Lesson 2

Graphing Data

High School Algebra 1

Bar graph activity

Strategies:

Study Guides

Sample

Before Reading

I. Motivation:

"Researchers work a great deal with data, as are you with your research projects. You have all collected your data now, and need a nice way to organize it. Yesterday we saw how to find the central measures of tendency and place the data in a frequency table. A frequency table is hard to understand just by looking at the graph. Researchers also have these problems, and need ways to organize their data. They want people who read their results to understand their data easily. There are numerous types of graphs but today we will be focusing on three. These three are bar, line and circle. We want to stay away from misleading graphs to insure our data can be easily read. "

II. Prior Knowledge

"Speaking of graphs...have you ever heard the phrase a picture is worth a thousand words?"

"I have, my mom says it all the time."

"Pictures come in many different varieties. One example that we will be looking at is a graph. A graph can convey a great deal of information, as we will see in a bit with the newspapers and magazines you all brought in. People use graphs to help convey information to people. Before we can talk anymore about graphs, I need to know what prior knowledge you all already have. To do this we are going to take a short warm-up. Please take out your math journals, and copy down the six problems off the overhead. Show all of your work, and remember there is no talking during the math warm-up."

Students will begin by taking a quick warm-up. This will help to assess where the students' knowledge currently is and what they understand about graphing. It is necessary that students understand how to switch from decimals to percents and vice versa for the lesson. Students will need to know how to evaluate the expressions for making circle graphs.

Write as a decimal. Round to the nearest thousandth if necessary.
1. 68% ____________	2. 3.2% ___________	3. 37/50 ___________	4. 37/150 __________
Evaluate each expression
5. 0.21 X 40,000 _______________________		6. 42% of 7300 ________________________

Answers:
1. 0.68

2. 0.032

3. 0.74

4. 0.25

5. 8,400

6. 3,066

"Time is up. Please stop working on the problems, and listen closely as I read off the answers. You may correct your own work. The answer to number one is 0.68. The answer to number two is 0.032. The answer to number three is 0.74. The answer to number four is 0.25. The answer to number five is 8,400. The answer to number six is 3,066. Are there any questions on those six problems, or how to find the answer?"

"How did you find the answer for number six?"

"Wonderful question Tommy. Would someone like to explain to Tommy how to find the answer to number six?"

"You take 42% and convert it to decimal form. This is .42. Then you multiply .42 times 7300. This gives you 3,066."

"Great job Amy. If there are no more questions, then I have a question for you. What types of graphs have you seen before?"

"I have seen a number line that was used to graph inequalities. Does that count?"

"Yes that is exactly right Daniel. What other types of graphs have you all seen before?"

"I have seen the x-y plot. My brother had some homework working with the x-y plot last night."

"Very good Timmy. Has anyone seen any other graphs?" No other suggestions are given. "How do we use these graphs to help us convey data?"

"Graphs are like pictures of words."

"That is exactly right Emily. Graphs can be like pictures of words. Does anyone know how to make a bar graph, or a line graph, or a circle graph?" No one raises his or her hand. This is expected since they have not worked with the graphs before. This will be taught in the lesson, so a lack of prior knowledge with this is expected. Students also will not know how to set up the axis with a scale so that the graph is not misleading. This will be looked at in the next exercise. The students will struggle with reading a bar graph when the top of the bar falls between two grid lines. This will become evident later in the lesson once they have mastered making bar graphs. To help students once the problem is apparent, I will suggest they use a ruler or the edge of the paper to visually determine where the top of the bar would intersect a grid line if it were extended.

"To investigate graphs further, we are going to look at ones we might see every day. To do this, I asked you all to bring in a current newspaper or magazine. Please take out your newspaper or magazine that you brought with you today. Again if you forgot or did not have one at home I have a few. Please wait until we break up into groups to choose one of mine. I want you all to get into groups of four when I say GO. In your group, there should be one leader, one timekeeper, on recorder, and one reporter. If you have forgotten what the rolls stand for, there is a poster on the wall describing the rolls. As a group, you are going to look at the magazines and newspapers that your group members brought. Find as many different graphs as possible in the magazines or newspapers. As a group, you then are going to share the graphs you found and explain what is being represented with the graph. Also, tell us if you feel the graph is misleading and explain your reasoning. I want students to determine for themselves what a misleading graph is. We will investigate the idea later in the lesson. GO."

Class will work in groups for 10 minutes. Then they will be brought back together to present their findings.

"Alright, we are going to go around the room and present your findings. When it is your group's turn, I would like all of the group members to come to the front of the classroom. The leader of the group needs to then show the graphs you found. The time keeper will explain what each graph is showing us, the recorder will tell us what type of graph each on is, and the reporter will tell us which if any of the graphs are misleading and why."

III. Purpose

"Thank you for all of the wonderful presentations. Misleading graphs can be difficult to work with, so we want to avoid them when presenting our information. Today we will be reading and creating graphs. We will read and create a study guide on the topic of graphing. The information gained through this lesson will allow all of you to present your data on your project to the class in a graph. All of you will also use the information to make judgments on whether a graph is misleading or not. Today we are reading to perform a task, the text structure is description.

During Reading

Schultz, J., Kennedy, P., Ellis, W., & Hollowell, K. (2003). Algebra I. Austin: Holt, Rinehart and Winston.

Readability: 9.1 grade level using the Flesch-Kincaid; which is exactly where it would need to be for high school students taking algebra one. All of the vocabulary for the lesson is included in the text. I believe this brings up the readability slightly since the definitions are not clearly provided to the students. The drawbacks of using a readability formula are well documented, so the cuing system and other readability factors should be considered. For this text, there are pictures of the three types of graphs in the text that would help the reader. Bold letters and different colored boxes help to set the text apart, which decreases readability slightly. The semantics and schematic would work best for someone who has some knowledge of math and statistics, however any concept that is presented in the text is well explained. In addition, given that this section falls in the middle of a chapter, the students would have the necessary knowledge to understand the information. This decreases the readability. The syntax of the reading follows the same pattern as the rest of the book. The students would be accustom to this layout, which would also decrease the readability. The text structure from this section provides information on the three types of graphing and vocabulary words. This would also decrease the readability. Overall, this section would work well for students reading at their grade level.

IV. Text Structure

Expository – The text structure is description, since the reading describes how to make graphs and when each type would be the best choice. The text also describes what a misleading graph is. The reading includes three vocabulary words and their definitions.

V. Vocabulary

"Before we can begin looking at the text and creating the study guide we need to look at the vocabulary. Please use the Cornell Note system we used yesterday. It is important to take notes, and I will be checking the notebooks this Friday when you take the test."

The vocabulary words and sentences will be written on the chalkboard before the start of class. Students will just need to copy down the words, definitions, and sentences into their notes.

Bar Graphs: A method of displaying data that uses rectangular bars or objects to represent data.
Sentence: The types of shoes worn and the amount of each type can be demonstrated using a bar graph, which has rectangular bars to represent the amount of each kind of shoe.

Line Graph: A type of graph that uses line segments between known data points to show changes that have occurred over time. Sentence: The changes in stock prices over time can be demonstrated using a line graph.

Circle Graph: A method of displaying data in which the data are represented by parts of a circle.
Sentence: Time spent in school is split between different activities which can be shown using a circle graph..

VI. Content

"Graphs make interpreting information easier. By looking at the newspapers and magazines, we have seen how researchers have used graphs to illustrate data. For your projects, you will need to put your data into graphs. I want you all to read section 4.5 on pages 193 through 197. If you find something confusing the first time thought, reread the challenging section. Leave room on the right side of your paper for examples that we will work through."

Students are given time to read the section and take notes using the Cornell Note system.

"Before we look at the examples, I would like to go a bit more into misleading graphs. In the beginning of the lesson, I let you all determine what a misleading graph was. Most of you were close if not right about what a misleading graph is. However, I would like to clarify because the topic of a misleading graph will be on the test on Friday. Insure to include this in your notes. A misleading graph is one with distracting pictures, which make it difficult to read. Another form, would be to have an axis that is poorly labeled making it difficult to interpret."

Example one:

Don invested $100 each in four companies in 1995. Find the approximate values of Don's MRI and MMart stock in 1999. Construct a bar graph that shows the approximate value of Don's WMB stock from 1995 to 1999.

"Now lets look at example 1. Please look at the line graph on page 195, on the right side. The line graph shows the values of four different stocks over time. Can anyone tell me the approximate value of Don's MRI stock in 1999?"

"I think it is about $290."

"Erica that is correct. Does anyone know how Erica found that answer?"

"Didn't she just have to look at the graph?"

"Yes that is all you need to do Matthew. Can someone tell me the approximate value of Don's MMart stock in 1999?"

"I got about $140 in 1999."

"Wonderful Randy. How did Randy get that answer?"

"I think he just used the line graph."

"Excellent Rachel. Does everyone understand how we found the values of Don's MRI and MMart stock in 1999?" Students shake their heads yes. "Now in your notes I want all of you to construct a bar graph that shows the approximate value of Don's WMB stock from 1995 to 1999. To do this you will need to use the line graph that is shown to you in the book. Make sure to label all of your axis to insure it is not misleading. If you want to use grid paper, there is some on the back counter. You can staple it into your notebooks once you have created the graph."

After students have been given some time to work on their graphs, we will go over the graphs. "The approximate value of Don's WMB stock was $100 in 1995, $260 in 1996, $220 in 1997, $200 in 1998 and $140 in 1999. Are there any questions on how the bar graph was made?" No hands are raised. "If their are no questions, then we will move on to example two."

Example two:

The circle graph at right shows how the average American family spends its money. Find how much money, on average, a family with an annual income of $32,000 spends on housing, transportation, and food. Please look at this graph now on page 196.

"Please look at the circle graph on page 196. It shows how the average American family spends its money. Can someone tell me, on average, how much money a family with an annual income of $32,000 would spend on housing?

"The graph shows us the average family spends 31% of their income."

"That is correct Ben. Lets take it a step further though. How would we figure out how much money they spend?"

"You have to turn 31% into a decimal to give you .31. Then you have to multiply .31 by 32,000. This gives you 9920."

"Very nice Nik. We looked at this skill in our warm-up. Are their any questions on how Nik found the answer?"

"So all we have to do is multiply the decimal by the dollar amount?"

"That is correct Nate! That is a wonderful way of simplifying what we are trying to say. Make sure you all have that in your notes. How much money would a family with an annual income of $32,000 spend on transportation?"

"If you turn 18% into a decimal you get .18, and I then multiplied by 32,000. I got 5,760."

"Exactly correct Timmy. How much money would this average family spend on food?"

"I multiplied .15 by 32,000 and got 4,800."

"Correct Amy. Are there any questions on example three and how to read a circle graph?" No one raises his or her hands. "Does everyone understand how we found the answers to this problem?" Again, no hands are raised. "Alright, lets move on to example three."

Example three/Checking for Understanding:

Use the information in the table to make a circle graph for the number of cats in each breed that are entered in the Geline Fanciers cat show.

Breed	Persian	Abyssinian	Balinese	Manx	Siamese	Total
Number of cats	99	30	66	41	106	342

"Please make a chart in your notes of the data in the table on the top of page 197. Look at the data presented in your table. With a partner decide which type of graph would work best for this type of information."

"We believe a circle graph will work best for the data because of the large range in the number of cats. A bar graph would have to be really large to insure it was not misleading."

"Emily and Andrew you are correct. Today we will be making a circle graph of the material in the table. Does anyone know how we are going to make a circle graph?"

"Can't we just find the percent of each cat in the show, and put that into a circle graph?"

"That will work, but how are we going to find the percent of each cat in the show?"

"We need to take the total number of cats entered in the show and divide it by the number of a particular kind of cat."

"Yes, that is the first step. Now after we have the percent of each type of cat, how do we know how much area of the circle to give that cat? In other words, how do we know what degree to give each cat?"

"I think you just make it the amount the percent is for. So if it is 3 percent, its 3 degrees."

"That is a good guess, but not exactly right. If we did it that way, there would mean the circle only has 100 degrees. How many degrees are in a circle?"

"360"

"Wonderful. So to find out how many degrees each cat gets, we need to multiply 360 by the percent we found for the cat. Now I want you all to make a table like the one I have made on the board. (See table with answers) Complete this table. Then make a circle and find the exact degrees each cat gets. Then finish your circle graph by labeling each section."

Breed	Number of cats	Percent	Number of degrees
Persian	99	29%	104.4 degrees
Abyssinian	3	9%	32.4 degrees
Balinese	66	19%	68.4 degrees
Manx	41	12%	43.2 degrees
Siamese	106	31%	111.6 degrees
Total	342	100%	360 degrees

"Now that you all have finished your circle graph, I would like you to compare it to your neighbors. If they do not match up, go back to the table to make sure you both have the same answers. If it is just rounding errors, that is all right. Now we are going to do a short bar graph activity. This will insure we know how to make a bar graph."

Bar graph activity

VII. Explicit/strategy instructions

A. Study Guides.

Study guides involve words, pictures or graphs. Students will be using key terms and ideas along with examples to formulate a study guide to help them study the material when it comes time for a test. Students need to use whatever works best for them.

"The bar graph activity and example three will be very helpful tonight when working on homework. Are there any questions on how to make a line graph, a bar graph, or a circle graph?" No one raises his or her hand. "I want you all to count off to six. When I say GO, I want you find the rest of your group members. You are going to make a study guide for the material we have just learned. This study guide needs to include both the reading from the chapter, and the examples. Make sure to include the key terms. The study guide will help you study the material for our test on Friday."

B. Steps

First analyze the content and identify major concepts and important details which will be done through taking notes using the Cornell note system.. The teacher will give the students a few key points to be sure to include in the study guide. The second step is to develop questions at multiple levels of understanding. This will involve questions that are explicit, implicit and experience based. Explicit questions mean the answer is right on the page, where as Implicit require the student to think and search for the answer. Experience based means the student will need to rely on their own experience to come up with an answer. Students will be responsible for having at least two questions of each type. The next step will be to bring the class back together and discuss the questions. Here we will discuss what questions groups had in common and which one was different. The class will come to a consensus as to which questions should be included on the class study guide. Once a study guide has been decided upon, the students will copy down the questions off the chalkboard. These questions will become part of the student's homework.

"In these groups you will need to analyze the section and identify the major concepts and important details. Your Cornell Notes will help you find the important parts of the section. Be sure to include bar graph, circle graph and line graph. You should have questions that look at multiple levels of understanding. That means there should be questions that are explicit, implicit, and experience based. Explicit questions are ones where the answer is right on the page. An example would be what is the definition of a circle graph? Implicit questions require the students to think and search for the answer. This would be how do you make a circle graph? Experience based means you will have to rely on your past experience to come up with an answer. This might be when would use a bar graph?"

C. Modeling

Sample

To model this skill the teacher will work with students to come up with some possible questions. They will use the previous day's lesson as material to make a study guide with. The students will already know the key terms, which are: Mean, Median, Mode, Range and Frequency Table. With these terms and the topics of central measures of tendency and frequency tables, they will be able to formulate questions.

"To make sure that we understand how to make a study guide, we are going to make a sample one for yesterday's lesson. You all already read the section and took notes Cornell style. Please take those notes out now. I want you all to review them with a partner for 5 minutes. We had five key words yesterday: Mean, Median, Mode, Range, and Frequency Table. Our topic from yesterday was central measures of tendency and frequency tables."

"What are examples of explicit questions?"

Sample: What is the definition of mean?

"What are some examples of implicit questions?"

Sample: How can we use measures of central tendency?

"What are some examples of experience based questions?"

Sample: How can a frequency table make it easier to compute the mean, median, and mode of a set of data?

 

D. Guided practice

Sample

The class will form one of each type of question based off the reading from section 4.5.

"What are examples of explicit questions?"

Sample: What is the definition of a bar graph?

"What are some examples of implicit questions?"

Sample: How do we make a circle graph?

"What are some examples of experience based questions?"

Sample: Which type of graph would work best for temperatures in Iowa over a 12 month period?

The students will now make their own study guide for section 4.5. They need to be sure to include the different types of questions.

"In your study guides you need to have two types of each question. I want you to insure everyone in your group of four is working on the study guide. You will have roles within your group, as we have in the past. So make sure before you start you have one person who will record, one who will reporter, and one to be the timekeeper. You all need to actively supply ideas, as this will help you study for your upcoming test. You will be given 15 minutes, in your group to make your study guides. After 15 minutes we will all come back together as a class and discuss the study guides. In this discussion we will look at what questions the groups had in common and which ones were different. As a class, we will come to a consensus as to which questions should be included on the class study guide. You all will then be responsible for copying down the class study guide. GO. "

After making the study guide, students will be brought back together for a large group discussion. This discussion will allow the students to correct any errors in their study guide and connect ideas.

"Will the reporter for each group please come and put your study guide questions on the chalkboard. Looking at the questions that are on the board, does anyone notice any similarities?"

"They all have questions using the key terms: bar graph, line graph, and circle graph"

"Any differences?"

"The questions are all phrased differently."

"What questions do you think would be important to answer before taking a test?"

"I think implicit questions would be helpful when preparing to take a test."

"Do we want to have questions that ask us to complete problems like the examples?"

"Yes, because those are the types of questions are on the test."

"Are there any questions anyone wants to add after looking at the ones on the board?"

"No, I think everything we need is on the board in one way or another."

"How are we going to decide which questions should be on our study guide?"

"I think it would work if we just kept one question on each topic. We have a few repeats up there, and we can remove those. Then we should just make sure each question is represented in some way in the final study guide. We should choose the best question that is going to help us prepare for the test the best."

This discussion will allow the students to decide exactly what is important on a study guide. Through this discussion, they will gain a deeper understanding of the material. After the discussion, the students will be asked the following questions.

Completed Study Guide Sample

After Reading

VIII. Questioning

"That was a wonderful discussion class. I just have a few more questions for you all. Why are graphs an important statistical tool?"

"They help us easily display information in a way that can be understood by the majority of people not just researchers."

"Exactly Nik. How can graphs be misleading?"

"Graphs are misleading when they are hard to understand. This happens if there are a lot of pictures in the graph or if the axis is not labeled very well."

"Great answer Nate. What are the differences between a bar graph and a line graph?"

"A bar graph has vertical lines going up from the x-axis. The vertical lines correspond to the y-axis. A line graph has points plotted, and the points are then connected with a line."

"Very nice Emily. Are there any other questions on anything we have learned today?"

IX. Writing to learn activity

"Alright, since none of you have questions, please take out your math journals. I want you all to copy down the following question from the overhead. Then in your journals describe what kind of information would work best in a circle graph, a line graph and a bar graph. If you do not finish, it will be part of your homework. You will have ten minutes right now."

1. Certain kinds of information are represented better with certain types of graphs. Describe the kind of information that would be best represented by each type of graph below. Circle graph:________________________________________________________________________ Sample answer: Circle graphs are excellent for showing data that represents different parts of a whole. They are specifically useful for situations involving percents, such as what percent of voters voted for each candidate in an election or how much of each dollar is spent in a certain category. Line graph:_________________________________________________________________________ Sample answer: Line graphs are great fro showing how to measure changes over time, such as the cost of living or a population. Bar graph:__________________________________________________________________________ Sample answer: Bar graphs are good for comparing data on the same scale, such as the top five grossing movies or the sales figures of different companies.

"Alright, time is up. Please finish your thought and look up front."

X. Closure

The independent practice will be questions 5-8, 9-35 odd. The students also need to answer the questions on the study guide. The answers will be gone over tomorrow, because it is imperative that all students have the correct answer. The study guides can be used for studying for the upcoming test. Students also need to decide which graph would be the best for their data. They do not need to make the graph, just decide which one to use.

" Graphs are very important to researchers when they need an easy way to present their data. With the data on your projects, graphs will also be important to you. The graphs for your project will be due in two days, so if you have any questions on graphing or which graph to use come and seek help. Also remember the test is on Friday, so if you need help with any of the homework or the study guide please come and get help. I will be in early tomorrow and I encourage anyone with questions to come in. Remember the research projects are due on Friday as well. The project needs to be typed up according to the assignment rubric I gave to you last week. You need to be sure to include a chart with the data, the central measures of tendency computed, and a graph of the data made."

"Tonight for homework do questions 5-8 and 9-35 odd. This needs to be done independently. We have about five minutes right now, where you can get started on this. You also need to answer all of the questions on the study guide that we made. The answers should be in the reading or notes that you took in class. Remember some of the questions will require you to look between the lines or use experience. We will go over the answers to both the homework questions and the study guides. You will want to make sure you have correct answers on your study guide so that you can use it to study for the test on Friday. You also need to decide which type of graph would be best for your data in your project. Make sure to use a graph that will not be misleading. As we saw at the beginning of the lesson, misleading graphs can cause our data to be misinterpreted. You do not need to make the graph tonight, just make sure you know which one you are going to use."