Lesson 10.3Math is Playful
Start hereWords, worked example, and sentence starters
Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions.
Why This Matters The Tower of Hanoi is a puzzle invented in 1883. It has three rods and a stack of discs, smallest on top to largest on the bottom. The number of discs can vary, but the number of rods does not — and hiding inside it is one of the prettiest patterns in mathematics.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| puzzleSpanish: rompecabezas | A problem designed to be played with — it has rules, a goal, and room to imagine solutions. | Tower of Hanoi: three rods, a stack of discs, and two rules. |
| patternSpanish: patrón | Something that repeats or changes in a predictable way. | Circle, square, triangle, circle, square, triangle … |
| pattern ruleSpanish: regla del patrón | A rule that tells you how to get the next value in a pattern from the one before it. | 1 → 3 → 7 → 15: the rule is “double it and add 1”. |
| predictSpanish: predecir | To say what will happen before it happens, using a pattern or reasoning. | We counted towers of 1, 2 and 3 discs. Reach PAST them: the rule says 5 discs takes 31 steps — before moving one. |
| organizeSpanish: organizar | To arrange information — often in a table — so patterns become easier to see. | Discs 1, 2, 3 in one column and steps 1, 3, 7 in the next. |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Before moving anything, I describe what the solved puzzle will look like: the discs in the same order — smallest on top to largest on bottom — but on a different rod.
- Then I play the small cases: a one-disc tower takes 1 move, and a two-disc tower takes 3 moves.
- A three-disc tower takes at least 7 moves. Small cases are how mathematicians play.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Discs: 1, 2, 3 — Steps: 1, 3, 7. What is happening from each row to the next?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- The rule is , so is legal / breaks it because .
- The number of never changes.
- There are always .
- Being playful with math means .
Word bank puzzlepatternpattern rulepredictorganize
Watch out!A common mistake
Doubling the previous number of steps but forgetting to add 1
Lesson 10.3Math is Playful
Version ASupported practice
Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions.
Why This Matters The Tower of Hanoi is a puzzle invented in 1883. It has three rods and a stack of discs, smallest on top to largest on the bottom. The number of discs can vary, but the number of rods does not — and hiding inside it is one of the prettiest patterns in mathematics.
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1Circle the letter of the best answer. Show how you know.
What is the fewest number of moves needed to solve a three-disc Tower of Hanoi puzzle?
- A3 moves
- B6 moves
- C7 moves
- D9 moves
Hint It is more than one move per disc.
Show your work -
2Circle the letter of the best answer. Show how you know.
Which move is against the rules of the Tower of Hanoi?
- AMoving one disc at a time
- BMoving the top disc from a stack
- CPlacing a larger disc on top of a smaller disc
- DPlacing a smaller disc on top of a larger disc
Hint Two of these are the rules themselves.
Show your work -
3Circle the letter of the best answer. Show how you know.
What will the solved puzzle look like?
- AThe discs in the same order — smallest on top to largest on bottom — but on a different rod
- BThe discs in reverse order, largest on top
- CThe discs spread out with one disc on each rod
- DThe discs back on the starting rod in any order
Hint Imagine the ending before you start.
Show your work -
4Circle the letter of the best answer. Show how you know.
How many moves does a one-disc tower take? A two-disc tower?
- A1 move and 3 moves
- B1 move and 2 moves
- C1 move and 4 moves
- D2 moves and 4 moves
Hint Play the one-disc game in your head first.
Show your work
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5Answer in complete sentences.
The number of discs in a Tower of Hanoi puzzle can vary. What NEVER varies?
Sentence starter The number of ___ never changes.
There are always ___ .Write your answer -
6Answer in complete sentences.
What does it mean to be playful with math? Use the Tower of Hanoi as your example.
Sentence starter Being playful with math means ___ .
With the Tower of Hanoi, I ___ .Write your answer
Explain your thinkingTake a move you called legal and your partner called a rule-break. What is the rule in your own words, and does it still hold on a move you both agree about?
Sentence starter The rule is ___, so ___ is legal / breaks it because ___.
Lesson 10.3Math is Playful
Version BCore practice
Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions.
Why This Matters The Tower of Hanoi is a puzzle invented in 1883. It has three rods and a stack of discs, smallest on top to largest on the bottom. The number of discs can vary, but the number of rods does not — and hiding inside it is one of the prettiest patterns in mathematics.
-
1Circle the letter of the best answer. Show how you know.
The table shows 1 disc → 1 step, 2 discs → 3 steps, 3 discs → 7 steps. What is the pattern rule for the number of steps when you add one more disc?
- AAdd 2 to the number of steps
- BAdd the number of discs to the steps
- CMultiply the number of steps by 3
- DDouble the number of steps and add 1
Show your work -
2Circle the letter of the best answer. Show how you know.
Use the pattern rule to complete the table: how many steps for a four-disc tower?
- A9 steps
- B14 steps
- C15 steps
- D28 steps
Show your work -
3Circle the letter of the best answer. Show how you know.
What is the minimum number of steps to solve a FIVE-disc Tower of Hanoi puzzle?
- A15 steps
- B30 steps
- C31 steps
- D35 steps
Show your work -
4Circle the letter of the best answer. Show how you know.
Why is finding a pattern MORE powerful than just solving the three-disc puzzle by hand?
- AThe pattern lets you skip the rules of the puzzle
- BThe pattern predicts the answer for towers you have never played, like five or six discs
- CPatterns make the discs move by themselves
- DA pattern proves the puzzle is impossible
Show your work
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5Answer in complete sentences.
How can you ORGANIZE the information about discs and moves to help you see patterns?
Write your answer -
6Answer in complete sentences.
Before solving, the lesson asked you to DESCRIBE what the solution will look like. Why is imagining the ending a useful way to start a puzzle?
Write your answer
Explain your thinkingTake a move you called legal and your partner called a rule-break. What is the rule in your own words, and does it still hold on a move you both agree about?
Lesson 10.3Math is Playful
ChallengeExtension
Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions.
Why This Matters The Tower of Hanoi is a puzzle invented in 1883. It has three rods and a stack of discs, smallest on top to largest on the bottom. The number of discs can vary, but the number of rods does not — and hiding inside it is one of the prettiest patterns in mathematics.
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1Write the letter of the matching item on each line.
Match each tower to the calculation that gives its fewest moves.
- 4-disc tower
- 5-disc tower
- 6-disc tower
- Any n-disc tower
- A2 × 7 + 1 = 15
- B2 × 15 + 1 = 31
- C2 × 31 + 1 = 63
- DDouble the (n − 1)-disc answer, add 1
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2Circle the letter of the best answer. Show how you know.
A five-disc tower takes 31 steps. Using the pattern rule, how many steps would a SIX-disc tower take?
- A32 steps
- B36 steps
- C62 steps
- D63 steps
Show your work -
3Circle the letter of the best answer. Show how you know.
Look at the table a different way: 1 disc → 1, 2 discs → 3, 3 discs → 7, 4 discs → 15, 5 discs → 31. Which describes the relationship between the number of discs and the number of steps?
- AThe steps are the number of discs multiplied by itself
- BThe steps are 6 times the number of discs, plus or minus 1
- CThe steps are the two previous step counts added together
- DThe steps are 1 less than a doubling pattern: 2, 4, 8, 16, 32 each minus 1
Show your work
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4Answer in complete sentences.
The 15 Puzzle is a sliding puzzle with 15 tiles in a frame 4 tiles high by 4 tiles wide. You arrange the tiles from 1 to 15 by sliding one tile at a time into the open space. Sometimes it is solvable and sometimes it is not. How would you make an argument about whether a particular 15 Puzzle is solvable, using what this lesson taught about being playful?
Write your answer -
5Answer in complete sentences.
The summary says playing with math lets us think differently by looking for patterns AND by imagining ways to visualize a problem. Describe how you used BOTH of those with the Tower of Hanoi.
Write your answer