Application of Algebra

Lesson notes

Curriculum details

Strand

Modelling with Algebra (Strand 1)

Sub-strand

Application of Algebra (1.1)

Content standard

2.1.1.CS.3 - Demonstrate understanding of the laws and properties of indices and apply the ideas to solve problems. 2.1.1.LO.1 Investigate De Morgan's law on sets algebraically and graphically, formulate and solve real life problems up to three sets. 2.1.1.LO.2 Model sequence recursively and explicitly, and establish the relationship between the two forms, as well as solve real life problems involving linear and exponential sequences and series. 2.1.1.LO.3 Apply indices and logarithms to solve real life problems, including logarithms with different bases, and sketch and interpret logarithmic functions. 2.1.1.LO.4 Formulate and derive appropriate strategies to solve quadratic inequalities. 2.1.1.LO.5 Graph systems of given inequality and identify the region that provides the feasible solution and apply it to real life situations. 2.1.1.LO.6 Determine the set of values for which a rational function is defined and resolve rational functions into partial fractions. 2.1.1.LO.7 Multiply matrices, determine the inverse of a 2 x 2 matrix, find the determinant up to a 3 x 3 matrix and represent matrices in linear transformations.

Indicator

2.1.1.LI.2 - Use the properties of logarithms and indices to solve equations involving logarithms, including changing the base.

Suggested placement

Semester 1, Week 6 (Week 6 of the year)

Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.

Source document note

The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

• exemplars - p.290: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page

Curriculum reference

NaCCA curriculum document, p. 290

Exemplars (from the NaCCA curriculum)

Collaborative Learning, Experiential Learning, Problem-based Learning, Project-based Learning and Talk for Learning
Activity 1: Rules of logarithms
Using Talk for Learning, Problem-based, Experiential and Collaborative learning approaches, learners state and prove the rules of logarithms and make deductions. Learners working in pairs and/or small groups discuss and investigate the relationship between indices and logarithms, state the laws of logarithms and make deductions from the laws.
NB: A logarithm is the power to which a number must be raised in order to get some other number.
Example
- The logarithm of a number 𝑥 to the given base 𝑏 is the power to which 𝑏 will be raised such that it equals 𝑥. i.e., if 𝑥 = 𝑏 0 , then 𝑥 = 𝑟
- If 𝑒 0 = 8, then 𝑥 = 8 !
- Express in terms of y if (𝑦 − 5) = 𝑥 )
\W
- Rewrite 𝑉 = " (𝑙 + ℎ)(𝑙 − ℎ) in the simplest log form.
Activity 2: Apply the laws of logarithms to solve problems.
Using Talk for Learning, problem-based, Experiential and Collaborative Learning Approaches, learners use a combination of rules of logarithms to solve problems.
- Learners working with a partner or in small groups apply a combination of rules of logarithms to solve problems.
Example
- 𝑥 and 𝑦 are real numbers. If 3𝑥 = 𝑦 and 4𝑥 = 𝑦 + 4, find the values of 𝑥 and 𝑦
- Find, simultaneously, the solution to the system of equations: 1. 2𝑥 − 3𝑦 = 7 2. 𝑥 − 2𝑦 = 4
- Solve the equation 6. 9 0 + 3 0 − 2 = 0 and justify your solution.
- Learners solve problems involving change of the base of logarithms.
Example !
- If 𝑛 = 𝑝, 3𝑛 = 𝑞, and 𝑝 − 𝑞 = 2, find the possible values of n "
- Find, correct to three significant figures the value of 𝑥, if 3 0$! = 4 )0*! If 𝑥 and 𝑦 are any two positive real numbers. Show that (𝑎) ∙ (𝑎) = 𝑎 + 𝑎
Teaching and Learning Resources:
- Graph paper
- Ruler
- A scientific calculator
Assessment (2.1.1.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.