Jessika Richter
(Lund  Sweden)The short and sweet: I have been a passionate teacher since 2001. I first worked with the National Park Service in Washington (state), then moved to Australia where I completed my DipEd at the University of Melbourne and then taught at Hailebury ...
Patterns, Variables and Functions: Integrated, ProjectBased, Science and Mathematics Inquiries for Diverse Fifth and Sixth G...
Description:Projectbased learning stations that help students bridge observations to algebraic concepts through semesterlong inquiries (each inquiry takes about a week). Once the class has completed the eight inquiries, the teacher may guide them through the inquiries again, exploring new patterns, variables and functions at a more complex level.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 elementary
 3rd
 4th
 5th
 middle
 6th
 7th
 8th
 tween
 Curriculum: Unit
 Contributed By: Sarah Zainfeld
Introduction
Description:Students learn science and math through opportunities to see and describe patterns in the world using learning stations, group work, and selfpaced reflection journals
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 elementary
 3rd
 4th
 5th
 middle
 6th
 7th
 8th
 tween
 Curriculum: Scope & Sequence
Evaluation of student success in learning the concepts presented in the learning stations can be accomplished through the analysis of student Observation and Reflection Journals for accurate collection, use and explanation of data and events. Language minority students’ understanding may be assessed in the same way as students in the rest of the class; through their use of graphs, pictures and symbols to describe mathematic and scientific concepts.
Themes are big ideas. Instead of isolated facts and concepts, they are links to the theoretical structures of the various scientific disciplines. According to the California State Science Framework, a theme is a way to give students “basic knowledge” while introducing them to a way of thinking. The standards refer to this as the “essence of science.” It is the aim of this integrated thematic instruction that students who are involved in exploring the essence of science will have a systematic way of appreciating, describing and symbolically representing patterns, variables and functions in the world. Students who are scientifically literate can contribute to the well being of the planet with informed decisions and actions based on observed phenomena.
Goals
Description:Design philosophy of the Patterns, Variables and Functions Unit
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Curriculum: Scope & Sequence
 Use math in the context of science
 Systematically acquire and record data
 Develop an intuitive sense of the relationship between two variables in a variety of contexts
 Recognize the change in relationship between two variables as described by a function
 Use spreadsheet software to input data and produce a graph
 Interpret and describe observed phenomena
Overview
Description:Selfpaced, inquirybased learning
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Curriculum: Scope & Sequence
In the first inquiry, students will explore the concept of "the infinite" as a way to grasp the idea that some events, such as a ball of tinfoil being launched from a catapult made of Popsicle sticks, would go on forever unless something gets in its way (e.g., gravity). Students' background knowledge is first activated through opportunities to brainstorm what they already know about the infinite. Through group discussion of interdisciplinary paradoxes of of the infinite (in home and expert jigsaw groups), students will discover there is often "something which keeps something else" from going on forever. Those "somethings" become the foundation for the vocabulary to be introduced in the next inquiry (e.g., the weight of an object, force of the catapult, or air resistance).
Inquiry Two: Discovering relationships among variables
The next inquiry presents students with opportunities to use their intuitive definitions of patterns, the infinite, and variables, while exploring the properties and relationships of objects in a variety of contexts. Through this guided, but selfpaced discovery, students will move between the following stations exploring the relationship between:
 Buoyancy and weight in the Lifeboat Inspection
 Gravity and weight in the Flipsticks activity
 Amount of liquid and pitch in the Physics of Sound activity
 Length of string and weight of an object in the Pendulum activity (above activities adapted from Foss kits)
InquiryThree: Describing relationships using TCharts, an introduction to functions
Students are encouraged to move from actual or physical experience with objects (variables), and their relationships (functions), toward a symbolic notion of the concepts. Students rotate in groups through the following stations in order to gain more experience with relationships in the real world.
 Length of the side of a square: Number of squares in the Squares from Squares activity.
 Number of triangles: Distance of perimeter in the Row of Triangles activity
 Number of pentagons: distance of perimeter in the Row of Pentagons activity
 Number of hexagons: distance of perimeter in the Row of Hexagons activity
(from Marilyn Burns, About Teaching Mathematics)
Inquiry Four: Introduction to Exponents Exponents are a key idea used in math and science to describe patterns of change between variables. Students are presented with a chart that shows the relationship between the powers of ten and ways of writing 10, 100, 1000, etc. Students may later draw on their experience with exponents when describing patterns (functions) they discover in the graphs they'll later make of observed phenomena.
Inquiry Five: Describing Factors and Using Variables
Using tiles, markers, and "hundreds" charts, students create patterns which "grow" by the same factor. Students write a rule to describe their pattern or function. Repeated exposure to the concepts of patterns, variables and functions provides multiple opportunities for students to see relationships. Students practice computing patterns of numbers by completing "textbook" problems.
Inquiry Six: Magic Squares
Students are individually presented with magic squares in which variables appear. They replace the variable with a number, which then creates an equal sum, whether one adds the numbers horizontally, vertically, or diagonally. Students complete "textbook" problems as practice in finding the value for the missing variables.
Inquiry Seven: Variables and functions we notice in the world
Students brainstorm possible sets of objects or events whose relationships cause an observable pattern, or function. Examples are given from the teacher during the brainstorming class brainstorming time, such as the size of an image on a screen to the distance of the projector from the screen, or the number of cars to the amount of carbon dioxide pollution. Students create collages that describe relationships they discover in the world through looking at newspapers and scientific magazines.
Inquiry Eight:Using graphs, words and pictures to describe functions
Using spreadsheet software, students enter data they collected from one or two experiments they conducted in Inquiry Three. Each person generates one or two graphs, a written explanation in English, other pictures if they wish as well as mathematical/symbolic descriptions.
Rationale
Description:Fifth and sixth grade students benefit from activities that allow them to build on their own intuitive sense of the way the world works. …from the Patterns, Variables and Functions Unit
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Curriculum: Scope & Sequence
The Patterns, Variables, and Functions thematic instruction aims to contribute to social justice and was designed to be culturally responsive by offering multiple opportunities for students to reflect. Handson activities help to create conditions that "enable subordinated students to move from their usual passive position to one of active [handson] and critical [reflective] engagement" (Bartolome, qtd. in Zainfeld, 1999).
The thematic instruction has been designed from the philosophical and political standpoint that students are capable of using one another as sources of knowledge to grapple with and solve complex problems. Rather than operating from a "deficiency model," where students "need to be repaired," the activities in these inquiries allow diverse learners access to complex and challenging subject matter. Through opportunities to explore materials, use knowledge and intuition, students represent their process of learning and knowledge gained using a variety of modes: Logical, symbolic, verbal, musical, spatial and kinesthetic.
The use of pictorial representations of data gives limited English proficient students in particular, access to the subject matter, since graphs help make explicit the specific phenomena and relationships between objects being observed. Group work is emphasized in order to create "a multiethnic education, which emerges from the realization and participation in the social construction of knowledge" (Banks, qtd. in Zainfeld, 1999). In contrast, outdated models of teaching and learning have students compete to make individual gains.
Heterogeneous grouping defies the "deficiency model" of grouping students according to ability. The activities in these inquiries provide room for learners at all levels to observe, record and find patterns. Ultimately, the goal of instruction is to create equity through learning communities and activities that value each student as a contributor and recipient of knowledge for the whole classroom. Equitable gains of subject matter knowledge and value of student contributions are enhanced with the opportunity to reflect. Writing in their Observation and Reflection Journals, students will have time to problemsolve in a way that isn't always possible during the excitement of an experiment. The process of intrapersonal reflection allows students who don't want to raise their hands in class an opportunity to grapple with important concepts and to receive feedback from the teacher. It is hoped questions a teacher will ask during the inquiries will be openended, thematic, and integrating, so that students may form deep, personal notions of the subject matter based on their own understanding of and experience in the world (at the same time their misconceptions are unraveled).
The Patterns, Variables and Functions theme has been created as an interdisciplinary math and science study. Interdisciplinary organization of subject matter provides a context for a student to relate more accurately to the ways in which the disciplines are actually used, thought about and developed by mathematicians and scientists, themselves. Interdisciplinary studies emerge from the use of themes such as "patterns", the "infinite", or "cause and effect," in which the crossdisciplinary connections are a means in the inquiry of phenomena in the world, as opposed to a direct focus on the theme, itself.
Context
Description:Throughout the school year, the concept of the "infinite" may have arisen across the disciplines …from the Patterns, Variables and Functions Unit
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Curriculum: Scope & Sequence
Assessment
Description:...assessment for this unit is built into each lesson...…from the Patterns, Variables and Functions Unit
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 elementary
 3rd
 4th
 5th
 middle
 6th
 7th
 8th
 tween
 Curriculum: Scope & Sequence
Inquiry One: The "Infinite" Across the Disciplines
Description:It is the goal of this inquiry to introduce students to many instances within the disciplines (besides math, alone), in which the concept of "infinite" appears. After students have completed all eight inquiries, they cycle back and start on this one again,choosing a new question and group from which to base the rest of the eight inquiries that follow.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 elementary
 3rd
 4th
 5th
 middle
 6th
 7th
 8th
 tween
 Other
It is the goal of this inquiry to introduce students to many instances within the disciplines (besides math, alone), in which the concept of "infinite" appears. Students begin to connect the notion of "infinite" as foundational to other concepts to which they may have been exposed in previous years, including ideas such as Divisibility, the Law of Big Numbers, Repeating Decimals, Statistical Samples, The Endless Expanding Universe and Generations.
Objectives
 Understand in a developmentally appropriate way, the concept of "infinite"
 Practice listening, speaking and asking questions in a jigsaw "home group," and "expert group"
 Gain experience in discussing philosophical questions and thinking logically
 Write or draw a representation to show the relationship between their topic (see activities, below) and the notion of "infinite"
 Develop curiosity and/or questions about the "infinite"
Materials
 Thirty Cards (or however many students are in your class), each with one of five stamped symbols (used to determine home group)
 Five drawings by M.C. Escher cut into 30 puzzle pieces (used to determine expert group): Mobius Strip, Reptiles, and Waterfall.
 Handouts with the questions listed under "Activities," below
Activities
Students are told they will be thinking about the concept of the infinite. They are reminded of experiences during which they've already encountered the infinite, such as in reading fiction about space exploration, counting numbers and multiples, thinking about African Ancestors and legacies, looking at patterns which go on forever, and dividing fractions on and on.
Students are given a card stamped with a symbol to organize them into groups of five for their "home groups." Each group discusses and draws a semantic web around the word "infinite." Each student uses a distinctly colored marker (which will help the teacher assess participation) and its responsible for writing down his or her own idea(s) within the web. Semantic webs are posted throughout the room, followed by a brief discussion during which a class semantic web is drawn on the board. Here is an example of a simple semantic web or idea map created using the Inspiration software.
Each student is given one puzzle piece of an M.C. Escher drawing and asked to find the people who have the pieces needed to complete the pictures. As students assemble complete pictures, they discover the people in their expert group.
One of the following questions is written on a handout for each group (each person receives a copy). Students are told they should use their own piece of paper to take notes that will help them explain their discussion of the question to their home group.
 Take a look at the following drawings by M.C. Escher, Mobius Strip, Reptiles, and Waterfall. What do you notice about these drawings? What makes them unique? [students use VisualSpatial, Interpersonal intelligences (see Howard Gardner's work)]
 What are some patterns you've seen in math, which could go on forever? [students use LogicalMathematic, Interpersonal intelligences]
 Has anyone invented a machine that moves or works forever? Does your group think this is possible? [students use Logical, Interpersonal intelligences]
Closure
Students next regroup as a class and share their findings, as individuals. During the debriefing, the teacher asks, "Were there any obstacles that came up for you in talking about the 'infinite?' What have we learned about the 'infinite?' What have we learned about working together?
Assessment
Since students each use a corresponding color to their name, the teacher can see by looking at the semantic webs and posters, if each child contributed. The structure of the home groups and expert groups provides a venue in which each student is accountable to bring back new information to their home group. By comparing the semantic webs each home group made to their second poster representing their definition of infinity, the teacher may determine whether or not the group developed new ideas about the concept.
Extensions
Students discuss or write about why companies may choose to name their products, "Infinity."
Plan B
If too much confusion arises as a result of having two jigsaw groups and six questions, the class can meet in "simple" jigsaw groups to discuss the same questions, debriefing with a semantic web as a class.
Inquiry Two: Discovering Relationships Among Variables
Description:This inquiry presents students with opportunities to use their intuitive definitions of patterns, the infinite, and variables, while exploring the properties and relationships of objects in a variety of contexts.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Other
This inquiry presents students with opportunities to use their intuitive definitions of patterns, the infinite, and variables, while exploring the properties and relationships of objects in a variety of contexts. Through this guided discovery, students will move between the four learning stations while exploring the relationship between objects and events. The following activities were excerpted from FOSS (Full Option Science Solutions) kits. Please visit the FOSSWeb site for informative video segments that detail the activities below.
Objectives
 Discover the relationship between gravity and weight
 Notice patterns in the swing of a pendulum and the trajectory of an object
 Discover the relationship between amount of liquid in a container and the audio pitch produced
 Discover the relationship between the length of a piece of string and weight of an object
Time
About an hour for the teacher to model the stations. Students will probably spend at least a week exploring the stations as a group and as their own, while documenting their findings in their math journals. This inquiry as well as the others in the Integrated Infinity Project are meant to coincide, complement and give realworld meaning to the work being done in their traditional math and science texts.
Materials
 Observation and Reflection Journal for each student
 Cards on which each of the four following jobs are written:
(1) Reader/Reporter: Reads instructions, keeps group members on task, reports findings back to class
(2) Recorder: The person to make sure the data is written down and that each student writes in his or her journal
(3) Materials Coordinator: Responsible for getting keeping inventory, and returning materials
(4) Starter: The person who initiates the station activity and makes sure everyone participates
Materials for Station #1: Lifeboat Inspection
 100 Pennies
 Three paper cups, three of which have 3cm, 5cm and 7cm, respectfully, of the tops cutoff. Label each cup, #1, #2, #3)
 A bucket filled with water
Materials for Station #2: Flipsticks
 1 flipper base (from Foss kit)
 1 flipper angle brace (from Foss kit)
 3 Popsicle sticks
 4 pieces of popsicle stick, 2cm
 1 rubber stopper
 1 cork
 1 metertape
 1 piece of aluminum foil, 10cm
 1 piece of aluminum foil, 20 cm
 1 plastic zip bag
 6 flipsticks
 1 mallet
 Ruler
 four glasses, filled with water at different levels
 xylophone to match pitches to notes
 6 pennies
 6 paper clips
 6 strings: (2)20cm, 2(30)cm, (2)40 cm
 1 meter tape
 A classroom clock with a second hand
 Pencils
 Masking tape on which to number pendulums:
 Pendulum #1, 20 cm, one penny
 Pendulum #2, 20 cm, two pennies
 Pendulum #3, 30cm, one penny
 Pendulum #4, 30 cm, two pennies
 Pendulum #5 40 cm, one penny
 Pendulum #6, 40 cm, two pennies
Activities
(1) Students are reminded by the teacher of the previous activity about "the infinite," where students discovered there is something that often keeps events from being infinite. Teacher explains that today the activities at the stations will allow students to think about what keeps things from being infinite.
(2) Model for the students each of the stations. Teacher reminds students to pay attention to patterns students will see in each of the experiments.
(3) Students rotate in groups of four through the stations in order to gain more experience with relationships in the real world. Groups are determined randomly, by passing out cards on which a job is written in a specific color. Groups meet by matching colors.
(4) Remind each student to perform his or her job while gathering data in their journals: Reader/Reporter reads instructions, keeps group members on task, reports findings back to class); Recorder (the person to make sure the data is written down and that each student writes in his or her journal); Materials Coordinator (responsible for getting, keeping inventory, and returning materials); and Starter (the person who initiates the station activity and makes sure everyone participates).
Closure
Students return to seats and use the graphic organizer to reinforce the concepts and vocabulary:
Name of Station

Variables

Factors

Functions

Physics of Sound

Water in container

2cms more in each container

For every change of 2cms in a container, the pitch of the sound increased by two notes on the xylophone

Lifeboat Inspection
 
Pendulums
 
Flipsticks

Individual reflection in journal: What is a variable? What are patterns you notice?
Discuss: What are some relationships you noticed at each station?
Assessment
Although students work in groups, each person is responsible for writing down their own data and intrapersonal reflections.
Extensions
Students may write their own "flipper game."
Plan B
Class meets as a whole to conduct one experiment at a time as a demonstration (use this method if you chose to teach this as a one time lesson as opposed to a weekorlonger inquiry).
Inquiry Three: Describing Change Using TCharts, an Introduction to Functions
Description:Students move from actual physical experience with objects (variables) and their relationships (functions), toward a symbolic notion of the concepts.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Other
Students move from actual or physical experience with objects (variables), and their relationships (functions), toward a symbolic notion of the concepts.
Objectives
 Students see patterns when represented numerically
 Understand relationships between shapes and perimeter
 Describe the patterns of change among variables
 Call those patterns of change a function
About an hour of direct instruction from the teacher followed by up to week of selfpaced exploration and expansion of the initial idea by students.
Materials
 Student Observation and Reflection Journals
 About 50 triangular tiles for Row of Triangles station
 About 50 square tiles for Row of Squares station
 About 50 pentagon tiles for Row of Pentagons station
Students rotate in groups through the following stations in order to gain more experience with relationships in the real world.
 Length of the side of a square to number of squares: Squares from squares
 Number of triangles to distance of perimeter: Row of triangles
 Number of pentagons to distance of perimeter: Row of pentagons
Students make a TChart in each of their journals for each of the stations while gathering data.
Closure
Students respond to the following questions in their observation and reflection journals: What are the variables for each activity? What are the factors for each activity? What is the pattern or function (the rule you wrote) in each activity? What are some other patterns you noticed?
Assessment
Teacher reads and responds to journals in order to determine whether or not students are able to collect data in an organized way and if they are beginning to use the vocabulary in an accurate way. This is an opportunity for the teacher to catch misconceptions students may have and write questions to the students which may help eliminate those misconceptions.
Plan B
Students rotate one person at a time through the stations while others read silently if the groups do not work well together.
Inquiry Four: Introduction to Exponents
Description:Students learn the function of an exponent
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Other
Students understand the function of an exponent.
Objective
Students see the connection between exponents in their written form and in an observed form
Materials
 Student Reflection and Observation Journal
 Classroom Math Textbook
 Online access to Secret Worlds, The Universe Within
Time
About one half hour to start the discussion of the basic themes. Students then selfpace their exploration of the online film, exponents chart and traditional textbook "homework" throughout the week.
Activities
Explain: Exponents are a key idea used in math and science to describe patterns of change between variables.
Students copy and continue exponents chart in their Observation and Reflection Journal
Students view, then discuss in pairs: Secret Worlds, The Universe Within
Students complete activities from the classroom math textbook chapter on exponents.
Closure
Reflect in journal: Define exponent.
Discuss: What do you know about exponents?
Assessment
In having each student complete
problems from the textbook, the teacher will be able to monitor whether
or not students are able to apply the vocabulary to the manipulation of
symbols.
Inquiry Five: Describing Functions and Solving for Variables
Description:Students will see relationships between variables and functions in order to solve equations.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Other
Students will see relationships between variables and functions in order to solve equations.
Objective
Provide repeated exposure to the concepts of patterns, variables, and functions in order to provide multiple opportunities for students to see and describe relationships.
Materials
 About 30 tiles for each student
 One "hundreds" chart for each student: Here is blank to printout for students, in .pdf format
or use an online, interactive version  Student Observation and Reflection Journal
 a computer with online access
 equations to solve (see below)
Up to one hour of direct instruction , then students selfpace their exploration of the hundreds chart and online practice/quizzes throughout the week.
Activities
 Using tiles, markers and "hundreds" charts, students create patterns which "grow" by the same factor.
 See Prentice Chapter 2: Algebra: Patterns and Variables practice/quizzes
Closure
Share patterns
Discuss problems completed.
Assessment
In having each student complete online activities, the teacher will
be able to monitor whether or not students are able to apply the
vocabulary to the manipulation of symbols.
Extensions
Students invent more patterns and describe them in their journals
Plan B
Teacher uses transparency tiles to make patterns on the overhead projector. Class discusses patterns.
Inquiry Six: Magic Squares
Description:Understand and use variables in a symbolic way.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 elementary
 3rd
 4th
 5th
 middle
 6th
 7th
 8th
 tween
 Other
Understand and use variables in a symbolic way
Objective
Gain practice in solving for a variable using given numbers
Materials
Magic Square handout
Time
About an hour of direct instruction to explain the concept of magic squares. Students use the rest of the week to practice solving existing magic squares and experiment with making their own.
Activities
Students are individually presented with
magic squares in which variables appear. They replace the variable with
a number that creates an equal sum, whether one adds the numbers
horizontally, vertically, or diagonally. Students complete handouts
that provide practice in finding the value for the missing variables:
Handout
"A magic square is an arrangement of numbers in a
square in which the rows, columns, and diagonals each have the same
sum. A magic square is shown below:
7

2


1

5


8

4

To find the missing values in a magic square first identify a row, column, or diagonal in which all the values appear. The sum of the completed diagonal in the magic square shown above can be represented by the numerical expression 8 + 5 + 2. A numerical expression contains only numbers and mathematical symbols.
1. What is the sum of each row, column, and diagonal of the magic square shown above? (answer: 15)
You can represent the missing value in each square by a variable, as shown below. A variable is a symbol, usually a letter, that stands for a number.
a

7

2

1

5

b

8

c

4

2. Name the variables in the magic square. (answer: a,b,c)
The variable expression a + 7 + 2 represents the sum of the entries in the first row. A variable expression is an expression that contains at least one variable.
3. What is the value of a? (answer: 6)
(handout adapted from Prentice Hall Mathematics, 6th Grade Edition, 2000)
Closure
Discuss problems and answers.
Assessment
In having each student complete problems
from the handouts, the teacher will be able to monitor whether or not
students are able to apply the vocabulary to the manipulation of
symbols.
Extensions
Students create their own magic square
Inquiry Seven: Variables and Functions We Notice in the World
Description:Make connections between observed phenomena in the world and math vocabulary
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 35 / Ages 810
 Grades 68 / Ages 1113
 Other
Make connections between observed phenomena in the world and in math vocabulary.
Objective
Brainstorm relationships
Materials
 Observation and Reflection Journal for each student
 Newspapers or scientific magazines
About one half hour to explain this week's project and show examples. As with the other inquiries, students selfpace their work for the rest of the week, coming back together as a class at the end of the inquiry for a "gallery walk" and discussion.
Activities
1. Students brainstorm possible sets of objects or events whose relationships cause an observable pattern, or function. The teacher gives examples during the brainstorming session such as: The size of an image on a screen to the distance of the projector from the screen, or the number of cars to the amount of carbon dioxide pollution.
2. Students make a list of possible relationships in their journals
3. Students create collages made from photographs and/or drawings cutout from newspapers and/or magazines throughout the week.
Closure
At the end of the week, class has a “Gallery Walk” to view
one another’s artwork and to discuss ideas about relationships
Assessment
Students demonstrate their
understanding of relationships through the use of pictures.
Extensions
Students interview other members of the
community about objects and events that have relationships
Plan B
Students write letters to scientists
asking them about phenomena such as weather, earthquakes or
transportation that have variables, factors and functions.
Inquiry Eight: Using Graphs, Words and Pictures to Describe Functions
Description:Demonstrate relationships between objects or events and the variables, factors, functions or patterns that describe them.
Last Updated:
Subject(s): Mathematics
 Mathematics > Algebra
 ...
 Grades 68 / Ages 1113
 middle
 6th
 7th
 8th
 tween
 Other
Demonstrate relationships between objects or events and the variables, factors, functions or patterns that describe them.
Objectives
 Represent data in an organized, graphic way in order to see patterns between variables and factors
 Use graphs, words, mathematic symbols and/or pictures to describe relationships between observed phenomena
 Gain experience using spreadsheet software to input data and generate a graph or graphs
 Student Observation and Reflection Journals with data from InquiriesTwo and Three
 Computer(s) with Microsoft Excel or Open Office spreadsheet software
(Depends upon number of computers available and how many projects each student decides to graph). At the end of the week, the class meets as an entire group to discuss the findings and give one another feedback.
Activities
During class discussion, students
create criteria for “what makes an informative graph.” The
criteria will be used to assess student work. An example of a
criterion: “A graph must produce a straight or curved line to show
a patterns or function.” Another example might be: “A written
explanation must help describe the graph,” or for students who are
up to a challenge: “Mathematic notation of variables must work with
any number.”
Students enter data they collected from one or two experiments they conducted in Inquiry Two (or data from Inquiry Three) into Microsoft Excel, Open Office Calc, or another spreadsheet program, while referring to their Observation and Reflection Journals. Each person generates one or two graphs, a written explanation in English, other pictures if they wish, and a mathematical/symbolic description as well.
Closure
Each student posts his or her graphs on
the bulletin board. Students take a “Gallery Walk” in a line to
look at other people’s data and graphs.
Here are some samples of student work:
Student Work Sample 4 (before graphed on computer)
Discuss
What did you notice in looking at your
colleagues’ graphs and explanations?
Why do some lines on graphs curve while
others are straight?
Assessment
Students gain feedback from one another
during and after the Gallery Walk. Teacher uses criteria generated by
students when giving evaluation and feedback.
Extensions
An optional culminating activity would
be to challenge students to find objects or events in the world
they’ve actually seen, experienced, or heard about in the media.
Examples include:
 The best size of a tire for a race car
 The most absorbent paper towel
 The longest burning candle
 The best recipe for lemonade
 The best fabric for a raincoat
Students draw their findings on graph paper instead of using software.