# Lesson 10

Fossils and Flags

## 10.1: A Fossil Puzzle (40 minutes)

### Optional activity

The mathematical purpose of this activity is for students to collect, summarize, interpret, and draw conclusions from bivariate data using scatter plots, best fit lines, residuals and correlation coefficients. Students measure the approximate lengths of the humerus bone and height of their classmates to collect data and create a linear model. The model is then used to approximate the height of an ancient human based on the length of a found humerus bone.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). By collecting their own data and using a best fit line to find additional information, students are modeling with mathematics (MP4).

### Launch

Arrange students in groups of 2 to 4. Present the task to students and ask students to brainstorm different ways that they could answer the question. After 2 minutes of quiet think time, ask students to share their ideas with the class. The remaining time should be used by students to collect, analyze, summarize, and interpret the data.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students further interpret and increase awareness of the language used to talk about scatter plots, best fit lines, residuals and correlation coefficients. Display the first sentence of this problem (“An anthropologist finds a fossilized humerus bone of an ancient human ancestor.”), and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions involving comparing bivariate data.
Design Principle(s): Maximize meta-awareness; Support sense-making

### Student Facing

An anthropologist finds a fossilized humerus bone of an ancient human ancestor. The humerus is an arm bone running from the shoulder to the elbow. It is 24 centimeters in length. Use data from your classmates to estimate the height of this ancient human.

### Anticipated Misconceptions

Students may find it difficult to start to answer the question. Ask students what are the variables given in the situation. Ask students if there is a way we could collect information from people in the classroom to help answer the question.

### Activity Synthesis

The purpose of this discussion is for students to communicate how they used mathematics to justify their findings.

• “How confident are you in your answer? What information helped you determine your confidence?” (Not very confident. Since the correlation coefficient is near 0.6, there is only a moderate relationship between height and humerus length. Additionally, this ancient human ancestor may have a different anatomy—for example, apes tend to have proportionally longer arms than humans do.)
• “How did you use mathematics to estimate the height of the ancient human?” (First, I collected data on the height and the approximate length of the humerus of my classmates. I then made a scatter plot to determine whether or not a linear model was appropriate, and then computed a line of best fit. I substituted 24 centimeters into my line of best fit and obtained my answer.)
• “Do you think that the way you measured the humerus of your classmates impacted your choice of linear model? Explain your reasoning.” (Yes, I think that I probably overestimated the bone length, because I was measuring from the outside and not the inside. I think that my model likely overestimates height.)

## 10.2: Playing Dirty (40 minutes)

### Optional activity

The mathematical purpose of this activity is for students to collect, summarize, interpret, and draw conclusions from bivariate data using scatter plots, best fit lines, residuals and correlation coefficients.

Students could examine number of penalties (like number of flags thrown against a football team) or severity of penalties (like penalty yards assessed against a football team).

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). By selecting which variables to use for their model and finding additional data to support a claim, students are modeling with mathematics (MP4).

### Launch

Arrange students in groups of 2 to 4. Give students 2 minutes to work through the first question, then pause for a whole-group discussion. Tell students, “Data is often kept about fouls in basketball, flags in football, or penalty box minutes in hockey.” If using the blackline master, then distribute it to students and tell them to begin questions 2 and 3. If you are not using the blackline master, then tell students to use the internet to find data about fouls, flags, penalties, or other similar data about rules violations and sports results. Ask them to record two questions for the data that they research that are similar to questions 2 and 3. Then ask them to answer their questions using mathematical work.

A blackline master is included with data for hockey and football if students do not have access to technology to find their own data.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Display sentence frames to support student conversation such as: “I predict _____ because . . .”, “We will need to know . . .”, “Why did you . . .?”, “I agree/disagree because . . .”
Supports accessibility for: Language; Social-emotional skills

### Student Facing

1. Before researching data, do you think the number of penalties a team gets in a season is correlated to the number of wins the team gets in that season? Do you think it is a positive or negative relationship? Do you think the relationship is weak or strong? Explain your reasoning.
2. Is there a relationship between the number of penalties a team gets in a season and the number of wins in that season? Show any mathematical work that leads to your answer.
3. Do penalties cause a change in wins or wins cause a change in penalties or neither? Explain your reasoning.

### Anticipated Misconceptions

Students may struggle with finding a method to determine the presence of a relationship. Ask students, “What data could you collect to begin answering the question? What representations could you create with that data to help illustrate a relationship?”

### Activity Synthesis

The purpose of this discussion is for students to communicate how they used mathematics to justify their findings.

If students are able to research their own data, tell the groups to create a visual display to present their findings. Some questions for discussion are listed here:

• “If there is a negative correlation, do you think the penalties would cause fewer wins?” (Although it may be part of the cause, it is generally not the main reason a team loses several games.)
• “Were there any interesting data points you would like to research more about?” (Some points were very far away from the rest of the data and might be interesting to look at more closely.)
• “How did you reach a conclusion? Explain your reasoning.” (I created a scatter plot from data that I collected. I then plotted the residuals to verify that a linear model would be appropriate. I computed a line of best fit and a correlation coefficient and found that there was a weak relationship between penalties and wins because the correlation coefficient was between 0 and 0.5.)

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion.

• “What did you find important, challenging, or otherwise notable about today’s lesson?” (It was really interesting to collect data to answer a question. I really felt like I was figuring something out.)
• “What mathematical work from this unit did you use in today’s activity?” (I used scatter plots, lines of best fit, and the correlation coefficient. I also had to determine whether or not the two variables showed a causal relationship.)
• “How might the mathematical tools that you have learned in this unit help you outside of mathematics class?” (I could use these tools to think about graphs or claims that I see being made in the media. It would help me to determine what to believe and what not to believe.)

## Student Lesson Summary

### Student Facing

Scatter plots can be useful to display possible relationships between two variables. Once a pattern is recognized, fitting a function to the data and then recognizing how well the function represents the relationship in the data can quantify your intuition about relationships between variables. The best fit line, for example, can be used to predict the value of one variable based on the value of a second variable. Although technology can aid in finding best fit lines and evaluating the strength of the line’s fit, human understanding of the variables and of how the data were collected is still required to determine whether the relationship is merely a relationship or whether there is a causal relationship.