Lesson 3 of 5
Model proportional change over equal input intervals.
Subject to changeOptional future study, outside the current course sequence. This lesson and its homework preview are unassigned and may change.
One idea at a time. Try the example, then keep going.
Worked example · 1 of 8
Worked example
Do this: Try this on paper, then compare with the worked check.
Start with 40 units. Increase by 50% once. Check: the added amount is 20 and the retained multiplier is , so the new amount is 60.
Big ideas
Do this: Read one idea, then explain it in your own words.
An exponential model multiplies by the same positive factor b each equal input step. For a nonconstant model, , and a is the output at .
Growth by decimal rate r uses . Decay by rate r uses . An 8% increase uses 1.08, while an 8% decrease uses 0.92.
The exponent counts periods. If a quantity doubles every 3 hours, use when t is measured in hours.
A constant factor is a modeling assumption, not a promise. Name units, the useful input domain and any required rounding before interpreting a prediction.
Worked example
Do this: Cover each worked step and try it on paper before reading on.
A fictional material begins at 96 units and retains each hour. After two hours, units remain. Check in steps: 96 becomes 72, then 54. Use this only over the modeled interval.
Step-through
Do this: Step through the example. Predict the next step before revealing it.
Worked example: repeated growth
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Do this: Cover the next step and try it yourself.
This assumes the same proportional rate each day.
Do this: Cover the next step and try it yourself.
Do this: Explain why the tempting shortcut fails.
Wrong idea: a 10% increase twice means multiplying by 1.20. Successive increases compound: . The second percentage uses the changed amount.
Watch
Do this: Watch the public lesson, pause to try a step, then return to your practice.
The linked video is free to watch. The creator may offer optional paid resources; they are not needed for this lesson. If YouTube is blocked, use the lesson notes and worked examples here.
Ready when you are.
Activity 1 of 7 · 0 of 7 activities done
Flashcards
Do this: Say the answer and a reason before you flip.
Card 1 of 8
0 Know it0 Still learning
Add the decimal rate to 1.
On the card: Enter or Space flips. Left and Right move. After revealing, 1 marks Still learning and 2 marks Know it. Tab reaches every control.
What factor means 6% growth per period?
1.06.
What factor means 15% decay per period?
0.85.
What is in ?
a.
How do exponential and linear patterns differ?
Exponential patterns have constant ratios; linear patterns have constant differences over equal steps.
What exponent counts five-hour periods when t is in hours?
.
For , what does describe?
Decay as the input increases.
What is the factor in outputs 5, 15, 45, 135 at equally spaced inputs?
3.
Why state a model interval?
A constant proportional pattern may not continue outside justified conditions.
True or false
Do this: Choose true or false; explain what would make a false statement correct.
Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Self-test
Do this: Try each problem without a timer. Read the feedback before moving on.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Write two or three sentences, then compare with the model.
Start here if it helps: I would first identify...
Your answer is not saved anywhere. It disappears when you leave the page.
Write it
Do this: Work on paper, explain your reasoning, then compare with the model. This practice does not award homework points.
Start here if it helps: I can connect the conditions by...
Your answer is not saved anywhere. It disappears when you leave the page.
Match
Do this: Choose a word, then choose its meaning.
Connect the ideas
0 / 4 connectedChoose a term, then its meaning. Or drag its grip. Use Tab and Enter or Space; Escape clears.
Subject to changeOptional preview. This page does not mean this homework is assigned. Only turn it in if your teacher asks.
Optional practice preview · 3 points if assigned
Opening your homework…
Practice it: You did 0 of 7 activities.
You decide when you have finished. This is your study progress, not a grade.
Not marked finished yet.
