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You did these with your group. Now do them on your own — the model above is yours to copy.

1. 1

   Same stepsFinish what the group started. Use the model above, step for step.

   Try it together — then prove it: The smallest rear gear has 12 teeth, and the pedal gear still turns 48 teeth per rotation. One full rotation of the 12-tooth gear takes 12 teeth. 48 ÷ 12 = 4, so the rear wheel makes 4 rotations for each pedal rotation. This gear makes it easier to pedal at faster speeds or go downhill. Now prove it: say why that move had to work at all — not just that it did.

   Show your work
2. 2

   Warm restartIn the Tower of Hanoi, what is the fewest moves needed for 3 disks?

   1. A7
   2. B6
   3. C8
   4. D3

   How do you know?
3. 3

   Warm restartLook at the pattern 1, 3, 7, 15. What is the rule?

   1. ADouble the term and add 1
   2. BAdd 2 each time
   3. CDouble the term
   4. DAdd the two terms before it

   How do you know?
4. 4

   Warm restartWhat is the next term in 1, 3, 7, 15, ___?

   1. A31
   2. B23
   3. C30
   4. D17

   How do you know?

You answered these out loud. Now write them.

Answer, then say how you know — the reason is the part that counts.

1. 5

   Think it throughA future vehicle's design uses a motor gear with 40 teeth turning a wheel gear with 10 teeth. Following the gear-ratio math from this lesson, how many wheel rotations happen for each motor gear rotation?

   1. A4 rotations
   2. B0.25 rotations
   3. C30 rotations
   4. D400 rotations

   Why is that the answer?
2. 6

   Think it throughWhy does a complete connect answer need to name a specific piece of math instead of just saying 'it uses math'?

   1. ANaming the actual ratio, measurement, or rate shows the math really shaped the design, not just that math exists somewhere
   2. BBecause grading requires exactly one number in every answer
   3. CBecause the word 'math' cannot be used in a written answer
   4. DBecause a vehicle description never needs any math at all

   How do you know? Give a second reason as well.
3. 7

   Think it throughA student's answer says only, 'My vehicle is powered by a battery.' Which addition would make this a complete connect answer, based on this lesson's check?

   1. ANaming a specific ratio or rate in the design, like a gear ratio between a motor and a wheel
   2. BSaying the vehicle looks very futuristic
   3. CSaying the vehicle is extremely fast
   4. DRepeating that it is powered by a battery a second time

   Explain your thinking. What would have to change for a different choice to be right?
4. 8

   Back to the modelThe Penny Farthing's pedals are attached directly to the front wheel — a 1 to 1 ratio. Why did inventors make the front wheel so large?

Sort it again — on paper this time

Write the letter of the group each one belongs in.

• The rear wheel makes 1.5 rotations per pedal rotation
• The rear wheel makes 4 rotations per pedal rotation
• Makes it easier to pedal at slower speeds or go uphill
• Makes it easier to pedal at faster speeds or go downhill
• Needs to turn 32 teeth to make one full rotation
• Needs to turn 12 teeth to make one full rotation
• The bike travels the shortest distance per pedal rotation
• The bike travels the greatest distance per pedal rotation

Use the words

Fill each blank with a word from the bank.

• Coming up with a different, sometimes unique solution is ___.
• One full turn of a wheel, gear, or pedal is a ___.
• A toothed wheel that passes motion through a chain is a ___.
• A comparison of two quantities, like pedal turns to wheel turns, is a ___.

Explain the thinking

1. 9

   Find the mistakeAnother student made this exact mistake in this lesson. Explain why it is wrong, then write what they should have done.

   The mistake: Thinking a rear gear with MORE teeth makes the wheel spin more — it is the opposite: the 32-tooth gear gives only 1.
2. 10

   Say moreWhen you sorted facts about the 32-tooth gear and the 12-tooth gear, why does the SMALLER gear give MORE wheel rotations per pedal rotation?
3. 11

   Write your ownWrite your own problem about this situation, then solve it and show the answer. The situation: An old photograph shows a Penny Farthing bicycle from the late 1800s — a rider perched high on an enormous front wheel, with a tiny rear wheel trailing behind.

   Show your work

Last check

These two come from Lesson 10.3. If they are shaky, that is the lesson to revisit — not this one.

1. 12

   Show what you knowExplain your thinking — Which statement best captures what this lesson means by 'Math is Ingenuity'?

   1. AMath can help us come up with creative solutions to everyday problems
   2. BMath is only useful for inventors who lived in the 1800s
   3. CIngenuity means avoiding math and just guessing
   4. DEvery bicycle must have a 1 to 1 pedal ratio

   Explain your choice.
2. 13

   From Lesson 10.3Explain your thinking — Which statement best shows what this lesson means by 'math is playful'?

   1. AWe play with math by imagining possible solutions and looking for patterns that help us think logically about puzzles
   2. BMath is playful only when there are no rules to follow
   3. CPuzzles like the Tower of Hanoi can only be solved by trial and error
   4. DPlaying with math means guessing until something works

   How do you know?
3. 14

   From Lesson 10.3Continuing the pattern from 15 steps for a 4-disc tower, how many steps does a 5-disc tower take?

   1. A16
   2. B30
   3. C31
   4. D45

   How do you know?

Be honest — this tells your teacher what to do next