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Toll Bridge Puzzle


Alignments to Content Standards: 2.NBT.B.6

Task

The picture shows islands connected by bridges. To cross a bridge, you must pay a toll in coins. If you start on the island marked in blue with 100 coins, how can you make it to the island marked in red?

Toll_puzzle_a446ec6c368fc0871cd00b21e6f9bca9

IM Commentary

This task is intended to assess adding of four numbers as given in the standard while still being placed in a problem-solving context.

As written this task is instructional; due to the random aspect regarding when the correct route is found, it is not appropriate for assessment.

This puzzle works well as a physical re-enactment, with paper plates marking the islands and strings with papers attached for the tolls.

Students will often be tempted by the single digit numbers to assume the route has to pass that way. They may also miss the "crossover" using the central island, finding the 23 + 25 + 29 + 24 route and the 32 + 15 + 40 + 38 route but not the 23 + 25 + 40 + 38 route or the 32 + 15 + 29 + 24 one.

Technically, paths that run from right to left along some bridges could be considered as well (for example, 32 + 5 + 41 + 40 + 29 + 24). However, for this particular example, such paths can be ignored since students will find a cheap enough path by only investigating paths that run from left to right. One might be tempted to modify the problem by having the student start with 99 coins, and ask if it is possible to reach the red island. However, such a modification would be inappropriate since a mathematically valid solution would require that the student consider all paths, including those that cross some bridges from right to left.

Solution

Toll_puzzle_solve_dd7761e28935782d3b6dd23402a7ed01

32 + 15 + 29 + 24 = 100

Other possible routes:

23 + 51 + 3 + 24 = 101
23 + 25 + 29 + 24 = 101
23 + 25 + 40 + 38 = 126
32 + 15 + 40 + 38 = 116
32 + 5 + 41 + 38 = 116

Laura says:

almost 3 years

After working the toll bridge puzzle, as given, my students enjoyed creating their own puzzles and having their classmates solve them. To differentiate, students who could handle more of a challenge, used 3 digit numbers and started with 200 or 500 coins to go from start to finish. This activity was highly engaging!

Teresa says:

over 5 years

This activity also lends itself to reasoning about the number 100. What does 100 look like compared to these paths? 100 is four 25s or two 50s. Can you see paths that would equal about those amounts?