Lesson 1 of 4
Build a linear rule from a starting amount and constant rate.
One idea at a time. Try the example, then keep going.
Big ideas · 1 of 8
Big ideas
Do this: Read one idea, then explain it in your own words.
A model needs a named input and output. Write their units before deciding how numbers fit the rule.
A linear model uses a constant rate m. Increasing the input by h changes the output by mh, even when the starting output is negative.
The value b is the output at input zero. It is a starting amount only when input zero is meaningful in the situation.
A model can have a restricted or discrete domain. State what the model assumes and do not extend it beyond useful conditions without checking.
Step-through
Do this: Step through the example. Predict the next step before revealing it.
Worked example: cost with a start fee
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.
Fractional negative kit counts do not belong to the model domain.
Do this: Cover the next step and try it yourself.
Do this: Explain why the tempting shortcut fails.
Wrong idea: a per-item charge is the intercept. The intercept is what remains when the item count is zero. A charge applied once belongs there; a charge for every item multiplies the count.
Worked example
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Worked example
Do this: Predict the next step on paper, then compare the worked reasoning.
Substitution gives intercept 5, hence .
Watch
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Ready when you are.
Activity 1 of 7 · 0 of 7 activities done
Flashcards
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Card 1 of 8
0 Know it0 Still learning
Its units divide output units by input units.
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What is the slope in ?
m, the output change per input unit.
What is the intercept in ?
b, the output at .
What signals a linear pattern in an equally spaced table?
Constant differences in outputs.
A 9-dollar start fee plus 4 dollars per item gives which rule?
.
What is the meaning of a negative slope?
The output decreases as the input increases.
Why might a count model use only whole inputs?
A count of whole objects cannot include fractions or negative counts.
How do you find when a falling-volume model empties?
Set its volume output to zero and solve for time.
Does a linear formula prove the real process stays linear forever?
No; constant change is an assumption that needs a valid interval.
True or false
Do this: Choose true or false; explain what would make a false statement correct.
Self-test
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6 questions. One at a time.
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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...
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Write it
Do this: Write a mathematical argument, then compare its reasoning with the model.
Start here if it helps: My argument starts with...
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Match
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Connect the ideas
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Homework · 3 points
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Homework
Worth 3 points.
https://gradebridge-beryl.vercel.app/learn/precalculus/linear-functions/linear-models#homework
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