# Lesson 9

Technological Graphing

## 9.1: It Begins With Data (10 minutes)

### Warm-up

The mathematical purpose of this activity is to gain familiarity with entering data into a spreadsheet and to prepare students for finding statistics using technology.

### Launch

Arrange students in groups of 2. If students are using the digital version of the materials, show them how to open the GeoGebra spreadsheet app in the math tools. If students are using the print version of the materials, they can access the GeoGebra spreadsheet app at www.geogebra.org/spreadsheet. If they use a different technology, you may need to provide them with alternate instructions.

Make sure students input the data in one column, even though the data is represented in two columns in the task statement.

### Student Facing

Open a spreadsheet window and enter the data so that each value is in its own cell in column A.

2. If you entered the data in the order that the values are listed, the number 7 is in the cell at position A1 and the number 5 is in cell A5. List all of the cells that contain the number 13.
3. In cell C1 type the word “Sum”, in C2 type “Mean”, and in C3 type “Median”. You may wish to double-click or drag the vertical line between columns C and D to allow the entire words to be seen.
A
1 7
2 8
3 4
4 13
5 5
6 15
7 14
8 8
9 12
10 2
A
11 8
12 13
13 12
14 13
15 6
16 1
17 9
18 4
19 9
20 15

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Activity Synthesis

The goal is to make sure that students know how to type data into a spreadsheet and to locate values in the spreadsheet by row and column. The locations will be referenced with spreadsheet functions in upcoming activities. Here are some questions for discussion.

• “What value is in cell A7?” (14)
• “What was interesting or challenging about this activity?” (I never knew that you could describe each cell in a spreadsheet using the row and column labels.)

## 9.2: Finding Spreadsheet Statistics (15 minutes)

### Activity

The mathematical purpose of this activity is to calculate statistics, create data displays, and to investigate how those change when values are added or removed from the data set. Monitor for students discussing the relationship between outliers and the measure of center.

### Launch

Keep students in the same groups. They will continue working using the spreadsheet they started in the previous activity.

Tell students that statistics are values that are calculated from data, such as the mean, median, or interquartile range.

Tell students that after they change the value in A1 to change the mean in the first set of questions, they should continue to use the changed value for the second set of questions rather than reset them to the values from the warm-up.

Note that GeoGebra is like any other computer program. It needs directions written in a specific way for it to execute a command. For example, if students forget to type the = symbol or don’t capitalize “Sum,” the formula won’t work. Ask students to pause after typing the formulas and ensure that cells D1, D2, and D3 display numbers for each statistic. If not, ask students to delete the contents of the cell and retype the formula, ensuring that they start with an = symbol and capitalize Sum, Mean, and Median.

Conversing: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise their initial draft of a response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language. For example, “Can you explain how the values changed when…?”, “You should expand on the relationship between the outliers and….”, etc. Afterward, invite students to build on their response, incorporating any feedback they received. Design Principle(s): Cultivate conversation
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one set of questions at a time.
Supports accessibility for: Organization; Attention

### Student Facing

Using the data from the warm-up, we can calculate a few statistics and look at the data.

• Next to the word Sum, in cell D1, type =Sum(A1:A20)
• Next to the word Mean, in cell D2, type =Mean(A1:A20)
• Next to the word Median, in cell D3, type =Median(A1:A20)
1. What are the values for each of the statistics?
2. Change the value in A1 to 8. How does that change the statistics?
3. What value can be put into A1 to change the mean to 10.05 and the median to 9?

We can also use Geogebra to create data displays.

• Click on the letter A for the first column so that the entire column is highlighted.
• Click on the button that looks like a histogram to get a new window labeled One Variable Analysis .
• Click Analyze to see a histogram of the data.
1. Click the button $$\Sigma x$$ to see many of the statistics.
1. What does the value for n represent?
2. What does the value for $$\Sigma x$$ represent?
3. What other statistics do you recognize?
2. Adjust the slider next to the word Histogram. What changes?
3. Click on the button to the right of the slider to bring in another window with more options. Then, click the box next to Set Classes Manually and set the Width to 5. What does this do to the histogram?
4. Click the word Histogram and look at a box plot and dot plot of the data. When looking at the box plot, notice there is an x on the right side of box plot. This represents a data point that is considered an outlier. Click on the button to the right of the slider and uncheck the box labeled Show Outliers to include this point in the box plot. What changes? Why might you want to show outliers? Why might you want to include or exclude outliers?

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Activity Synthesis

The purpose of this discussion is for students to create data displays using technology and to analyze what happens to the displays and the statistics when changes are made to the data set. Here are some questions for discussion.

• What happened to the statistics when you changed the value for A1 to 8 in the spreadsheet?” (When it was changed to 8, the mean increased slightly but the median stayed the same.)
• “Why did the mean increase?” (The sum of the data increased but the number of numbers stayed the same so the mean had to increase.)
• “Why did the median stay the same?” (Changing a 7 to an 8 in the data set did not change the middle numbers, 8 and 9, in the data set).
• “What did you notice when you changed the width of the classes for the histogram?” (This changed the intervals for each bar to a width of 5 and the data was resorted into those intervals.)

Select students who were previously identified as discussing the relationship between outliers and the measures of center. Ask, “what is the relationship between outliers and the measures of center?” (When outliers are present the median is the preferred measure of center because it is less impacted by outliers than the mean.)

## 9.3: Making Digital Displays (10 minutes)

### Activity

The mathematical purpose of this activity is for students to create data displays and calculate statistics using technology. Students plot the survey data they collected from a statistical question in a previous lesson.

### Launch

Arrange students in groups of two. Tell them that they will be using technology to create data displays and calculate statistics for data they collected from a survey question in a previous lesson.

Conversing, Representing, Writing: MLR 2 Collect and Display. As pairs are working, listen for and collect vocabulary, phrases, and methods students use for creating dot plots, boxplots, and histograms using technology, such as data set, skewed distribution, median, the box plot for the IQR. Consider grouping language for each data display in different areas, and be sure to include the spreadsheet formulas. Continue to update collected student language throughout the entire activity and the entire unit. Remind students to borrow language from the display as needed. This will help students to expand their notation of recursive definitions. Design Principle(s): Support sense-making; Maximize meta-awareness
Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, create an exemplar dot plot highlighting specific features, such as the scale and labels, as well as calculations for the mean and median.
Supports accessibility for: Attention; Social-emotional skill

### Student Facing

Use the data you collected from the numerical, statistical question from a previous lesson. Use technology to create a dot plot, boxplot, and histogram for your data. Then find the mean, median, and interquartile range for the data.

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Student Facing

#### Are you ready for more?

A stem and leaf plot is a table where each data point is indicated by writing the first digit(s) on the left (the stem) and the last digit(s) on the right (the leaves). Each stem is written only once and shared by all data points with the same first digit(s). For example, the values 31, 32, and 45 might be represented like:

$$\displaystyle \begin{array}{r|l l} 3 & 1 & 2\\ 4 & 5 \end{array}$$
Key: 3 | 1 means 31

A class took an exam and earned the scores:

86, 73, 85, 86, 72, 94, 88, 98, 87, 86, 85, 93, 75, 64, 82, 95, 99, 76, 84, 68

1. Use technology to create a stem and leaf plot for this data set.

2. How can we see the shape of the distribution from this plot?

3. What information can we see from a stem and leaf plot that we cannot see from a histogram?

4. What do we have more control of in a histogram than in a stem and leaf plot?

### Student Response

Student responses to this activity are available at one of our IM Certified Partners

### Anticipated Misconceptions

Students may lose one data display when they begin to create the next one. Explain to students that it is important to copy their solutions into a more permanent place so they can refer to it later.

### Activity Synthesis

The goal of this activity was for students to create graphs and find statistics using technology. Here are some questions for discussion.

• “What were some challenges that you faced using technology and how did you overcome them?” (I was not sure what buttons to press to get to the spreadsheet. I checked with my partner and figured it out.)
• “What width did you use for you histogram? Why?” (I used 5 because my data set has values ranging from 1 to 42. I could have used 10 but then I would have only had 5 bars.)
• “What is the appropriate measure of center for your data set?” (The median was appropriate because my data set has a skewed distribution.)
• “Which display allows you to calculate the IQR the most easily?” (The box plot because it displays Q1 and Q3.)
• “Can you find the median using your histogram?” (No, the data is grouped into intervals, so a histogram cannot be used to find the middle value for the median.)

## Lesson Synthesis

### Lesson Synthesis

The goal of this lesson is for students to display and investigate data using technology. Here are some questions for discussion.

• “How do you create data displays using technology?” (You type the data into the spreadsheet and then click the appropriate buttons.)
• “What are some advantages of using technology to display data and calculate statistics?” (You can easily switch between different data displays and you can change the intervals on histograms without having to sort through the data again. The advantage of having the technology calculate the statistics is that I can see how the statistics change as I enter or make changes to the data.)
• “When do you think it is appropriate to use technology to display data or to calculate statistics?” (Graphing technology makes it easier to determine the shape of a distribution. I might use it to determine the most appropriate measure of center for a data set. Using technology to calculate statistics makes sense to do in most situations because statistics are calculated using algorithms that can get complicated when there are many values in the data set. The chance of making a mistake while calculating statistics by hand makes using technology a good choice.)

## 9.4: Cool-down - What Are These Values? (5 minutes)

### Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

## Student Lesson Summary

### Student Facing

Data displays (like histograms or box plots) are very useful for quickly understanding a large amount of information, but often take a long time to construct accurately using pencil and paper. Technology can help create these displays as well as calculate useful statistics much faster than doing the same tasks by hand. Especially with very large data sets (in some experiments, millions of pieces of data are collected), technology is essential for putting the information into forms that are more easily understood.

A statistic is a quantity that is calculated from sample data as a measure of a distribution. Mean and median are examples of statistics that are measures of center. Mean absolute deviation (MAD) and interquartile range (IQR) are examples of statistics that are measures of variability. Although the interpretation must still be done by people, using the tools available can improve the accuracy and speed of doing computations and creating graphs.