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

Lesson 10.4Math is Ingenuity

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.

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 later bicycle design uses a chain: the pedal gear has 48 teeth, and the largest rear gear has 32 teeth.

  1. For each rotation of the 48-tooth pedal gear, the wheel gear also turns 48 teeth.
  2. The 32-tooth gear needs to turn 32 teeth to make one full rotation, so I find the equivalent ratio: 48 ÷ 32 = 1.5.
  3. The rear wheel makes 1.5 rotations for each pedal rotation — this gear makes it easier to pedal at slower speeds or go uphill.

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, and the pedal gear still turns 48 teeth per rotation.

Answer:

Say it and write itSentence starters

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

  • With the -tooth gear the ratio is , so the wheel .
  • A bigger wheel travels with each pedal rotation because .
  • A smaller wheel would travel .
  • One ingenious solution is .

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 10.4Math is Ingenuity

Version ASupported 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.

Part 1Apply itreal-world problems
  1. 1Circle the letter of the best answer. Show how you know.

    On the Penny Farthing, what is the ratio of pedal rotations to front-wheel rotations?

    1. A1 to 1
    2. B1 to 2
    3. C1 to 4
    4. D2 to 1

    Hint The pedals are attached directly to the front wheel.

    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    How far does the Penny Farthing travel with each pedal rotation?

    1. AThe height of the rider
    2. BThe length of the perimeter of the front wheel
    3. CThe length of the pedal
    4. DIt does not move a fixed distance

    Hint Picture the wheel rolling one full turn on the ground.

    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    Why was the front wheel of the Penny Farthing so large?

    1. ASo the bicycle is easier to stop
    2. BSo the rider sits closer to the ground
    3. CSo the bicycle travels a greater distance with one pedal rotation
    4. DSo the rear wheel spins faster

    Hint Each pedal rotation moves the bike one perimeter of the wheel.

    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    For each rotation of the 48-tooth pedal gear, how many teeth does the wheel gear turn?

    1. A1 tooth
    2. B12 teeth
    3. C32 teeth
    4. D48 teeth

    Hint The chain moves one link for each tooth on the pedal gear.

    Show your work
Part 2Explain and correctreasoning
  1. 5Answer in complete sentences.

    How does the size of the front wheel affect the distance the bicycle travels?

    Sentence starter A bigger wheel travels ___ with each pedal rotation because ___ .
    A smaller wheel would travel ___ .
    Write your answer
  2. 6Answer in complete sentences.

    Name one ingenious solution — from this lesson or from your own life — where math helped solve an everyday problem.

    Sentence starter One ingenious solution is ___ .
    The math behind it is ___ .
    Write your answer

Explain your thinkingYour partner says the bigger rear gear makes the wheel turn more. Is that right, and what stays equivalent in the ratio of teeth when you switch gears?

Sentence starter With the ___-tooth gear the ratio is ___, so the wheel ___.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.4

Lesson 10.4Math is Ingenuity

Version BCore 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.

Part 1Apply itreal-world problems
  1. 1Circle the letter of the best answer. Show how you know.

    The pedal gear turns 48 teeth per rotation. The largest rear gear needs 32 teeth for one full rotation. How many rotations does the rear wheel make for each pedal rotation?

    1. A0.5 rotations
    2. B1.5 rotations
    3. C2 rotations
    4. D16 rotations
    per 1
  2. 2Circle the letter of the best answer. Show how you know.

    The smallest rear gear has 12 teeth. How many rotations will the rear wheel make for each rotation of the 48-tooth pedal gear?

    1. A0.25 rotations
    2. B3 rotations
    3. C4 rotations
    4. D36 rotations
    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    You are about to ride up a steep hill. Which rear gear should the chain be on, and why?

    1. AThe 12-tooth gear, because 4 rotations per pedal is always better
    2. BThe 32-tooth gear, because it makes the bike go faster
    3. CIt does not matter which gear you use on a hill
    4. DThe 32-tooth gear, because 1.5 rotations per pedal makes pedaling easier at slow speeds
    Show your work
  4. 4Circle the letter of the best answer. Show how you know.

    With the chain on the 12-tooth gear, the rider makes 10 pedal rotations. How many rotations does the rear wheel make?

    1. A14 rotations
    2. B22 rotations
    3. C40 rotations
    4. D120 rotations
    Show your work
Part 2Explain and correctreasoning
  1. 5Answer in complete sentences.

    Why do you think the larger gear makes it easier to pedal at slower speeds?

    Write your answer
  2. 6Answer in complete sentences.

    When the chain is on the largest gear the rear wheel makes 1.5 rotations per pedal rotation; on the smallest gear it makes 4. What do you notice about the size of the wheel gears and the distance the bike travels?

    Write your answer

Explain your thinkingYour partner says the bigger rear gear makes the wheel turn more. Is that right, and what stays equivalent in the ratio of teeth when you switch gears?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.4

Lesson 10.4Math is Ingenuity

ChallengeExtension

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.

Part 1Understand the ideamodels and meaning
  1. 1Write the letter of the matching item on each line.

    Distance = rotations × circumference. Match each ride to its distance.

    1. 4 wheel rotations, 7 ft circumference
    2. 3 wheel rotations, 6 ft circumference
    3. 2 wheel rotations, 7 ft circumference
    4. 48-tooth pedal gear with a 12-tooth rear gear
    • A28 feet
    • B18 feet
    • C14 feet
    • D4 rotations per pedal turn
Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    The rider makes 3 full pedal rotations with the chain on the 32-tooth gear. How many rotations does the rear wheel make?

    1. A3 rotations
    2. B4.5 rotations
    3. C6 rotations
    4. D96 rotations
    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    A designer adds a middle gear so the rear wheel makes exactly 2 rotations per rotation of the 48-tooth pedal gear. How many teeth must the new gear have?

    1. A16 teeth
    2. B22 teeth
    3. C24 teeth
    4. D96 teeth
    per 1
Part 3Explain and correctreasoning
  1. 4Answer in complete sentences.

    How would the distance traveled differ with a smaller front wheel on the Penny Farthing? Use the 1 to 1 pedal ratio in your answer.

    Write your answer
  2. 5Answer in complete sentences.

    How did math help the bicycle's designers come up with ingenious solutions — first with the Penny Farthing, then with chains and gears?

    Write your answer

Explain your thinkingYour partner says the bigger rear gear makes the wheel turn more. Is that right, and what stays equivalent in the ratio of teeth when you switch gears?

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