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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.

   Let's do one together: 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? How many teeth make one wheel turn? (12) One pedal turn moves 48 teeth. What scale factor takes 12 to 48? (× 4) Do the same to the wheel turns. How many turns for one pedal turn? (4) Is this gear better for going fast or for climbing a hill? (For going fast or downhill. The wheel turns 4 times for each pedal turn.)

   My first step is ___ , because the problem asks for ___ .

   Show your work
2. 2

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

   1. A3
   2. B6
   3. C7
   4. D8
3. 3

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

   1. AAdd 2 each time
   2. BDouble the term
   3. CAdd the two terms before it
   4. DDouble the term and add 1
4. 4

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

   1. A17
   2. B23
   3. C30
   4. D31

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. A0.25 rotations
   2. B4 rotations
   3. C30 rotations
   4. D400 rotations

   Why is that the answer?

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

   How do you know?

   I chose ___ because ___ .
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. ASaying the vehicle looks very futuristic
   2. BSaying the vehicle is extremely fast
   3. CRepeating that it is powered by a battery a second time
   4. DNaming a specific ratio or rate in the design, like a gear ratio between a motor and a wheel

   Explain your thinking.

   I chose ___ because ___ .
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?

   The front wheel is large because it ___ the distance traveled per pedal rotation.

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?

   The 12-tooth gear gives more rotations because ___.
3. 11

   Change one numberGo back to Part 1 and pick one problem you already solved. Change one number in it, then solve your new version. Show every step.

   If ___ changed to ___ , then ___ .

   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 knowQuick check — you've got this: Which statement best captures what this lesson means by 'Math is Ingenuity'?

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

   Explain your choice.

   I know it is ___ because ___ .
2. 13

   From Lesson 10.3Quick check — you've got this: 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
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

Be honest — this tells your teacher what to do next