Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.7

Lesson 10.310.3 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 10.3. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
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.

A tower took 127 steps to solve. For each new disc, the rule is “double the steps, then add 1.” How many discs were in the tower?

  1. Undo the rule: subtract 1, then halve: (127 − 1) ÷ 2 = 63
  2. Keep undoing the rule: (63 − 1) ÷ 2 = 31; (31 − 1) ÷ 2 = 15
  3. Stop when you reach 1 step: (15 − 1) ÷ 2 = 7; (7 − 1) ÷ 2 = 3; (3 − 1) ÷ 2 = 1
  4. Count the numbers in the list: 1, 3, 7, 15, 31, 63, 127 → 7 numbers
  5. The tower had 7 discs.

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. Find the pattern rule and the steps for 4 and 5 discs. Then explain why the rule works.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

Word bank puzzlepatternpattern rulepredictorganize

Watch out!A common mistake

Doubling the previous number of steps but forgetting to add 1

Grade 6 Mathematics · Unit 10: Math Is...Standard MPP.7

Lesson 10.310.3 Small Group · Group 2

Group 2Challenge practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can use patterns and relationships to make sense of the Tower of Hanoi puzzle and think logically about its solutions, explain why the method works, and use it on a problem I have not seen before.

Why This Matters This is your challenge small group for Lesson 10.3. You can already get the answer — so in here the answer is the starting point. We ask when the method holds, when it breaks, and how you would convince someone who disagrees.

  1. 1Complete each step, then write the final answer.
    DiscsMoves
    11
    23
    ?255
    Undo the rule until you reach 1 move.

    A tower took 255 moves to solve. For each new disc, the rule is “double the moves, then add 1.” Undo the rule to find how many discs the tower had.

    1. 1Undo the rule once: subtract 1, then halve. What do you get?
    2. 2Keep undoing the rule until you reach 1. List the numbers.
    3. 3Count the numbers in your list, from 1 to 255. How many discs?
    Final answer:
  2. 2Complete each step, then write the final answer.
    PositionNumber
    1?
    2?
    3?
    454

    A pattern uses the rule “double, then add 2.” The 4th number is 54. What is the 1st number?

    1. 1Undo the rule: subtract 2, then halve. What is the 3rd number?
    2. 2Undo the rule again. What is the 2nd number?
    3. 3Undo it once more. What is the 1st number?
    4. 4Why do you subtract 2 before you halve?
    Final answer:
  3. 3Complete each step, then write the final answer.

    A 10-disc tower takes 1,023 moves. How many moves does an 11-disc tower take?

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.

    A pattern uses the rule “double, then add 1.” The 4th number is 95. What is the 1st number?

    Show your work
    Final answer:
  5. 5Complete each step, then write the final answer.

    A robot moves 1 disc every 2 seconds. A 4-disc tower takes 15 moves. How many seconds does the robot need for a 5-disc tower?

    Show your work
    Final answer:
  6. 6Complete each step, then write the final answer.

    A 3-disc tower takes 7 moves. Jay says a 6-disc tower takes 7 × 2 = 14 moves, because 6 is double 3. How many moves does a 6-disc tower take?

    Show your work
    Final answer:

Explain your thinkingWill a tower ever take an even number of moves? Use the rule “double, then add 1” to prove your answer.

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help