Unit 1 · Essential SkillsE3 · Rates · Ground Ops

Rates, Speed & Work

Connect distance, rate, and time, including average speed and combined work.

29 min
Unlocks: Launch Crane
Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

Unit rate: A rate with denominator 1.
Average speed: Total distance divided by total time.
Work rate: The fraction of one job completed per unit time.

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Try it first

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Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

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The big idea

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See the structure

Every rate carries units. In $d=rt$ , the units of $r$ must turn the time unit into the distance unit.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

Average speed is total distance divided by total time. Do not average two speeds unless the corresponding times are equal.

Explore in Desmos

Check distance from rate and time

A train travels 45 miles per hour for 4 hours. Read 180 miles on line 1, then use line 2 to recover 45 miles per hour. The units explain why multiplication answers the distance question.

The calculator is live: change anything and see what happens.

Check distance from rate and time

Degree mode

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Stretch to a 36

For a fixed job, rates add: if one worker completes $1/a$ job per hour and another completes $1/b$ , together they complete $1/a+1/b$ job per hour.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Use units to choose multiplication

Problem

A train travels 58 miles per hour for 3.5 hours. How far does it travel?

Use

d=rt

because hours cancel:

(58\text{ mi/hr})(3.5\text{ hr})

Which rate relationship keeps the units consistent?

58(3.5)=203

miles.

Which calculation recovers the stated distance, time, or rate?

Worked example

Example 2: Weight average speed by time

Problem

A driver goes 60 miles in 1 hour, then 90 miles in 2 hours. What is the trip average speed?

Total distance is

60+90=150

miles; total time is

1+2=3

hours.

Which rate relationship keeps the units consistent?

150/3=50

mph.

Which calculation recovers the stated distance, time, or rate?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

How many hours do machines with individual completion times of 6 hours and 3 hours need when working together?

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Combine work rates before finding time

For machines that finish one job in 6 hours and 3 hours, type 1/6 and 1/3. These are job-per-hour rates, not completion times to average.

Type 1/6+1/3 and read

1/2

job per hour. Then type 1/(1/6+1/3) and read 2 hours for one complete job.

Combine work rates before finding time

Degree mode

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.