How can different stages of a growing pattern be determined?
Kimiko ran a total of 6 miles in 4 days. Each day, she ran ½ mile more than she had run the day before.
- How many miles did Kimiko run on the first day?
- If Kimiko continues this pattern, how many miles will she run on the eighth day?
- How could you represent Kimiko’s running?
- How far could Kimiko have run on Day 1? What would that mean for her distance on Day 2?
- Kimiko’s Aunt Leslie ran 25 miles in 5 days. Each day, Aunt Leslie ran 1 ¼ miles more than she had the day before. How far did she run on the first day? If she kept increasing her distance in this way past 5 days, how many days would it take before she ran more than the distance of a marathon (26.2 miles)?
- What if Kimiko started with the same number of miles but doubled the distance she ran each day? How many days would it be until she ran more than 10 miles in a day? More than 50? More than 100?
This task asks students to analyze a growing pattern involving addition and subtraction of fractions. The task is made more challenging by varying the information often provided for a pattern. Students are provided the rate of growth each day and the total distance, but not the starting amount. However students approach the problem, they will add (or subtract) fractions, look for and use structure, and model a real-world problem with mathematics.
Students may approach this problem in different ways. One possible way to work backward would be to realize that the growth occurs in increasing amounts over the second, third, and fourth days. Students may consider how many first-day distances and ½ mile increases occur over the four days to determine how to partition the 6 miles. They might choose to use the available information to estimate the distance run on the first day, then adjust that estimate based on the outcome. Alternatively, students may determine how many miles would be run each day if the miles were equally distributed, and then adjust that answer to fit the rate of growth outlined in the problem.
Students may choose to use an app to create a representation of the situation.
- In the Number Line app, students may begin at the total of 6 miles and work backward. They will have to choose an appropriate scale for the number line, and also realize that the distance on the first day is repeated on all subsequent days (in addition to new ½ mile increases). One possible beginning to a solution is shown here.
- Students may also use the Number Line app to construct a model of the problem using unknown jumps, but showing the ½ mile increases where appropriate. Here is the beginning of a possible strategy.
- In the Fractions app, students may arrange the fraction pieces in such a way to easily show the ½ mile increase per day. If they use estimation to determine that Day 1 will be less than 1 ½ miles, they might choose 1 mile as a possible starting distance. This example of student work shows the first few steps taken to solve the problem using this initial estimate.
To draw out students’ mathematical thinking about the pattern in this activity, you may ask: How many ½ mile distances happened over these four days? Recognizing and defining the structure of a growing pattern is mathematically important. Students may not realize that each day’s ½ mile increase happens in addition to the previous day’s increase. Other questions may support students’ thinking about the relationships among numbers in the pattern. Consider asking, Is Kimiko’s 8th day distance double her 4th day distance? Why or why not? Doubling is a common strategy for students with multiplication. Exploring when doubling does and does not work can be informative for students. Finally, asking Can you explain how to determine the number of miles Kimiko would run on any day? invites students to generalize this pattern. Some may be able to verbally explain a generalizable rule, while others might even be able to use variables to describe their rule.