Applications of Algebra

Collaborative Learning: Learners will work in convenient groups (ability, mixed-ability, mixed gender etc.) to solve problems and present answers.
Talk for Learning Approaches: Learners will be engaged through think-pair-share in brainstorming, build on what others say to justify and provide feedback on sequences created.
Project-based Learning: Engages learners to come up with a sequence that is relevant and applicable to real life situations. Experiential Learning: Engages learners in hands-on activities (learning by doing).
Problem-based Learning: Learners inquire into what happens when terms of sequences are summed up to infinity.
Activity 1: Use rules to find a specified term of the sequence
Collaborative Learning: Have groups work together to recall and explain sequences and their features. Learners investigate linear and exponential:
- Learners work in pairs or groups to create at least two different numeric and/or non-numeric sequences (creators must know the patterns that define the sequences). Swap between groups and determine the pattern that defines each of the other sequences and give reasons for the pattern form.
- Pairs or groups work with a sequence of numbers to establish the general rule for arithmetic and geometric sequence, using the first terms and relationship between consecutive terms (simple recursions) and the notations of sequences, and determine if a sequence is arithmetic or geometric and identify key terms of the sequence such as the first, the common difference and/or ratios, and the nth-term.
Note:
- In an arithmetic sequence, the difference between one term and the next term is a constant (e.g., 1, 4, 7, 10, 13, 16, 19, . . . ) It has the general rule of 𝑈𝑛 = 𝑎 + 𝑑(𝑛 − 1) where a is the first term, d is the common difference, and n is the number of terms.
- For geometric sequence, each term is found by multiplying the previous term by a constant (e.g. 2, 4, 8, 16, 32, 64, ...) it has a general rule, 𝑈𝑛 = 𝑎 (n-1) where 𝒂 is the first term, 𝒓, the common ratio and n is the number of terms in the sequence.
Activity 2: Using a given rule or formula (input-output tables) Learners determine the first few elements of the sequence given the algebraic relationship.
Example: Complete the table for the sequence given by 𝑢 % = (2) %*! ; 𝑛 > 0 ! "
𝑛 1 2 3 4 5 6 𝑢 % = " (2) %*! ! 1 2 4 8 16 32 3 3 3 3 3 3
Alternatively, learners will be required to determine the rule for values given in a table.
Activity 3: Learners use other conventional strategies to find the nth term of a sequence.
Activity 4: Learners work in groups to describe linear and exponential sequences and use their understanding to solve real life problems.
Example: The second term of an A.P. is 15, and the fifth term is 21. Find the common difference and the sum of the first 5 terms.
Solution: 15 = 𝑎 + 𝑑 .......... (1) and 21 = 𝑎 + 4𝑑 ......... (2) Solving simultaneously, we have 𝑑 = 2:
The sum of the first five terms is 85
More examples:
- A man starts savings on September 1 st . He saves 10 pesewas the first day, 20 pesewas the second day, 40 pesewas the third, and so on, doubling the amount every day. How much would he save if he managed to keep saving under this system until the end of the month (September 30 th )? Leave your answer in cedis correct to 2 decimal places.
- The set of whole numbers is partitioned into subsets, with the first number in the first subset, the next two numbers in the second subset, and the next three numbers in the third subset and so on. Find, in terms of 𝑛, a formula for the last member of the 𝑛 𝑡ℎ subset.
Note: Start with the number of whole number(s) in the first partition, second partition, etc.
Teaching and Learning Resources:
- Cut out shapes of different colours and orientation
- Maths Technology learning apps, tools and devices
- Chalkboard illustrations
- Worksheets
- Cut-out geometrical shapes of different colours and orientation
Assessment (1.1.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 4 Extended critical thinking and reasoning.
1. Find the 10𝑡ℎ term of the arithmetic sequence: 2, 5, 8, 11, . .. 2. Given the arithmetic sequence: 3, 7, 11, 15, . .. Find an expression for the nth term of this sequence. 3. For the geometric sequence: 4, 12, 36, 108, . .. Find the formula for the nth term of the sequence. The set of whole numbers is partitioned into subsets, with the first number in the first subset, the next two numbers in the second subset, and the next three numbers in the third subset and so on. Find in terms of n a formula for the first member of the 𝑛ℎ subset. NB: Start with a number of whole number(s) in the first partition, second partition, etc