# GPS: Working Backward with Data

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• 1 Dana Hall School in Wellesley, Massachusetts
• 2 Wayne State University in Detroit, Michigan
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This month's Growing Problem Solvers focuses on Data Analysis across all grades beginning with visual representations of categorical data and moving to measures of central tendency using a “working backwards” approach.

Growing Problem Solvers this month focuses on data analysis across all grades, beginning with visual representations of categorical data and moving to measures of central tendency. Using a “working backward” approach, learners in the early grades will start with pictographs and bar graphs and reflect on the data set. Learners in the upper grade bands begin with the measures of central tendency and reflect on the data set that yielded those values.

The PK–grade 2 task asks students what changes would need to be made—using basic addition and subtraction skills—to the given data set so that all categories are equal. After asking the first question but before posing a new question, the teacher could monitor the classroom to see if students are coming up with potential ways of doing so. The teacher could extend this task by using a data set with more categories for more challenging computations.

In the 3–5 grade band task, students are expected to work with a frequency bar graph and reflect on the data organization and categorization. Similar to the K–2 task, the 3–5 task is open ended and allows multiple answers as students start with aggregated data and reflect on other ways of representing the data set, such as animals with scales or household pets. Some possibilities of data categorization to fit the given graph could be 4 feet, 2 feet, or no feet; or big, medium, and small.

For the 6–8 grade band task, students should be familiar with the concepts of mean, median, and range. The questions asked do not necessarily have correct answers; all answers offered by students should come with justifications, which the class can then discuss and evaluate. Question 4 tries to home in on the difference between labeling a claim as false versus labeling it unsupported by the evidence (making its truth value as yet unknown). Question 6 is meant to pick at the limitations of the range as a statistic. Discussion could lead to talk about quartiles, the interquartile range, or even standard deviation.

For the 9–12 grade band task, students should be familiar with the concepts of mean, median, and standard deviation. Scenario B is impossible to create; the others are possible. This naturally leads to discussions and further investigations of either how far from the mean the median can be with a given standard deviation, or how small the standard deviation can be, given a particular median. In general, the median is always within

$3/5≈0.7746$
standard deviations of the mean, but we would not expect students to arrive at this. They may make conjectures of their own, and we would press them to give some kind of justification.

Jenny's class created a picture graph of their pets.

• Which animals must they add so the class graph has the same number of each kind of animal?
• How many of each animal must they add?
• Which animals must they subtract so the class graph has the same number of each kind of animal?
• How many of each animal must they subtract?
• How many animals in all must they add so the class graph has the same number of each kind of animal?
• How many animals in all must they subtract so their class graph has the same number of each kind of animal?

Jenny realized that they forgot one group of animals from their data-6 hamsters.

• Which animals must they add so the class graph has the same number of each kind of animal?
• How many of each animal must they add?
• Which animals must they subtract so the class graph has the same number of each kind of animal?
• How many of each animal must they subtract?
• How many animals in all must they add so the class graph has the same number of each kind of animal?
• How many animals in all must they subtract so their class graph has the same number of each kind of animal?

Jenny's class collected data on their favorite animals as 5 dogs, 3 turkeys, 4 cows, 5 snakes, 3 fish, 2 birds, 1 cat, and 1 whale. They sorted the favorite animals into three categories of animals. When they created a bar graph to represent their sorted animals, they forgot to label the categories.

1. What are three categories that the animals could have been sorted into?
2. What is another way the animals could have been sorted into three categories?
3. What are three categories that could be used to sort the animals into three equal groups?
4. What are three categories that could be used to sort the animals into three groups with two of the groups having the same number of animals?
Table 1

Summary Statistics of Age Distribution in Two Fictional Counties

AberdeenBothwell
Median Age4427
Mean Age4331
Range88 years90 years
Total Population35,0266,915
1. In which county would you expect there to be more elementary schools? Why?
2. Which county would you expect to have more golf courses? Why?
3. A classmate looks at these statistics and makes the claim that “there must be more 27-year-olds in Bothwell County than there are in Aberdeen County.” Is his claim supported by the statistics? If so, what makes you say so? If not, why not?
4. What is the difference between saying that a claim is false and saying that it is not supported by the available evidence?
5. What are some other claims one might be tempted to make that are not supported by these numbers?
6. Is the age range a good indicator of how spread out the ages are? Why or why not?

The Desmos plot shows a set of 10 data points that have a mean of 100. The data points are adjustable. Can you adjust them so that the median and standard deviation have the given values?

• A. Median = 80, Standard Deviation = 20
• B. Median = 80, Standard Deviation = 40
• C. Median = 90, Standard Deviation = 20
• D. Median = 90, Standard Deviation = 40

Matt Enlow, matt.enlow@danahall.org, teaches upper-school mathematics at the Dana Hall School in Wellesley, Massachusetts. He is interested both in the art of problem posing and in mathematical art. He tweets at @CmonMattTHINK.

S. Asli Özgün-Koca, aokoca@wayne.edu, teaches mathematics and mathematics education courses at Wayne State University in Detroit, Michigan. She is interested in effective and appropriate use of technology in secondary mathematics classrooms along with research in mathematics teacher education.

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## Footnotes

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners' growth as problem solvers across their years of school mathematics.

### Desmos Activity

Mathematics Teacher: Learning and Teaching PK-12