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Problem Solving for the 21st Century: Built for the NY NextGen Standards

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
Domain:
Standard:
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.

View all Grade 7 tasks

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

  • Reasoning about ratios and rates
  • Finding unit rates
  • Comparing and operations with rational numbers

Possible Problem-Solving Strategies

  • Find and compare unit rate
  • Equivalent ratios
  • Ratio table
  • Double number line
  • Graph

Formal Mathematical Language and Symbolic Representation

  • Complex fraction
  • Unit rate
  • Quotient
  • Scaling
  • Scale factor
  • Proportional relationship
  • Rate
  • Ratio
  • Equivalent ratio
  • Independent variable
  • Dependent variable
  • Coordinate point
  • Rational number
  • Benchmark
  • Greater than (>)
  • Less than (<)

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.

Click image to enlarge

Task-Specific Evidence

This task requires students to find and compare rates in order to determine which of three dragons will be the first to finish a race between two islands and back.

Printer-Friendly Planning Sheet
Printer-Friendly Exemplars Rubric

Possible Solutions

  • Task Solution (active tab)
  • More Accessible Solution
  • More Challenging Solution
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
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.

OR
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.

More Accessible 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 use a variety of strategies to decide who will win the race.

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 use a variety of strategies to decide who will win the race.

Manic completes `sf text (3)/sf text(8)` of the race per hour.

Crater completes `sf text (2)/sf text(3)` of the race per hour.

Comet completes `sf text (1)/sf text(2)` of the race per hour.

More Challenging Solution

This problem can be solved a number of ways. Answers may vary based on student decisions. Possible solutions may include:

Based on the dragons measurements, students may predict Comet to be the fastest because Comet weighs the least. Alternatively, they may predict Manic will win because he has less weight for every foot of height.

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

Manic completes `sf text (9)/sf text(14)` or 0.643 of the race per hour.

Crater completes `sf text (2)/sf text(3)` or about 0.667 of the race per hour.

Comet completes `sf text (27)/sf text(40)` or 0.675 of the race per hour.

Possible Connections

Below are some examples of mathematical connections. Your students may discover some that are not on this list.

  • A common quantity for either the time or distance the dragons travel allows you to compare their rates.
  • When a dragon flies the full distance to Otter Island, this is equivalent to half of the race.
  • The dragon that reaches Otter Island first is expected to win the race assuming they continue at their current speeds.
  • Crater is in the lead from the beginning of the race, and will hold the lead the entire race.
  • The three dragons will never be at the same location other than at the start and end of the race.
  • Solve more than one way to verify the answer.
  • Relate to a similar task and state a math link.

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