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Kevin Driscoll
(Los Angeles - United States)I taught Computer Science and Mathematics from 2004-2007 at Prospect Hill Academy Charter School in Cambridge, MA. Since then, I completed a Master's degree in Comparative Media Studies at MIT and am now a PhD candidate in Communication at University of ...
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- Contributed By: Sarah Zainfeld
Students learn science and math through opportunities to see and describe patterns in the world. The Patterns, Variables and Functions inquiry provides diverse learners access to see patterns, learn vocabulary, as well as make and describe models through repeated exposure to concepts. Throughout the process, they engage in mathematic and scientific reasoning not unlike mathematicians and scientists themselves, thereby giving students authentic and explicit exposure to the fields. Students start out with an exploration of materials, then move toward a use of vocabulary and symbolic notation. Students access their background knowledge from the first hands-on projects to build new understandings so that by the end of the semester they create and describe functions with graphs, words, mathematic notations and pictures.
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.
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.
Mathematics and science are closely linked disciplines. The essence of mathematics is the study of patterns in the world and the essence of science is the process of making sense of observed phenomena in the world. Because math and science complement and depend upon one another, the goals of this thematic instruction on patterns, variables and functions are for students to be able to:
- 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
Inquiry One: The "infinite" across the disciplines
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 self-paced discovery, students will move between the following stations exploring the relationship between:
InquiryThree: Describing relationships using T-Charts, 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.
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.
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 self-paced 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 T-Charts, 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.
Fifth and sixth grade students benefit from activities that allow them to build on their own intuitive sense of the way the world works. It is also of vital importance that misconceptions be identified and corrected by the teacher. Opportunities for teachers to correct misconceptions happen when students conduct systematic observations of properties of objects and when they have opportunities to record, hypothesize and draw conclusions. Students should be encouraged to ask and develop new questions. "Some of the most important scientific concepts students learn are the results of their ability to see relationships between objects and events. Relationships always involve interactions, dependencies, and cause-and-effect events" (Foss Variables Module 11). Due the developmental level of fifth and sixth graders, they are able to weigh multiple events, but in a limited way, therefore the scope of the activities is narrowed to emphasize the relationship of two variables, when possible.
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. Hands-on activities help to create conditions that "enable subordinated students to move from their usual passive position to one of active [hands-on] 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 multi-ethnic 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 problem-solve 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 open-ended, 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 cross-disciplinary connections are a means in the inquiry of phenomena in the world, as opposed to a direct focus on the theme, itself.
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. Hands-on activities help to create conditions that "enable subordinated students to move from their usual passive position to one of active [hands-on] 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 multi-ethnic 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 problem-solve 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 open-ended, 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 cross-disciplinary connections are a means in the inquiry of phenomena in the world, as opposed to a direct focus on the theme, itself.
The following inquiries are designed for a fifth or sixth grade class. Some English Language Learners may have "tested out" or special classes, but still benefit from sheltering. Throughout the school year, the concept of the "infinite" may have arisen across the disciplines, usually most often during math. Students have been told, "numbers go on forever." Their basic understanding of the word, "infinite" can be used to build other knowledge such as the ability to see patterns, identify variables, relationships and factors in change.
Most assessment for this instruction is embedded in each inquiry. Through the use of student Observation and Reflection Journals, the teacher may see if (1) students learn and collect data in a systemic way; (2) use intuitive notions of concepts to build vocabulary; (3) draw on accurate use of vocabulary to (3) gain symbolic representations of data; and (4) understand the connection between symbols and real-world events. A student's ability to use symbols to represent data will be assessed primarily through their assignment in Inquiry Eight, where each will make a graph and explain it using words, mathematic notation, and/or other pictures. Students will develop rubrics for "what makes an informative graph." When students generate their own criteria and/or rubrics for good work, they are more likely to be self-regulated learners who truly understand and can benefit from the ways they are being assessed
Goal
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
Materials
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.
Closure
Students next re-group 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.
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 Visual-Spatial, Interpersonal intelligences (see Howard Gardner's work)]
- What are some patterns you've seen in math, which could go on forever? [students use Logical-Mathematic, 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 re-group 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.
Goal
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
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 real-world meaning to the work being done in their traditional math and science texts. Materials
Materials for Station #1: Lifeboat Inspection
Materials for Station #2: Flipsticks
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:
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 week-or-longer inquiry).
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 real-world 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 cut-off. 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 meter-tape
- 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 (names of objects) |
Factors (patterns you noticed for this variable only) |
Functions (patterns you noticed between all the variables) |
|
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 week-or-longer inquiry).
Goal
Students move from actual or physical experience with objects (variables), and their relationships (functions), toward a symbolic notion of the concepts.
Objectives
About an hour of direct instruction from the teacher followed by up to week of self-paced exploration and expansion of the initial idea by students. Materials
Students rotate in groups through the following stations in order to gain more experience with relationships in the real world.
Students make a T-Chart 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.
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 self-paced 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 T-Chart 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.
Goal
Students understand the function of an exponent.
Objective
Students see the connection between exponents in their written form and in an observed form Materials
Time
About one half hour to start the discussion of the basic themes. Students then self-pace 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.
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 self-pace 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.
Goal
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
Up to one hour of direct instruction , then students self-pace their exploration of the hundreds chart and online practice/quizzes throughout the week. Activities
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.
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 print-out 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 self-pace 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.
Goal
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:
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.
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
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
|
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
Goal
Make connections between observed phenomena in the world and in math vocabulary. Objective
Brainstorm relationships Materials
About one half hour to explain this week's project and show examples. As with the other inquiries, students self-pace 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.
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 self-pace 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.
Goal
Demonstrate relationships between objects or events and the variables, factors, functions or patterns that describe them.
Objectives
(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 1 Student Work Sample 3 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:
Students draw their findings on graph paper instead of using software.
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 1 Student Work Sample 3 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.
