Standard 2: Know the content and how to teach it
These are strategies I have used in the classroom, from a unit I taught on Indices:
- Syllabus outcome: A student simplifies algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases. (MA5-IND-C-01)
- Each lesson begins with a Settling Activity (i.e. Do Now problem-solving question, Mathdrills.com / Numeracy Ninjas sheet, etc.)
Lesson 1
- Material: index notation, convert between expanded and index forms, express prime factorisation
Structure
- Diagnostic test on indices
- Introduction to indices
- I explain notation and work examples, incorporating review of individual student knowledge
- I explain prime factorising in index form: connection to past concepts, then demonstrate an example
- Students work an example of factorisation in index form This structure accommodated the large amount of material in the first topic, focusing on skill review.
Strategies
- Connecting learning: I reference past concepts when introducing the concept. This demonstration of a progression helps students develop a big-picture view. Contextualising a topic in this way helps develop relational knowledge, for future use of the material in different topics.
- I used images of number lines and graphs in my slides to connect the concept of exponentiation to previous concepts of repeated counting, addition, and multiplication. I also included images of place value blocks, to link the concept of powers of ten to the more general idea of powers of different numbers. These images provide a more concrete conceptualisation of an abstract idea. They also make the slides more visually appealing, which improves engagement over 'walls' of text/math.
Lesson 2
- Material: index laws for multiplication and division
Structure
- I demonstrate Multiplicative/Divisive Law of Indices, then provides general formula; students note down formula
- Collaborative worked examples
- Pair/small-group activity for fluency with converting numbers to index form (Tarsia puzzle) The smaller topic allowed space for an activity chosen to boost engagement, which drove the lesson structure.
Strategies
- The Tarsia activity: provides a variety of questions, which allows for differentiation in that lower-ability students are able to make progress in joining pieces by finding easier questions while higher ability students can approach more difficult questions. It also 'lowers the floor' by allowing students of different abilities to choose questions of appropriate difficulty and work with their friends who help them across the Zone of Proximal Development.
- The activity also builds on conceptual understanding established in the previous lesson. It develops fluency with the concept and calculations involved. All questions practice the same skills and the hands-on nature of the puzzle engages students regardless of ability level.
Lesson 3
- Material: raising a power to another power, the zero index
Structure
- I explain the power of a power and zero index concept
- I present a GeoGebra visualisation of Indices to revise material and extend to the present content
- Collaborative worked examples
- Individual activity (zero index and multiplication index laws maze) building on Lesson 2 with Lesson 3 material The structure front-loaded more complex material, with the visualisation providing a break in a wall of numbers. The activity linked the previous class with the new material.
Strategies
- Multiple Exposures: When teaching the concept of a zero index, I draw upon the multiplication and division laws taught previously. The approach I take in connecting the concepts through specific examples is differentiated, as I varied the depth of my review the previous laws depending on the ability of the student I am interacting with (through questioning or support during seatwork).
- Visualisation: I used a visualisation which provides a multimodal learning experience. GeoGebra specifically has demonstrated increased student engagement. This strategy also supports learners who prefer visuals, as many students struggle with a notation-dominated lesson.
Lesson 4
- Material: index laws for removing brackets, simplifying expressions both simple and complex through the index laws
Structure
- I explicitly teach the index laws for distribution
- I review common mistakes and past index laws, leaving both on the board for students to self-refer
- Collaborative worked examples
- Students attempt differentiated exercises This structure allowed students to consolidate knowledge through the index laws reference being provided in advance of the examples and exercises, which combined multiple index laws.
Strategies
- Worked Examples: I work an example problem on the board, using questioning for engagement, e.g. "What are a and b here?" when expanding (ab)^n. This specifically aids students with mathematics difficulties. I differentiate by providing more or fewer hints when asking questions, allowing lower ability students to experience success.
- Students practiced questions from the textbook, printed out in a booklet format. The use of exercises in mathematics is a key strategy for developing fluency and confidence. An iterative approach which alternates practice with conceptual explanations has been shown to be most beneficial and has been employed in the booklet design, suiting it well for classroom use.
Lesson 5
- Material: negative indices
Structure
- Quick whole-class discussion of what negative indices could mean, exposing potential misconceptions
- I explain the relevant rules and work examples
- Students use the worked examples to practice similar problems
- Revision and consolidation of current and previous material through Blooket activity This structure introduced new material first but later focused on review, bearing in mind that students had likely forgotten material over a long weekend.
Strategies
- The Blooket activity: engages all students while allowing me to provide additional support to particular students. I also chose the answers to the Blooket questions to target common misconceptions (i.e. if they apply the wrong law, another option seems correct). To support all students, I wrote all the index laws, for their easy reference.
- The Blooket operates as a revision of concepts at a midway point in the chapter, which is especially important because of a time gap between lessons due to a public holiday and the previous class being an exam. Digitized quizzing is a valuable tool which allows for consolidation and reinforces retrieval of learned information. The competitive and timed elements excite students and raise engagement.