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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: Discs: 1, 2, 3 — Steps: 1, 3, 7. What is happening from each row to the next? Double the number of steps and add 1: 1 doubled plus 1 is 3, and 3 doubled plus 1 is 7. So 4 discs take 7 × 2 + 1 = 15 steps, and 5 discs take 15 × 2 + 1 = 31 steps. The pattern just solved a puzzle we never touched. Now prove it: say why that move had to work at all — not just that it did.

   Show your work
2. 2

   Warm restartHow many lines of symmetry does a square have?

   1. A4
   2. B2
   3. C1
   4. D8

   How do you know?
3. 3

   Warm restartA shape has bilateral symmetry. This means…

   1. Aone line splits it into two mirror halves
   2. Bit has four equal sides
   3. Cit can be turned in a circle
   4. Dit has no straight edges

   How do you know?
4. 4

   Warm restartWhich capital letter has a vertical line of symmetry?

   1. AA
   2. BR
   3. CP
   4. DF

   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 throughThe table shows: 1 disc = 1 step, 2 discs = 3 steps, 3 discs = 7 steps. What is the pattern rule from one row to the next?

   1. AAdd 2 to the previous number of steps
   2. BDouble the previous number of steps and add 1
   3. CMultiply the previous number of steps by 3
   4. DAdd the disc number to the previous steps

   Why is that the answer?
2. 6

   Think it throughUsing the rule "double the steps and add 1," how many steps does a 4-disc tower take, starting from the 3-disc value of 7 steps?

   1. A8
   2. B14
   3. C15
   4. D21

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

   Think it throughContinuing 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

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

   Back to the modelA one-disc Tower of Hanoi takes 1 move, a two-disc tower takes 3 moves, and a three-disc tower takes 7 moves. What pattern rule takes you from one row to the next?

Sort it again — on paper this time

Write the letter of the group each one belongs in.

• Move the top disc of a stack to an empty rod
• Place a small disc on top of a larger disc
• Move one disc at a time
• Move the smallest disc back to the rod it came from earlier
• Place a larger disc on top of a smaller disc
• Move two discs together in one move
• Pull out a disc from the middle of a stack
• Add a fourth rod to make the puzzle easier

Use the words

Fill each blank with a word from the bank.

• A problem designed to be played with — rules, a goal, room to imagine — is a ___.
• Objects or shapes repeated in a regular order form a ___.
• The rule for getting the next value from the one before is the ___ ___.
• To say what will happen before it happens, using a pattern, is to ___.

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: Doubling the previous number of steps but forgetting to add 1 — answering 14 instead of 15 for four discs, or 30 instead of 31 for five.
2. 10

   Say moreWhen you sorted moves into 'legal move' and 'breaks a rule,' what made 'placing a smaller disc on top of a larger disc' legal but the reverse illegal?
3. 11

   Write your ownWrite your own problem about this situation, then solve it and show the answer. The situation: Three rods stand on a wooden base. On the left rod, a stack of discs is arranged with the smallest on top and the largest on the bottom.

   Show your work

Last check

These two come from Lesson 10.2. If they are shaky, that is the lesson to revisit — not this one.

1. 12

   Show what you knowExplain 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

   Explain your choice.
2. 13

   From Lesson 10.2Explain your thinking — Which statement best shows what this lesson means by 'math is beauty'?

   1. AMath creates beautiful designs like bilateral symmetry, found in nature and used in human art, music, and architecture
   2. BOnly things made by mathematicians can be beautiful
   3. CButterflies are the only living things with symmetry
   4. DBeauty and math are opposites — one is feeling and one is numbers

   How do you know?
3. 14

   From Lesson 10.2Your friend says only one of two flowers has bilateral symmetry. You test both by drawing a line through each flower's center and find both sides match on both flowers. What is most likely true?

   1. AThe friend is right — only one flower truly has symmetry
   2. BBoth flowers have bilateral symmetry, but the second flower's line of symmetry likely runs in a direction the friend didn't try
   3. CSymmetry cannot be tested in flowers
   4. DThe two flowers must be identical in shape to both be symmetric

   How do you know?

Be honest — this tells your teacher what to do next