Counting Heads and Feet
How many animals could there be?
Chickens and sheep live in a field next to Fernando’s school. Looking over the fence, Fernando counted 8 heads. Looking under the fence, Fernando counted 22 legs. How many of each animal could there be?
- How many legs do chickens and sheep have?
- How might you represent the relationships in this story?
- Is your solution the only solution? How can you be sure?
- Flamingos are birds that have two legs but often stand on only one. How could your answer change if the animals were flamingos and sheep?
- Can you write a problem like this of your own?
Students consider a combination of two values—the number of heads and the number of feet—and the number of chicken and sheep that will result in the given combination. Students may engage in a range of mathematical skills as they engage in this task, including counting, addition fact practice, and equivalent quantities as they consider exchanges between chickens and sheep. This task also begins to set some foundational understandings of multiplication.
Since chickens have 2 feet and sheep have 4 feet, yet both animals have 1 head, only one possible combination of 8 chickens and sheep has 22 total feet. Students might determine the number of feet found on different combinations (e.g., 1 chicken and 7 sheep, and make adjustments to the combination until there are 22 feet. Or, students might model and record the number of feet found on 1 through 7 chickens and sheep and look for the combination of 8 animals with 22 total feet. Tasks like this one are systems of equations problems often found later in algebra courses. However, they are fun and accessible for elementary students, especially when they have various tools available for their exploration.
Various apps may be used to represent the context of this problem.
- In the Number Frames app, students might use different-colored markers representing chicken and sheep feet in groups of 2 and 4 to fill 22 total spaces (as shown here). These markers are easily deleted, replaced, and rearranged if adjustments need to be made.
- Alternatively, students could use pattern shapes to represent the animals’ heads and feet, grouping them appropriately (as shown here). These representations could be deleted or duplicated to adjust for the target quantities.
- The Number Line app could be used to represent the number of feet as a single jump for a chicken or a sheep (as shown here). These jumps of 2 and 4 could be iterated until there are 22 feet accounted for with a total of 8 jumps.
Students might solve this problem by modeling the context and working with groups of 2 and 4 until they have satisfied the requirements for heads and feet. However, there may be opportunities for students to reason about exchanges in quantity. For example, if there are 22 feet but only 6 heads, ask: How could you keep the same number of feet using more animals? If there are 8 heads but only 18 feet, ask: How could you keep the same number of animals but use more feet? Students may reason about exchanges in the context that keep one quantity the same while adjusting the other. This is foundational algebraic reasoning!