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Turning a story into $y = mx + b$, and saying what each number means in it.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you build a linear function from a description or from two measurements, and then say what its two numbers mean in the situation. The slope is a rate, such as dollars per hour or inches per minute, and the intercept is what was there at the start. Saying that in the words of the problem, not just computing it, is what separates modeling from arithmetic.
You know the form $y = mx + b$: $m$ is the rate of change (the slope) and $b$ is the initial value (the $y$-intercept). You can find a rate from two points by dividing the change in $y$ by the change in $x$, and you can tell whether a table or story has a constant rate. In this lesson you go the other way: you start from a real situation, described in words or by a few measurements, build the equation yourself, and then explain what each number in it means.
| Term | What it means |
|---|---|
| Model | An equation that describes a real situation, so you can calculate with it and make predictions. |
| Input and output | The quantity you choose or know (such as hours, $x$) and the quantity that depends on it (such as cost, $y$). |
| Rate of change (slope) | How much the output changes for each one unit of input, in units such as dollars per hour or miles per gallon. |
| Initial value (intercept) | The output when the input is $0$: a starting amount, a fixed fee or a flat charge. |
| Interpret | Explain what a number means in the words and units of the situation. |
| Prediction | An output worked out from the model for an input you have not measured. |
Many situations share one shape: something starts at a fixed amount and then changes by the same amount for every unit of something else. A plumber charges a trip fee and then a price per hour. A bathtub holds some water and then fills at a steady number of gallons per minute. Any situation like that is a linear model:
$$\text{output} = (\text{rate}) \times (\text{input}) + (\text{starting amount}), \qquad y = mx + b.$$
Building the model means finding the two numbers, $m$ and $b$. Interpreting it means saying what they mean in the story, with units:
For example, renting a bike costs $\$8$ plus $\$3$ an hour. The model is $C = 3h + 8$. The slope $3$ means the cost goes up $\$3$ for each extra hour. The intercept $8$ means you pay $\$8$ even before riding. Saying it in those words, not just "the slope is $3$", is what turns arithmetic into modeling.
Another way: picture
On a graph, the intercept is where the line starts on the vertical axis, and the slope is how much the line climbs for each step to the right. For the bike, the line starts at $8$ and climbs $3$ for every hour.
Another way: table
Make a table for the bike: $0$ hours costs $\$8$, $1$ hour $\$11$, $2$ hours $\$14$, $3$ hours $\$17$. The first entry is the intercept, and the equal jumps of $3$ are the slope.
Words give the numbers away. Words like per, each, every and an (as in "$\$12$ an hour") signal a rate: the amount is paid or added again for each unit, so it multiplies the input. Words like fee, flat, to start, up front, already and initially signal the initial value: it happens once, so it is added (or subtracted) on its own.
Read this: "A school club has $\$45$ in its account and sells bake-sale cookies at $\$2$ each." The rate is $2$ dollars per cookie, and the start is $45$ dollars. With $n$ cookies sold, the money is $M = 2n + 45$.
Watch for rates that make the output go down. "A $\$25$ gift card, with $\$4$ spent on every visit" gives $G = 25 - 4v$, or $G = -4v + 25$. The slope is $-4$: the card loses $\$4$ per visit. And a one-time amount can be taken away, as with a coupon: "$\$9$ per ticket, $\$5$ off the order" is $C = 9t - 5$.
Sometimes a story does not tell you the rate or the start. It gives you two measurements instead. Then you find the rate first and the start second.
Suppose a tree was $14$ feet tall $3$ years after it was planted and $20$ feet tall $7$ years after, growing steadily. The height rose $20 - 14 = 6$ feet in $7 - 3 = 4$ years, so the rate is $6 \div 4 = 1.5$ feet per year. Now step back from year $3$ to year $0$: $14 - 3 \times 1.5 = 9.5$. The tree was $9.5$ feet tall when it was planted, and the model is $H = 1.5y + 9.5$.
Test the model on the measurement you did not use: $1.5 \times 7 + 9.5 = 20$. It matches. The same method works when the measurements come from a table or from two points read off a graph.
You can step back from either measurement; you will get the same starting value both times, because the rate is the same all along the line. From year $7$: $20 - 7 \times 1.5 = 9.5$ again. Pick the one with the smaller input, since it needs fewer steps and less arithmetic. And do not be surprised by a rate that is a decimal or a fraction, like $1.5$ feet per year: real rates are rarely whole numbers, and the model works just the same.
A good interpretation names the quantity, the amount and the units. Two sentence frames help:
For $H = 1.5y + 9.5$: for each extra year, the tree grows $1.5$ feet; when it was planted, it was $9.5$ feet tall.
Sometimes the intercept does not make sense in the story. If a model links a person's shoe size to their height using adults' data, the height at shoe size $0$ is not a real person. The model is only trusted over the inputs it was built from, and a little beyond. It also stops working when the situation changes: a draining pool cannot have a negative depth, so the model for it ends when the pool is empty.
Why each move is allowed. A constant rate means every unit of input changes the output by the same amount, so the total change is the rate times the number of units. Adding that to the starting amount gives the output. Stepping back to $0$ subtracts the rate once per unit, which undoes the growth.
How to check. Put each given measurement back into your model: every one must come out right. Check the units: the slope must be "output per input". Check the sign: a cost that grows needs a positive slope. And ask whether your prediction is sensible: a two-hour repair should not cost less than the trip fee.
Crickets chirp faster when it is warm. In 1897 the physicist Amos Dolbear published a rule for snowy tree crickets: $T = 50 + \frac{N - 40}{4}$, where $N$ is the number of chirps in a minute and $T$ is the temperature in degrees Fahrenheit. Simplified, it is a linear model:
$$T = \frac{1}{4}N + 40.$$
The slope $\frac{1}{4}$ means each extra chirp per minute signals a quarter of a degree warmer, or $1$ degree for every $4$ chirps. If you count $120$ chirps in a minute, $T = 30 + 40 = 70\,^{\circ}$F. The intercept $40$ would be the temperature at $0$ chirps, but that part does not make sense: crickets stop chirping in the cold long before that, so the model only works on warm evenings.
The air usually gets colder as you go up a mountain, by about $3.5\,^{\circ}$F for every $1000$ feet of height. If it is $70\,^{\circ}$F at sea level, a model for the temperature $h$ thousand feet up is
$$T = 70 - 3.5h.$$
The slope is $-3.5$ degrees per thousand feet: negative, because climbing makes it colder. The intercept $70$ is the sea-level temperature. Denver, the "Mile High City", is about $5.3$ thousand feet up: $70 - 3.5 \times 5.3 \approx 51\,^{\circ}$F. The top of Mount Whitney in California, about $14.5$ thousand feet up, would be around $70 - 50.75 \approx 19\,^{\circ}$F, which is why hikers there pack warm clothes even in summer. Real weather varies, so this is an estimate, not a forecast.
Many electric companies charge a fixed monthly customer charge plus a price for each kilowatt-hour (kWh) of electricity used. Suppose a family paid $\$74$ in a month when they used $400$ kWh and $\$116$ in a month when they used $700$ kWh. The rate is $(116 - 74) \div (700 - 400) = 42 \div 300 = 0.14$ dollars per kWh: $14$ cents. Stepping back to $0$ kWh gives $74 - 400 \times 0.14 = 18$ dollars. So the bill is $B = 0.14k + 18$.
Now the family can predict: running a space heater that adds $150$ kWh a month would add $150 \times 0.14 = \$21$ to the bill, while the $\$18$ charge is paid no matter what.
Swapping the slope and the intercept. In "$\$3$ to start and $\$2$ a mile", the $2$ multiplies the miles. $F = 3x + 2$ is wrong.
Using a measurement as the intercept. If the data start at year $3$, the value at year $3$ is not $b$. Step back to $0$.
Forgetting to divide by the change in input. A change of $\$195$ over $3$ hours is $\$65$ an hour, not $\$195$.
Losing the minus sign. Something that drains, cools or is spent has a negative slope.
Interpreting without units. "The slope is $65$" is not an interpretation. "The cost rises $\$65$ for each hour" is.
A bowling alley charges $\$4$ for shoe rental plus $\$6$ for each game. Name the input and the output.
$g = \text{games}, \qquad C = \text{cost in dollars}$
The cost depends on how many games you bowl.
Find the rate.
$m = 6 \text{ dollars per game}$
"For each game" is paid again for every game.
Find the initial value.
$b = 4 \text{ dollars}$
Shoe rental is paid once, even before the first game.
Write the model.
$C = 6g + 4$
Rate times input, plus the starting amount.
Use it for $3$ games.
$C = 6 \times 3 + 4 = 22 \text{ dollars}$
Three games at $\$6$ is $\$18$, and the shoes add $\$4$.
A plumber's $2$-hour job cost $\$190$ and a $5$-hour job cost $\$385$. Find the change in cost.
$385 - 190 = 195$
The extra hours are why the longer job cost more.
Find the change in hours.
$5 - 2 = 3$
Subtract in the same order as the costs.
Divide to find the hourly rate.
$m = 195 \div 3 = 65 \text{ dollars per hour}$
The rate is the change in cost for each hour.
Step back from $2$ hours to $0$ hours.
$b = 190 - 2 \times 65 = 60$
Taking away two hours of work leaves the fixed fee.
Write the model and interpret it.
$C = 65h + 60$
The plumber charges a $\$60$ trip fee plus $\$65$ for each hour of work.
Check with the $5$-hour job.
$65 \times 5 + 60 = 325 + 60 = 385$
The model reproduces the measurement it was not built from.
A phone shows $90\%$ battery after $1$ hour of gaming and $66\%$ after $4$ hours, draining steadily. Find the change in charge.
$66 - 90 = -24$
The charge went down, so the change is negative.
Divide by the change in hours.
$m = \dfrac{-24}{4 - 1} = -8$
The phone loses $8$ percentage points per hour.
Step back from $1$ hour to $0$ hours.
$b = 90 + 1 \times 8 = 98$
Going back in time adds back the charge used.
Write the model.
$B = 98 - 8t$
Start at $98\%$ and lose $8$ points each hour.
Interpret the slope and the intercept.
$-8 \text{ points per hour}; \quad 98\% \text{ at the start}$
The slope is negative because gaming drains the battery.
Set up the question: when will it show $18\%$?
$98 - 8t = 18$
The model's output is the battery level, so set it equal to $18$.
Solve for $t$.
$8t = 80 \quad\Rightarrow\quad t = 10$
Subtract $18$ and add $8t$ to both sides, then divide by $8$.
Check that the answer makes sense.
$98 - 8 \times 10 = 18; \qquad 10 < 98 \div 8 \approx 12.25$
It matches, and it comes before the battery would reach $0\%$, where the model ends.
Find the rate.
$m = 30 \text{ dollars per month}$
"A month" means paid every month.
Find the initial value.
$b = 25 \text{ dollars}$
The joining fee is paid once.
Write the model.
Find the cost of $6$ months.
A pool is being drained. The depth of the water is $d = 89 - 9t$ inches after $t$ minutes. By how many inches does the depth change each minute? (Use a negative number for a decrease.)
The depth changes by answer inches each minute.
A bean plant is $22$ mm tall on day $2$ and $31$ mm tall on day $5$, growing steadily. Complete the worked solution to model its height and predict it on day $7$.
Divide the change in height by the change in days.
$\dfrac{31 - 22}{5 - 2} =$ r mm per day
A steady growth rate is the slope of the model.
Step back from day $2$ to day $0$.
$b =$ s mm
Two days back takes off two days of growth.
Use the model $h = md + b$ for day $7$.
$h =$ t mm
Put the day number in for $d$.
Jada saves the same amount every week. The money in her account after $w$ weeks is $A = 11w + 130$ dollars. How much money was in the account when she started saving?
She started with answer dollars.
A taxi charges $5$ dollars as soon as you get in, plus $2$ dollars for each mile. Write the fare $F$, in dollars, for a ride of $x$ miles. Type the expression for $F$.
Answer:
An electrician charges a fixed fee for coming out plus the same amount for every hour of work. A $1$-hour job cost $132$ dollars and a $4$-hour job cost $366$ dollars. Write the cost $C$ in dollars for a job of $h$ hours.
C = mh + b
A team store sells T-shirts for $15$ dollars each. A coupon takes $7$ dollars off the whole order. Fill in the model for the total cost $C$ of $n$ shirts.
C = mn - b
A tablet's battery drains at a steady rate while it plays video. It shows $73\%$ after $3$ hours and $37\%$ after $7$ hours. After how many hours of video will it show $28\%$?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
An electrician charges a fixed fee for coming out plus the same amount for every hour of work. A $1$-hour job cost $146$ dollars and a $3$-hour job cost $322$ dollars. Write the cost $C$ in dollars for a job of $h$ hours.
C = mh + b
You can build a linear model from a story and interpret its slope and intercept. Without looking: for a taxi fare of 3 dollars plus 2 dollars a mile, which number is the slope and what are its units?
16. Your turn: a gym costs $\$25$ to join plus $\$30$ a month. Model the cost of $n$ months., step 3
$C = 30n + 25$
Rate times months plus the fee.
16. Your turn: a gym costs $\$25$ to join plus $\$30$ a month. Model the cost of $n$ months., step 4
$C = 30 \times 6 + 25 = 205 \text{ dollars}$
Put $n = 6$ into the model.