### Mathematical models of a dog’s age

**Directions for teachers:**

Ask students to read the online *Science News* article “Calculating a dog’s age in human years is harder than you think” and answer the questions below. Questions relate to defining, applying and analyzing the new dog age model described in the article. Have students partner up to discuss the last two prompts, which ask them to think about examples of mathematical models in other fields and about those models’ benefits and limitations. Bring the class back together as a group and discuss questions of your choice.

See the Discussion exercise “Go beyond Mendeleev’s model” for additional questions about the principles and purposes of scientific models in all areas of science and engineering.

Want to make it a virtual lesson? Post the online *Science News* article “Calculating a dog’s age in human years is harder than you think” to your virtual classroom. When you’re ready to pair students up, have them discuss and answer the final two prompts using a video-conferencing platform, or talking by phone. They can collaborate in a shared document during the conversation. After posting the answers to an online discussion board, have students give feedback on another pair’s responses.

**Define the model**

1. The article describes a new mathematical model for determining a dog’s age in human years. Use the information from the article to write an equation for the new relationship between a dog’s age in dog years and a dog’s age in equivalent human years. Define the variables used.

*As defined in the article y = 16(ln(x)) + 31, where y is a dog’s age in equivalent human years and x is age in dog years. *

2. Use the article to define the general steps taken by scientists to determine the equation above.

*Since methylation state data has been used to track biological age in animals, scientists first had to gather methylation state data from humans and dogs of different ages. Next, the methylation state data for humans and dogs were compared. Finally, an equation defining the relationship was determined. *

**Apply the model**

3. Based on the new dog age equation, if a dog is 14 years old, how old is the dog in human years (round your answer to the nearest whole year)? Use the graph in the online article to verify your answer.

*From the equation y = 16(ln(x)) + 31, if x = 14 years, then y = 73 years. So, the dog would be 73 years old in human years.*

4. Based on the new dog age equation, if a human is 14 years old, how old is the human in dog years (round your answer to the hundredths place)? Use the graph in the online article to verify your answer. Then, convert your answer to a number of months (round to the nearest whole number of months).

*From the equation y = 16(ln(x)) + 31, if y = 14 years, x = 0.35 years. So the human would be 0.35 years old in dog years, which is equal to about 4 months old.*

**Analyze the model**

5. What does the new canine age model tell you about the rate of a dog’s development compared with that of a human’s over their respective life spans? Use the equation and the graph to explain your answer.

*The rate of a dog’s development compared with a human’s does not stay constant over time. This can be seen by the change in the slope of the graphed equation. Dogs develop much faster than humans in early life stages than in later stages of life. The slope of the line decreases, or the curve flattens, as dogs age.*

6. Explain why you think this model was created. Why is it helpful?

*The life spans of dogs and humans are very different. Relating a dog’s age to the equivalent age of a human may be helpful for better understanding a dog’s stage of life and thus the care and nutrition it needs.*

7. What factors might affect the rate at which a dog ages? How do these factors limit the new mathematical model that the article describes?

*Specific breeds of dog may have different life spans. Since the canine data collected was based only on Labrador retrievers, the model might not predict the correct human age for other dog breeds. Other factors that could affect the rate at which a dog ages are a dog’s health, underlying medical issues and living conditions.*

**Compare models**

8. The existing “rule of thumb” for determining dog age in equivalent human years was to multiply a dog’s age by seven. Either print out or redraw the graph from the online version of the article. Then, plot at least five points that fit the “rule of thumb” equation on the dog age graph and draw the resulting line that represents the formula. Write the appropriate equation below.

*The graph should include points that fit the equation y = 7x, **where y is a dog’s age in equivalent human years and x is age in dog years.*

9. Now compare the new and old dog age graphs. At what age in dog years do the two equations predict the most similar age in human years (only use whole number dog years unless otherwise instructed by your teacher)? What is happening to the two graphed equations near this point?

*When a dog is 10 years old, the old model predicts that the dog is 70 years old in human years. The new model predicts that the dog is 68 years old in human years. The equations will converge and intersect near this point before diverging again.*

10. According to the graphs, does the old formula predict a human age that is too young or too old relative to the new formula for dogs under the age you mentioned in the answer above. What about for dogs older than the age you mentioned above?

*Before the point of intersection, the old dog age formula represents a human age that is too young, and after the point, the old dog age formula represents a human age that is too old, according to the new dog age formula.*

11. What does the old dog age model indicate about the rate of a dog’s development compared with that of a human’s over their respective life spans? How does that compare with the new dog age formula? Use the graphed equations to explain your answer.

*Unlike the new dog age formula, the old model suggests that the rate of a dog’s development compared with a human’s stays constant over their respective life spans.*

**Final prompts**

12. What additional questions might you ask based on the current dog age model? What other pet-related questions could be asked and answered using a mathematical model? Give a few examples of mathematical models used in other fields of study.

*I could ask how the methylation state data for additional breeds compare to the Labrador retriever data. Questions about a pet’s nutrition, health or its ability to be trained could also be modeled. Meteorologists use mathematical modeling to predict the weather, epidemiologists use models to track and predict the spread of disease, data scientists use math modeling in sports to make future performance projections, astronomers use modeling to determine the projected path of objects in space, economists use modeling to predict market ups and downs, and so on.*

13. Why are mathematical models beneficial? What are some limitations of models?

*A mathematical model allows for the behavior of a system to be predicted, which can be helpful in a lot of situations. It can help people prepare for upcoming weather, stop the spread of disease or pick the best sports player for a team. Ultimately, such predictions can help save resources such as time, money and energy, and keep people safe. Models do not always provide perfect answers though. Rather they provide an estimated output from an input. As with the dog age formula, models are generally based on a particular dataset. The output of a model is not necessarily meaningful if the input isn’t representative of the model’s dataset.*