Dragon Races

The 479th Annual Dragon Race is today on the Island of Barnacle. The race is out to Otter Island and back. The three fastest dragons, Manic, Crater, and Comet, have been invited to compete.

After `sf text(1)/sf text(3)` of an hour Manic is seen `sf text(1)/sf text(4)` of the way to Otter Island. Crater is seen turning around at Otter Island after 45 minutes. Comet is seen returning from Otter Island halfway back to the Island of Barnacle after 1`sf text(1)/sf text(3)` hours.

Based on the different sightings of the dragons, who do you predict will return to the finish line first? Provide mathematical evidence to support your prediction.

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Instructional

Mathematical Practices:

MP.1

MP.1

Make sense of problems and persevere in solving them.

MP.2

MP.2

Reason abstractly and quantitatively.

MP.3

MP.3

Construct viable arguments and critique the reasoning of others.

MP.4

MP.4

Model with mathematics.

MP.5

MP.5

Use appropriate tools strategically.

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More Accessible Version

The 479th Annual Dragon Race is today on the Island of Barnacle. The three fastest dragons have been invited to compete. The race is out to Otter Island and back.

After `sf text(1)/sf text(3)` of an hour Manic is seen `sf text (1)/sf text(4)` of the way to Otter Island.

After `sf text(3)/sf text(4)` of an hour Crater is seen turning around at Otter Island.

After 1`sf text(1)/sf text(2)` hours Comet is seen returning from Otter Island halfway back to the Island of Barnacle.

Based on the different sightings of the dragons, who do you predict will return home first? Provide mathematical evidence to support your prediction.

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More Challenging Version

The 479th Annual Dragon Race is today on the Island of Barnacle. The race is out to Otter Island and back. The three fastest dragons have been invited to compete. Based on the dragons measurements, provided below, which would you predict to be the fastest? Why?

Manic is 20 feet tall, with a wingspan of 40 feet and weighs 720 pounds. Crater is 45 feet tall, with a wingspan of 90 feet and weighs 3,780 pounds. Comet is 8 feet tall, with a wingspan of 16 feet and weighs 352 pounds.

Once the race is underway, officials report out the following information.

After `sf text (1)/sf text(3)` of an hour Manic is seen `sf text (3)/sf text (7)` of the way to Otter Island. Crater is seen turning around at Otter Island after 45 minutes. Comet is finally seen halfway back to the Island of Barnacle after 1`sf text (1)/sf text (9)` hours.

Based on the different sightings of the dragons, who do you predict will return home first? How does this compare to your original prediction? Provide mathematical evidence to support your thinking.

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Plan

Underlying Mathematical Concepts

Possible Problem-Solving Strategies

Formal Mathematical Language and Symbolic Notation

Teacher Notes:

Possible questions to pose when launching this task:

What units can be used to represent speed?
How can you simplify a complex fraction where a fraction is in the numerator and in the denominator?
What does the dragon race look like? Is there a model you can use to show what’s happening during the race?

Possible misconceptions that students may encounter with this task:

Students may misinterpret the meaning of fractional quantities in relation to the whole they decide to use i.e., `sf text(1)/sf text(2)` way to Otter Island is `sf text (1)/sf text(4)` of the race.

Suggested materials

Engagement Image:

Teachers may project the image below to launch this task for their students, define nouns, promote discussion, access prior knowledge, and inspire engagement and problem solving.

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Possible Solutions

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Task Solution

Crater will return to the finish line first based on where each dragon was seen and assuming they continue the race at that same rate.

Students may decide to use the distance from the Island of Barnacle to Otter Island as the unit or the whole race as the unit.

Where each dragon was seen:

Manic was seen `sf text (1)/sf text (4)` of the way to Otter Island which is `sf text(1)/sf text (8)` of the race.
Crater was seen at Otter Island which is `sf text (1)/sf text (2)` of the race.
Comet was seen returning from Otter Island halfway back to the Island of Barnacle which is `sf text (3)/sf text (4)` of the race.

Find and Compare Unit Rate (based on whole race distance as the unit)

Manic:

`sf (1/8)/(1/3)` = `sf text(1)/sf text(8)` ÷ `sf text(1)/sf text(3)` = `sf text(1)/sf text(8)` x 3 = `sf text (3)/sf text(8)` of the race per hour
Manic’s speed is less than `sf text (4)/sf text (8)` or `sf text (1)/sf text(2)`.

Crater: 45 minutes = `sf text (3)/sf text (4)` of an hour

`sf (1/2)/(3/4)` = `sf text (1)/sf text (2)` ÷ `sf text (3)/sf text (4)` = `sf text (1)/sf text (2)` x `sf text (4)/sf text (3)` = `sf text (4)/sf text (6)` of the race per hour
Crater’s speed is greater than `sf text (3)/sf text (6)` or `sf text (1)/sf text (2)` by `sf text (1)/sf text (6)` (or 0.1666...) which means Crater is faster than Manic.

Comet:

`sf (6/8)/(4/3)` = `sf text (6)/sf text (8)` ÷ `sf text (4)/sf text (3)` = `sf text (6)/sf text (8)` x `sf text (3)/sf text (4)` = `sf text (18)/sf text (32)` of the race per hour
Comet’s speed is greater than `sf text (16)/sf text (32)` or `sf text (1)/sf text (2)` by `sf text (2)/sf text (32)` or `sf text (1)/sf text (16)` (or 0.0625).

Comet is also faster than Manic, but slower than Crater, since Comet is a smaller amount greater than `sf text (1)/sf text (2)`.

Crater is traveling the fastest and will return to the finish line first.

Equivalent Ratios (time : portion of the race)

Manic:

Therefore, it will take Manic 1 hour to complete `sf text (3)/sf text (8)` of the race. This is less than half of the race.

Crater:

Therefore, it will take Crater 1 hour to complete `sf text (1)/sf text (2)` of the race. `sf text (2)/sf text (3)` is more than `sf text (1)/sf text (2)` of the race, so Crater is faster than Manic.

Comet:

Therefore, it will take Comet 1 hour to complete `sf text (9)/sf text (16)` of the race. `sf text (9)/sf text (16)` is less than `sf text (2)/sf text (3)` of the race, so Crater is faster than Comet.

Ratio Table

Students may look for and use relationships in their table to compare the dragons’ rates. In each table, the highlighted information is provided in the task. Every cell does not need to be filled in to compare the rates of the dragons.

Time (minutes)	Distance into the Race
Time (minutes)	Manic	Crater	Comet
20	`sf (1/8)`
40	`sf (2/8)`
45		`sf (4/8)`
80	`sf (4/8)`		`sf (3/4)`

Crater completes `sf text (1)/sf text (2)` of the race in less time than Manic’s so Crater is faster Manic. Comet flies farther than Manic in 80 minutes so Comet is faster than Manic.

Distance to Otter Island	Manic's Time (hours)	Crater's Time (hours)	Comet's Time (hours)
`sf (1/4)`	`sf (1/3)`
`sf (1/2)`	`sf (2/3)`	`sf (3/8)`
1	`sf (4/3)` = 1`sf(1/3)`	`sf (3/4)`
1`sf(1/2)`	2	`sf (3/8)` + `sf (3/4)` or `sf (3/4)` ⋅ 1`sf(1/2)` = 1`sf(1/8)`	1`sf(1/3)`

Based on the last row in the table, it will take Crater the least amount of time to go the same distance as Manic and Comet.

Double Number Line

Compare each dragon’s time to the portion of the race they’ve completed. Any of the common points after the start of the race can be used to compare the rates of the dragons.

The number lines can also be used to provide a visual model to help find the time it takes each dragon to complete the whole race:

Manic: `sf text(1)/sf text(3)` x 8 = `sf text (8)/sf text(3)` = 2`sf text(2)/sf text(3)` hours

Crater: `sf text (3)/sf text(4)` x 2 = `sf text (6)/sf text(4)` = 1`sf text(1)/sf text(2)` hours

Comet: 1`sf text(1)/sf text(3)` + (`sf text (1)/sf text(3)` x `sf (1 1/3)`) = 1`sf text(7)/sf text(9)` hours

2`sf text(2)/sf text(3)` > 1`sf(1/2)` < 1`sf text(7)/sf text(9)` Crater needs the least amount of time to finish the race and will return to the finish line first.

Graph

The beginning of the race can be plotted at (0, 0) for each dragon and connected with a line to the point in the race they were seen.

The graph shows that Crater is moving at a faster rate and will return to the finish line first.