If the cartoons maintain their current speed, the Wicked Witch is expected to make it around the world first and win the race.
Two different unit rates can be used to solve this problem: the distance around the world per hour or time per trip around the world.
Unit Rate - Distance around the world per hour
Peter Pan: `sf text(4)/sf text(7)` of the Distance Around the World Per Hour
(`sf text(2)/sf text(7)` around the world ÷ `sf text(1)/sf text(2)` hour = `sf text(2)/sf text(7)` x 2 = `sf text(4)/sf text(7)`)
Aladdin: `sf text(5)/sf text(8)` of the Distance Around the World Per Hour
(`sf text(1)/sf text(4)` around the world ÷ `sf2/sf5` hour = `sf text(1)/sf text(4)` x `sf text(5)/sf text(2)` = `sf text(5)/sf text(8)`)
Wicked Witch: `sf text(6)/sf text(9)` of the Distance Around the World Per Hour
(`sf text(2)/sf text(9)` around the world ÷ `sf text(1)/sf text(3)` hour = `sf text(2)/sf text(9)` x 3 = `sf text(6)/sf text(9)` = `sf text(2)/sf text(3)`)
The cartoon's rates can all be compared to the benchmark of `sf text(1)/sf text(2)`
Peter Pan: `sf text(1)/sf text(2)` = `sf text(3.5)/sf text(7)` so `sf text(4)/sf text(7)` is half of `sf text(1)/sf text(7)` (or `sf text(1)/sf text(14)`) more than `sf text(1)/sf text(2)`.
Aladdin: `sf text (1)/sf text(2)` = `sf text(4)/sf text(8)` so `sf text(5)/sf text(8)` is `sf text(1)/sf text(8)` more than `sf text(1)/sf text(2)`.
Wicked Witch: `sf text(1)/sf text(2)` = `sf text(1.5)/sf text(3)` so `sf text(2)/sf text(3)` is half of `sf text(1)/sf text(3)` (or `sf text(1)/sf text(6)`) more than `sf text(1)/sf text(2)`.
`sf text(1)/sf text(14)` < `sf text(1)/sf text(8)` < `sf text(1)/sf text(6)` so `sf text(4)/sf text(7)` < `sf text(5)/sf text(8)` < `sf text(6)/sf text(9)`
An open number line can be used to show this relationship:
`sf text(2)/sf text(7)` x 2 = `sf text(4)/sf text(7)` of the distance around the world per hour
Aladdin:
`sf text(1)/sf text(4)` x 2 + `sf text(1)/sf text(8)` = `sf text(5)/sf text(8)` of the distance around the world per hour
Wicked Witch:
`sf text(2)/sf text(9)` x 3 = `sf text(6)/sf text(9)` of the distance around the world per hour
Unit Rate - Time to travel around the world once
It will take Peter Pan 1`sf text(3)/sf text(4)` hours to travel around the world
(`sf text(1)/sf text(2)` hour ÷ `sf text(2)/sf text(7)` around the world = `sf text(1)/sf text(2)` x `sf text(7)/sf text(2)` = `sf text(7)/sf text(4)` = 1`sf text(3)/sf text(4)`)
It will take Aladdin 1`sf text(3)/sf text(5)` hours to travel around the world
(`sf text(2)/sf text(5)` hour ÷ `sf text(1)/sf text(4)` around the world = `sf(2/5)` x 4 = `sf text(8)/sf text(5)` = 1`sf text(3)/sf text(5)`)
It will take the Wicked Witch 1`sf text(1)/sf text(3)` hours to travel around the world
(`sf text(1)/sf text(3)` hour ÷ `sf text(2)/sf text(9)` around the world = `sf text(1)/sf text(3)` x `sf text(9)/sf text(2)` = `sf text(9)/sf text(6)` = 1`sf text(3)/sf text(6)`)
Since 1`sf text(3)/sf text(6)` 1`sf text(3)/sf text(5)` 1`sf text(3)/sf text(4)` it will take the Wicked Witch the least amount of time to finish the race.
Tape Diagram - Can be used to find and represent each unit rate
Peter Pan:
`sf text(1)/sf text(2)` hour x 3 + `sf text(1)/sf text(4)` hour = 1 `sf text(3)/sf text(4)` hours
Aladdin:
`sf text(2)/sf text(5)` hour x 4 + `sf text(8)/sf text(5)` = 1 `sf text(3)/sf text(5)` hours
Wicked Witch:
`sf text(1)/sf text(3)` hour x 4 + `sf text(1)/sf text(6)` hour = 1`sf text(3)/sf text(6)` hours
Ratio Table
After 2 hours, Peter Pan could travel 1`sf text(1)/sf text(7)` times around the world; Aladdin could travel 1`sf text(1)/sf text(4)` times around the world; the Wicked Witch could travel 1`sf text(3)/sf text(9)` times around the world.
1`sf text(1)/sf text(7)` 1`sf text(1)/sf text(4)` 1`sf text(3)/sf text(9)` so the Wicked Witch is the traveling the fastest. A variety of strategies can be used to compare these rates.
Aladdin is traveling faster than Peter Pan because 1`sf text(1)/sf text(4)` > 1`sf text (1)/sf text(7)`.
The Wicked Witch is traveling faster than Aladdin because 1`sf text(3)/sf text(9)` > 1`sf text(1)/sf text(4)`.
Graph
The information given for each character can be graphed as points on a graph as (time, distance) along with the point (0, 0) for each which represents the start of the race.
The two points for each character can be connected with a line. Each character’s speed is represented by the steepness of their line. Since the Wicked Witch’s line is the steepest, she is traveling at the fastest rate.
Scaling
Each characters' time can be scaled up to 1 hour to find how far each character has traveled per hour.
Peter Pan: `sf text(1)/sf text(2)` hour x 2 = 1 hour. `sf text(2)/sf text(7)` of the distance x 2 = `sf text(4)/sf text(7)`.
Aladdin: `sf text(2)/sf text(5)` hour x 2.5 = 1 hour. `sf text(1)/sf text(4)` of the distance x 2.5 = `sf text(2.5)/sf text(4)` or `sf text(5)/sf text(8)`.
Wicked Witch: `sf text(1)/sf text(3)` hour x 3 = 1 hour. `sf text(2)/sf text(9)` of the distance x 3 = `sf text(6)/sf text(9)`.
MP.1
Make sense of problems and persevere in solving them.