Making Predictions with Data

Collaborative Learning, Initiate Talk for Learning and Talk for Learning Approaches, Problem-based learning,
Learning Experience: Learners to discuss the types and features of different permutations (circular arrangements, identical elements etc.), and what happens when terms of ordered and un-ordered arrangements and special arrangements are used.
Activity 1: Ordered arrangements: Initiate Talk for Learning, Experiential and Collaborative learning Approaches to support learners to explore how ordered arrangements can be counted.
Learners think-pair-square to brainstorm and discuss situations where they have to arrange a set of objects in which the order of appearance of elements of the set matters. For instance, APC differs from PAC.
Example: Three students, Esi, Abiba and Makafui, are contesting for 1st, 2nd and 3rd positions in a marathon race.
Using the first letters of their names, the possible arrangements are: EAM, AEM, EMA, MEA, MAE, and AME. Learners, in convenient groups, work together on different examples to establish that permutation is an ordered arrangement of objects in a row. It is a selection of groups of objects in which order is important.
Example: Kweku has 4 different mathematisc books on his shelf. Find the number of orders in which the 4 books can be arranged on the shelf.
Solution: There will be 4 choices in the first slot. There will be 3 choices in the first slot. There will be 2 choices in the first slot. There will be 1 choice in the first slot. ∴ There are 4 × 3× 2 × 1=24 possible arrangements of Kweku's books on the shelf.
Learners recognise that the number of permutations of n objects is denoted n! and use it to solve related problems.
Example: Kweku has 4 different mathematics books on his shelf. Find the number of orders in which the 4 books can be arranged on the shelf.
Solution: There are 4! Ways to arrange the four books hence 4! = 4 ×3× 2× 1=24 ways Kweku can arrange the mathematics books.
Activity 2: Permutation formula Initiate Talk for Learning, Experiential and Collaborative learning Approaches to support learners to use the permutation formula to solve related problems. Learners think-pair-square to brainstorm and discuss facts about permutation, selection of items out of a given set and unordered arrangement of things. Learners establish that the number of ways in which r items can be selected out from a set n items in which order matters is given by the formula: nPr =P(n,r)=n!/(n-r)! = n(n-1)(n-2)(n-3)...[n-(r-1)]
Example: Evaluate the permutation 5 P 3 Solution: 5 P 3 = P(5,3)=5!/(5-3)!=60
Activity 3: Permutation with identical/repeated objects Initiate Talk for Learning, Experiential and Collaborative learning Approaches to support learners to use the permutation formula to solve related problems. Learners establish that the number of ways of arranging n objects where different r's of them are of the same kind is given by: n!/(r1!.r2!.r3!.....rk!)
Example: In how many ways can the letters in the word SECRETS be arranged?
Solution: The number of letters is 5, so n=5; S and E are repeated, so the permutation will be affected by 2×2! Therefore, the required number of ways is 5!/(2! ×2!)= 30 ways.
Activity 4: Permutation with identical/repeated Objects Initiate Talk for Learning, Experiential and Collaborative learning Approaches to support learners to use the permutation formula to solve related problems.
Learners work in groups and, through role play (acting out), establish/discover that the number of ways of arranging objects in a circle is (n-1)! Example: In how many different ways can 6 people sit at a circular table? Solution: The number of persons is 6, so n=6; since they are to sit in a circle, we have (6- 1)!=5!=5×4×3×2×1=120 ways.
Teaching and Learning Resources:
- Ludo dice, coins, playing cards, objects in a sac or bowl.
- YEAR TWO
Assessment (1.4.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.