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Chapter 13 · Period 1 of 6 · 45 min

Unpacking the Magic: How Algebra Explains Number Tricks

Algebra Play (Introduction) · Inductive

Watch for students who substitute only one number to 'verify' a trick and consider it proved — the key insight is that the algebra must hold for any starting value, not just the one they chose. A quick counter: ask them to test with a fraction or a large number, then show that the variable expression makes the general case obvious. Math Talk, section 6.2 p.137 ("Can you change the steps in this trick and still find the original date?") is a productive self-study pointer for students who want to push further.

0–8Pose the classic five-step 'Think of a Number' trick to the class: think of a number, add 5, double it, subtract 4, halve it, subtract the original number. Students perform the steps mentally and share their result. Collect several responses to surface the surprise that everyone lands on the same answer — algebra as a frame has not yet entered; the trick speaks for itself first.
8–22Introduce the variable x for the unknown number and trace each of the five steps as an algebraic expression on the board, arriving at the identity x + 2 − x = 2. Students follow the substitution in their notebooks. Then present the date-guessing trick from section 6.2 (100M + 165 + D) and walk through how multiplying and adding encodes two values. Figure it Out, section 6.2 p.137 ("Find the dates if the final answers are the following: (i) 1269 (ii) 394 (iii) 296…") is attempted in pairs: students subtract 165 and read off the month and day.
22–35Class discussion: students share decoded dates and verify them. Shift to Figure it Out, section 6.2 p.136 ("How would you change this game to make the final answer 3? What about 5?") — students adjust constants in the original five-step trick and test whether their modified version works by substituting two or three different starting numbers.
35–45Whole-class share-out: one pair presents a modified trick; others check the algebra on their own page. Close by naming the two ideas established — a trick collapses to a constant when variable terms cancel, and an encoding trick embeds two unknowns into place-value position — laying the ground for inventing original tricks.

Homework

Can you come up with more complicated steps that always lead to the same final value? (**Math Talk, section 6.2 p.136**)

Assessment

Figure it Out, section 6.2 p.136 — How would you change this game to make the final answer 3? What about 5?

In a 'Think of a Number' trick, Priya follows these steps: think of a number, add 7, multiply by 2, subtract 4, divide by 2, subtract the original number. What is the final result, regardless of the starting number?

  • A 14
  • B 5
  • C 7
  • D The original number

B. 5

What each wrong option reveals

  • A The student doubled the 7 (getting 14) but forgot to apply the subsequent subtraction and division steps.
  • C The student stopped after the 'add 7' step and did not continue simplifying through all the operations.
  • D The student believes the trick returns the original number, confusing a constant-output trick with an identity function.
Chapter 13 · 6 periods of 45 min

Also saved as 4 periods.

  • 2Inventing Your Own Algebraic Number Trick
    Thinking about 'Think of a Number' Tricks
  • 3Building and Generalising Number Pyramids
    Number Pyramids
  • 4Pyramids, Calendars, and Algebra Grids in Action
    Number Pyramids
  • 5Proving Optimisation and Divisibility Rules Algebraically
    The Largest Product
  • 6Algebra at Work: Four Problems, Four Methods
    Synthesis
Ganita Prakash · Class 8 · 14 chapters

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