Unit 1 Organizer:
|Grade 4 Mathematics Frameworks |
|Unit 4 |
|Geometric Figures, Plane Coordinates, and Data |
Unit 4
GEOMETRIC FIGURES, PLANE COORDINATES, AND DATA
(6 weeks)
TABLE OF CONTENTS
Overview 3
Key Standards & Related Standards 4
Enduring Understandings 5
Essential Questions 6
Concepts & Skills to Maintain 7
Selected Terms and Symbols 7
Classroom Routines 9
Strategies for Teaching and Learning 9
Evidence of Learning 9
Tasks 10
• Cube It! 11
• Be an Expert! 16
• Quadrilateral Challenge 26
• My Many Triangles 34
• Polygon Challenge 41
• Shoo Fly 46
• Measuring Up the Data 54
• And the Survey Says... 60
• Is Something Missing? 65
• Tell Me About My State! 71
Culminating Task - Geometry Town 76
OVERVIEW
In this unit students will:
• Identify and classify angles
• Distinguish between parallel and perpendicular lines and use them in geometric figures
• Identify differences among quadrilaterals
• Locate points in the coordinate plane
• Graph ordered pairs in the first quadrant
• Identify different types of graphs
• Define a graph as either a bar, line, or pictograph
• Collect data and create line, line plot, bar, and pictographs
• Compare and contrast line, line plot, bar, and pictographs
• Show evidence of finding missing data in graphs
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 develop their understanding of number and computational fluency.
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
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
a. Examine and compare angles in order to classify and identify triangles by their angles.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4G2. Students will understand fundamental solid figures.
a. Compare and contrast a cube and a rectangle prism in terms of the number and shape of their faces, edges, and vertices.
b. Describe parallel and perpendicular lines and planes in connection with the rectangular prism.
c. Build/collect models for solid geometric figures (cubes, prisms, cylinders, pyramids, spheres, and cones) using nets and other representations.
M4G3. Students will use the coordinate system.
a. Understand and apply ordered pairs in the first quadrant of the coordinate system.
b. Locate a point in the first quadrant in the coordinate plane and name the ordered pair.
c. Graph ordered pairs in the first quadrant.
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
a. Construct and interpret line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs.
b. Investigate the features and tendencies of graphs.
c. Compare different graphical representations for a given set of data.
d. Identify missing information and duplications in data.
e. Determine and justify the range, mode, and median of a set of data.
RELATED STANDARDS
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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
• Figures are classified by their properties.
• The analysis and interpretation of data relates to the type of graph used to display the information.
• On the coordinate plane, a point represents the two facets of information associated with an ordered pair.
• Graphical representations can be used to make predictions and interpretations about real world situations.
ESSENTIAL QUESTIONS
• How do other shapes and their attributes play a part in the composition of a cube?
• What is the connection between a cube and a rectangular prism?
• What properties do solid figures have in common?
• How are solid figures different from plane figures?
• How can solid figures be categorized and classified?
• What is a quadrilateral?
• How can you create different types of quadrilaterals?
• How are quadrilaterals alike and different?
• What are the properties of quadrilaterals?
• How can angle and side measures help us to create and classify triangles?
• How can plane figures be combined to create new figures?
• How does combing figures affect the attributes of those figures?
• How does the coordinate system work?
• How can the coordinate system help you better understand other map systems?
• What are some ways we make sense of data?
• How do we choose the best graph to represent our data?
• Why are there different types of graphs?
• How are different graphs alike and different?
• How are the labels of graphs determined?
• When can events be displayed on a line graph?
• How do we use data and graphs to answer questions?
• When can information be displayed on more than one type of graph?
• How do we use data and graphs to answer questions?
• Which graph should you choose to represent data?
• Where is geometry found in your everyday world?
• How do coordinate grids help you organize information?
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/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.
Correctly name figures by number of sides and vertices.
Identify the properties of fundamental geometric figures.
Transfer data from charts to graphs and graphs to charts.
Understand that graphs are a visual representation of information called data.
Identify line plot, bar, or pictograph by name.
Interpret data from graphs.
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.
Definitions for these and other terms can be found on the Intermath website, a great resource for teachers. Because Intermath is geared towards middle and high school, grade 3-5 students should be directed to specific information and activities.
← Bar Graph: A way of displaying categorical data using horizontal or vertical bars so that the height or length of the bars indicates its value.
← Chart: A visual representation of data, in which "the data are represented by symbols, such as bars in a bar chart or lines in a line chart. (from: )
← Data: Facts or numbers collected to describe something.
← Edges: The intersection of two surfaces of a 3-D geometric figure.
← Face: The plane figures that make up a solid figure.
← Hexagon: A six-sided polygon.
← Irregular Polygon: A polygon that has sides and/or angles of differing sizes.
← Line Graph: A graph that uses line segments to connect data points.
← Parallel Lines: Two lines in a plane that do not intersect (are always the same distance apart).
← Pentagon: A five-sided polygon.
← Perpendicular Lines: Two lines that intersect to form a right angle.
← Pictograph: A graph that displays data in a table using symbols. (It uses the same symbol to represent all of the categories. Each picture may represent more than one item.
← Plane figure: A figure of which all points lie in the same plane. (Plane figures included in the Grade 4 GPS are quadrilaterals – parallelograms, squares, rectangles, trapezoids,
rhombi – as well as triangles, pentagons, hexagons, and irregular polygons.)
← Polygon: A closed plane figure having three or more straight sides.
← Quadrilateral: A four-sided polygon.
← Regular Polygon: A polygon that is equiangular (all of the angles are congruent) and equilateral (all of the sides are congruent).
← Solid Figure: A three-dimensional geometrical figure. Solid figures in the Grade 4 GPS include cubes, prisms, cylinders, pyramids, cones, and spheres.
← Sphere: A three-dimensional figure with all points equidistant from a point called the center.
← Table: A way to organize data into columns and/or rows.
← Trapezoid: A quadrilateral with only one pair of parallel sides.
← Vertex: A corner of a geometric figure. (Plural: Vertices)
CLASSROOM ROUTINES
The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities as graphing class achievements, noticing parallel and perpendicular lines within the school, finding quadrilaterals and other plane figures, and interpreting data from various graphs posted in school. They should also include less obvious routines, such as how to work cooperatively, how to use materials in a productive manner, and how to put materials away. 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 the routines is important to the development of students’ understanding of expectations which will support students’ performances on the tasks in this unit.
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 numbers, 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:
• Graph data in the first quadrant.
• Name quadrilaterals (square, rectangle, parallelogram, trapezoid, and rhombus).
• Identify parallel and perpendicular lines in plane geometric figures.
• Use nets to create 3-D figures.
• Sort geometric figures by their properties.
• Identify, create and interpret line graphs, line plot graphs, bar graphs, Venn diagrams, and pictographs.
• Interpret data from different types of graphs.
• Make logical predictions from information presented in a graph.
TASKS
The following tasks represent the level of depth, rigor, and complexity expected of all fourth grade students. These tasks or tasks 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).
|Task Name |Task Type |Content Addressed |
| |Grouping Strategy | |
|Cube It! |Learning Task |Build a cube and |
| |Partner Task |identify related shapes and attributes |
|Be an Expert! |Learning Task |Analyze solid figures |
| |Small Group Task | |
|Quadrilateral Challenge |Performance Task |Create and compare quadrilaterals |
| |Partner/Small Group Task | |
|My Many Triangles |Learning Task |Classify triangles by sides and angles |
| |Partner/Small Group Task | |
|Polygon Challenge |Performance Task |Construct and compare new polygons by combining simple polygons|
| |Individual/Partner Task | |
|Shoo Fly |Learning Task |Locate points on a coordinate grid |
| |Partner/Small Group Task | |
|Measuring Up the Data |Learning Task Partner/Small Group Task |Determine the meaning of median, mode, and range for a set of |
| | |data |
|And the Survey Says… |Performance Task |Use a survey to create the most appropriate graphs for specific|
| |Partner/Small Group |uses |
|Is Something Missing? |Performance Task |Analyze graphs to determine missing information |
| |Individual/Partner Task | |
|Tell Me about My State! |Performance Task |Create and compare types of graphs to analyze data |
| |Partner Task | |
|Culminating Task: |Performance Task |Design a map on a coordinate grid using multiple geometric |
|Geometry Town |Individual Task |figures |
LEARNING TASK: Cube It!
Cube illustrations for this task are from:
STANDARDS ADDRESSED
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4G2. Students will understand fundamental solid figures.
a. Compare and contrast a cube and a rectangle prism in terms of the number and shape of their faces, edges, and vertices.
b. Describe parallel and perpendicular lines and planes in connection with the rectangular prism.
c. Build/collect models for solid geometric figures (cubes, prisms, cylinders, pyramids, spheres, and cones) using nets and other representations.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 other shapes and their attributes play a part in the composition of a cube?
• What is the connection between a cube and a rectangular prism?
MATERIALS
• Six square pieces of colored paper (each piece a different color) for folding (4", 5", or 6" sizes of origami paper work best)
• “Cube It!” student sheet
GROUPING
Whole Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will explore a cube through the construction process using origami methods. Students are introduced or re-introduced to plane figures and their attributes through the folding process.
Comments
“Cube It!” can be an introductory lesson in learning the attributes of different quadrilaterals. As students work with origami paper, they start with a square, fold rectangles and triangles, and end with a parallelogram. When one triangle is folded down from the parallelogram a trapezoid is folded. Students can also discuss the fact that a square is a special type of rhombus with four right angles. In this way, all of the quadrilaterals students should be aware of can be discussed through this task.
All six squares must be folded identically, so that the students will be able to create a cube. It may be helpful to have students fold the first square as the steps are modeled to the class. This will allow the students the opportunity to discuss the types of quadrilaterals that are formed as the square is folded and to describe the properties of the quadrilaterals. Keep a list of properties on the board or chart paper and encourage students to make connections among the shapes throughout the process. When students become familiar with the folds, they can complete the other square pieces of paper on their own.
Once all six squares have been folded, the students can assemble cube as the process is being modeled for the class. Once each student has created a cube, they can discuss the properties of a cube including the number of edges, vertices, and faces. Students can also consider which pairs of edges or which pairs of faces of their cube are parallel or perpendicular. Finally, given a rectangular prism, students can compare/contrast the two figures.
Background Knowledge
Following the directions from the sources below, teachers should create the cube themselves prior to teaching the lesson. In addition to the directions provided on the “Cube It!” student sheet (), teachers may also want to view the following video:
Students should have experience identifying and exploring plane and solid figures and their properties. Also students should be able to identify perpendicular and parallel lines.
Task Directions
Students will follow the steps as they are modeled for them, and then use the directions from the “Cube It!” student sheet to remind them of the steps when folding subsequent squares.
Once each student has created a cube, they can discuss the properties of a cube including the number of edges, vertices, and faces. Students can also consider which pairs of edges or which pairs of faces of their cube are parallel or perpendicular. Finally, given a rectangular prism, students can compare/contrast the two figures.
Questions/Prompts for Formative Student Assessment
• How does this shape relate to the previous one before the last fold?
• Which quadrilaterals do you see?
• What are the most common properties of the shapes throughout the folding process?
• Can this process apply to any other solid figure? Why or why not?
• Can you identify two edges (or faces) that are parallel (or perpendicular)? How do you know?
Questions for Teacher Reflection
• Could students readily identify the properties of a cube as well as other quadrilaterals recognized in the folding process?
• Were students able to make connections between the different quadrilaterals folded?
DIFFERENTIATION
Extension
Have students research the Origami method and create additional folded figures outlining the shapes created throughout the process. There are several websites students could visit in the “Technology Connection” section below.
Intervention
Allow students to be in a small group where the folds can be modeled more than once and immediate help is available if needed.
TECHNOLOGY CONNECTION
• Short video shows the cube being made.
• Written directions with illustrations for creating this origami cube.
• Very clear photographs showing the steps required to make the cube, but the method is slightly different.
• The Origami Club provides over 300 examples of foldable figures and has animated videos showing the process as well.
• Eric’s Origami Page provides more information about paper folding and its connection to math. Detailed history and current uses are also provided.
• Provides pictures of solid figures and polyhedrons created, based on the Sonobe Origami module.
Name _________________________________________ Date __________________________
Cube It!
Directions for making an origami cube are given below. The following illustrations and full directions can be found at the following web site:
LEARNING TASK: Be an Expert!
STANDARDS ADDRESSED
M4G2. Students will understand fundamental solid figures.
a. Compare and contrast a cube and a rectangle prism in terms of the number and shape of their faces, edges, and vertices.
b. Describe parallel and perpendicular lines and planes in connection with the rectangular prism.
c. Build/collect models for solid geometric figures (cube, prisms, cylinder, pyramids, spheres, and cones) using nets and other representations.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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
• What properties do solid figures have in common?
• How are solid figures different from plane figures?
• How can solid figures be categorized and classified?
MATERIALS
• “Be an Expert! Geometric Characteristics Graphic Organizer” student recording sheet
• Electronic version or poster of “Be an Expert! Geometric Characteristics Graphic Organizer” student recording sheet
• A collection of geosolids and/or items that are solid figures (15-20 would work best)
• Nets of each solid (cube, cylinder, cone, prism, pyramid)
GROUPING
Small group task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students will review solid figures introduced in previous grades and further explore their properties through categorization, classification, deconstruction, and properties description.
Comments
As an introduction, each group of students can be given a set of geometric solids. Students can sort the solids into groups. They may also be asked to identify additional items in or out of the classroom that might fit into each group they create. Students can describe their sort to their classmates, defending their placement of each figure. (Students could draw a circle around each group so that other students can see the figures and how they were sorted.)
Once each group of students has chosen (or has been assigned) a particular figure, make the copies of the nets for available to the students. Allow students to choose the net they think belongs to their figure. Students can cut and fold the nets. (Any part of the figures shaded gray is a glue tab, even the large surface on the cone.) If a student chooses a net that doesn’t match their figure, ask the student about his choice and talk about what will be required in the net before the student goes to choose another net.
Once groups have finished their graphic organizer, allow each group to share what they learned about their figure and post their work in the classroom as a reference for the students.
Background Knowledge
In previous grade levels, students should have been introduced to the basic solid figures (cube, cone, sphere, cylinder, prism, and pyramid). Therefore, they should be able to identify an example of each. Student should also be able to sort and classify figures and use simple graphic organizers.
Task Directions
Students will follow directions below from the “Be an Expert! Geometric Characteristics Graphic Organizer” student recording sheet.
Your task is to become an expert on one geometric solid. Each group will have a different solid. You will need to complete the following parts of this task in order to become an expert on your solid. Then you will need to share your expertise with your classmates.
You will be given a model of your solid and a net for your figure. With your materials determine the following:
□ Write the name (names) of your figure in the center of your graphic organizer.
□ Complete the graphic organizer for your figure.
□ Make your figure by folding and gluing or taping the net closed. Attach it to the “Examples section of your graphic organizer.
□ Keep one net unfolded and attach it to the “Examples” section of your graphic organizer.
□ For “Examples” and “Non-examples” think about objects in the real world.
□ Be able to defend any information on your graphic organizer.
□ Post your graphic organizer in the classroom, plan how you will share your expertise with your classmates.
Geometric Characteristics Graphic Organizer:
Questions/Prompts for Formative Student Assessment
• What characteristics did you use to group your solid figures?
• What other items could be added to this group? Why?
• What are the properties of your figure? What properties does your figure have that all (prisms, pyramids) don’t have?
• Where do you see your figure in the real world?
• Would a (triangle, rectangle, circle) be an example of your figure? Why? Why not?
Questions for Teacher Reflection
• Can students consider more than one attribute at a time?
• Can students justify the placement of the solids in their groups?
• Which students can complete the graphic organizer accurately?
• Which students cans show how their figure is similar to/different from other figures?
• Are students able to recognize the difference between essential and non-essential properties for their figure?
DIFFERENTIATION
Extension
• Have students create the solid figures with nets. Then they may combine the solids to create new solid figures and name and define the figure by its attributes.
Intervention
• Have students trace each side of a solid figure before providing nets. Compare the tracings to sort the figures.
• Ask students to build 3-D figures using straws and pipe cleaners. (Cut the pipe cleaners in half. Each pipe cleaner half can be pushed into the ends of two straws to connect two edges.) Use these figures to identify edges and vertices. Ask students to trace around the faces of the figure to identify the shapes that make up the faces.
TECHNOLOGY CONNECTION
• This applet allows students to “fold” nets (triangular pyramid and cube) to form geometric solids. Some are not required for fourth grade but interesting to view.
• In addition to very complex figures, basic nets of figures such as a triangular prism and a hexagonal pyramid are provided.
• A lesson from NCTM which asks students to create nets using dot paper and analyze the properties of 3-D figures. Part of a unit on geometric solids .
• Printable nets, used below.
• Teacher background information on solids and nets.
Name _________________________________________ Date __________________________
Be an Expert!
Task Directions
Your task is to become an expert on one geometric solid. Each group will have a different solid. You will need to complete the following parts of this task in order to become an expert on your solid. Then you will need to share your expertise with your classmates.
You will be given a model of your solid and a net for your figure. With your materials determine the following:
□ Write the name (names) of your figure in the center of your graphic organizer.
□ Complete the graphic organizer for your figure.
□ Make your figure by folding and gluing or taping the net closed. Attach it to the “Examples section of your graphic organizer.
□ Keep one net unfolded and attach it to the “Examples” section of your graphic organizer.
□ For “Examples” and “Non-examples” think about objects in the real world.
□ Be able to defend any information on your graphic organizer.
□ Post your graphic organizer in the classroom, plan how you will share your expertise with your classmates.
Geometric Characteristics Graphic Organizer:
Group Members _____________________________________________________________________ Date _________________________________________
Be an Expert!
Be an Expert!
3-D Figure Nets
Be an Expert!
3-D Figure Nets
Be an Expert!
3-D Figure Nets
Be an Expert!
3-D Figure Nets
PERFIORMANCE TASK: Quadrilateral Challenge
Adapted from a lesson by Amanda Grant, Eagle Springs ES, Houston County Schools
A video of this lesson can be found at:
STANDARDS ADDRESSED
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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
• What is a quadrilateral?
• How can you create different types of quadrilaterals?
• How are quadrilaterals alike and different?
• What are the properties of quadrilaterals?
MATERIALS
For Each Group:
• “Quadrilateral Challenge, Quadrilateral Mat” student recording sheet (print on legal-size paper)
• “Quadrilateral Challenge, Quadrilateral Properties” student sheet (One sheet for every two groups; cut along dotted line; print on colored paper)
• Rulers
• Protractors
• Index cards
• Scissors
• Glue sticks
• Wikki Sticks (optional, approximately five sticks per group) OR dot paper
• “Quadrilateral Challenge, Dot Paper” student recording sheet (optional)
Comments
Students will need either Wikki Sticks (or other materials with which to create their quadrilaterals) OR the “Quadrilateral Challenge, Dot Paper” student recording sheet so that students can use to draw, cut out, and glue their quadrilaterals to their “Quadrilateral Mat.”
Copy the “Quadrilateral Mat” student recording sheet onto white legal-size paper. To contrast the white paper, the “Quadrilateral Properties” student sheet should be copied onto colored paper; otherwise crayons need to be provided so that students can color the quadrilaterals before they cut them out.
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
The purpose of this task is for students to become familiar with the properties of quadrilaterals. Working in pairs, students will create the following quadrilaterals: parallelogram, rhombus, square, rectangle, trapezoid. They will identify the attributes of each quadrilateral, then compare and contrast the attributes of different quadrilaterals.
Comments
Students will work in pairs. They will first construct an example of each quadrilateral (parallelogram, square, rectangle, trapezoid, and rhombus) on their quadrilateral mats. After they have created each quadrilateral, the students will determine the properties of each figure by sorting the property cards. Students need to sort all of the property cards and match them to the correct quadrilateral before they glue the cards underneath the figures that they have created.
Once students finish their Quadrilateral Mats, students should choose 2 quadrilaterals and write to compare and contrast their properties.
Background Knowledge
Students should have the following background knowledge.
• Be able to use a straight edge or ruler to draw a straight line.
• Know how to use a protractor, a ruler, and how to identify right angles (90 degrees), obtuse angles, and acute angles (using a protractor or the corner of an index card).
• Understand that opposite sides can not touch each other; they are on opposite sides of the quadrilateral.
• Know parallel means that lines will never intersect or cross over each other no matter how long they are extended. (Students may prove that lines are parallel by laying down 2 straight objects, such as rulers, on the parallel sides of the quadrilateral, extending those sides. This will show how the line segments do not intersect even if they are extended.)
• Understand that perpendicular means lines or segments intersect or cross forming a right angle. (Some students may use a protractor, while others may use the corner of an index card or the corner of a sheet of paper to show an angle is a right angle.)
• Know that a property is an attribute of a shape that is always going to be true. It describes the shape.
• Be able to use a ruler to measure sides to verify they are the same length.
Students should have some experience with the properties of quadrilaterals as shown in the diagram below.
Some properties of quadrilaterals that should be discussed are included below. As students draw conclusions about the relationships between different figures, be sure they are able to explain their thinking and defend their conclusions.
• A shape is a quadrilateral when it has exactly 4 sides and is a polygon. (To be a polygon the figure must be a closed plane figure with at least three straight sides.)
• A square is always a rectangle because a square will always have 4 right angles like a rectangle.
• A rectangle does not have to have 4 equal sides like a square. It can have 4 right angles without 4 equal sides. Therefore, rectangle is not always a square.
• A square is always a rhombus because it has 4 equal sides like a rhombus and it is also a rectangle because it has 4 right angles like a rectangle.
• A rhombus does not have to have right angles like a square. It can have 4 equal sides without having 4 right angles. Therefore a rhombus is not always a square.
• A trapezoid can never be a parallelogram because a trapezoid has only one pair of opposite sides that are parallel and a parallelogram has two pairs of opposite sides parallel.
• A parallelogram can be a rectangle if it has 4 right angles.
Task Directions
Students will create at least one quadrilateral (using a material such as Wikki Sticks or dot paper) for each section of the “Quadrilateral Mat” Then students will cut, sort, and glue three properties for each quadrilateral from the “Quadrilateral Properties” student sheet. Finally students will write to compare and contrast the properties of two quadrilaterals.
Questions/Prompts for Formative Student Assessment
• How do you know this quadrilateral is a ________ (square, rectangle, parallelogram, trapezoid, or rhombus)?
• How did you create your quadrilaterals?
• Is there a quadrilateral that was easier/harder to create than others? Why?
• How do you know these are quadrilaterals?
• How are quadrilaterals different from other 2-D shapes? How are they the same?
• What is meant by the term “opposite sides”?
• What does “parallel” mean? How can you show that those sides parallel?
• What does “perpendicular” mean? How can you show that those sides are perpendicular?
• How can you show that 2 sides are equal?
• What are some ways we can show an angle is a right angle?
Questions for Teacher Reflection
• Were students able to easily create the five quadrilaterals?
• Were students able to identify similarities and differences between two quadrilaterals?
• Were students able to identify right angles, parallel and perpendicular lines, and equal sides in a figure?
• Did students accurately complete the “Quadrilateral Mat”?
DIFFERENTIATION
Extension
• Ask students to create a Venn diagram which contains a comparison of the properties of two quadrilaterals.
Intervention
• Allow students to list similarities and differences of two quadrilaterals rather than write a paragraph.
• Help students organize the quadrilateral properties before placing them on the mat. There are several that are the same. Have students place like properties in a pile and then decide which shape has that particular property. Place all of one property on the mat before moving to another property.
TECHNOLOGY CONNECTION
• A classroom video showing the Quadrilateral Challenge task developed into a lesson.
• A summary of the properties of several quadrilaterals.
• A virtual “Quest” where students match properties with quadrilaterals.
|Square |My Quadrilateral Sort Map |Rectangle |
| | | |
| | | |
| | | |
| |Names _________________________________________ | |
| | | |
|Rhombus |Trapezoid |Parallelogram |
| | | |
Quadrilateral Challenge
Quadrilateral Properties
Quadrilateral Challenge
Dot Paper
LEARNING TASK: My Many Triangles
STANDARDS ADDRESSED
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
a. Examine and compare angles in order to classify and identify triangles by their angles.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 QUESTION
• How can angle and side measures help us to create and classify triangles?
MATERIALS
• “My Many Triangles” student recording sheet
• “My Many Triangles, Triangles to Cut and Sort” student sheet
• White construction paper (one sheet per student or per pair of students)
• Colored construction paper cut into strips[pic] wide (each student will need approximately 10 strips of paper)
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
This task requires students to sort triangles according to common attributes and then create triangles according to two properties.
Part 1
Adapted from Van De Walle, J.A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and Middle School Mathematics: Teaching Developmentally 7th Ed. Boston: Pearson Education, Inc., p. 413-414.
Comments
As an introduction to this task, students can be asked to fold different types of triangles. Using a piece of plain paper, ask students if they can fold to create any of the following triangles. (Small pieces of plain paper can be used, approximately 4” x 4”.)
• Equilateral
• Right
• Acute
• Obtuse
Discuss how students know their triangle belongs to one or more of the categories listed above.
(For some children paper folding can be a little challenging at first. Reassure children that it is okay to make mistakes when folding and to persevere until they are successful. It will be necessary to have several extra pieces of paper available for all students. This task helps students become more confident in their spatial abilities.)
Background Knowledge
Students should be able to identify triangles by the lengths of their sides (isosceles, equilateral, and scalene) as well as by the measure of their angles (right, obtuse, and acute).
The type of each triangle on the “My Many Triangles, Triangles to Cut and Sort” student sheet are shown below.
Allow students to struggle a little bit with this part of the task. Students may need to try out a few possibilities before finding that lengths of sides and measures of angles are two ways to sort these triangles so that each triangle belongs to exactly one group when sorted.
Sorted according to side lengths Sorted according to angle measures
Equilateral triangles: 8, 12 Acute triangles: 3, 4, 8, 9, 12, 13
Isosceles triangles: 2, 3, 5, 6, 9, 14 Right triangles: 2, 5, 7, 10
Scalene triangles: 1, 4, 7, 10, 11, 13 Obtuse triangles: 1, 6, 11, 14
Task Directions
Cut out the triangles below. Sort the triangles into groups where there are no triangles that do not fit into a group and there are no triangles that belong to more than one group. Then sort the triangles in a different way. Again, there should be no triangles that do not fit into a group and no triangles that belong to more than one group. Record how you sorted the triangles and the number of the triangles in each group. Be able to share how you sorted the triangles.
Questions/Prompts for Formative Student Assessment
• How do you know this is a(n) ______ (isosceles, right, equilateral, etc.) triangle?
• Are there any triangles that don’t belong in a group?
• Are there any triangles that belong to more than one group?
• Can you think of another way to sort the triangles?
• What are some properties of this triangle? Can you use one of those properties to think of a way to group all of your triangles?
Part 2
Comments
Students may need some assistance using the chart to identify the triangles they need to create. Be sure students understand they need to attempt to make nine different types of triangles, two of which are not possible to create. Encourage students to try to make an equilateral obtuse angle and an equilateral right triangle so that they can see that it is not possible to create a three-sided closed figure with two obtuse angles or two right angles. (See below.)
Background Knowledge
Students will need to be able measure the sides and the angles in order to create the required triangles. Also, they will need to know that the sum of the measures of the angles of a triangle is 180( (see Unit 3).
Of the nine triangles, two are not possible.
• An equilateral right triangle is not possible because an equilateral triangle also has equal angle measures (equiangular). A triangle can have no more than 180( however, [pic] which is more than 180(.
• An equilateral obtuse triangle is not possible because an equilateral triangle has equal angle measures (equiangular). A triangle can have no more than 180( however, by definition an obtuse angle is greater than[pic]. Multiplying a number greater than[pic] by 3 will be greater than 180(.
Task Directions
Use the strips of construction paper to create the triangles described in each box below. Use the row label and the column label to identify the properties required for each triangle. For example, the box labeled “A” needs to be acute and isosceles because the row label is “Acute” and the column label is “Isosceles.”
Two triangles are not possible; for those, explain why each triangle is not possible on the lines below.
Glue each triangle onto the construction paper and label it.
Questions/Prompts for Formative Student Assessment
• Can you create an equilateral right triangle? An equilateral obtuse triangle? How do you know?
• Is there a scalene equilateral triangle? How do you know?
• How do you know this is a ___________ (i.e. scalene obtuse) triangle?
• How can you prove to use that this is a ___________ (i.e. scalene obtuse) triangle?
• If it is a ___________ (i.e. scalene obtuse) triangle, what is true about the length of its sides? The measures of its angles? Prove that the triangle you created has those attributes,
Questions for Teacher Reflection
• Are students able to identify the seven different types of triangles?
• Are students able to identify the attributes of the “Triangles to Cut and Sort” and use that information to sort them accurately?
• Are students able to describe why an obtuse equilateral triangle and a right equilateral triangle are not possible? Can they use what they know about the sum of the measures of the angles of a triangle to explain their thinking?
• Which students were successful at making the seven triangles with the strips of paper?
• Which students were able to measure segments and angles accurately?
DIFFERENTIATION
Extension
Challenge students to write directions for a triangle that they chose so that someone else could follow their directions and create the same triangle. Allow a partner to try these directions with to see how successful they were at describing how to create their triangle.
Intervention
Allow students to use a picture glossary or the triangles from Part 1 of this task to help them create the triangles for Part 2.
TECHNOLOGY CONNECTION
Gives basic definitions for the types of triangles. Note: this web site contains advertising.
My Many Triangles
Triangles to Cut and Sort
Cut out the triangles below. Sort the triangles into groups where there are no triangles that do not fit into a group and there are no triangles that belong to more than one group. Then sort the triangles in a different way. Again, there should be no
triangles that do not fit into a group and no triangles that belong to more than one group. Record how you sorted the triangles and the number of the triangles in each group. Be able to share how you sorted the triangles.
Name _________________________________________ Date __________________________
My Many Triangles
Use the strips of construction paper to create the triangles described in each box below. Use the row label and the column label to identify the properties required for
each triangle. For example, the box labeled “A” needs to be acute and isosceles because the row label is “Acute” and the column label is “Isosceles.”
Two triangles are not possible; for those, explain why each triangle is not possible on the lines below.
Glue each triangle onto the construction paper and label it.
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
PERFORMANCE TASK: Polygon Challenge
STANDARDS ADDRESSED
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
a. Examine and compare angles in order to classify and identify triangles by their angles.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 can plane figures be combined to create new figures?
• How does combing figures affect the attributes of those figures?
MATERIALS
• Pattern blocks (trapezoid, triangle, hexagon, and rhombus only)
• “Polygon Challenge” student recording sheet (two per student)
• Crayons or colored pencils
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
This task directs students to creating polygons using 8 pattern blocks. Students should begin moving toward creating regular polygons through the process.
Comments
Encourage students to create several polygons before choosing one to record on the student sheet. Students can explore the different polygons they can make by rearranging the pattern blocks within the same shape or by moving the same pattern blocks in different ways to make a variety of polygons.
Have students repeat this process for a second polygon. When finished, provide an opportunity for children to share their solutions for this task and to describe their problem solving strategies. Also, students can write a description of each polygon underneath the picture of it. Encourage students to list all of the properties of their polygon and to describe the process used to create it. Finally, students can create a Venn diagram for their two polygons allowing them to compare the polygons, listing their similarities and differences.
Teachers can facilitate the creation process by monitoring students closely as they work. The use of pattern blocks is encouraged as the materials allow kinesthetic learners to process their environment as well as scaffold their learning.
Background Knowledge
Students should be able to identify basic, regular and irregular, convex and concave polygons with ease. Students should also be familiar with pattern blocks and the expectations for their use.
To solve this problem, students should be able to trade two blocks for one block that is the same size and shape. They may need to do this when they have made a polygon, but used too many pattern blocks.
Possible solutions for a triangle (equilateral), quadrilateral (trapezoid), quadrilateral (parallelogram), pentagon, regular hexagon, irregular hexagon, and concave octagon are shown below.
Task Directions
Students will follow the directions below from the “Polygon Challenge” student task sheet.
Follow the requirements below to build a polygon using pattern blocks.
• Use only the hexagon, rhombus, trapezoid, and triangle pattern block shapes.
• Select 8 pattern blocks to use. (They can be in any combination.)
• Create a polygon using all 8 blocks.
• Draw and color it on the triangle paper below.
Questions/Prompts for Formative Student Assessment
• How are the two polygons you created alike? Different?
• Which eight pattern blocks are you using? How did you choose them?
• Can you use at least one of each of the four shapes to make a polygon?
• What problem solving strategy/strategies did you use to make your polygon?
Questions for Teacher Reflection
• Which students able to successfully complete this task?
• Are students able to flexibly replace two blocks with one or one block with two so that they have the correct number of pattern blocks?
• Which students showed an understanding of the properties of the polygon they created?
DIFFERENTIATION
Extension
• Have students create a pictograph or a bar graph to demonstrate the number and type of pattern blocks they used for their design. This would emphasize that the solution can be correct and not necessarily look the same.
• Using the length of the equilateral triangle (green triangle) as one unit, ask students to find the perimeter of their polygon.
Intervention
• Assign the regular polygons and pattern blocks you would like the student to use.
• Provide an outline of a polygon and allow the student to manipulate the pattern blocks to fill the outline.
TECHNOLOGY CONNECTION
Students could use either of the web sites below to manipulate Pattern Blocks virtually.
•
•
• Allows user to create different types of graph paper.
Name _________________________________________ Date __________________________
Polygon Challenge
Follow the requirements below to build a polygon using pattern blocks.
• Use only the hexagon, rhombus, trapezoid, and triangle pattern block shapes.
• Select 8 pattern blocks to use. (They can be in any combination.)
• Create a polygon using all 8 blocks.
• Draw and color it on the triangle paper below.
LEARNING TASK: Shoo Fly
Adapted from “Fly on the Ceiling” Lesson:
STANDARDS ADDRESSED
M4G3. Students will use the coordinate system.
a. Understand and apply ordered pairs in the first quadrant of the coordinate system.
b. Locate a point in the first quadrant in the coordinate plane and name the ordered pair.
c. Graph ordered pairs in the first quadrant.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 does the coordinate system work?
• How can the coordinate system help you better understand other map systems?
MATERIALS
• The Fly on the Ceiling by Julie Glass or similar book
• “Fly Tic-Tac-Toe, Directions” student sheet
• “Fly Tic-Tac-Toe, Game board,” student recording sheet
• “Shoo Fly” game board (laminated) for each student
• Markers (wet erase/dry erase)
• Flashlight
GROUPING
Partner Game
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
The two games in this task require students to locate points on a coordinate grid and name ordered pairs.
Comments
Identifying points on a coordinate grid is important in understanding how the coordinate system works and in constructing simple line graphs to display data or to plot points. These skills further help us to examine algebraic functions and relationships. The skills developed in this lesson can be applied cross-curricular to reading latitude and longitude on a map and to plotting data points.
One way to introduce this task is to read the book Fly on the Ceiling by Julie Glass or a similar book about plotting points on in the first quadrant of a coordinate plane.
Another introductory activity is to ask students to look at the ceiling and ask them what they see. (In most schools, you will have a modified grid system on the ceiling from the ceiling tiles. If you do not have this, skip this.) If you have a metal frame supporting the ceiling tiles, use these to create a coordinate grid. You might want to label them just below the ceiling on the wall. (If no metal frame is visible, you may need to point out the grid that is created where the ceiling tiles meet.) Be sure to label the lines created by the grid and not the tiles themselves. Turn the lights out and pretend you found a fly. Using a flashlight, shine the light on an intersection in the ceiling grid. Ask students to identify the ordered pair. Continue on until the class has grasped the concept. Then give the students flashlights and call out different ordered pairs for students to identify with the flashlight.
The game boards used for this task can be laminated and used with water-based, fine-tip markers (such as Vis-à-Vis® markers) so the game boards can be reused.
Background Knowledge
Students need to know the difference between vertical and horizontal lines and how locate and name points in the first quadrant of the coordinate plane.
Task Directions
Students will follow the directions below from the “Shoo Fly, Directions” student sheet.
Shoo Fly
Materials: 2 “Shoo Fly, Game Board” student recording sheets
2 water-based Vis-à-Vis® markers
“Shoo Fly, Directions” student sheet
Number of Players: 2
Objective: To “swat” all of the opponent’s flies by calling out the coordinates that identify the location of the “fly families.”
Directions:
(This game is similar to Battleship.)
• Each player has five fly families: one (1) family of two, two (2) families of three and two (2) families of four.
• Provide each player with a “Shoo Fly, Game Board” student recording sheet. Have them draw their fly families on the top grid using a water based Vis-à-Vis® marker. They can draw the families vertically or horizontally. Each family member must be placed where two lines intersect.
• On a turn, a player calls out the location of a point, (e.g. (3,2)). The opponent responds with “hit” if the point is located where one of the members of a fly family is hidden and “miss” if no fly is on that point. On the bottom grid the player records an “O” for a miss and an “X” for a hit on that point. (Recording on the bottom grid helps to prevent calling out the same location twice during a game.)
• The opponent will also mark a “hit” on his/her grid so s/he will know when all members of the fly family have been hit. When a player has hit all of the flies in a fly family, the opponent calls out “swatted” to signal all flies in a family have been hit.
• Play proceeds until one of the players has “swatted” all his/her opponent’s fly families.
• The first player to locate and “swat” all of their opponent’s fly families wins the game.
Students will follow the directions below from the “Fly Tic-Tac-Toe, Directions” student sheet.
Fly Tic-Tac-Toe
Materials: “Fly Tic-Tac-Toe, Directions” student sheet
“Fly Tic-Tac-Toe, Game board,” student recording sheet
Pencil
Number of Players: 2
Objective: To mark four points in a row
Directions:
• Players choose to be the “X” or the “O” and choose who will go first.
• The first player chooses a point and describes it using an ordered pair of numbers to describe it, e.g., (2,3). Mark the point on the “Fly Tic-Tac-Toe” game board and record the correct ordered pair on the Player 1 list.
Remember:
← The first number of the ordered pair tells how far to go across, the second number tells how far to go up.
← Points are marked at intersections of a grid.
← The size of the grid is 4 x 4 with corners at (0,0), (0,4), (4,4), and (4,0).
• If a player states the wrong coordinates, their turn ends.
• Players take turns choosing and plotting points on the game board.
• To win, a player must get four coordinate points in an uninterrupted straight line —horizontally, vertically, or diagonally.
Questions/Prompts for Formative Student Assessment
• What is the coordinate for the horizontal axis?
• What is the coordinate for the vertical axis?
• Why do you need to plot your point where two lines intersect?
• How do you graph a point on the coordinate plane?
• How do you name a point on the coordinate plane?
• How do you use an ordered pair to identify a point on the coordinate plane?
• How do you use an ordered pair to locate a point on the coordinate plane?
Questions for Teacher Reflection
• Are student plotting points on the intersection of the lines and not in the spaces between the lines?
• Are students able to fluently identify and locate points on a coordinate plane?
DIFFERENTIATION
Extension
• Play a variation of the Fly Tic-Tac-Toe game by using a 5 x 5 grid and a die labeled with the numbers 0-5. Instead of choosing a point, students need to roll the die using the number rolled as the first coordinate (the x value) of the ordered pair. Students are able to choose (if possible) a point whose coordinates start with the rolled number. This limits the students’ choice a little bit and focuses on the meaning of the coordinates of an ordered pair.
• Have students create a picture on a coordinate grid. List the ordered pairs of the points that need to be plotted to complete the mystery picture on a separate sheet of paper. Have a partner try to recreate the mystery picture following the coordinates given.
Intervention
• Ask students to plot coordinate points in order to create a mystery picture. Visual students will be able to see their mistakes when working in the context of a picture.
TECHNOLOGY CONNECTION
• A quick game where students need to direct Billy Bug to ten locations of food on the coordinate plane (first quadrant) as quickly as possible.
• . Simple Coordinates computer game for practice identifying and plotting points in the first quadrant.
Name _________________________________________ Date __________________________
Shoo Fly
Directions
Materials: 2 “Shoo Fly, Game Board” student recording sheets
2 water-based Vis-à-Vis® markers
“Shoo Fly, Directions” student sheet
Number of Players: 2
Objective: To “swat” all of the opponent’s flies by calling out the coordinates that locate the “fly families.”
Directions:
(This game is similar to Battleship.)
• Each player has five fly families: one (1) family of two, two (2) families of three and two (2) families of four.
• Provide each player with a “Shoo Fly, Game Board” student recording sheet. Have them draw their fly families on the top grid using a water based Vis-à-Vis® marker. They can draw the families vertically or horizontally. Each family member must be placed where two lines intersect.
• On a turn, a player calls out the location of a point, (e.g. (3,2)). The opponent responds with “hit” if the point is located where one of the members of a fly family is hidden and “miss” if no fly is on that point. On the bottom grid the player records an “O” for a miss and an “X” for a hit on that point. (Recording on the bottom grid helps to prevent calling out the same location twice during a game.)
• The opponent will also mark a “hit” on his/her grid so s/he will know when all members of the fly family have been hit. When a player has hit all of the flies in a fly family, the opponent calls out “swatted” to signal all flies in a family have been hit.
• Play proceeds until one of the players has “swatted” all his/her opponent’s fly families.
• The first player to locate and “swat” all of their opponent’s fly families wins the game.
Name _________________________________________ Date __________________________
Shoo Fly
Game Board
Source: and
Name _________________________________________ Date __________________________
Fly Tic-Tac-Toe
Directions
Materials: “Fly Tic-Tac-Toe, Directions” student sheet
“Fly Tic-Tac-Toe, Game board,” student recording sheet
Pencil
Number of Players: 2
Objective: To mark four points in a row
Directions:
• Players choose to be the “X” or the “O” and choose who will go first.
• The first player chooses a point and describes it using an ordered pair of numbers to describe it, e.g., (2,3). Mark the point on the “Fly Tic-Tac-Toe” game board and record the correct ordered pair on the Player 1 list.
• Remember:
← The first number of the ordered pair tells how far to go across, the second number tells how far to go up.
← Points are marked at intersections of a grid.
← The size of the grid is 4 x 4 with corners at (0,0), (0,4), (4,4), and (4,0).
• If a player states the wrong coordinates, their turn ends.
• Players take turns choosing and plotting points on the game board.
• To win, a player must get four coordinate points in an uninterrupted straight line —horizontally, vertically, or diagonally.
Name _________________________________________ Date __________________________
Fly Tic-Tac-Toe
Game Board
Source: and
LEARNING TASK: Measuring Up the Data
Adapted from Van de Walle, J.A. & Lovin, L.H. (2006) Teaching Student-Centered Mathematics: Grades 3-5. Boston: Pearson Education, Inc. p. 324-325.
STANDARDS ADDRESSED
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
b. Investigate the features and tendencies of graphs.
d. Identify missing information and duplications in data.
e. Determine and justify the range, mode, and median of a set of data.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
ESSENTIAL QUESTIONS
• What are some ways we make sense of data?
MATERIALS
• “Measuring Up the Data, Data Sets” student recording sheet
• “Measuring Up the Data” student recording sheet
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Students will analyze several sets of data and make a conjecture for the meaning of, median, mode, and range.
Comments
This task allows students to develop their own understanding of median, mode, and range based on information provided in sets of data. According to Van de Walle and Lovin (2006):
Having students examine these statistics over several data sets and create their own definitions is much more powerful and meaningful way to help them understand what these statistics are about than simply providing the definition to them...the conventional definition will make much more sense to students after attempting to generate their own. (p. 325)
Allow students to work through the questions on the “Measuring Up the Data” student recording sheet and then summarize the students’ conjectures. The result from the class should closely resemble the conventional definition.
It is important for students to consider the following when thinking about median, mode, and range.
• How would the _______ (median, mode, or range) be affected if another piece of data were added to the data set?
• How would the _______ (median, mode, or range) be affected if one very large or very small value (outside the range of the rest of the data set) was added to the data set?
• How well does the _______ (median, mode, or range) describe the data?
After students have completed this task, a follow-up might be for students create a “human” line plot graph. The median for the data can be determined by having pairs of students sit down, one from each end of the data. When all of the students have sat down, except for the one in the middle, the median has been determined. For further details about this activity and more follow-up activities see the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12 Curriculum Framework, by Christine Franklin, et. al. The document can be downloaded from
.
Background Knowledge
The meaning of each term is as follows.
Conjecture – a proposition (as in mathematics) before it has been proved or disproved.
conjecture. (2010). In Merriam-Webster Online Dictionary. Retrieved February 10, 2010, from .
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 in the middle of the two middle numbers.
The median is one of the measures of central tendency. It is not affected by outliers – values much larger or smaller than the range of the rest of the data set.
Mode – The number (or numbers) that occurs most frequently in a numerical data set.
The mode is not always descriptive of the data because the mode can occur anywhere in the data and can easily change with small changes in the data. For example, for the data set
2, 2, 2, 3, 7, 8, 10, 10
The mode of the data is 2. However, if one of the twos was 10, the mode would be 10 and if one of the twos was a one there would be two modes, 2 and 10.
Range – The difference between the largest and smallest values in a numerical data set.
One way to use range in a discussion about data is to compare the range of two groups. For example, looking at the range of data for the heights of girls and boys in the class, one might find that the range for the heights of the boys is smaller than the range for the heights of the girls in the class. This would indicate that the height for the boys in the class is more consistent than the height for girls.
Source: , see Level A, p. 29-30.
Task Directions
Students will follow the directions below from the “Measuring Up the Data” student recording sheet.
Your task is to make a conjecture about the meaning of the terms median, mode, and range. Be prepared to share your final conjectures with the class.
1. Look at Data Set A on the “Measuring Up the Data, Data Sets” student sheet. Make a conjecture based on the information in Set A for these terms.
a. Median
b. Mode
c. Range
2. Look at Data Set B on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
3. Look at Data Set C on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
4. Look at Data Set D on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
Questions/Prompts for Formative Student Assessment
• Where do you find the value for the median in each data set?
• What do you notice about the mode for each set of data?
• What do you notice about the range for each set of data?
• What in the data makes you think as you do about the meaning of median? Mode? Range?
• Is your conjecture about _________ (median, mode, range) true when you look at the next set of data?
Questions for Teacher Reflection
• Which students were able to revise or confirm their conjectures based with the additional data sets?
• Which students’ conjectures accurately represented the median, mode, and range for a set of data?
DIFFERENTIATION
Extension
• Once students have completed the task give them a new set of data either in a graph or as a data set. Ask students to determine the median, mode, and range for the set of data. Then provide the correct values for median, mode, and range to the students. This allows students to verify their conjecture holds true for a new set of data and to rethink their conjectures if the values are not true.
Intervention
• Allow students to complete one of the activities on the websites below. These both provide an introduction to median and the bbc.co.uk website also introduces mode.
• Ask students identify and then highlight (using different colors) the values in each set of data that represents the median and mode. Then look for commonalities in each graph. For example, students should recognize that when the mode is highlighted, there is more than one of these data values. And when the median is highlighted, it always seems to be near the middle of the data values. Once students articulate this, help them write their thinking as a conjecture. A teacher working with a small group may work best for this type of support.
TECHNOLOGY CONNECTION
• Could be used as an Intervention. Students need to find the troll with the median height. There is no direct connection between the middle height and the median height, so that connection will need to be made.
• There is a median and mode activity on this site. Stop after finding the mode, the next task is to find the mean which is not a fourth grade standard.
Name _________________________________________ Date __________________________
Measuring Up the Data
Your task is to make a conjecture about the meaning of the terms median, mode, and range. Be prepared to share your final conjectures with the class.
1. Look at Data Set A on the “Measuring Up the Data, Data Sets” student sheet. Make a conjecture based on the information in Set A for these terms.
a. Median _____________________________________________________________________________________________________________________________________________________________________________________________________________________
b. Mode
_____________________________________________________________________________________________________________________________________________________________________________________________________________________
c. Range
_____________________________________________________________________________________________________________________________________________________________________________________________________________________
2. Look at Data Set B on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
3. Look at Data Set C on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
4. Look at Data Set D on the “Measuring Up the Data, Data Sets” student sheet. Do you need to revise any of your conjectures? Explain your thinking below.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Name _________________________________________ Date __________________________
Measuring Up the Data
Data Sets
1. Data Set A
Median 5
Mode 1, 5, 9
Range 9
2. Data Set B
Median 11
Mode None
Range 24
3. Data Set C
10, 12, 17, 24, 25, 32, 34, 34, 42, 53, 54, 68, 70, 79, 81, 85, 86, 91, 97, 99
Median 53[pic]
Mode 34
Range 89
4. Data Set D
0, 2, 2, 2, 2.5, 3, 4, 4.5, 6
Median 2.5
Mode 2
Range 6
PERFORMANCE TASK: And the Survey Says…
STANDARDS ADDRESSED
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
a. Construct and interpret line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs.
b. Investigate the features and tendencies of graphs.
c. Compare different graphical representations for a given set of data.
d. Identify missing information and duplications in data.
e. Determine and justify the range, mode, and median of a set of data.
This task aligns with English-Language Arts standards ELA4W1 and ELA4W2.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 choose the best graph to represent our data?
• Why are there different types of graphs?
• How are different graphs alike and different?
MATERIALS
• “And the Survey Says...” student sheet
• Poster board or chart paper
• Tools to create graphs (markers, yardsticks, etc.)
GROUPING
Partner/Small Group Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
This task allows students to collect data through the use of a survey and to analyze the data in order to display it in a variety of graphical representations. Students will also show the best use for each kind of graph. (This task will take more than one day.)
Comments
To introduce this task, work as a class to complete the following:
Create a list of 5 common entrées served in your school cafeteria.
Design a survey to distribute to all the 4th graders in your school or in a small school two or more grade levels in your school. The survey should ask students to select their favorite cafeteria lunch entrée from a list.
• Decide on a deadline for surveys to be completed and returned.
• Designate a place for surveys to be returned.
• Elect representatives from your class to deliver the surveys to the other teachers and ask for their class to participate in the survey. Be sure the representatives can accurately explain the surveys, the deadline, and the designated return area.
• Compile the data from the survey by counting the votes for each lunch entrée.
Background Knowledge
Students will need to have prior knowledge of how to create and understand charts, bar graphs, and pictographs.
Task Directions
Students will follow the directions below from the “And the Survey Says...” student sheet.
You are putting together a news report for your local TV station. This report will show what 4th grade students from your school think about the lunch choices in the cafeteria.
To work on this task, your four-person group needs to be divided into two partner pairs.
Each pair within your group needs to choose one of the following graphs. (Both pairs may not create the same type of graph.)
Bar graph
Pictograph
1. Create your graphs using the data your class collected about lunch entrées. Make sure it is clearly written and easy to read.
2. When you finish creating your graph, show and explain your graph to your group. Be willing to accept their recommendations for improvement. Also, be a polite and courteous group member who only shares comments that are helpful – not hurtful. Take turns so that each pair in the group gets a chance to share their work and comment on others’ work.
3. Prepare a news broadcast script. The script should include an opportunity for each person to present their graph in an appropriate, informative way and allow the group’s presentation to be complete, but not repetitive.
4. When everyone is finished, each group will present their graphs to the class.
Questions/Prompts for Formative Student Assessment
• What increments will you use on your bar graph? Why?
• How many students will each symbol represent in your pictograph? Why?
• How did you choose your labels and title for your graph?
• How does having two graphs that represent the same data help you better understand the data?
• Which graph is easiest to create? To read? To share? Why do you think so?
Questions for Teacher Reflection
• Are students able to accurately create the graphs?
• Are students able to represent the data using appropriate increments (bar graph) or a symbol to represent more than one student (pictograph)?
• Are the parts of the graph clearly labeled and appropriate?
• Did the students’ description of their graphs indicate they understood the data they presented?
DIFFERENTIATION
Extension
• Students could videotape their news report and use digital video editing to create a report to present in a broadcast to the school.
Intervention
• Allow students to create at least one of the graphs using the website below, MS Excel, or another computer program.
• For students who have difficulty speaking publicly, allow them to video record their presentation and play it for the class.
TECHNOLOGY CONNECTION
• A student-friendly web site with which several types of graphs can be created.
Name _________________________________________ Date __________________________
And the Survery Says...
You are putting together a news report for your local TV station. This report will show what 4th grade students from your school think about the lunch choices in the cafeteria.
To work on this task, your four-person group needs to be divided into two partner pairs.
Each pair within your group needs to choose one of the following graphs. (Both pairs may not create the same type of graph.)
Bar graph
Pictograph
1. Create your graphs using the data your class collected about lunch entrées. Make sure it is clearly written and easy to read.
2. When you finish creating your graph, show and explain your graph to your group. Be willing to accept their recommendations for improvement. Also, be a polite and courteous group member who only shares comments that are helpful – not hurtful. Take turns so that each pair in the group gets a chance to share their work and comment on others’ work.
3. Prepare a news broadcast script. The script should include an opportunity for each person to present their graph in an appropriate, informative way and allow the group’s presentation to be complete, but not repetitive.
4. When everyone is finished, each group will present their graphs to the class.
PERFORMANCE TASK: Is Something Missing?
STANDARDS ADDRESSED
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
a. Construct and interpret line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs.
b. Investigate the features and tendencies of graphs.
c. Compare different graphical representations for a given set of data.
d. Identify missing information and duplications in data.
e. Determine and justify the range, mode, and median of a set of data.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 the labels of graphs determined?
• When can events be displayed on a line graph?
• When can information be displayed on more than one type of graph?
MATERIALS
• “Is Something Missing? Part I” student recording sheet
• “Is Something Missing? Part II” student recording sheet
• Rulers
GROUPING
Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task students interpret different types of graphs and add any missing information.
Background Knowledge
Students need to know how to read and interpret line graphs, line plot graphs, and bar graphs. Additionally, students need to know how to name a segment by naming its endpoints . Finally, students need to know that each graph requires labels, units, and a title.
Students need to have experiences with tasks such as what is given in “Is Something Missing? Part I” student recording sheet before using giving this task as a performance task. Sample line graph problems, similar to the ones presented in this task, can be found at the following link. Click on “go to the current page” to see the actual PowerPoint.
Comments
On the “Is Something Missing? Part II” student recording sheet (question #3, “List the values that would be needed to describe the number of pets owned by the students in the class”), students may need further clarification. They are to list the values for which students in the class have that number of pets. (e.g. If no one in the class has three pets then the number three (3) would not be part of the list. If four students have two pets, then the number two (2) would be part of the list, but only listed once.)
Before beginning the “Is Something Missing? Part II” student recording sheet, collect class data for the number of pets each student has at home. Data can be collected in a chart by asking students to raise their hands when the number of pets they have is called. Alternatively, a class list could be displayed with the number of pets for each person listed next to their name.
Questions/Prompts for Formative Student Assessment
• How do you know what each segment of the line graph represents?
• What parts of the graph are missing? How do you know?
• How did you find the median for your data? Mode? Range?
• In what situations is a line graph appropriate? Would this data be appropriately displayed in a line graph? How do you know?
• Were students able to explain when the use of line graphs is appropriate and when it is not appropriate?
Questions for Teacher Reflection
• Which students are able to identify events as represented on a line graph?
• Which students are able to represent collected data in more than one way?
• Which students are able to interpret line, line plot, and bar graphs accurately?
DIFFERENTIATION
Extension
• Ask students to add a line to represent Terrence’s trip to school that morning. Use a different color to add a line for Terrence’s trip on the line graph provided on “Is Something Missing? Part I” student recording sheet.
Intervention
• Help students connect the story at the top of the page to the line graph below. Encourage students to circle parts of the text and draw an arrow to the corresponding part of the graph or make a small sketch (and/or quick notes) to represent what is happening during each segment of the graph.
TECHNOLOGY CONNECTION
• Information on the parts of a bar graph. Links to the left provide information on the parts of line graphs.
• Provides a definition plus several links on the left to more information about pictographs.
Name _________________________________________ Date __________________________
Is Something Missing?
Part I
The line graph below shows Manuel’s trip to school. He walked toward school for about 4 minutes before he arrived at Terrence’s house. He waited for about
6 minutes for Terrence to finish his breakfast. Then both boys walked together towards school. After about 4 minutes of walking, Manuel realized he left his homework at home. While Terrence continued to walk to school, Manuel ran home. When he got home, it took him about 6 minutes to find his homework and to convince his mom to drive him to school so he wouldn’t be late.
1. What information is missing from this graph? How do you know? Add all missing information to the graph.
________________________________________________________________________________
________________________________________________________________________________________________________________________________________________________________
2. Which segment represents Manuel riding in the car to school? How do you know?
________________________________________________________________________________
________________________________________________________________________________________________________________________________________________________________
3. Which segment represents Manuel waiting for Terrence? How do you know?
________________________________________________________________________________
________________________________________________________________________________________________________________________________________________________________
Name _________________________________________ Date __________________________
Is Something Missing?
Part II
1. Look at the “number of pets” data recorded for the students in your class. Use the data to create a line plot graph below.
2. Find the mode, median, and range for the data. Explain how you found each.
________________________________________________________________________________
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
3. List the values that would be needed to describe the number of pets owned by the students in the class.
________________________________________________________________________________
________________________________________________________________________________
4. On back of this paper, create a horizontal bar graph or a pictograph to represent this data.
5. Would you be able to create a line graph to display this data? Explain your thinking below.
________________________________________________________________________________
________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Name _________________________________________ Date __________________________
Is Something Missing?
Part III
1. Look at the graph to the right. This graph represents a person filling a tub with water, getting undressed, getting into the tub, taking a bath, and then getting out of the tub, drying off, and finally, draining the tub. Label the horizontal and vertical axes of this graph and title it appropriately.
2. On a winter day, the temperature slowly rises as the sun comes up and warms the air. By late afternoon, when the sun starts to sink, the air starts to cool. Once the sun sets, the temperature drops quickly. Label the horizontal and vertical axes of this graph and title it appropriately.
3. A fourth grade class was surveyed to determine the types of pets owned by students in the class. More students owned dogs than any other type of pet. Only a couple of students owned reptiles. Cats were the second most popular type of pet. Label the horizontal and vertical axes of this graph and title it appropriately.
4. Each student in our class recorded the number of hours they spent playing video games last week. Some students didn’t play video games at all, while two students played video games the most, for 5 hours that week. Label the horizontal axis of this graph and title it appropriately.
PERFORMANCE TASK: Tell Me About My State!
Adapted from the Illuminations lesson, “State Population Projections.”
STANDARDS ADDRESSED
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
a. Construct and interpret line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs.
b. Investigate the features and tendencies of graphs.
c. Compare different graphical representations for a given set of data.
d. Identify missing information and duplications in data.
e. Determine and justify the range, mode, and median of a set of data.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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 use data and graphs to answer questions?
• Which graph should you choose to represent data?
MATERIALS
• “Tell Me About My State!” student recording sheet
• Copy of “Projections of the total Population of States: 1995 to 2025 (Series A)” (one per pair of students) available at:
• Grid paper (have both large and small grid sizes available)
Large grid:
Small grid:
• Plain paper (2-3 sheets per pair of students)
• Rulers and colored pencils to create graphs
GROUPING
Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
In this task, students examine census information to investigate projections of the total population of states from 1995-2025. Using the provided data, students will analyze statistics and create graphs to compare and contrast the information and to make predictions.
Background Knowledge
Students should have previous experience recording data in a table and creating and interpreting bar and line graphs. Students should also understand the process of using data and graphs to answer a question.
Comments
One way to introduce this task is to discuss the purpose of conducting the US Census. There is information available on the census on the following web page.
For information about the census developed for schools, see the following web page.
For facts about Georgia go to:
Ask students questions about the census such as:
• What is the census?
• Why is it important?
• How can the census data be used?
• What types of questions can be answered using the census data?
Before asking students to complete the “Tell Me About My State!” student recording sheet, give each pair of students the “Projections of the total Population of States: 1995 to 2025 (Series A).” Discuss with students how the table is organized and how to read it. Be sure students understand the title, the column headings, and the use of “Numbers in the Thousands.” Ask students to find information in the table (e.g. What state has the largest population in 1995?). Then convert some of the values in the table to standard form (e.g. 602 represents 602,000 and 4,324 represents 4,323,000).
The student sheet can be broken into three parts: (Numbers 1-3) Population for the state of Georgia, (Numbers 4-6) Comparison of populations between five states, (Number 7) Student generated question. After students have completed a part of the student recording sheet, allow them to share their work and to compare their findings with the findings of other students in the class. Discuss why student pairs might draw different conclusions with the same data.
Questions/Prompts for Formative Student Assessment
• Which type of graph would be/is most helpful in displaying an answer to your question? Why?
• How could these graphs help you make population predictions?
• What trends or tendencies do you see in the data?
• How did you represent numbers in the thousands on your graph?
• Describe the parts of a graph for a bar graph? A line graph?
Questions for Teacher Reflection
• Which students are able to create line graphs and bar graphs accurately?
• Which students are able to accurately find required data in a table? Are students able to represent it in a graph?
• Which students are able to write a question that can be answered with given data?
• Which students are able to interpret data and graphs to answer a question and justify their answer?
DIFFERENTIATION
Extension
• Allow students to write and answer an additional question that they would like to explore using the Census bureau data.
Intervention
• Allow students to work in small groups and have each student create only on graph.
• Scaffold students when creating graphs by allowing them to use a computer program (see links below) or provide assistance as students determine the scale and increments of the graph.
TECHNOLOGY CONNECTION
• A link to the lesson from which this task was adapted.
• An applet that allows students to create a bar graph from NCTM.
• A program that allows students to create several types of graphs from the National Center for Education Statistics (NCES).
Name _________________________________________ Date __________________________
Tell Me About My State!
You will be using data from the US Census Bureau population projections to find out more about your state and other states in the United States.
1. List the population data below for the state of Georgia.
|Georgia |1995 |2000 |2005 |2015 |2025 |
2. On a separate sheet of paper, create a line graph to display the Georgia population data.
3. Based on your line graph, make a prediction for the population of Georgia in the year 2035. Record your thinking on the back of your line graph and then justify your prediction using the data and line graph as support.
4. For five states, choose three years of data from the US Census Bureau population projections table (see ). Record the population projections data in the chart below. (Use “Series A” for this task.)
|STATE NAME |YEAR |YEAR |YEAR |
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
5. On a separate sheet of paper, create three bar graphs, one for each year, to represent this data.
6. Look at the three graphs you created and determine which state (among the five) will have the highest population when the next census is taken. Justify your answer using your data and graphs as support.
7. Write one question that you would like to explore using the Census bureau data. Find data to help you answer the question. If applicable, record the data in a table, and draw a graph which represents that data (on a separate sheet of paper). Answer the question you wrote. Explain your answer, using your data and graph as support.
Unit 4 Culminating Task
PERFORMANCE TASK: Geometry Town
STANDARDS ADDRESSED
M4G1. Students will define and identify the characteristics of geometric figures through examination and construction.
a. Examine and compare angles in order to classify and identify triangles by their angles.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi) by their properties.
d. Compare and contrast the relationships among quadrilaterals.
M4G2. Students will understand fundamental solid figures.
a. Compare and contrast a cube and a rectangle prism in terms of the number and shape of their faces, edges, and vertices.
b. Describe parallel and perpendicular lines and planes in connection with the rectangular prism.
c. Build/collect models for solid geometric figures (cubes, prisms, cylinders, pyramids, spheres, and cones) using nets and other representations.
M4G3. Students will use the coordinate system.
a. Understand and apply ordered pairs in the first quadrant of the coordinate system.
b. Locate a point in the first quadrant in the coordinate plane and name the ordered pair.
c. Graph ordered pairs in the first quadrant.
M4D1. Students will gather, organize, and display data according to the situation and compare related features.
a. Construct and interpret line graphs, line plot graphs, pictographs, Venn diagrams, and bar graphs.
b. Investigate the features and tendencies of graphs.
d. Compare different graphical representations for a given set of data.
a. Identify missing information and duplications in data.
b. Determine and justify the range, mode, and median of a set of data.
M4P1. 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.
M4P2. 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.
M4P3. 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. Analyze and evaluate the mathematical thinking and strategies of others.
c. Use the language of mathematics to express mathematical ideas precisely.
M4P4. 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.
M4P5. 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
• Where is geometry found in your everyday world?
• How do coordinate grids help you organize information?
• Which graph should you choose to represent data?
MATERIALS
• “Geometry Town” student sheet.
• Copies of nets for rectangular prisms, cubes, and cylinders
• Copies of “Geometry Town, 2-D shapes” (optional intervention)
• Poster paper or chart paper with 1 inch grid
• Notebook or copy paper
• 1” x 24” Strips of black or brown construction paper for streets, avenues, and roads (approximately 12 strips per city model)
• Markers, crayons, and/or colored pencils
• Protractors, rulers, yardsticks
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Students create a plan for a city using geometric figures. Also, students represent the features of the town graphically.
Comments
A review of vocabulary would be an effective way of leading into this culminating task. One of the most important aspects of this task is for the children to demonstrate the mastery of the meaning of each term and show how to use and recognize these terms in their everyday lives.
Students may need extra time getting started on this task because it requires planning and cooperation. This task does take a considerable amount of time to complete; therefore, teachers should allow students the time required to discuss their project as they plan and create their model.
It may be helpful to create a rubric that can be used to assess the city model students will develop and describe in this task.
Background Knowledge
As a culminating task, students will need to utilize the understanding and skills developed during this unit. Grade level teachers can create the rubric, or students can participate in the creation of the assessment tool.
Task Directions
Students will follow the directions below from the “Geometry Town” student sheet.
In your role as city planner, you have been asked to plan a new part of your city. Create a model of your plan, including 3-D models of the buildings, to present to the committee. You are required to meet the following specifications.
□ 4 streets that are parallel to each other
□ 1 road that is perpendicular to the 4 parallel streets
□ 1 avenue that intersects at least 2 streets but is not perpendicular to them
□ 2 rectangular prism-shaped buildings
□ 2 cube-shaped buildings
□ 1 cylindrical building
□ 1 park shaped like a trapezoid with the following features:
□ A circular swimming pool
□ A right, isosceles triangular sandbox
□ 2 rectangular basketball courts
□ Name the park and the streets, the road, and the avenue.
Plan your city on a sheet of paper first. Once your plan is complete, create your model. Build your model on 1” grid chart paper. Use paper strips to create the streets, road, and avenue, use nets to create the buildings and draw a trapezoidal park. Add the required features to the park by creating the appropriate 2-D shapes for your park. When your model is finished, create a poster with the following requirements.
• Draw and label an x-axis and y-axis on the grid paper to create a coordinate plane.
• Identify all of the buildings by listing an ordered pair for the location of each building.
• Create a graph that displays the number of pairs of parallel lines, perpendicular lines, and intersecting lines in your plan.
• Create a different type of graph to display the number of square, rectangular, and circular faces on your buildings.
• Describe the properties of each type of building in your plan.
• Using a Venn diagram, compare the properties of a trapezoid and a rectangle.
Questions/Prompts for Formative Student Assessment
• How do you know the basketball courts are rectangular? The park is trapezoidal?
• What are the properties of a rectangle? A trapezoid? A triangle?
• How can you prove to me that the park is trapezoidal?
• What ordered pairs would you use to describe the locations of the cubes? Rectangular prisms?
• Describe the faces of a rectangular prism? A cube? A cylinder?
• How do you know the line segments are perpendicular? Parallel? Intersecting?
• What graph would be appropriate to display _________ (parallel, perpendicular, intersecting; or faces of figures)? Why?
Questions for Teacher Reflection
• Which students accurately completed all parts of the task?
• Which student demonstrated an understanding of:
~ Parallel, perpendicular, intersecting.
~ Naming locations on a coordinate plane (first quadrant).
~ Graphing data collected.
~ Describing properties of figures.
DIFFERENTIATION
Extension
• Students may add a new part to the city using their own rules for things to add to the map.
• Invite an architect to the classroom to talk about planning and the models they build in their work.
• Encourage students to prepare a presentation to the committee regarding their city plan. Students should try to persuade city planning committee members to choose their plan.
Intervention
• Pre-made 2-D shapes could be made available to students.
• Students could be provided the opportunity to create graphs using an online software program such as the ones listed below.
• Students could be given the attribute cards from the “Quadrilateral Challenge” task. They can use these cards to create a Venn diagram rather than writing out the attributes.
TECHNOLOGY CONNECTION
• An applet that allows students to create a bar graph from NCTM.
• A program that allows students to create several types of graphs from the National Center for Education Statistics (NCES).
Name _________________________________________ Date __________________________
Geometry Town
In your role as city planner, you have been asked to plan a new part of your city. Create a model of your plan, including 3-D models of the buildings, to present to the committee. You are required to meet the following specifications.
□ 4 streets that are parallel to each other
□ 1 road that is perpendicular to the 4 parallel streets
□ 1 avenue that intersects at least 2 streets but is not perpendicular to them
□ 2 rectangular prism-shaped buildings
□ 2 cube-shaped buildings
□ 1 cylindrical building
□ 1 park shaped like a trapezoid with the following features:
□ A circular swimming pool
□ A right, isosceles triangular sandbox
□ 2 rectangular basketball courts
□ Name the park and the streets, the road, and the avenue.
Plan your city on a sheet of paper first. Once your plan is complete, create your model. Build your model on 1” grid chart paper. Use paper strips to create the streets, road, and avenue, use nets to create the buildings and draw a trapezoidal park. Add the required features to the park by creating the appropriate 2-D shapes for your park. When your model is finished, create a poster with the following requirements.
• Draw and label an x-axis and y-axis on the grid paper to create a coordinate plane.
• Identify all of the buildings by listing an ordered pair for the location of each building.
• Create a graph that displays the number of pairs of parallel lines, perpendicular lines, and intersecting lines in your plan.
• Create a different type of graph to display the number of square, rectangular, and circular faces on your buildings.
• Describe the properties of each type of building in your plan.
• Using a Venn diagram, compare the properties of a trapezoid and a rectangle.
Geometry Town
Building Nets
Geometry Town
Building Nets
Geometry Town
Building Nets
Geometry Town
2-D Shapes (Optional)
-----------------------
MATHEMATICS
It is important for students to compare the rectangular prism and the cube (see M4G2a). Ask student groups to compare the two figures using a Venn diagram. Next, groups can share their comparison of the two figures, allowing groups to edit their Venn diagram as needed. Finally, each student should summarize the information in their Venn diagram in paragraph form.
Essential (Must Have) Properties Nonessential (Might Have) Properties
Examples Non-examples
Essential (Must Have) Properties
Nonessential (Might Have) Properties
Non-examples
Examples
#1, #11 – obtuse scalene
#2, #7 – right scalene
#4, #13 – acute scalene
#5, #10 – right isosceles
#8, #12 – acute equilateral
#3, #9 – acute isosceles
#6, #14 – obtuse isosceles
or
A
1
4
9
14
11
12
6
10
13
5
8
7
3
2
A
................
................
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