Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.4

Lesson 10.410.4 Small Group · Group 1

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target With my small group, I can use ratios and models to explain how the Penny Farthing and gear-driven bicycles were ingenious solutions to a real problem — one step at a time, with support.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

Word	What it means	Example
ingenuitySpanish: ingenio	Coming up with different, and sometimes unique, solutions to problems.	The Penny Farthing's huge wheel, then chains and gears — two clever fixes.
rotationSpanish: rotación	One full turn of a wheel, gear, or pedal all the way around.	One full turn of the pedal — 360° all the way around.
gearSpanish: engranaje	A toothed wheel that passes motion to another toothed wheel through a chain.	A 48-tooth pedal gear pulling a 12-tooth wheel gear through the chain.
ratioSpanish: razón	A comparison of two quantities, like pedal rotations to wheel rotations.	48 pedal teeth to 12 wheel teeth — 4 wheel turns for every pedal turn.
equivalent ratiosSpanish: razones equivalentes	Ratios that make the same comparison using different numbers.	1 : 4 and 3 : 12 describe the same gear, at different sizes.

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

A bike's pedal gear has 48 teeth. The chain is on a rear gear with 32 teeth. How many rotations (full turns) does the rear wheel make for each pedal rotation?

Write the rear gear: 32 teeth make 1 turn: 32 : 1
Divide both by 2 to find 16 teeth: 1 ÷ 2 = 0.5
Multiply both by 3 to reach 48 teeth: 0.5 × 3 = 1.5
Check with division: 48 ÷ 32 = 1.5
The rear wheel makes 1.5 rotations for each pedal rotation. This gear makes it easier to pedal slowly or uphill.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

The chain moves to the smallest rear gear, with 12 teeth. The pedal gear still has 48 teeth. How many rotations does the rear wheel make for each pedal rotation?

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

A bigger wheel travels with each pedal rotation because .
A smaller wheel would travel .

Word bank ingenuityrotationgearratioequivalent ratios

Watch outA common mistake

Thinking a rear gear with MORE teeth makes the wheel spin more — it is the opposite: the 32-tooth gear gives only 1.5 rotations per pedal rotation, while the 12-tooth gear gives 4.

Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.4

Lesson 10.410.4 Small Group · Group 1

Group 1Supported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

1Complete each step, then write the final answer.
20 teeth make one wheel turn.
A bike's pedal gear has 40 teeth. The rear gear has 20 teeth. How many times does the rear wheel turn for each pedal turn?
1. 1How many teeth make one wheel turn?
2. 2One pedal turn moves 40 teeth. What scale factor takes 20 to 40?
3. 3Do the same to the wheel turns. How many turns for one pedal turn?
Final answer:
2Complete each step, then write the final answer.
Use 8 teeth, half a turn, as a bridge.
The chain moves to a rear gear with 16 teeth. The pedal gear still has 40 teeth. How many times does the rear wheel turn for each pedal turn?
1. 1How many teeth make one wheel turn?
2. 2Divide both by 2. How many wheel turns go with 8 teeth?
3. 3What do you multiply 8 teeth by to reach 40?
4. 4Do the same to the wheel turns. How many turns for one pedal turn?
Final answer:
3Complete each step, then write the final answer.

The pedal gear has 30 teeth. The rear gear has 10 teeth. How many times does the rear wheel turn for each pedal turn?
Show your work
Hint Put 10 teeth and 1 turn in the first column. What do you multiply 10 by to get 30?
Final answer:
4Complete each step, then write the final answer.

A racing bike has a 54-tooth pedal gear and a 36-tooth rear gear. How many times does the rear wheel turn for each pedal turn?
Show your work
Hint Put 36 teeth and 1 turn in the first column. Find the turns for 18 teeth first. Then scale up to 54.
Final answer:
5Complete each step, then write the final answer.

Each pedal turn moves 42 teeth.
Leo's bike has a 42-tooth pedal gear and a 14-tooth rear gear. He pedals 5 full turns. How many times does the rear wheel turn?
Show your work
Hint Put 14 teeth and 1 turn in the first column. Scale to 42 teeth for one pedal turn. Then multiply by 5.
Final answer:
6Circle the letter of the best answer. Show how you know.
The 32-tooth gear. Make one for the 10-tooth gear too.
The pedal gear has 40 teeth. A rider must climb a steep hill. Which rear gear makes pedaling easier: one with 32 teeth or one with 10 teeth?
1. AThe 32-tooth gear
2. BThe 10-tooth gear
3. CThey are the same
Hint Make a ratio table for each gear. Scale to 40 teeth. Fewer wheel turns per pedal turn are easier on a hill.
Show your work

Explain your thinkingWhy do we divide the pedal teeth by the rear teeth? Use a 40-tooth pedal gear and a 20-tooth rear gear.

Sentence starter One pedal turn moves ___ teeth on the rear gear. · The scale factor from ___ teeth to ___ teeth is the wheel turns.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help