Unit 1 Organizer: “GROOVY GRAPHING”
|Grade 5 Mathematics Frameworks |
|Unit 1 |
|Data Analysis and Graphing |
Unit 1
DATA ANALYSIS AND GRAPHING
(4 weeks)
TABLE OF CONTENTS
Overview………………………………………………………………………………………... 3
Key Standards & Related Standards……………………………………………………………. 4
Enduring Understandings……………………………………………………………………….. 6
Essential Questions……………………………………………………………………………... 6
Concepts & Skills to Maintain …………………………………………………………………. 6
Selected Terms and Symbols.….. ……………………………………………………………… 7
Classroom Routines…………………………………………………………………………….. 8
Strategies for Teaching and Learning…………………………………………………………... 8
Evidence of Learning………………………………………………………………..………….. 8
Tasks……………………………………………………………………………………………. 9
• What Does Mean Mean? ………………………………………………………………. 10
• A Balanced Mean ……………………………………………………………………… 15
• What Does a Combo Cost? ……………………………………………………………. 20
• What’s the Story? ......………………………………………………………………….. 25
• Weathering the Data …………………………………………………………………... 29
• Tater Head ……..…………………………………………………………………….… 33
• Loose Marbles …....……………………………………………………………………. 37
• Candy Bars …………………………………………………………………………….. 44
• Building Houses ……….………………………………………………………………. 47
• Survey Says …….……………………………………………………………………… 55
Culminating Task
• Solo Graphing…………………………………………………………………………... 59
OVERVIEW
In this unit students will:
• continue to develop their understanding of data analysis
• read, interpret, and analyze given set(s) of data
• collect and display data in various ways
• understand how to determine the most appropriate means of displaying data
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the five process standards; problem solving, reasoning, connections, communication, and representation, should be addressed constantly as well. The first unit should establish these routines, allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.
To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
KEY STANDARDS:
M5A1. Students will represent and interpret the relationships between quantities algebraically.
b. Investigate simple algebraic expressions by substituting numbers for the unknown.
c. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimals) substituted for the variable.
M5N5. Students will understand the meaning of percentage.
a. Apply percents to circle graphs.
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
c. Determine and justify the mean, range, mode, and median of a set of data.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
RELATED STANDARDS
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ENDURING UNDERSTANDINGS
• Data can be collected from experiments and surveys.
• Data are gathered for the purpose of explaining characteristics of the sample from which the data were collected.
• There are various ways to display data; the most appropriate way should be determined by the type of data collected and the purpose it serves.
• Tendencies of data can be described using mean, mode, median, and range.
• Sample size influences the accuracy of population representation.
• A complete circle graph must represent 100% of the data collected. Percent can be represented as a fraction with one-hundred as the denominator.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we conduct an experiment or survey?
• How do we determine who should take our survey and what our survey should be about?
• What is a sample?
• How do we determine the most appropriate graph to use?
• How do we choose whether to orient our bar graph horizontally or vertically?
• How will we interpret a set of data?
• Are there patterns in a set of data?
• How do graphs help explain real-world situations?
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge and experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
• Whole number computations
• Comparing, ordering, and estimating
• Analyzing and interpreting data
• Measuring using a ruler
• Data representations: tally marks, Venn diagrams, bar graph, line graphs and pictographs
• Mode, median, and range
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.
The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures or numbers.
← Bar graph: A visual display (graph) of data, used to show data using horizontal or vertical bars.
← Circle graph: A graph that displays data in the form of a circle. The circular region is divided into a number of pie-shaped sectors to represent all portions of the data.
← Data: Information gathered; facts or figures from which conclusions may be drawn.
← Frequency table: A table that organizes the number of times something occurs in an interval or set of data.
← Line graph: A visual display (graph) of data, shown by using lines to show change over time, contiguous data.
← Line plot: A graph that uses symbols above a number line to represent data.
← Mean: The sum of the numbers in a set of data divided by the number of pieces of data.
← Median: The number in the middle of a set of data when the data are arranged in order. When there are two middle numbers, the median is the number that is halfway between the two middle numbers.
← Mode: The number that occurs most frequently in a set of numbers.
← Percent: Per hundred. A special ratio that compares a number to 100 using the symbol %.
← Pictograph: A visual display (graph) of data shown by using symbols. Also may be referred to as a picture graph.
← Range: The difference between the largest and smallest values in a numerical data set
← Tally mark: A mark used in keeping track of acts or objects. The marks consist of four vertical lines bundled diagonally or horizontally by a fifth line.
← Venn diagram: Venn diagrams use circles to show relationships among sets. If sets contain the same element(s) the circles overlap or intersect. If sets do not contain the same elements, there is no intersection or overlap.
CLASSROOM ROUTINES
The importance of continuing established classroom routines cannot be overstated. Daily routines must include such obvious activities such as taking attendance and lunch count, doing daily graphs, questions, and calendar activities as whole group instruction. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills and communication. These routines contribute to a rich, hands-on standards based classroom and will support students’ performances on the tasks in this unit and throughout the school year.
STRATEGIES FOR TEACHING AND LEARNING
• Students should be actively engaged by developing their own understanding.
• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words.
• Appropriate manipulatives and technology should be used to enhance student learning.
• Students should be given opportunities to revise their work based on teacher feedback, peer feedback and metacognition which includes self-assessment and reflection.
• Students should write about the mathematical ideas and concepts they are learning.
EVIDENCE OF LEARNING
By the conclusion of this unit, students should be able to demonstrate the following competencies:
• Conduct an experiment
• Create and conduct a survey to gather data on a particular subject
• Interpret, read, and analyze a graph
• Determine the most appropriate graph for a set of data
• Use percents as related to circle graphs
• Establish a plausible situation to describe a given set of data
• Construct an appropriate graph for a given data set
• Determine and justify the mean of a set of data
TASKS
|Task Name |Task Type |Content Addressed |
|What Does Mean Mean? |Learning Task |Find mean using different strategies |
| |Partner/Small Group Task | |
|A Balanced Mean |Learning Task |Determine a set of numbers for a given mean |
| |Partner/Small Group Task | |
|What Does a Combo Cost? |Performance Task |Find mean using decimals |
| |Individual/Partner Task | |
|What’s the Story? |Performance Task |Create a context for data, describe a possible collection |
| |Individual/Partner Task |technique, and represent data in a graph |
|Weathering the Data |Performance Task |Collect and display data using different representations |
| |Individual/Partner Task | |
|Tater Head |Learning Task |Collect data by conducting an experiment and representing the |
| |Partner/Small Group Task |data in a line graph |
|Loose Marbles |Learning Task |Collect data and create bar and circle graphs |
| |Partner/Small Group Task | |
|Candy Bars |Learning Task |Compare vertical and horizontal bar graphs |
| |Partner/Small Group Task | |
|Building Houses |Performance Task |Extend a pattern, determine relationships, graph by plotting |
| |Individual/Partner Task |points |
|Survey Says |Performance Task |Collect using a survey, use multiple representations to display |
| |Individual/Partner Task |data |
|Culminating Task: |Performance Task |Create a way to collect date and display data in multiple formats|
|Solo Graphing |Individual/Partner Task | |
The following tasks represent the level of depth, rigor, and complexity expected of all third grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).
LEARNING TASK: What Does Mean Mean?
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
c. Determine and justify the mean, range, mode, and median of a set of data.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How do we determine the central tendencies of data?
• What do central tendencies of data tell us?
• How do the central tendencies of data help to explain real-world situations?
MATERIALS
• “What Does Mean Mean?” Recording Sheet (one per student)
• Interlocking blocks or other manipulative
• Markers, colored pencils, or crayons
• Graph paper or chart paper
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will explore the measures of central tendency for a set of data (mean, median, mode, and range).
Comments
Data will be collected for each student in the class. Consider the student population when choosing what type of data to collect. Some alternatives to the data collected in the task below include:
• number of vowels in each student’s name
• number of children in each student’s family
• number of people who live in each student’s home
• number of pets in each student’s home
• number of books in each student’s desk
• number of TVs (doors, chairs, windows, etc.) in each student’s home
• number of pockets in each student’s clothes
• number of servings of fruit eaten yesterday
• number of times each student has eaten fast food in the past week
Background Knowledge
Students studied median, mode, and range in fourth grade. In fifth grade, students should develop a conceptual understanding of mean and understand the algorithm for finding mean.
Task Directions
1. Allow students to explore mean in a small group setting (groups of 5) using the “What Does Mean Mean?” student recording sheet. As students are working, some students may be identified as needing more instruction in finding median, mode, and range. When students are finished, discuss the results. In particular, discuss the questions under item 6.
2. Have students bring their towers to the front of the room. Display all of the towers on a table or whiteboard tray. Ask the students how they could find the class mean for the data. Students may suggest that the towers could be “evened out.” Further probe the students by asking if they can think of a way to do that mathematically, because there are so many towers. If students don’t suggest a method, the following questions may be used:
• How many towers do we have?
• How many do you estimate will be in each tower if there were “evened out”?
• What if we connected all the towers?
• How could we share the blocks equally among _____ towers?
• Could we do it mathematically? What operation would we use? (Division)
• What would we divide? (The dividend would be the total number of blocks; the divisor would be the number of towers.)
• What did you get for a quotient?
• What do you do with the remainder?
• What does the quotient tell you?
3. Have students summarize how they found the mean for the class data. Generate a class algorithm for finding the mean for a set of data. (The sum of the numbers in a set of data divided by the number of pieces of data.)
Questions/Prompts for Formative Student Assessment
• What does the dividend represent?
• What does the divisor represent?
• What does the quotient represent?
• Can you think of a context in which you would want to report the mean as a decimal or fraction?
Questions for Teacher Reflection
• Are students able to explain how the algorithm can be represented by making the towers of equal heights
• Are students able to explain how the algorithm can be represented by making one tower and breaking it into towers of equal height?
DIFFERENTIATION
Extension
• In this situation, whole numbers are appropriate to represent the mean of the data. Ask students to consider contexts in which decimals or fractions would be appropriate. In small groups, have students collect the data and find the mode, median, range, and mean.
• Students can create a set of data that satisfies given criteria for the data’s mean and median at .
Intervention
• Before determining the mean for the class data, go through the same process with a small group using just the group’s data. Have students put all of the blocks together and break the tower into equal towers, based on the number of students in the group, to model the addition and division steps of the algorithm.
TECHNOLOGY CONNECTION
• An NCTM Illuminations applet that give the mean and median for data students enter.
Name __________________________________ Date ______________________
What Does Mean Mean?
1. Collect data about the number of letters in the first name of each student in your group.
a. Take one interlocking block for each letter in your first name
b. Make a tower with your blocks
c. Place all towers in the middle of your group to create a bar graph
d. Fill in the frequency table based on the data collected
2. Order your group’s data points from smallest to largest below:
|Name |Number of |
| |Letters |
| | |
| | |
| | |
| | |
| | |
| | |
3. Find the median of the data. ______
(Remember, the median is the number in the middle of a set of data when the data are arranged in order. When there are two middle numbers, the median is the number that is halfway between the two middle numbers.)
4. Find the mode of the data. ______
(Remember, the mode is the number that occurs most frequently in a set of numbers.)
5. Find the range of the data. ______
(Remember, the range is the difference between the largest and smallest values in a numerical data set.)
6. Find the mean of the data. ______
a. To find the mean, make all of the towers of your group’s bar graph the same height. Place any extras to the side. Record the mean as the number of blocks in each tower.
b. Does your group have enough extras to make at least half of the towers one block taller? If so you can round your mean up by one. Can you explain why?
c. Why does the mean for this data make more sense as a whole number?
LEARNING TASK: A Balanced Mean
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
d. Determine and justify the mean, range, mode, and median of a set of data.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How do we create a set of data with a given mean?
• What do central tendencies of data tell us?
MATERIALS
• “A Balanced Mean” Recording Sheet (one per student)
• 1 cm cubes, beans, or other small manipulative (5 per group)
• Calculators (1 available for each group)
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will explore the sets of numbers that have a given mean. The understanding of mean in this task is based on numbers being balanced around the mean.
Background Knowledge
The example below is one way students may work to find a given mean. Have students put all of the sticky notes at the desired mean. Then move the post-it notes two (or more) at a time, keeping the distance from the mean the same each time. In this way, students are able to create a different set of data with the same mean. (See example below.)
Data Set {3, 3, 3, 3, 3} Data Set {0, 3, 3, 4, 5} Data Set {0, 2, 4, 4, 5}
As students work with mean, they should recognize that the sum of the numbers needs to equal the number of pieces of data times the mean. In the example above, the sum of the data in each case must be 15 because there are five pieces of data and the mean is 3 (5 x 3 = 15). This works because the inverse of the operation used to find the mean (15 ( 5 = 3) is the product of the number of pieces of data in the set and the mean (5 x 3 = 15).
Task Directions
1. To activate this lesson ask students if it is possible to create two different sets of data that have the same mean. Give students a number line from 0-5 on poster paper (with the numbers far enough apart so that square sticky notes can be placed above each number on the number line) and 5 square sticky notes. In small groups ask students create a set of data that has a mean of 3. Allow students to experiment with different combinations of numbers until they have found a set of data that has a mean of 5. Allow students to use a calculator; although some students will find it easier not to use a calculator because the numbers are small.
2. Once each group has found at least one set of data that has a mean of 3, invite students with different data sets to share their findings and how they determined their set of numbers has a mean of 3.
3. Give students the recording sheet, “A Balanced Mean” to work on in small groups or pairs. Allow students to use 1 cm blocks to try different sets of numbers. Once a solution has been determined, students can use an “x” in place of the blocks to record their solution.
4. Once all groups have completed “A Balanced Mean” again have students share their solutions and thinking strategies.
Questions/Prompts for Formative Student Assessment
• What is the mean? How many members of the set are needed? What does that tell you?
• Do the elements of the set need to add to a certain amount? Why? How do you know?
• What are the limitations for your set of data? Have you considered them? How does your set of data need to change?
• When you divide, what dividend do you need in order to get the required mean? How do you find that dividend?
• What have you tried? Why didn’t it work?
• What do you need to do next? How do you know?
Questions for Teacher Reflection
• When considering which groups should share their solutions:
▪ Which group has a different solution than others?
▪ Which group solved the problem in a unique, efficient, or effective way?
▪ What strategies, solutions, or thought processes would be important to highlight?
▪ In what order should students present their solution strategies?
• What connections have students made regarding the mean as a balance point and the relationship between the number of elements of a set, the mean, and the sum of numbers in the set?
DIFFERENTIATION
Extension
• Allow students to find a given mean using a larger set of numbers and/or a number line that is more extensive. Students can explore these extensions on paper or use the following applet with which to explore different sets of data. .
Intervention
• Give students two or three of the five elements for a set of data and ask students to work with the remaining two pieces of data to find the desired mean. This will limit the possible variations and make the task more manageable for some students.
• Allow students to use the computer applet below, which immediately determines the mean for a given set of data. Once the students plot five points on the number line, they can change the values of the numbers. This will give students an opportunity to receive immediate feedback about the mean for a set of data.
TECHNOLOGY CONNECTION
• An NCTM Illuminations applet that give the mean and median for data students enter.
Name _________________________________________ Date __________________________
A Balanced Mean
Think of the mean of a set of data as the balancing point of the data. Create a set of 6 pieces of data given the requirements below. Place 6 centimeter cubes above the number line to represent your data.
Problem # 1
• Create a set of 6 numbers whose mean is 5.
• No more than one member of the set can be the number 5.
• Show how you know your answer is correct.
Problem # 2
• Create a set of 6 numbers whose mean is 5.
• All members of the set must be greater than 7 or less than 3.
• Show how you know your answer is correct.
Problem # 3
• Create a set of 6 numbers whose mean is 5.
• Your set must include the following numbers: 5, 5, 4, 9.
• Show how you know your answer is correct.
PERFORMANCE TASK: What Does a Combo Cost?
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
e. Determine and justify the mean, range, mode, and median of a set of data.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS
• How much does a typical hamburger meal cost?
• What does the mean for a set of data tell us?
• How do we use the mean for a set of data to explain real-world situations?
MATERIALS
• “What Does a Combo Cost?” Recording Sheet (one per student)
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will explore the mean for sets of data. Also, students will compare prices for combo meals and buying items separately.
Comments
To activate this task, ask students how much money they think they’ll need to buy a meal at a fast food restaurant. Ask them the items they typically buy and the amount they typically pay.
Throughout the school year, students can be given many opportunities to find the mean for a set of data. Finding the mean of their grades gives students experience in finding mean, rounding decimals to the nearest whole number, and recognizing how a new piece of data effects the mean.
Background Knowledge
This activity gives students experience with finding the mean of decimal numbers. Students will need to be comfortable with the algorithm for finding the mean for a set of data.
In fourth grade students were expected to divide two-digit decimals by a whole number. This task will require students to use that skill to find the mean of decimal numbers.
Task Directions
Students will follow the directions on the student recording sheet.
1. Explore the costs for items purchased individually and in a combo.
• What is the mean cost of a ¼ lb burger?
• What is the mean cost of large fries?
• What is the mean cost of a large drink?
2. Complete the chart.
• Find the total at each restaurant for a ¼ lb burger, large fries, and a large drink.
• Find the mean of the totals found in the previous step.
• Find the mean cost for a large combo at each restaurant.
3. What’s the difference between the mean cost of purchasing the three items separately and purchasing the three items in a combo? (The combo is $1.01 less than purchasing the three items separately.)
4. Why do you think restaurants offer combo prices?
5. Do you buy items in a combo? Why? Why not?
Questions/Prompts for Formative Student Assessment
• For what set of data do you need to find the mean?
• How many items are in your set?
• How do you find the mean for a set of data?
• How do you divide a decimal by a whole number?
Questions for Teacher Reflection
• Are students using the algorithm for finding mean with fluency?
• Do students recognize that the $5.83 on the chart can be found in two ways, by finding the sum of the means for each item and by finding the mean of the total cost for the three items (the sum of the first three numbers in the “Mean Cost” column and the mean of the “Total” row)?
DIFFERENTIATION
Extension
• Ask students to explore the results when a higher priced restaurant is added to the set. What happens to each mean cost when a ¼ lb burger costs $9.58 (it comes with fries) and a drink costs $2.29? Have students find
1. The change in the mean cost for burgers and the change in the mean cost for burgers and fries.
2. The change in the mean cost for drinks.
3. The change in the mean cost for the total of a burger, fries, and a drink.
• Ask students to create a restaurant and determine prices so that the total mean cost (for a ¼ lb. burger, a large fries, and a large drink) for all five restaurants is greater than $6.00 or less than $5.00.
Intervention
• Break the chart down into individual problems as shown.
1. Find the mean price for a ¼ lb burger at the given four hamburger restaurants.
2. Find the mean price for a large fries at the given four hamburger restaurants.
TECHNOLOGY CONNECTION
• An NCTM Illuminations applet that gives the mean and median for data students enter.
Name __________________________________ Date ______________________
What Does a Combo Cost?
1. Explore the costs for items purchased individually and in a combo.
• What is the mean cost of a ¼ lb burger?
• What is the mean cost of large fries?
• What is the mean cost of a large drink?
|Items |Burger Barn |Bibb Burgers |Burke Burgers |Burger Bistro |Mean Cost |
|Large Fries |$1.47 |$1.89 |$1.45 |$1.19 | |
|Large Drink |$1.17 |$1.39 |$1.25 |$0.99 | |
|Total Cost | | | | | |
|(1/4 lb burger, large | | | | | |
|fries, large drink) | | | | | |
|Large Combo |$4.99 |$4.50 |$5.50 |$4.29 | |
2. Complete the chart.
• Find the total at each restaurant for a ¼ lb burger, large fries, and a large drink.
• Find the mean of the totals found in the previous step.
• Find the mean cost for a large combo at each restaurant.
3. What’s the difference between the mean cost of purchasing the three items separately and purchasing the three items in a combo?
4. Why do you think restaurants offer combo prices?
5. Do you buy items in a combo? Why? Why not?
PERFORMANCE TASK: What’s the Story?
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we determine the most appropriate graph to use to display the data?
• How will we interpret a set of data?
• How do graphs help explain real-world situations?
MATERIALS
• Set of data that is teacher or student generated or “What’s the Story” Recording Sheet (one per student)
• Markers, colored pencils, or crayons
• Graph paper or chart paper
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will be working with graphs and data sets. Given a set of data, students will create a graph, describe a context for the data, explain a possible collection method, and report what they learn from the data. The set of data used can be student or teacher created.
Comments
You may want to demonstrate this type of activity as a whole class before assigning this task. The students should have graph paper or chart paper available in case they choose to use it. Students should also have the opportunity to share their solutions. The set of data can be determined by the teacher so that the data set can apply to different learning levels, and degrees of difficulty. The teacher also has the option of allowing students to create their own data set.
A sample set of data could be as follows: 24, 25.5, 32, 27, 28.5, 29, 25, 26.5, 30, 24, 25, 29, 29.5, 24, 28.5, 29, 27, 26, 26.5, 32, 25, 32, 29, 26.5, 29, 32, 26.5, 29.5, 25.5, 29, 25, 32
Background Knowledge
Students need a good understanding of the different ways to represent data in a graph in order to choose the most appropriate graph for the data presented. The data presented above are most appropriate for a bar graph or circle graph. Data for a line graph would need to be given as an ordered pair or with two pieces of information (in a t-chart for example) for each data point. Line graphs are frequently used to display data over time.
Be sure students understand that the bars in bar graphs should not be attached to one another. A small space must be placed between each bar within the graph. Histograms are similar to bar graphs in that they use bars, but represent continuous data; therefore they do not have spaces between each bar. (This will be discussed in sixth grade.)
Task Directions
Have students follow the directions below:
Use your set of data to:
• display the data on a graph that is appropriate
• label your graph appropriately
• create a situation that would fit the set of data given
• explain how the set of data was/might have been collected
• give at least five real-world interpretations from the given set of data on your Recording Sheet
Questions/Prompts for Formative Student Assessment
• Why did you select this type of graph?
• How did you decide what kind of situation would be appropriate to describe the data in your graph?
• What are ways in which these data could have been collected?
• Would the data be appropriate on another type of graph? If so which graph(s)?
• Is there another way that your data could have been collected?
• Are there other interpretations you can make from your graph?
• Have you labeled your graph appropriately?
Questions for Teacher Reflection
• Are students able to explain and justify their choice of a graph?
• Do students understand that different types of data require different kinds of graphs?
• Have students labeled their graphs clearly and correctly?
• Can students determine real-world situations that would appropriately describe their data set?
• Do students understand various ways to collect data?
DIFFERENTIATION
Extension
• Have students repeat the activity using data that require a different type of graph than the one they selected for the initial task.
• Have students research and describe situations in which data are collected and displayed routinely.
Intervention
• Have students work with a smaller data set for the task.
• Allow students to collect or create data for their project.
TECHNOLOGY CONNECTION
• An NCTM weather data task using a spreadsheet.
• How to create a bar graph from an excel worksheet.
• Allows students to create charts and graphs.
• Interactive student and teacher website with activities and lessons.
• Mathematics Help Central-grid/graph paper
• Guilford High School website that has grid/graph paper
Name __________________________________ Date ______________________
What’s the Story?
Using the set of data below:
• display the data on a graph that is appropriate
• label your graph appropriately
• create a situation that would fit the set of data given
• explain how the set of data was/might have been collected
• give at least five interpretations from the given set of data
24, 25.5, 32, 27, 28.5, 29, 25, 26.5, 30, 24, 25, 29, 29.5, 24, 28.5, 29,
27, 26, 26.5, 32, 25, 32, 29, 26.5, 29, 32, 26.5, 29.5, 25.5, 29, 25, 32
PERFORMANCE TASK: Weathering the Data
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we determine the most appropriate graph to use?
• How will we interpret a set of data?
• How do graphs help explain real-world situations?
• Can the same data be accurately represented on more than one kind of graph?
MATERIALS
• Thermometer to measure temperature or Internet access to find current temperature
• Chart paper or graph paper
• Markers, colored pencils, and/or crayons
• “Weathering the Data” Recording Sheet (one per student)
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This task involves the students in the collection and display of data on daily temperatures. The class will measure the temperature outside the school building and inside the classroom for one week using a thermometer, making every effort to record the temperature at the same time each day for consistency. Students will also record the daily high temperature of their city/town from a news channel or the Internet. Daily temperatures should be kept on a large chart in the classroom. At the end of the week students should record their findings on the “Weathering the Data” student handout provided. Students write a paragraph to explain the data and display the data different ways on different types of graphs (the use of a line graph is highly recommended). How many different ways the data is displayed should be determined by the level of the students and how much of the unit has been covered. Graph paper or chart paper should be available to those who choose to use it.
Comments
The teacher may need to assess the student’s ability to read a thermometer and provide a quick review if needed. To complete this activity in one day, give students temperature data for a given week or have students record five temperature readings during the course of one day.
Students should routinely be encouraged to write about mathematical ideas and procedures. If students are just beginning to write in mathematics class, you may want to give them some structure or guidelines for their writing. This could be as comprehensive as an outline or as simple as a prompt. For this particular writing assignment, the points students should cover are included in the Task Directions.
Background Knowledge
Students should know how to create a line graph including creating an appropriate scale and/or labels for both the x and y axis, plotting points in the first quadrant, and an awareness of when points can be connected and when they need to be considered discrete points.
Task Directions
Have students follow the directions below:
Collect the temperatures for inside your classroom, outside your building and a daily high temperature for your town/city for one week and complete the following:
- Record the data on your “Weathering the Data” Recording Sheet.
- Record five observations as instructed on the student handout.
- Create at least two different graphs from your data (one should be a line graph).
Write a paragraph that:
- explains your data collection process
- presents information you gathered from the data
- describes why the graphs you chose are appropriate to display your information
Questions/Prompts for Formative Student Assessment
• Describe the process by which you collected your data.
• Why is it important to collect daily temperature data at about the same time each day?
• Is there another way your data could have been collected?
• Which ways did you choose to display your data?
• Explain how these data can be shown in more than one way.
• What do the data tell you?
• What were some of your observations about your data?
• How many days was the outside temperature above 70 degrees?
Questions for Teacher Reflection
• Do students understand the purpose of a line graph?
• Do students understand when and how data can appropriately be shown in more than one type of graph?
• Can students write effectively about their graph in this task?
DIFFERENTIATION
Extension
• Have students select different types of graphs for students to create to display other aspects of the week’s weather.
• Use an LCD projector/TV and the Internet to compare the temperatures collected by the class outside the classroom to the official temperature that was recorded for your area that week. Discuss the differences and prompt students to speculate about why any differences might have occurred.
• Have students measure and record the temperatures at multiple times during the day and make observations about their findings.
• Students should be able to analyze and compare the graphs of the two different sets of data.
Intervention
• Have students display the data on a line graph only.
• Have students record only one set of data, either outside or inside temperatures.
• Provide written prompts in the form of sentence starters for the written portion of the task.
TECHNOLOGY CONNECTION
• Allows you and the students to create charts and graphs.
• The Weather Channel lists weather by zip code or city
• Website allows you to find the weather in your area.
Name ________________________________________ Date ___________________________
|Day of the week |Outside Temperature |Inside Temperature |Internet Temperature for your |
| | | |location |
|Monday | | | |
|Tuesday | | | |
|Wednesday | | | |
|Thursday | | | |
|Friday | | | |
Weathering the Data
Write five observations about the weather that you were able to determine by analyzing the data.
1.______________________________________________________________________
________________________________________________________________________
2.______________________________________________________________________
________________________________________________________________________
3.______________________________________________________________________
_______________________________________________________________________
4.______________________________________________________________________
________________________________________________________________________
5.______________________________________________________________________
________________________________________________________________________
LEARNING TASK: Tater Head
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we conduct an experiment?
• How do we determine the most appropriate graph to use?
• How do graphs help explain real-world situations?
MATERIALS
• Potato (one per student)
• Rye grass seeds (about one teaspoon per student)
• Spoon or melon scoop for making hole in top of potato
• “Tater Head” Recording Sheet (one per student)
• Graph paper, chart paper, markers, colored pencils, crayons, etc.
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will conduct an experiment from which they will collect data. Students will cover the top end of a potato with grass seed. They will measure the grass as it grows, collecting the data in a chart and graphing their data on a line graph.
Comments
You may want to slice a small section from the bottom of the potato in order for the Tater Heads to stand upright. If potatoes aren’t readily available, the task can be accomplished using small Styrofoam containers. Graph paper or chart paper can be used for this activity.
Read the directions on the bag of grass seed carefully. Some grass seed requires overnight soaking before use.
SPECIAL NOTE: Use EXTREME CAUTION when using any cutting instrument in the classroom. An option is to prepare the potatoes ahead of time.
Background Knowledge
Students should have prior knowledge of constructing and analyzing line graphs and know that they are used to show data over time. The students should also know how to measure to the nearest centimeter using a ruler.
Task Directions
Have students follow the directions below:
• Using markers, create a face on your potato.
• With adult supervision or assistance, slice the top off the potato and make a shallow hole for the grass seeds.
• Insert about a teaspoon of Rye grass seeds into the hole.
• Each day measure your Tater Head’s longest individual “hair” to the nearest centimeter and record your data on your “Tater Head” Recording Sheet.
• Use the daily measurements to create a line graph to display your data.
• Create a poster to present your information. This poster should include:
1. a title
2. daily measurement data in a chart
3. line graph appropriately labeled
4. a paragraph describing your graph
5. other observations about your graph
Questions/Prompts for Formative Student Assessment
• How long did it take for “hair” to grow on your potato?
• Did your “hair” grow rapidly or slowly? What do you think caused that rate of growth?
• Explain how you measured your Tater Head’s “hair.” What was the final length of your potato “hair”?
• Compare Tater Head “hair” lengths with your classmates. What do you notice? Why is it important to use a line graph to measure the results of this experiment?
• Can you make any general conclusions about your data?
• Do you think your results would be the same if you repeated the experiment? Why or why not?
Questions for Teacher Reflection
• Do students understand the purpose of a line graph?
• Do students understand how to measure to the nearest centimeter?
• Can students write effectively about their graph in this task?
• Can students accurately analyze the data in their graphs?
• Do students understand that the same experiment, if repeated, may not yield the same results?
DIFFERENTIATION
Extension
• Have students measure the Tater Head “hair” under different circumstances and compare the growth. For example, put one potato in a dark closet, put another potato on a sunny windowsill, or put a lot of grass seeds in one potato and only a few seeds in another.
• Have students graph the results of several Tater Heads on the same line graph and explain their process.
Intervention
• Have students work with a partner to develop their line graphs.
• Provide a template for recording and graphing data collected during this experiment.
• Record data and create a graph using a spreadsheet program or a webpage such as .
TECHNOLOGY CONNECTION
• How to create a bar graph from an excel worksheet.
• Allows you and the students to create charts and graphs.
• Interactive student and teacher website with activities and lessons.
• Mathematics Help Central-grid/graph paper
• Guilford High School website that has grid/graph paper
Name________________________________ Date__________________________
Tater Head Recording Sheet
Name of your Tater Head__________________________________
| |Date the “hair” was measured |Length of your Tater’s head’s |
| | |“hair” to the nearest |
| | |centimeter |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
|5 | | |
|6 | | |
|7 | | |
|8 | | |
|9 | | |
|10 | | |
|11 | | |
|12 | | |
|13 | | |
|14 | | |
|15 | | |
|16 | | |
|17 | | |
|18 | | |
|19 | | |
|20 | | |
|21 | | |
|22 | | |
|23 | | |
|24 | | |
|25 | | |
LEARNING TASK: Loose Marbles
STANDARDS ADDRESSED
M5N5. Students will understand the meaning of percentage.
b. Apply percents to circle graphs.
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we conduct an experiment or survey?
• How do we determine who should take our survey and what our survey should be about?
• What is a sample?
• How do we determine the most appropriate graph to use?
• How will we interpret our data?
• Are there patterns in our data?
• How do graphs help explain real-world situations?
• How can we use a circle graph to show percentages?
• How can the same data be shown in a bar graph and a circle graph?
MATERIALS
• One paper bag per group filled with 10 marbles, color tiles or 10 other small objects in 3 or 4 different colors
• Markers, crayons or colored pencils
• “Loose Marbles” Recording Sheet (one per student)
• “Circle Graph” Recording Sheet (one per student)
GROUPING
Partner/Small Group
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This task will involve students in a probability experiment that will ultimately result in the creation of a bar and a circle graph. Students select and record objects from a paper bag to use in the gathering of data to be used to create the graphs.
Comments
You may choose to have bags with a different ratio of colors, or use the same color composition in each bag to have uniform answers among the groups. A different ratio of colors for different groups can provide an opportunity for rich discussion about why the results are different for each group.
Before assigning this task, you may want to model creating a circle graph from data that are displayed in a linear fashion much like the data students will collect while completing this task. An effective way to accomplish this is to have each student select his/her favorite color of M&M’s™. Have students place their selection in a straight line, making sure all candies of the same color are grouped together. Then have them rearrange the straight line into a circle. Determine the center point of the circle and draw straight lines from the center to each point on the circle where the colors change.
Although students work in partners, have each student complete his/her own recording sheet to allow for the recording of individual ideas and work.
Background Knowledge
Students should have an understanding of different types of graphs and when each is appropriate to use. They should know that circle graphs represent 100% of the data collected and that these data groups can be represented as percentages.
Task Directions
Prepare a paper bag containing 10 colored marbles for each group of students (i.e., 3 blue, 1 white, 6 red). Have each student select from the bag 25 times and record the color marble after each selection on his/her own Loose Marbles Recording Sheet. Using crayons or markers, have students complete their own recording sheet to record the marble color. Then, using the back of their recording sheet, have students write at least five things they observed from the data.
Next, have students create a bar graph and a circle graph to represent the data. To create the circle graph, have students group like colors together in the last column provided on the chart entitled “Similar colors grouped together.” The last column will be cut apart and taped together at the ends. Students will use this paper circle to create the circle graph as indicated in the student directions.
Have students follow the directions below:
You will receive a paper bag containing different colored marbles.
• Remove a marble from the bag 25 times and record the color marble that you selected using crayons or markers to complete the Loose Marbles Recording Sheet. Use the back of the sheet to write at least five observations based on the data and to create a bar graph that represents your data, using appropriate labels. The colors in your graph should correspond to the colors of marbles recorded.
• Then use the data you collected to create a circle graph. To create the circle graph, use the data for your 25 selections to group the colors together in the last column (“Similar colors grouped together”) provided on the Loose Marbles Recording Sheet. You will be recording your selections a second time, this time in color order. For example, record all blue selections before recording any other color. Cut the “Similar colors grouped together” column from the Loose Marbles Recording Sheet and tape the long rectangle together at the ends so that it does not overlap, forming an open circle. This will help you create the circle graph.
• On your Circle Graph Recording Sheet, locate and mark a point that would be most near the center of the blank circle graph. Place the circular paper on the blank circle, much like a cookie cutter, matching the two circles as much as possible. Divide your circle into sections as they are divided on the circular strip and mark the outside of the blank circle at each color division. Use a ruler to connect the points of division to the center of the circle. Color and label your circle graph.
• With your partner, estimate the percentage each color represents on your circle graph and record it on your Circle Graph Recording Sheet.
After students have completed their graphs, select one of the graphs to lead students in a discussion about the application of percentages to the portions of the circle graph. Ask if there is any group that has found a way to determine the percentages of the various colors in the graph. One solution is that since there were 25 selections from the paper bag, each selection could be multiplied by 4 to determine its worth as a percent. Therefore, if there were 14 blue marbles selected, the blue portion on the graph would represent 14 x 4, or 56% of the total. Be sure students understand that the total must equal 100%.
Questions/Prompts for Formative Student Assessment
• What color marble was selected most? Least?
• Was any color marble selected the same amount of times?
• If there are ten marbles in your bag, how many of each color do you predict will be selected in 25 tries? Can you make an accurate prediction? Why or why not? Were any marble colors not picked at all? How do you explain that?
• What do you think will happen if we repeat this experiment?
• Explain how your data can be displayed in both a bar and a circle graph.
• Can all bar graphs become circle graphs and vice versa? How do you know?
• Would you use a line graph to display these data? Why or why not?
• How can you use a circle graph to show percentages?
Questions for Teacher Reflection
• Can students explain the relationship between a bar graph and a circle graph?
• Do students understand that circle graphs represent 100% of data collected and that the data can be expressed as percentages?
• Do students understand why the data in this task are not suitable for a line graph?
• Can students explain why the same experiment, if repeated, is not likely to yield the same data set?
Note: Although the focus of this task is not experimental or theoretical probability, students are naturally curious about the results of their data collections. These discussions can provide a meaningful preview of the introduction to probability in sixth grade.
DIFFERENTIATION
Extension
• Have students empty their bag and count how many marbles are in each color group. Ask students to explain whether or not the data collected by selecting objects 25 times accurately reflects what is in the bag.
• Have students collect data for 100 selections (or more) from the bag and determine whether or not the proportions in the data accurately reflect the proportions in the bag. Ask how they can draw conclusions about larger data samples.
• Have students write a journal entry describing the process of the data collection as well as the results. The journal entry should include pictures and/or examples.
• To extend the lesson using technology, have students use graphing calculators to simulate the choosing of the marbles.
Intervention
• Have students examine a graph similar to the one to be created in this task. Have them work with a partner to identify and explain the elements of the graph before collecting and recording data of their own.
TECHNOLOGY CONNECTION
• How to create a bar graph from an excel worksheet.
• Allows you and the students to create charts and graphs.
• Allows students to create bar graphs
• Mathematics Help Central-grid/graph paper
• Guilford High School website that has grid/graph paper
Name ________________________________________ Date____________________________
Loose Marbles Recording Sheet
| |Color of marble drawn |Similar colors grouped |
| | |together |
| | |(used to complete the circle |
| | |graph) |
|1 | | |
|2 | | |
|3 | | |
|4 | | |
|5 | | |
|6 | | |
|7 | | |
|8 | | |
|9 | | |
|10 | | |
|11 | | |
|12 | | |
|13 | | |
|14 | | |
|15 | | |
|16 | | |
|17 | | |
|18 | | |
|18 | | |
|19 | | |
|20 | | |
|21 | | |
|22 | | |
|23 | | |
|24 | | |
|25 | | |
Name __________________________________ Date_______________________
Circle Graph Recording Sheet
LEARNING TASK: Candy Bars
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS
• How do we choose whether to orient my bar graph horizontally or vertically?
• How are data collected?
• How do we conduct an experiment?
• How do we determine the most appropriate graph to use?
• How will we interpret a set of data?
• How do graphs help explain real-world situations?
MATERIALS
• Candy or colored paper squares, enough for two per student (Whole class version) or a handful per student (Individual/ Group/ Partner assignment)
• Box or container to hold the candy or squares
• Chart paper or graph paper
• Colored pencils, markers and/or crayons
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will use square candy wrappers or squares of paper to create both horizontal and vertical bar graphs and then examine similarities and differences between the two types of graphs.
Comments
Use candy with rectangular or square congruent wrappers (1 in/cm squares of paper can be substituted for the candy wrappers).
Background Knowledge
Students should be very familiar with both horizontal and vertical bar graphs. They should understand the characteristics of a bar graph such as scale increments, labels, titles, etc.
Task Directions
Whole class version
Have students select one or two pieces of candy/paper squares from the box. Students will then make whole class bar graphs by stacking their squares horizontally and then vertically. Students should discuss how to label their graphs. Students should be able to articulate which type of bar graphs they prefer and why. They should also be able to explain similarities and differences of the two graphs.
Directions for the teacher are as follows:
• Ask each student to select one or two pieces of candy from the box. Then have students place their wrappers on a chart, stacking the wrappers horizontally and then vertically. The wrapper of each candy bar should be placed neatly side by side. As a class, discuss how to label each bar graph created.
• Discuss at least three observations from the data.
Group/ Partner Activity
Each student will use a small bag or a handful of candy or colored paper squares. Students will then make bar graphs by stacking their squares horizontally and then vertically. Have students discuss how to label their graphs. They should be able to articulate which type of bar graphs they prefer and why as well as the similarities and differences of the two graphs.
Have students follow the directions below:
You and your group will have your own pack of candy or handful of paper squares. Place your wrappers on a chart, stacking the wrappers horizontally and then vertically. The wrapper of each candy bar should be placed neatly side by side. Choose which way you like better to display your data. Glue the wrappers or squares on your chart and label the bar graph appropriately.
• Write a paragraph explaining at least three observations from your data. Write another paragraph that tells which type of bar graph you prefer and why. Discuss the similarities and differences between the two graphs.
Questions/Prompts for Formative Student Assessment
• What was the purpose of the experiment?
• Describe the process you used to collect the data.
• Did you like creating the bar graph or the pictograph better? Why?
• What did the data tell us?
Questions for Teacher Reflection
• Did students accurately construct the bar graphs with appropriate labels?
• Can students explain the similarities and differences between horizontal and vertical bar graphs?
• Did students write an effective paragraph recording meaningful observations?
DIFFERENTIATION
Extension
• Have students represent the data using tally marks and a frequency table for the type of candy/color paper squares that were chosen.
Intervention
• Have students work with a partner and be sure they understand correct labeling of the graphs with both orientations.
TECHNOLOGY CONNECTION
• Allows you and the students to create charts and graphs.
• Graph paper that can be easily printed
PERFORMANCE TASK: Building Houses
STANDARDS ADDRESSED
M5A1. Students will represent and interpret the relationships between quantities algebraically.
b. Investigate simple algebraic expressions by substituting numbers for the unknown.
c. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimals) substituted for the variable.
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How do we determine the most appropriate graph to use?
• How will we interpret a set of data?
• Are there patterns in a set of data?
MATERIALS
• Pattern blocks (squares and triangles) or virtual manipulatives
• “Building Houses” Recording Sheet
• Graph paper
• Colored pencils, crayons, and/or markers
• Poster paper or chart paper (one per small group)
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students will explore numeric and geometric patterns with this task. Given the first four house styles, students are to continue the pattern to create the next house pattern. Then students will fill in building information for house styles 1-10, 20, 50, and 100.
As students work, they will be asked to think about how they see the pattern growing. What parts of the building stay the same, what parts of the building changes? When finished, students will work in small groups to share their thinking and solutions.
Comments
It may be easier to use virtual manipulatives with this task because most pattern block sets do not have an abundant supply of squares. One possible web site is .
Alternately, students could use colored square tiles with the green triangles from the pattern blocks.
Students will use the “Work Space” on the Building Houses Recording Sheet (see sample on right) to record the operations they used to find the total number of blocks in each style of house. Some possible solutions are shown below.
• If students see one row of 3 blocks and 4 more on top, they might describe the pattern as, “3 times the style number of the house plus 4 or 3x + 4 equals the number of blocks.”
• If students see 4 squares and 3 triangles they might describe the pattern as, “4 plus 3 triangles plus 3 square blocks added to each style.” Or Style l as 4 + 3, Style 2 as 4 + 3 + 3, Style 3 as 4 + 3 + 3 + 3, Style 4 as 4 + 3 + 3 + 3 + 3. Students may then recognize the pattern as 4 + 3 times the style number of the house or 4 + 3x = the number of blocks.
• If students see one row of 3 blocks times 3 minus two empty spaces as the picture above shows, they might describe the pattern as, “3 times the number of rows minus 2.” They might determine that the total number of rows is always equal to the style number plus 2. So their work space may show 3(x + 2) – 2.
Background Knowledge
Be aware that if you were to apply the rules of algebra to any of these representations of the pattern, you would find the same equation y = 3x + 4. If the subtraction method is used, 3(x + 2) – 2 = 3x + 6 – 2 which equals 3x + 4, the equation in simplest form is always the same.
As students plot their findings on the coordinate grid, please note that these points should not be connected to create a line graph. The points represent the number of blocks required to build the house. There is no house that uses a fraction of a block. Therefore, the graph would not be continuous; it would be made up of separate points. (In this case the graph is made of discrete points.)
Task Description
Have students follow the directions that appear on the Building Houses Recording Sheet:
• As a builder, you have several variations of a building as shown below.
Style 1 Style 2 Style 3 Style 4
• Following the pattern above, create house style 5. Then fill in the table for house styles.
• Describe how to find the number of blocks needed for any style house. (Use x to represent the style number of the house.)
• Using the values for x and y in the table above, plot the first ten ordered pairs in the grid below. Be sure to label the scale, the axes, and give your graph a title. (The table and grid are both shown on the Recording Sheet.)
• Describe what the graph tells you about the house styles and explain any patterns that you notice.
Questions/Prompts for Formative Student Assessment
• How do you see the pattern growing?
• How can you write a sentence describing the growth?
• What values can you substitute for the sentence you wrote?
• Which blocks are always the same?
• Which blocks are being added?
• Can you predict how the 7th style will look? The 15th?
• Is there more than one way to determine the total number of blocks? How?
• How is the coordinate grid different from a regular line graph?
• Why are there no lines connecting the points on the grid?
Questions for Teacher Reflection
• Are students able to determine a pattern?
• Can students describe how to find the total number of blocks for any style?
• Do students understand the difference in the graph on a coordinate grid and a traditional line graph?
• Do students understand that the graph on the coordinate grid can be used to predict data?
DIFFERENTIATION
Extension
• Have students create their own pattern to use as they follow the same steps as required in this task.
• Have students continue to find alternate ways to determine the total number of blocks in any given style house.
Intervention
• Not all students may complete all steps of this task. Some students may need to sketch or build each house style through 10. If this is the case, they may not be able to determine the expression to represent the number of blocks. Allow students to move to the graph if they have at least the first ten ordered pairs on their table.
• Have students work with a more simple pattern first.
TECHNOLOGY CONNECTION
Students are able to choose a grid and plot the points collected from the task. (Note: To allow the first ten ordered pairs to be plotted, right click on the graph and choose a 1:5 ratio of the x-axis to the y-axis so it will look like this:
Name _______________________________________ Date ____________________________
Building Houses Recording Sheet
As a builder, you have several variations of a building as shown below.
Style 1 Style 2 Style 3 Style 4
1. Following the pattern above, create house style 5. Then fill in the table for house styles.
|House Style Number |Work Space |Number of Blocks (y)|
|(x) |(How did you find the total number of | |
| |blocks?) | |
|1 | |7 |
|2 | |10 |
|3 | | |
|4 | | |
|5 | | |
|6 | | |
|7 | | |
|8 | | |
|9 | | |
|10 | | |
| | | |
|20 | | |
| | | |
|50 | | |
| | | |
|100 | | |
| | | |
|x | | |
2. Describe how to find the number of blocks needed for any style house. (Use x to represent the style number of the house.)
3. Using the values for x and y in the table above, plot the first ten ordered pairs in the grid below. Be sure to label the scale, the axes, and give your graph a title.
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
4. Describe what the graph tells you about the house styles and explain any patterns that you notice.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
PERFORMANCE TASK: Survey Says
STANDARDS ADDRESSED
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M5P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How are data collected?
• How do we conduct an experiment or survey?
• How do we determine who should take our survey and what our survey should be about?
• What is a sample?
• How will we interpret a set of data?
• How do graphs help explain real-world situations?
MATERIALS
• Poster-sized paper to display work
• Crayons, markers and/or colored pencils
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will create a survey question and collect data from at least ten other people. Using the responses given, students will display their data using tally marks, a frequency table, a bar graph and a pictograph. Students will also write a paragraph about the data that were collected. Have students illustrate the page containing the data and the paragraph.
Comments
Approve the question before the students begin collecting the data to insure that the survey question is appropriate and will yield adequate results.
You may want to create a rubric with which to grade student projects. Here are some web sites that might be helpful.
Background Knowledge
Students should have prior knowledge of a survey, population, and sample size.
Task Description
Have students follow the directions below:
• Create a survey question and collect data from at least ten other people.
• Using the responses given, display their data using tally marks, a frequency chart, a bar graph and a pictograph. Be sure to include all appropriate labels and a title.
• Write a paragraph about the data that was collected.
• Illustrate the page containing the data and the paragraph.
Questions/Prompts for Formative Student Assessment
• How did you decide what your survey question would be?
• How did you chose who would answer your survey?
• If you asked a different group of people, would your results be different? Explain.
• If you did another survey, would you rephrase your question? Why or why not?
• Which of the graphs do you think most effectively shows your data? Why?
• What observations can you make about your survey results?
Questions for Teacher Reflection
• Do students have a clear understanding of all types of graphs represented?
• Do students understand which graphs are appropriate for a given set of data?
• Do students understand what makes a good survey question?
DIFFERENTIATION
Extension
• Have students create a larger survey. (i.e., five questions) and analyze the results of that survey.
• Have students hypothesize what they think the results of their survey will be compared to their actual results.
• Show the students a segment of “The Family Feud” and discuss how we use and collect data in the real world.
• Use the available websites to play Family Feud as a class.
Intervention
• Have students work with a partner as they formulate their question and collect the data.
• Some students may need a review of the different types of graphs they will be required to construct.
TECHNOLOGY CONNECTION
• How to create a bar graph from an excel worksheet.
• Allows you and the students to create charts and graphs.
• Directions for creating a pictograph using Microsoft Excel
• Example of a pictograph
UNIT ONE CULMINATING TASK
PERFORMANCE TASK: Solo Graphing
This culminating task represents the level of depth, rigor and complexity expected of all fifth grade students to demonstrate evidence of learning.
STANDARDS ADDRESSED
M5A1. Students will represent and interpret the relationships between quantities algebraically.
c. Determine that a formula will be reliable regardless of the type of number (whole numbers or decimals) substituted for the variable.
M5N5. Students will understand the meaning of percentage.
b. Apply percents to circle graphs.
M5D1. Students will analyze graphs.
a. Analyze data presented in a graph.
b. Compare and contrast multiple graphic representations (circle graphs, line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs) for a single set of data and discuss the advantages/disadvantages of each.
M5D2. Students will collect, organize, and display data using the most appropriate graph.
M5P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M5P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M5P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
ESSENTIAL QUESTIONS
• How do graphs help explain real-world situations?
MATERIALS
• Graph paper
• Poster paper
• Markers, crayons and/or colored pencils
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this culminating task, students will choose an appropriate data collection procedure and conduct their own experiment. Students will then organize the data, display it in multiple ways, and discuss what they learned based on their analyses and observations of the data.
Comments
While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them.
Students should have the opportunity to brainstorm ideas for their project, with teacher direction. Students should answer questions pertaining to their project.
Background Knowledge
At this point students should have prior knowledge of the following means of collecting data: experiment, observation and surveys. Students should be able to analyze, interpret, compare, and display data.
Have students follow the directions below:
To complete this task, you must:
• Think of your own experiment, survey or observation about which you will collect your own data.
• Display your data in multiple ways.
• Describe the procedure you used to conduct the experiment.
• Explain as much as you can about the data collected.
Some examples of observations, survey questions and experiments are listed below. You may choose one of these examples, or you may create your own.
Observations
• How many times did someone use the pencil sharpener? Record whether it was a girl, boy or teacher.
• How does a pattern change? (i.e., create a design showing change over time like the “Building Houses” task earlier in this unit)
• How many times did the teacher say certain words ? (i.e. survey, data, math)
• How many times class was interrupted on a given day and by whom (i.e. teacher, students, phone, intercom, etc.)
Survey Questions (Have students create choices to limit response data)
• What is your favorite color? (i.e., blue, green, red, yellow, or black)
• How many siblings do you have?
• How many hours of the day do you talk on the phone, watch TV, play video games, play outside, do homework etc…?
• What is your favorite subject?
• What is your favorite food?
Experiments
• Create 3 different types of airplanes and describe the distance traveled between the 3 of them, and measure the distance of flight between the airplanes, be advised, you can fly the paper airplanes as many times as you like to obtain your answer.
• How many jumping jacks can 10 students do in 30 seconds?
• Measure the molding process between 3 pieces of bread, put one in the refrigerator, one in a cabinet, and one outside to determine which bread molds at a faster pace.
Questions/Prompts for Formative Student Assessment
• How did you conduct the experiment?
• How did you conduct the survey?
• How did you choose what your survey would be about?
• How did you decide who would take your survey?
• What did you learn from your data?
• How would you have done your experiment/survey differently if you had to do it again?
• Is your sample size sufficient to draw conclusions for the entire student body?
Questions for Teacher Reflection
• Do students have a clear understanding of all types of graphs?
• Do students understand which graphs are appropriate for a given set of data?
• Can students explain the advantages and disadvantages of different types of graphs?
• Can students explain the various elements of their graphs?
• Can students interpret data within a given graph?
DIFFERENTIATION
Extension
• Have students present their project, data and interpretations from the data to the class or grade level.
• Have students collect data and/or represent data as part of a persuasive argument
Remediation
• Depending on the level of student ability, it is at the teacher’s discretion whether or not to assign survey questions and or experiment as opposed to allowing the students to chose their own.
TECHNOLOGY CONNECTION
• Allows you and the students to create charts and graphs.
• Students are able to choose a grid and plot the points collected from the task.
-----------------------
MATHEMATICS
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- wyoming unit 1 elk
- physical science unit 1 test
- ap psychology unit 1 progress check frq
- physical science unit 1 quizlet
- entrepreneurship unit 1 notes
- entrepreneurship unit 1 test
- unit 1 ap psychology test
- biology unit 1 test review
- ap bio unit 1 test
- biology unit 1 test answers
- ib biology unit 1 test
- ap biology unit 1 notes