Students combine the index laws one step at a time, check that the bases match before joining powers, and use the laws with powers of 10 (AC9M8N02, node 2).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min
Do the Run board yourself first. Students need the index laws (previous lesson).
The one idea: apply one law at a time (powers of powers, then multiply and divide left to right), and only for powers of the same base. The common mistake is joining different bases (Lia's mistake).
Adding powers (2³ + 2³) is not covered by the laws: a good discussion.
Wider: dividing in scientific notation; a systematic order for long expressions.
Checking the bases first. Ask students to circle each base before they use any law.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student adds indices for different bases. | Circle the bases: are they the same? | Only the same base combines. |
| A student adds when there is a power of a power. | How many groups, of how many factors? | Powers of powers multiply the indices. |
| A student treats 3 as 3⁰. | How many 3s is 3? | 3 = 3¹. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Multiply (4 × 10⁵) × (5 × 10³) and give the answer in scientific notation (careful: 4 × 5 = 20).
Support: Write each expression out as factors for the first two problems.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Using the index laws". The idea: Applying exponent laws in combination: I can Students apply the exponent laws individually and in combination to simplify numerical expressions, checking that bases are the same before applying. What understanding looks like: - Correctly applies each exponent law individually. - Applies laws in combination without confusing different-base expressions. - Does not apply exponent laws when bases differ. Mistakes students often make here: - A student adds indices for different bases. - A student adds when there is a power of a power. - A student treats 3 as 3⁰. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - If I seem to be missing the idea that comes before this one (Establishing the exponent laws from expanded form), check it quickly, then come back. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Using the index laws".
Simplify (5³)² × 5⁴ ÷ 5⁸.
5⁶ × 5⁴ ÷ 5⁸ = 5¹⁰ ÷ 5⁸ = 5².
This completes AC9M8N02.
Where it leads: scientific notation in science, negative indices and algebraic powers (Year 9).