcomes. Provide for this. Walk around the classroom and check that learners are writing the correct notation (pairs in brackets). For learners who finish early, ask: 'Can you draw a picture showing the three colours of bottle tops next to each outcome? For example, next to (R,R), draw two red circles.' This creates a visual anchor. Differentiation: Struggling learners — copy only the first 3 pairs (R,R), (R,G), (R,P) and underline them; then pair with an average learner to complete the full list together. Average learners — copy the full list of 9. Fast finishers — after copying, create a table with three columns (First Draw, Second Draw, Outcome) and fill in all 9 rows
- 4Teacher Guidance: The key concept here is that 'with replacement' means the bottle top goes back, so the bag returns to its original state before the second draw. Many learners confuse this with 'without replacement' (where the object stays out). Use the phrase 'we PUT IT BACK' repeatedly and demonstrate the physical action of returning the bottle top each time. A common error is learners listing only 6 outcomes (forgetting (P,R), (P,G), (P,P) — the cases where pink is drawn first). Check each group's manila card before they share to catch this early. Struggling learners may think pink cannot be drawn twice because 'there is only one pink bottle top' — reassure them: 'Because we PUT THE PINK BACK, it can be drawn again.' For fast finishers, the table extension prepares them for the next day's probability calculation activity. Ensure all 35 learners have the correct complete list of 9 in their exercise books before moving to Phase 3, as this is the foundation for tomorrow's lesson on calculating probabilities.
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- 1Brown paper bag (or opaque cloth bag)
- 23 red bottle tops
- 32 green bottle tops
- 41 pink bottle top
- 5Textbook (Handling Data chapter on Probability, pages [insert page numbers])
- 6Exercise books (one per learner)
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- 1Consolidation Question — Ask the whole class to stand. Say: 'I am going to say an outcome. If it is in our sample space, clap once. If it is NOT in our sample space, stay silent.' Call out: '(R,R)' — learners clap. '(G,P)' — learners clap. '(R,G)' — learners clap. '(P,P)' — learners clap. '(R,B)' — silence (blue was never an option). '(G,R)' — learners clap. This playful activity confirms that learners understand which outcomes belong to the sample space and why. It also allows you to see at a glance who is uncertain (those who clap inconsistently)
- 2Reflection and Peer Check Activity — Learners return to their seats. Say: 'Turn to your neighbour. Ask them: "What is the sample space?" They must answer: "It is a list of all possible outcomes." Then ask: "Why do we put the bottle top back?" They must answer: "So the bag stays the same and we can draw the same colour again." Listen to two or three pairs answering aloud to confirm understanding. This cements the two key concepts: the definition of sample space and the meaning of 'with replacement.'
Exercise
- 1Write this question in learners' exercise books or read aloud: 'Kofi has a bag with 2 yellow and 1 blue bottle top. He draws once, puts it back, and draws again. List ALL the possible outcomes (pairs) in the sample space.' Model Answer Hint: The sample space should contain 9 outcomes: (Y,Y), (Y,B), (Y,Y), (B,Y), (B,B), (B,Y), (Y,Y), (Y,B), (Y,Y). [Note: This is intentionally repetitive because drawing from 3 bottle tops (2 yellow identical, 1 blue) with replacement creates 9 position-based outcomes, though only 3 are distinct by colour: (Y,Y), (Y,B), (B,Y).] OR simplified acceptable answer: The three distinct outcomes are (Y,Y), (Y,B), (B,Y). Learners should list these correctly. This exercise directly assesses the Phase 1 objective (identifying all possible outcomes) in the context of two independent events, which is the Day 1 foundation for Day 2's probability calculations
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- 1Learners will identify and list the sample space for a probability experiment involving two independent events. This objective is critical because understanding sample space (all possible outcomes) is the foundation for calculating and expressing probabilities in any form—without knowing all possible outcomes, learners cannot accurately determine probabilities as fractions, decimals, percentages, or ratios
- 2Activity 1 — Prior Knowledge Recall: Display a picture of a spinner divided into 4 equal sections (red, blue, yellow, green) on the board. Ask learners: 'If I spin this spinner once, what are all the possible colours I could get?' Write their answers as a list on the board. Ask: 'How many different outcomes are there?' (Answer: 4). This activates their memory of single-event sample spaces before moving to two independent events
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- BUILDING SAMPLE SPACE FOR TWO INDEPENDENT EVENTS AND CONVERTING PROBABILITIES TO MULTIPLE FORMS
- 1Main Activity — Tree Diagram Construction and Probability Expression: Write on the board: 'Kofi has two spinners. Spinner 1 is divided into 2 equal sections: Green and Yellow. Spinner 2 is divided into 2 equal sections: 1 and 2. Kofi spins both spinners once. What is the probability of getting Green AND 1?' Draw a tree diagram on the board step by step. First branch shows Spinner 1 outcomes (Green, Yellow), each with probability 1/2 written on the branch. Second set of branches from each first outcome shows Spinner 2 outcomes (1, 2), each with probability 1/2. List all outcomes: Green-1, Green-2, Yellow-1, Yellow-2 (4 total outcomes). Count the favourable outcomes for Green AND 1: only 1 outcome matches. Calculate probability: 1/4. Now convert this fraction to other forms on the board: (1) Fraction = 1/4 (write as a fraction using the textbook example format); (2) Decimal = 0.25 (divide 1 by 4 step by step, showing working); (3) Percentage = 25% (multiply decimal by 100); (4) Ratio = 1:4 (favourable outcomes to total outcomes). Have learners copy the complete tree diagram and all four probability forms into their exercise books. Teacher prompts as learners work: 'Does each branch represent an outcome from one spinner? How many branches do we have in total? Is 1/4 the same as 0.25? Show me on your calculator.' Use the textbook to verify that the four conversion forms match the worked example given
- 2Sub-Activity 1 — Table Method for Two Independent Events: Say: 'Some people find tables easier than tree diagrams. Let's use a table to show the same experiment.' Draw a 3×3 table on the board. Column headers: Spinner 2 outcomes (blank, 1, 2). Row headers: Spinner 1 outcomes (blank, Green, Yellow). Fill in cells with all four combined outcomes: G-1, G-2, Y-1, Y-2. Ask: 'How many total outcomes are in this table?' (4). Highlight the cell for Green AND 1. Ask: 'What is the probability of Green AND 1 as a fraction? As a decimal? As a percentage? As a ratio?' Learners answer using their calculation from the tree diagram. Reinforce: 'A table shows the same information as a tree diagram, just in a different layout. Both help us count all outcomes and identify which are favourable.' Learners draw the table and label it in their exercise books
- 3Sub-Activity 2 — Independent Practice with Two-Event Probability: Give each learner a manila card with a printed two-event scenario (use teacher-prepared cards or write on the board). Scenario example: 'Ama rolls a fair die (outcomes: 1, 2, 3, 4, 5, 6) and flips a coin (outcomes: Heads, Tails). Find the probability of getting an even number AND Heads. Express it as a fraction, decimal, percentage, and ratio.' Learners work individually. They must: (1) Draw a tree diagram OR table to show all outcomes; (2) Count total outcomes (12); (3) Count favourable outcomes (even numbers AND Heads: 2-H, 4-H, 6-H = 3 outcomes); (4) Write probability as 3/12 = 1/4; (5) Convert to decimal (0.25), percentage (25%), and ratio (1:4). Circulate and check working. Ask the first learner to finish: 'Show the class your tree diagram. Point to the branch that gives us Heads. Circle the even numbers.' This applies learners' understanding to a new two-event scenario and practises all four probability forms
- 4Differentiation and Support: Struggling learners: Provide a partially completed tree diagram (only first branches drawn). Learner completes the second set of branches and counts outcomes. Use a coin-flip scenario (simpler than dice) or reduce Spinner 2 to just two options. Pair struggling learner with a peer to check outcomes. Average learners: Use the full task as described with two spinners or coin + die. Provide a step-by-step checklist on their manila card: (1) Draw tree or table, (2) List all outcomes, (3) Count total, (4) Count favourable, (5) Write fraction, (6) Convert to decimal/percentage/ratio. Fast finishers: Challenge task — 'Create your own two-event experiment (e.g., choosing a sweet from Makola Market basket A and basket B). Draw the tree diagram, list all outcomes, and express the probability of at least three different events in all four forms. Then explain to a peer why the probabilities you calculated are correct.' Extension learners may also find probability of 'at least one Heads' or other compound events that require adding multiple outcomes. Use textbook examples as models for showing working clearly.
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- 1Textbook (sample worked examples for tree diagrams and probability conversion)
- 2Exercise book (for recording tree diagrams, tables, and probability calculations)
- 3Manila cards (pre-printed with two-event probability scenarios for Sub-Activity 2)
- 4Board and chalk/whiteboard marker (for drawing tree diagram and table step by step)
- 5Coins and spinners (physical or pictures, for demonstrations in Starter and warm-up)
- 6Calculator (optional, for learners to verify decimal conversions)
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- 1Plenary Activity 1 — Consolidation Discussion: Display the two spinners scenario from the main activity (Green/Yellow and 1/2). Ask the class three consolidation questions: (1) 'Without drawing anything, how many total outcomes should we have when we spin both spinners? Why?' (Answer: 2 × 2 = 4, because each outcome from Spinner 1 can pair with each outcome from Spinner 2.) (2) 'We found that the probability of Green AND 1 is 1/4. Is this the same as 0.25 and 25% and 1:4? Why are all these forms correct?' (Answer: Yes, they all represent the same probability, just written differently.) (3) 'If I asked for the probability of getting Yellow, would we count Yellow-1 and Yellow-2, or just one of them?' (Answer: Both, because both have Yellow.) Ask a learner who struggled with the starter to attempt Question 1 first, guiding them through the logic. Ask a confident learner to justify Question 3 to the class. Learners show understanding by raising their hand to contribute
- 2Plenary Activity 2 — Peer Explanation and Reflection: Pair learners (stronger with struggling, or mixed ability). Each pair receives one of the manila cards used in Sub-Activity 2 (a different scenario from the one they worked on). Pair reads the scenario aloud. The first partner explains: 'How many total outcomes do we have?' The second partner explains: 'How do we express the probability in all four forms?' Partners take turns explaining to check each other's understanding. Circulate and listen for correct use of terms: 'sample space,' 'total outcomes,' 'favourable outcomes,' 'fraction,' 'decimal,' 'percentage,' 'ratio.' Ask the pair that finishes first to share their scenario and explain to the class. This consolidates learning through peer teaching and reveals any remaining misconceptions before final assessment
Exercise
- 1Assessment Question (written in exercise book): 'Yakubu spins two spinners. Spinner A has 3 equal sections: Red, Blue, and Yellow. Spinner B has 2 equal sections: Even and Odd. (a) Draw a tree diagram or table to show all possible outcomes when both spinners are spun once. (b) How many total outcomes are there? (c) What is the probability of getting Blue AND Even? Express this probability as a fraction, a decimal, a percentage, and a ratio. Show all your working.' Model Answer Hint: (a) Tree diagram with 3 branches for Spinner A (each 1/3) and 2 branches from each (each 1/2), or 3×2 table with 6 cells; (b) 6 outcomes; (c) 1 outcome (Blue AND Even) out of 6 total, so fraction = 1/6, decimal = 0.167 (or 0.17), percentage ≈ 16.7%, ratio = 1:6. Learners must show tree diagram/table and conversion steps to receive full marks
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