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Teaching video
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Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Use a repeated retention factor to model proportional decay.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A model signal retains 80% of its strength each stage. Starting from 125 units, what remains after three stages?
Why this math matters
Model attenuation or repeated discounting while naming the stage and the fraction retained. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Model a constant fraction remaining
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A model signal retains 80% of its strength each stage. Starting from 125 units, what remains after three stages?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Retention factor = 0.80
A twenty-percent loss leaves eighty percent, not twenty percent.
Develop the calculation
S(n)=125(0.8)ⁿ
Each stage scales what remains from the previous stage.
Check and interpret
S(3)=125·0.512=64 units
The loss per stage becomes smaller in absolute size as the signal shrinks.
The result
S(3)=125·0.512=64 units
The loss per stage becomes smaller in absolute size as the signal shrinks.
Common mistakes to catch
- Loss rate and retention factor are complements.
- Do not subtract the original loss amount at every stage of proportional decay.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A model halves a 96-unit amount every cycle. What remains after five cycles?
Show a hint
Apply the half factor five times.
Reveal answer and explanation
3 units
96(1/2)⁵=96/32=3.
Practice 2
Does repeated twenty-percent loss reach exactly zero after five stages?
Show a hint
Every retention factor remains positive.
Reveal answer and explanation
No
125(0.8)⁵ is positive; the ideal sequence approaches zero without reaching it at a finite stage.
Take the idea with you
Model attenuation or repeated discounting while naming the stage and the fraction retained.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Combine polynomials by matching powers
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