The Story of Montarek (2-5th Grade) — Doubling, Tripling, Quadrupeling and Adding Five (Exponential Growth vs. Linear)

After a week focusing on exponential growth in the population of the world and most of its countries, I shared an ancient legend with the chlldren about The Story of Montarek, which better emphasizes the power of exponential growth. In the Montarek saga, a peasant girl saves the King’s daughter and is offered a reward. The peasant outlines a plan for a reward and then the King, Queen and Royal Financial Advisor proposes a reward strategy. I let the children make conjectures about which plan would yield the highest reward for the peasant. Many of them took the bait of the Financial Advisor’s plan but others were skeptical as they know how fast numbers grow when multiplied continuously:

Peasant Plan 1: Double the previous square for 64 squares starting with one Ruba (roughly equivalent to one penny); for example, the first square 1, then 2, then 4, then 8 and so on.

King Plan 2:  Triple the previous square for 16 squares starting with one Ruba; for example, the first square 1, then 3, then 9, then 27 and so on.

Queen Plan 3:  Quadrupel the previous square for 12 squares starting with one Ruba; for example, the first square 1, then 4, then 16, then 64 and so on.

Financial Advisor’s Plan 4: Start with 20 Rubas and add 5 on each square for 64 squares; for example, 20, then 25, then 30 and so on.

We created a spread sheet of each plan and then proceeded to estimate the reward payout for each, starting with Plan 4 which would have 335 Rubas on the last square, Plan 3 would have 4 million Rubas on the last square, Plan 4 would have 14 million Rubas on the last square, and Plan 1 would have 8 quintillion Rubas on the last square.

I taught them how to convert these numbers to dollars:

Plan 1: \$80,000,000,000,000,000

Plan 2: \$140,000

Plan 3: \$40,000

Plan 4: \$3.35

We used the power of estimation to develop “back of the envelop” calculations as to the total Rubas for each plan. For Plan 4, we looked at a linear equation of R = 63 x 5 + 20 = 335 and then converted that to \$ 3.35. I embellished a bit by stating that the Financial Advisor was beheaded because the King and Queen felt that he undervalued their daughter. The Queen’s Plan of 4,000,000 Rubas yielded \$ 40,000. The King’s Plan yielded 14,000.000 Rubas or \$140,000. The Queen’s plan yielded 4,000,000 Rubas or \$\$ 40,000. Finally, the Peasant’s plan yielded 80 Quintillion Dolllars.

Then I gave the children worksheets to calculate the actual numbers for each Plan, teaching them a simple method of doubling, tripling and quadrupling, using both repeated addition and the standard algorithm of multiplication. The worksheet had answers along the way so they could check their progress and find mistakes early. I included an answer key as well as an example of the different methods. I also gave the the Story of Montarek to share with their families.

Finally, I told them they might want to “trick” their parents the next time they did a good deed and were offered ice cream or a new game. “All I want is a penny and for 64 squares on a chessboard, double the number on the previous square.” Of course, after a week, you will be on to them and will stop this “nonsense.” Doubling adds up faster than we think.

Have fun.

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