Math is Playful
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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🎯 Content Objective / Objetivo de contenido
I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
A 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?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Unit 10 · Lesson 3
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.
Concept Launch
💡 Playing with math means playing with patterns
The objective is to move the whole stack of discs to another rod. The rules: move only one disc at a time, move only the top disc from a stack, and never place a larger disc on top of a smaller disc.
Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now you try
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| puzzle rompecabezas |
A problem designed to be played with — it has rules, a goal, and room to imagine solutions. Un problema diseñado para jugarse: tiene reglas, una meta y espacio para imaginar soluciones. |
The Tower of Hanoi is a puzzle that was invented in 1883. | Tower of Hanoi: three rods, a stack of discs, and two rules. |
| pattern patrón |
Something that repeats or changes in a predictable way. Algo que se repite o cambia de una manera predecible. |
1 disc takes 1 move, 2 discs take 3, 3 discs take 7 — the moves follow a pattern. | Circle, square, triangle, circle, square, triangle … |
| pattern rule regla del patrón |
A rule that tells you how to get the next value in a pattern from the one before it. Una regla que te dice cómo obtener el siguiente valor de un patrón a partir del anterior. |
Double the number of steps and add 1 to find the steps for a tower with one more disc. | 1 → 3 → 7 → 15: the rule is “double it and add 1”. |
| predict predecir |
To say what will happen before it happens, using a pattern or reasoning. Decir lo que va a pasar antes de que pase, usando un patrón o el razonamiento. |
The table let us predict that a five-disc tower takes 31 steps — without moving a single disc. | The rule says a five-disc tower needs 31 steps — before moving a disc. |
| organize organizar |
To arrange information — often in a table — so patterns become easier to see. Ordenar la información, muchas veces en una tabla, para que los patrones sean más fáciles de ver. |
We can create a table of discs and steps to help us identify any patterns. | Discs 1, 2, 3 in one column and steps 1, 3, 7 in the next. |
Which Word Fits?
A problem designed to be played with — rules, a goal, room to imagine — is a ___.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
A 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?
👂 Listen For
A strong answer states 'double and add 1' and checks it against both jumps (1→3 and 3→7). A weak answer only notices the numbers are growing without naming the rule.
Extend: Using the pattern rule, predict the number of moves for a four-disc tower — and justify why you don't need to actually play it out to know your answer is right.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
The rules are the whole game: one disc at a time, only the top disc of a stack, and a larger disc can never sit on a smaller disc. After sorting, tell a partner which rule is hardest to obey when you actually play.
✍️ Explore Discourse
Take 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?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
When 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?
👂 Listen For
Listen for students naming the size-order rule specifically — a larger disc may never sit on a smaller one — not just 'that one felt wrong.'
Extend: Someone claims 'moving two discs together in one move' should be allowed because it would make the puzzle faster. Argue against that claim using the actual rules.
Practice Check A
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?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
Use the pattern rule to complete the table: how many steps for a four-disc tower?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Net Fold Explorer
Complete the interactive activity using today's strategy.
✍️ Justify Your Thinking
Sort each move: is it allowed by the Tower of Hanoi rules, or against them?
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
No worked steps provided for this lesson.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes" — and it works because ___.
Because puzzle means ___, but a tricky part is ___, so I have to ___.
I can justify my method by showing ___, which proves my answer makes sense.
I can prove my answer is correct by ___, using pattern to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes because ___
Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes but ___
Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
The rules are the whole game: one disc at a time, only the top disc of a stack, and a larger disc can never sit on a smaller disc. After sorting, tell a partner which rule is hardest to obey when you actually play.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
What is the fewest number of moves needed to solve a three-disc Tower of Hanoi puzzle?
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?
A five-disc tower takes 31 steps. Using the pattern rule, how many steps would a SIX-disc tower take?
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
You solved the small towers by hand: 1 disc takes 1 step, 2 discs take 3 steps, 3 discs take 7 steps. Now someone hands you a five-disc tower and asks for the fewest steps — but you do not have time to play it out.
✍️ Connection Reasoning
How can a pattern help you imagine the solution?
The pattern rule for the Tower of Hanoi is to ___ the number of steps for the smaller tower and ___ for each new disc. Applying this rule twice starting from 7 steps for a three-disc tower gives a five-disc tower a minimum of ___ steps.
Turn & Talk — Connect
You know a three-disc tower takes 7 steps. Without playing it, how can the pattern rule tell you the minimum steps for a five-disc tower?
👂 Listen For
A strong answer shows both steps (7 → 15 → 31) and explains that a pattern rule predicts values you never directly tested. A weak answer just states 31 without showing the doubling steps.
Extend: Will the number of steps for a Tower of Hanoi puzzle always be odd, no matter how many discs? Justify your answer using the pattern rule.
Summarize: Math is Playful
Algorithmic Thinking & Recursive Patterns. 1. Deconstruct complex multi-step problems into smaller, repeatable sub-tasks. 2. Trace patterns from simple cases to formulate general mathematical rules. 3. Verify algorithmic rules across edge cases and increasing problem sizes
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
Which statement best shows what this lesson means by 'math is playful'?
Bonus Exit Check
What is the minimum number of steps to solve a FIVE-disc Tower of Hanoi puzzle?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• A strong answer states 'double and add 1' and checks it against both jumps (1→3 and 3→7). A weak answer only notices the numbers are growing without naming the rule.
• Listen for students naming the size-order rule specifically — a larger disc may never sit on a smaller one — not just 'that one felt wrong.'
• A strong answer shows both steps (7 → 15 → 31) and explains that a pattern rule predicts values you never directly tested. A weak answer just states 31 without showing the doubling steps.
• Listen for students naming a specific strategy — like building a table — tied to spotting the pattern, not just "I figured it out."
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.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: Double the number of steps and add 1 — 1 × 2 + 1 = 3 and 3 × 2 + 1 = 7, so each new disc doubles the steps and adds 1.
✓ Practice 2: 15 steps — A three-disc tower takes 7 steps, so a four-disc tower takes 7 × 2 + 1 = 15 steps.
✓ Practice 3: 31 steps — A four-disc tower takes 15 steps, so a five-disc tower takes 15 × 2 + 1 = 31 steps.
✓ Practice 4: The pattern predicts the answer for towers you have never played, like five or six discs — The pattern told us a five-disc tower needs a minimum of 31 steps — without moving a single disc.
✓ Exit ticket: We play with math by imagining possible solutions and looking for patterns that help us think logically about puzzles — The summary names both moves: imagining different ways to visualize the problem, and looking for patterns or relationships that help us think logically.