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.
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 outA common 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.
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.
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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.
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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.
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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