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

Lesson 10.410.4 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can use ratios and models to explain how the Penny Farthing and gear-driven bicycles were ingenious solutions to a real problem, explain why the method works, and use it on a problem I have not seen before.

Start hereWords for this lesson

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

WordWhat it meansExample
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 designer wants the rear wheel to make 2.4 rotations for each turn of the 48-tooth pedal gear. How many teeth must the rear gear have?

  1. Write one pedal turn: 48 teeth and 2.4 turns: 48 : 2.4
  2. Multiply both by 5 to make whole turns: 48 × 5 = 240; 2.4 × 5 = 12
  3. Divide both by 12 to reach 1 turn: 240 ÷ 12 = 20
  4. Check by multiplying: 20 × 2.4 = 48
  5. The rear gear must have 20 teeth.

Now you trySame steps, your turn

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

The smallest rear gear has 12 teeth. The pedal gear has 48 teeth. How many rotations does the rear wheel make for each pedal rotation? Compare it with the 32-tooth gear.

Answer:

Say it and write itSentence starters

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

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 2

Group 2Challenge practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can use ratios and models to explain how the Penny Farthing and gear-driven bicycles were ingenious solutions to a real problem, explain why the method works, and use it on a problem I have not seen before.

  1. 1Complete each step, then write the final answer.
    Teeth turned42??Wheel turns3.571× ?× ?÷ ?÷ ?
    One pedal turn: 42 teeth and 3.5 wheel turns.

    A designer wants the rear wheel to turn 3.5 times for each turn of a 42-tooth pedal gear. How many teeth must the rear gear have?

    1. 1How many teeth go with 3.5 wheel turns?
    2. 2Multiply both by 2. How many teeth go with 7 turns?
    3. 3What do you divide 7 turns by to get 1 turn?
    4. 4Do the same to the teeth. How many teeth must the rear gear have?
    Final answer:
  2. 2Complete each step, then write the final answer.
    Teeth turned15?Wheel turns12.4× ?× ?
    One wheel turn uses 15 teeth.

    A rear gear has 15 teeth. A designer wants the rear wheel to turn 2.4 times for each pedal turn. How many teeth must the pedal gear have?

    1. 1How many teeth make one wheel turn?
    2. 2What do you multiply 1 turn by to get 2.4 turns?
    3. 3Do the same to the teeth. How many teeth must the pedal gear have?
    4. 4Why do you multiply here, but divide in the last problem?
    Final answer:
  3. 3Complete each step, then write the final answer.
    Teeth turnedWheel turns________________________
    Each pedal turn moves 50 teeth.

    A pedal gear has 50 teeth. The rear gear has 20 teeth. The rider makes 8 pedal turns. How many times does the rear wheel turn?

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.
    Teeth turnedWheel turns________________________

    A pedal gear has 52 teeth. A designer wants the rear wheel to turn 3.25 times for each pedal turn. How many teeth must the rear gear have?

    Show your work
    Final answer:
  5. 5Complete each step, then write the final answer.
    Teeth turnedWheel turnsMeters________________________
    One wheel turn: 11 teeth and 2 meters.

    Ana's pedal gear has 44 teeth. Her rear gear has 11 teeth. Each turn of the rear wheel moves her 2 meters. How far does she go in 10 pedal turns?

    Show your work
    Final answer:
  6. 6Complete each step, then write the final answer.
    Teeth turnedWheel turns________________________
    Rico's Gear B. Make one for Gear A too.

    The pedal gear has 40 teeth. Gear A has 25 teeth. Gear B has 16 teeth. Rico says Gear B is easier for a hill. How many wheel turns per pedal turn does the easier gear give?

    Show your work
    Final answer:

Explain your thinkingA shop says a smaller rear gear always makes a bike better. When is a small rear gear better, and when is a big one better?

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

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