English Language Arts 10 – 2
MATHEMATICS 10C
LINEAR RELATIONS
High School collaborative venture with
Harry Ainlay, Jasper Place, McNally, Queen Elizabeth,
Ross Sheppard and Victoria Schools
[pic]
Harry Ainlay: David Cunningham, Christine Dibben
Jasper Place: Linda Aschenbrenner, Shelaine Kozakavich, Roy Riege, Nic Ryan
Ross Sheppard: Tim Gartke, Jeremy Klassen, Don Symes
Victoria: Kevin Bissoon, Elisha Pinter
Facilitator: John Scammell (Consulting Services)
Editor: Rosalie Mazurok (Contracted)
2009 - 2010
TABLE OF CONTENTS
| | |
|STAGE 1 DESIRED RESULTS |PAGE |
| | |
|Big Idea |4 |
| | |
|Enduring Understandings |4 |
| | |
|Essential Questions |4 |
|Knowledge |5 |
| | |
|Skills |6 |
| | |
|Stage 2 ASSESSMENT EVIDENCE | |
| | |
|Teacher Notes For Transfer Tasks |7 |
| | |
|Transfer Tasks | |
| The Biggest Loser | |
|Teacher Notes for The Biggest Loser and Rubric |8 - 9 |
|Transfer Task |10 - 11 |
|Rubric |12 - 13 |
| | |
|Height vs. Historical Period | |
|Teacher Notes for Height vs. Historical Period and Rubric |14 - 15 |
|Transfer Task |16 - 22 |
|Rubric |23 - 24 |
|Possible Solution |25 - 26 |
| | |
|Stage 3 LEARNING PLANS | |
| | |
|Lesson #1 Describing and Representing Linear and Non-Linear Relations |27 - 29 |
|Lesson #2 Properties of Linear Relations |30 - 32 |
| | |
|Lesson #3 Building a Personal Strategy to Find an Equation of a Linear Relation Given |33 - 35 |
|a Graph or Information about the Line | |
|Lesson #4 Slope y-intercept Form |36 - 38 |
|Lesson #5 Point-slope Form |39 - 41 |
|Lesson #6 General Form (Ax+By+C=0) |42 - 43 |
|Lesson #7 Finding an Equation Given Characteristics of a Linear Relation |44 - 45 |
| | |
|APPENDIX - Handouts | |
| | |
|Linear Relations Unit Handouts |47 |
Mathematics 10C
Linear Relations
| |
|STAGE 1 Desired Results |
[pic] Big Idea:
Linear relations provide the tools to communicate, model and explore the relationship between two sets of data.
[pic] Enduring Understandings:
Students will understand that…
• graphs and equations of linear relations have characteristics that can be analyzed and interpreted in context.
• there is a relationship between the equation of a linear relation and its graph.
o the equation of a linear relation can be written in various forms.
[pic] Essential Questions:
• Which form of a linear equation is appropriate in any particular contexts?
o How do the equivalent equation forms inter-relate?
• How can we identify and model linear relationships?
• What is the contextual significance of the intercepts and slope of a graph?
• What restrictions should be placed on a linear function, and in what contexts would such restrictions be necessary?
[pic] Knowledge:
|Enduring |Specific |Knowledge that applies to this Enduring Understanding |
|Understanding |Outcomes | |
| | | |
|Students will understand that… | |Students will know… |
| | | |
|graphs and equations of linear |*RF5 |characteristics of graphs of linear relations, including: |
|relations have characteristics that | |slope |
|can be analyzed and interpreted in | |intercepts |
|context. | |domain |
| | |range |
| | | |
| | |Students will know… |
| | | |
|there is a relationship between a | |a point must satisfy the equation. |
|linear relation and its graph. | | |
| | |the forms of linear equations: |
|the equation of a linear relation can|*RF6 |slope-intercept form, [pic] |
|be written in various forms. | |general form [pic] |
| |*RF7 |slope-point form [pic] |
| | | |
| | |that certain forms are more appropriately used in different |
| | |situations. |
| | | |
*RF= Relations and Functions
SS
[pic] Skills:
|Enduring |Specific |Skills that apply to this |
|Understanding |Outcomes |Enduring Understanding |
| | | |
|Students will understand that… | |Students will be able to… |
| | | |
|graphs and equations of linear | |determine slopes. |
|relations have characteristics that |*RF5 | |
|can be analyzed and interpreted in | |determine x– intercepts and |
|context. |*RF6 |y– intercepts. |
| | | |
| | |determine domain and range. |
| | | |
| | |interpret a graph in a given context. |
| | | |
|Students will understand that… | |Students will be able to… |
| | | |
|there is a relationship between a | |create the equation when given: |
|linear relation and its graph. | |a graph |
| | |a point and the slope |
|the equation of a linear relation can| |two points |
|be written in various forms. | |a point and the equation of a parallel or perpendicular line |
| | | |
| | |convert from one form of an equation to another, |
| | |[pic] |
| |*RF4 | |
| | |apply an appropriate strategy for creating an equation or graph. |
| |*RF6 | |
| | |write the equation, given a graph. |
| |*RF7 | |
| | |draw a graph, given the equation. |
| | | |
| | | |
*RF= Relations and Functions
| |
|STAGE 2 Assessment Evidence |
1 Desired Results Desired Results
[pic] The Biggest Loser or Height vs. Historical Period
Teacher Notes
There are two transfer tasks to evaluate student understanding of the concepts relating to linear relations. The teacher (or the student) will select one for completion. Photocopy-ready versions of the two transfer tasks and rubric are included in this section.
Each student will…
(The Biggest Loser)
• create data for each participant.
• analyze the data and plot graphs of the data.
• interpret the graphs.
(Height vs. Historical Period)
• create averages for each era.
• identify the restricted domain and range for each era and explain their significance.
• identify the intercepts and explain their significance.
Teacher Notes for The Biggest Loser
For this project you may assist students in the following ways:
• Teacher may lead students to draw graphs of week 0 to 5, if they have not already created this on their own.
• When students find the slope between weeks 2 and 5, the teacher may discuss the meaning of parallel lines (i.e., Jared and Arya)
• Students will find a linear equation for each contestant modeling their weight loss for weeks 0 to 5, if they have not already done so on their own. There are many ways to do this, for example students could use the line of best fit or use the point- slope formula with the slope from weeks 2 to 5.
• State the domain and range for each contestant for weeks 0 to 5.
Differentiated Instruction
If students require a further challenge, you may want to look at this.
Background
• The Biggest Loser requires you to plan a program that allows males to eat 2000 calories a day and train enough to burn 6000 calories a day through exercise. Females eat 1200 calories a day and burn 5000 calories per day through exercise.
• A deficit of approximately 3500 calories is required to lose one pound.
• The basal metabolic rate is what your body needs to maintain normal functions like breathing and digestion. This is the minimum number of calories you need to eat each day.
Some basal metabolic calculators can be found at:
Teacher Notes for Rubric
• No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.
• Limited is considered a pass. The only failures come from Insufficient/Blank.
• When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.
[pic] The Biggest Loser - Student Assessment Task
Situation:
The Biggest Loser is a television reality show that takes morbidly obese Americans and trains them full - time on an isolated campus. You have been hired by The Biggest Loser to train the following contestants:
• Jared is a 43 year old male who is 6’2’’ tall and 435 pounds.
• Salim is a 27 year old male who is 5’6” tall and 380 pounds.
• Hassan is a 41 year old male who is 6’8” tall and 513 pounds.
• Arya is a 28 year old female who is 5’7’’ tall and 370 pounds.
You will have them on campus for 10 weeks and your goal is to help them lose as much weight as possible. The winner will be the contestant who loses the greatest percentage of his/her body fat over the course of 10 weeks.
Task:
The producers of the show want to have an idea as to which contestant will win the show. They have asked you to make a prediction after the first 5 weeks, supported by mathematical models, so that they can prepare for the eventual winner. Track and analyze the weight loss of each contestant.
Data for contestants:
|Jared |Salim |Hassan |Arya |
|Week |Weight |Week |Weight |Week |Weight |
|Performs Calculations |Performs precise and |Performs focused and |Performs appropriate |Performs superficial |No score is awarded |
| |explicit |accurate |and generally accurate |and irrelevant |because there is no |
| |calculations. |calculations. |calculations. |calculations. |evidence of student |
| | | | | |performance. |
|Presents Data |Presentation of data is |Presentation of data |Presentation of data is|Presentation of data is|No data is presented. |
| |insightful and astute. |is logical and |simplistic and |vague and inaccurate. | |
| | |credible. |plausible. | | |
|Explains Choice |Shows a solution for the|Shows a solution for |Shows a solution for |Shows a solution for |No explanation is |
| |problem; provides an |the problem; provides|the problem; provides |the problem; provides |provided. |
| |insightful explanation. |a logical |explanations that are |explanations that are | |
| | |explanation. |complete but vague. |incomplete or | |
| | | | |confusing. | |
|Communicates findings |Develops a compelling |Develops a convincing|Develops a predictable |Develops an unclear |No findings are |
| |and precise presentation|and logical |presentation that |presentation with |communicated. |
| |that fully considers |presentation that |partially considers |little consideration of| |
| |purpose and audience; |mostly considers |purpose and audience; |purpose and audience; | |
| |uses appropriate |purpose and audience;|uses some appropriate |uses inappropriate | |
| |mathematical vocabulary,|uses appropriate |mathematical |mathematical | |
| |notation and symbolism. |mathematical |vocabulary, notation |vocabulary, notation | |
| | |vocabulary, notation |and symbolism. |and symbolism. | |
| | |and symbolism. | | | |
[pic]Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
Teacher Notes for Height vs. Historical Period
Use any or all of the questions #1-7 as discussion with students. Students can choose a format for their presentation such as a blog, research paper, PowerPoint, etc.
Students may need guidance with regard to labelling axes when graphing.
In order to complete this task students need not establish equations for each era. As an extension, they could create and compare the equations with respect to their
y-intercepts and slopes.
Direction should be given pertaining to the y-intercepts. There could be a y-intercept established for each time period “at the beginning of.”
The teacher has the choice of guiding students with respect to scales or allowing students to choose their own. Guidance will result in consistency but differences in personal choice will promote the development of strategies and discussion.
Differentiated Instruction
Research the current average height in your community. How does this compare to your extrapolation on the graph? What might account for the similarities / differences between your extrapolation and the actual averages you found in your research?
Is there anything that could be considered misleading when you compare your graphs of the three eras? If so, how could this be corrected? Are the slopes misleading?
Teacher Notes for Rubric
• No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.
• Limited is considered a pass. The only failures come from Insufficient/Blank.
• When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.
The proof may be found beneath the Leverhulme Centre for Human Evolutionary Studies in Cambridge where 20 000 ancient skeletons lie. There are rows of skulls kept in boxes with plastic windows at the front. As you walk along, you can peer into the great dark eye sockets of a skull from Sarawak or the eggshell-thin cranium of a child from New Guinea. "Bones are like a book, recording the history of each person," explains Stock as he slides out the boxes that contain the remains of ancient human beings. It was this collection that first proved to the world that humanity shares a common African ancestor and it is this collection that has demonstrated agriculture could be bad for us. To illustrate the impact of agriculture, Stock and one of his students, Anne Starling, examined a unique set of these skeletons - 9000 are from the Nile Valley in Egypt that span an extraordinary historical range from Neolithic hunter-gatherers through to the year 1500 BC. The researchers were looking for signs of malnutrition, which are reflected in a person's teeth. Just as tree-rings can indicate the health and age of a tree, so too, a defect in the layers of enamel called linear enamel hypoplasia (LEH) can indicate whether a person has been ill or deprived of food for several months.
What Stock and Starling discovered was that 40 percent of hunter-gatherers who lived 13,000 years ago had LEH. Fast-forward 1000 years to the Egyptians farmers and the figure rises dramatically to 70 percent. Originally, the hunter-gatherers were about 5ft 8in with robust skeletons. Yet once farming began, the average height decreased by four inches. Stock showed the bones of a man who lived 7000 years ago, which are so thin and delicate they look as if they might snap.
What caused this reduction in height besides malnutrition linked to an agriculture-based diet? One possibility is disease. Stock and Dr. Andrea Migliano of Cambridge demonstrate the link by comparing pygmy skeletons from the Andaman Islands whose already smaller body size shrank dramatically when they encountered Western colonialists, who brought with them diseases like influenza and syphilis to the skeletons of other tribes who kept their distance from the newcomers and actually grew taller during this period.
Yet agriculture did have advantages as well. The Egyptian skeletons reveal that around 4000 years ago farmers suddenly started to grow bigger and become healthier. The average height returned to 5ft. 8in., and only a fifth of the skeletons show signs of malnourishment.
In any case, says Dr. Stock, we are quite clearly at the point of no return. Agriculture has led to a surplus of food and this in turn allowed women to have more children (albeit initially unhealthy ones), leading to a global population of almost seven billion. "A lot of the problems we are facing today stem from the advent of agriculture," he says. "But we are ingenious enough to come up with technological solutions.”
Role: You are a forensic anthropologist.
Audience: A group of your peers at the conference.
Format: Create a presentation based upon the data provided.
Task: Given the data, create a graphical representation of the data provided that will
be presented at the American Anthropology Association (AAA) Conference.
Topic: Height as a function of historical time period.
The following data is adapted from average heights found in Dr. Diamond’s research. It has been creatively enhanced to facilitate the project.
11 000 BC – 10 000 BC 10 000 BC – 2 000 BC 2000 BC – 2010 AD
Procedure:
1. Using the graph paper provided create the three separate graphs based on the tables above. You may wish to consider how these graphs can be combined before you actually begin graphing
2. Assemble the three graphs in chronological order. You may wish to assemble the graphs in other ways for your presentation.
3. Based upon the reading and graphing of the data over the three time periods discuss what factors influence the potential heights in each era.
Conference Presentation
Consider the following questions:
1. Based upon the reading and graphing of the data over the three time periods discuss what factors influence the potential heights in each era.
2. Slope is commonly described as a rate of change. Determine the slope of each era. What is the meaning of each of these slopes? How would you know if growth rates are accelerating or decelerating by comparing the slopes?
3. Can the graphs be changed to better compare the slopes? If so, how?
4. Can you predict/extrapolate for current heights in your generation?
5. Does geographical location matter? Why or why not?
6. What effect does the import/export market on population heights?
7. Growth hormones emerged in the mid -1980’s in our food in Canadian food. What effect might this have on our present day heights?
8. The heights will likely plateau in our era. Comment on this and speculate what might/would/could occur to produce this result.
[pic] Assessment
Mathematics 10C
Linear Relations
Rubric
| | | | | | |
|Level |Excellent |Proficient |Adequate |Limited* |Insufficient / Blank* |
| |4 |3 |2 |1 | |
|Criteria | | | | | |
|Performs Calculations |Performs precise and |Performs focused and |Performs appropriate |Performs superficial |No score is awarded |
| |explicit |accurate |and generally accurate |and irrelevant |because there is no |
| |calculations. |calculations. |calculations. |calculations. |evidence of student |
| | | | | |performance. |
|Presents Data |Presentation of data is |Presentation of data |Presentation of data is|Presentation of data is|No data is presented. |
| |insightful and astute. |is logical and |simplistic and |vague and inaccurate. | |
| | |credible. |plausible. | | |
|Explains Choice |Shows a solution for the|Shows a solution for |Shows a solution for |Shows a solution for |No explanation is |
| |problem; provides an |the problem; provides|the problem; provides |the problem; provides |provided. |
| |insightful explanation. |a logical |explanations that are |explanations that are | |
| | |explanation. |complete but vague. |incomplete or | |
| | | | |confusing. | |
|Communicates findings |Develops a compelling |Develops a convincing|Develops a predictable |Develops an unclear |No findings are |
| |and precise presentation|and logical |presentation that |presentation with |communicated. |
| |that fully considers |presentation that |partially considers |little consideration of| |
| |purpose and audience; |mostly considers |purpose and audience; |purpose and audience; | |
| |uses appropriate |purpose and audience;|uses some appropriate |uses inappropriate | |
| |mathematical vocabulary,|uses appropriate |mathematical |mathematical | |
| |notation and symbolism. |mathematical |vocabulary, notation |vocabulary, notation | |
| | |vocabulary, notation |and symbolism. |and symbolism. | |
| | |and symbolism. | | | |
[pic]Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
[pic]
Calculations for Height vs. Historical Period
| |
|STAGE 3 Learning Plans |
Lesson 1
Describing and Representing Linear and Non-Linear Relations
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that… |Which form of a linear equation is appropriate in any particular |
| |contexts? |
|graphs and equations of linear relations have characteristics that |How do the equivalent equation forms inter-relate? |
|can be analyzed and interpreted in context. |How can we identify and model linear relationships? |
|there is a relationship between the equation of a linear relation and|What restrictions should be placed on a linear function, and in what |
|its graph. |contexts would such restrictions be necessary? |
|the equation of a linear relation can be written in various forms. | |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|characteristics of graphs of linear relations, including: |apply an appropriate strategy for creating an equation or graph. |
|slope |write the equation, given a graph. |
|intercepts |draw a graph, given the equation. |
|domain | |
|range | |
[pic]Lesson Summary
Students will be able to graph a variety of relations by focusing specifically on the characteristics of linear relations.
Students should be able to describe their observations.
[pic] Lesson Plan
Activity Suggestions
Split the class into two groups to work at stations.
Station #1 – Categorizing Relations
Given a variety of relations (tables of values, graphs, equations, etc) have students work in groups to categorize the relations (working towards linear vs. non-linear, slopes, etc).
For example:
Station #2 - Describing a Graph
Provide students with linear and non-linear graphs, some with scales, and some without. One person describes a graph to his/her partner describes and the partner attempts to re-create it.
Follow up discussion:
• What kinds of words did you use to describe the graph?
• What words would have been useful?
• What do you notice about the equations of the linear graphs?
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 5.6)
Math 10 (McGraw Hill: sec 6.2)
[pic]Glossary
linear – capable of being represented by a straight line
non-linear – not capable of being represented by a straight line
slope – incline or grade or [pic]
table of values – an arrangement of information or data into columns and rows
Lesson 2
Properties of Linear Relations
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that … |What is the contextual significance of the intercepts and slope of a |
| |graph? |
|graphs and equations of linear relations |What restrictions should be placed on a linear function, and in what |
|have characteristics that can be analyzed and interpreted in context.|contexts would such restrictions be necessary? |
| |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|characteristics of graphs of linear relations, including: |determine slopes. |
|slope |determine x– intercepts and y– intercepts. |
|intercepts |determine domain and range. |
|domain |interpret a graph in a given context. |
|range | |
| | |
[pic]Lesson Summary
Students will be able to identify and calculate slope of a line in a variety of situations.
Students will be able to identify and interpret
• the x- and y- intercepts.
• the domain and range.
[pic] Lesson Plan
Review of slope
Given graphs, equations, points, etc. find the slope.
(Shooting Balloons)
Introduction of x- and y-intercepts
• Given equations (including both positive and negative slopes and a non – linear equation) have students create a table. Plot points and identify where
o the x- and y- intercepts are on the graph.
o the x- and y- intercepts are in the table.
• Given a selection of graphs with equations – identify intercepts as ordered pairs.
• Discuss observations :
o Is there a more efficient method to determine the x- and y- intercepts?
o Develop a rule to calculate x- and y- intercepts given equations.
Real-life examples related to the context of slope, x and y- intercepts, and domain and range.
For example, the change in speed over time when driving, or the change in the
height of a burning candle.
Differentiation opportunities exist in the context and choice of word problems.
[pic] Going Beyond (possible assessment)
Activity
Divide the students into two groups and have students collect and graph data they collect.
• One group gathers data which is continuous (i.e., time – Some students time others while they answer a series of multiplication questions.)
• The other group gathers discrete data (i.e., number of students in a class).
Compare the two graphs in their contexts.
[pic]Assessment
Exit card:
Given the equation related to a specific situation, determine the slope, x- and y-intercepts and their significance in the context of the problem and the domain and range.
[pic]Resources
Math 10 (McGraw Hill: sec 6.2)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 5.4, 5.7)
[pic]Glossary
x-intercept – the value representing where a graph intercepts or crosses the x-axis
y-intercept – the value representing where a graph intercepts or crosses the y-axis
ordered pair – refers to the location of a point in the coordinate plane
Lesson 3
Building a Personal Strategy to Find an Equation of a Linear Relation Given a Graph or Information about the Line.
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that… | |
|graphs and equations of linear | |
|relations have characteristics that can be analyzed and interpreted |Which form of a linear equation is appropriate in any particular |
|in context. |contexts? |
|there is a relationship between the equation of a linear relation and|. |
|its graph. | |
|the equation of a linear relation can be written in various forms. | |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
|characteristics of graphs of linear relations, including: | |
|slope |create the equation when given: |
|intercepts |a graph |
|domain |a point and the slope |
|range |two points |
| |a point and the equation of a parallel or perpendicular line |
|that a point must satisfy the equation. | |
| |convert from one form of an equation to another, |
|forms of linear equations: |[pic] |
|slope-intercept form, [pic] | |
|general form [pic] |apply an appropriate strategy for creating an equation or graph. |
|slope-point form [pic] | |
| |write the equation, given a graph. |
|that certain forms are more appropriately used in different | |
|situations. |draw a graph, given the equation. |
[pic]Lesson Summary
Students will explore the communication of various forms of linear relations as rules or equations.
[pic] Lesson Plan
Introduction
• Given a real-life situation – create a rule or equation to represent this information. Use the rule to determine other points on the line.
• Class discussion to follow
• Matching worksheet – Students receive a worksheet on which one side has graphs and a list of equations in slope y- intercept form. The other side has the same graphs, but the equations are in the general form and the point – slope form.
• Discuss and compare equations.
o Why can some different equations result in the same graph?
Lesson:
• Students are given tables of values for linear relations and will determine a pattern and equation to represent the relation.
• Students are given graphs of linear relations and will determine an equation to represent the relation.
• Special cases where slope of the line is 0 or undefined.
Activity
Go back to different ways of describing relationships with respect to equations.
• Given a graph [pic] find the equation.
• Given a table [pic] find the equation.
• Given ordered pairs [pic] find the equation.
• Given words [pic] find the equation.
Differentiation is included in the above activity, as each level represents a different level of thinking.
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.4)
Math 10 (McGraw Hill: sec 7.1)
[pic]Glossary
slope y-intercept form of a linear equation – an equation written in the form [pic], where m is the slope and
b is the y-intercept
linear relation – a meaningful connection between two variables which would result in a straight line graph
undefined slope – slope of a vertical line
zero slope – slope of a horizontal line
Lesson 4
Slope y-intercept Form
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that… | |
| | |
|graphs and equations of linear relations have characteristics that | |
|can be analyzed and interpreted in context. |Which form of a linear equation is appropriate in any particular |
|there is a relationship between the equation of a linear relation and|contexts? |
|its graph. | |
|The equation of a linear relation can be written in various forms. | |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
| |Students will be able to… |
|Students will know … | |
| |apply an appropriate strategy for creating an equation or graph. |
|that a point must satisfy the equation. | |
|forms of linear equations: |write the equation, given a graph. |
|slope-intercept form, [pic] | |
|that certain forms are more appropriately used in different |draw a graph, given the equation. |
|situations. | |
[pic]Lesson Summary
Students will be able to graph a linear relation from corresponding equations given in slope y-intercept form.
Students will discover a connection between the slope-intercept form and certain characteristics of the graph of a linear relation.
[pic] Lesson Plan
Introductory Activity
• Given graphs and their equations labelled – determine their slope and y-intercept and identify a rule for determining equations.
Lesson:
• Given graphs (where y-intercept and slope are obvious) determine an equation.
• Given equation, draw the graph.
• Given real-life data, find equations.
• In partners - each partner presents an equation or a graph to their partner, and has their partner determine the other.
[pic] Going Beyond (possible assessment)
Give students in pairs the Matching Graphs and Equations Handout from the back of this unit.
On the grids, students should draw their graphs and below give the equation for each corresponding graph.
Cut up the page into the rectangles and place the pieces in a bag or box. The container should be passed to another pair, who will match each graph to its corresponding equation.
[pic]Resources
Math 10 (McGraw Hill: sec7.1)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.4)
scissors
[pic]Glossary
general form of a linear equation – an equation written in the form [pic]
Lesson 5
Point-slope Form
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that… | |
|graphs and equations of linear relations have characteristics that | |
|can be analyzed and interpreted in context. |Which form of a linear equation is appropriate in any particular |
|there is a relationship between the equation of a linear relation and|contexts? |
|its graph. | |
|the equation of a linear relation can be written in various forms. |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… |Students will be able to… |
| | |
|that a point must satisfy the equation. |create the equation when given: |
| |a graph |
|forms of linear equations: |a point and the slope |
|slope-point form [pic] |two points |
| |a point and the equation of a parallel or perpendicular line |
|that certain forms are more appropriately used in different | |
|situations. |convert from one form of an equation to another, |
| |[pic] |
| | |
| |apply an appropriate strategy for creating an equation or graph. |
| | |
| |write the equation, given a graph. |
| | |
| |draw a graph, given the equation. |
[pic]Lesson Summary
Students will be able to graph a linear relation from corresponding equations given in point - slope form.
Students will discover a connection between the point - slope form and certain characteristics of the graph of a linear relation.
[pic] Lesson Plan
Warm-up Activity
• Example to review Lesson #3
Diagnostic of algebra and formula manipulation
Provide students with a sheet of 8 – 10 questions ranging from simple to complex. Students could also submit a self – generated assessment showing their current learning.
Activity
Students are given a graph and an equation in point – slope form. They are asked to determine the equation of the line in slope y- intercept form.
Class discussion
• Compare the equation created to the one given.
• Have the students focus on the y- intercept. How does that relate in both equations?
• Extend to showing a graph, but NOT having the intercepts.
How do we find an equation with this information?
• What characteristics are important when writing the equation of a linear function?
o Point and slope
o Two points
o Intercepts
[pic] Going Beyond
Make a connection between slope y-intercept and point-slope form.
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.5)
Math 10 (McGraw Hill: sec 7.3)
[pic]Glossary
point-slope form of a linear equation – an equation written in the form [pic]
Lesson 6
General Form (Ax+By+C=0)
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand that… | |
|graphs and equations of linear relations have characteristics that | |
|can be analyzed and interpreted in context. |Which form of a linear equation is appropriate in certain contexts? |
|there is a relationship between the equation of a linear relation and|How do the equivalent equation forms inter-relate? |
|its graph. |. |
|the equation of a linear relation can be written in various forms. | |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
|Students will know… | |
| |Students will be able to… |
|that a point must satisfy the equation. | |
|forms of linear equations: |apply an appropriate strategy for creating an equation or graph. |
|general form [pic] | |
|that certain forms are more appropriately used in different |write the equation, given a graph. |
|situations. | |
| |draw a graph, given the equation. |
[pic]Lesson Summary
Students will be able to graph linear relations from corresponding equations given in general form.
Students will discover a connection between the general form and certain characteristics of the graph of a linear relation.
[pic] Lesson Plan
Introductory Activity
• Given graphs and their equations in standard form – determine x and y- intercepts and identify a rule for determining intercepts.
Lesson
General Form Activity
Students are given an equation in general form and are asked to graph the function.
• Discuss the various ways that students may have used to complete this task (going over table of values, intercepts, and rearranging the equation).
• Give students the equation in general form and have them graph the function (what different methods can you use to do this?)
Equations of Horizontal Lines Activity
Students are given a graph with various horizontal lines and are asked to, using what they know, create an equation for each.
• Lead into a discussion of a slope of 0.
• Lead into a discussion of vertical lines and their equations
[pic]Resources
Math 10 (McGraw Hill: sec 7.2)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.6)
Lesson 7
Finding an Equation Given Characteristics of a Linear Relation
|[pic]STAGE 1 |
| |
|BIG IDEA: |
| |
|Linear relations provide the tools to communicate, model and explore the relationship between two sets of data. |
| |
| | |
|ENDURING UNDERSTANDINGS: |ESSENTIAL QUESTIONS: |
| | |
|Students will understand… |Which form of a linear equation is appropriate in certain contexts? |
|that graphs and equations of linear relations have characteristics |How do the equivalent equation forms inter-relate? |
|that can be analyzed and interpreted in context. |In what ways is the real world linear and/or non-linear? Justify. |
|that there is a relationship between the equation of a linear |Can certain data be represented with a linear relation? |
|relation and its graph. |Is it appropriate to model time and space using a linear relation? |
|that the equation of a linear relation can be written in various |How can we identify and model linear relationships? |
|forms. |What restrictions should be placed on a linear function, and in what |
| |contexts would such restrictions be necessary? |
| |. |
| | |
|KNOWLEDGE: |SKILLS: |
| | |
| |Students will be able to… |
| | |
| |create the equation when given: |
|Students will know… |a graph |
| |a point and the slope |
|that certain forms are more appropriately used in different |two points |
|situations. |a point and the equation of a parallel or perpendicular line |
| | |
| |apply an appropriate strategy for creating an equation or graph. |
| | |
| |write the equation, given a graph. |
| | |
| |draw a graph, given the equation. |
[pic]Lesson Summary
Students will be able to determine the equation of a linear relation.
Students will re-examine their personal strategies.
[pic] Lesson Plan
Students work in partners/small groups to discuss specific real-life situations to determine what strategy they would use to create the equation of the line. Situations could be teacher or student generated. Students could present their strategies to the class and class discussion could follow.
Student generated review with teacher guidance:
o What are the types of linear equations that we have been working with?
o What kinds of scenarios would work well with each kind of equation?
o Given scenarios, what form of a linear equation would you use?
[pic]Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 5.6, 5.7, 6.1 – 6.6)
Math 10 (McGraw Hill: sec 7.1 – 7.4)
Appendix
Handouts
Matching Graphs and Equations
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|y=_____________ |y=_____________ |
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|y=_____________ |y=_____________ |
ACKNOWLEDGEMENTS
Pages 8 – 11
Pages 16 – 18
1O'Connell, Sanjida. “Is Farming the Root of All Evil?” Web. Telegraph.co.uk.
23 Jun 2009
-----------------------
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to it often.
Implementation note:
Ask students to consider one of the essential questions every lesson or two.
Has their thinking changed or evolved?
Implementation note:
Teachers need to continually ask
themselves, if their students are acquiring the knowledge and skills needed for the unit.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Implementation note:
Teachers need to consider what performances and products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Implementation note:
Teachers need to consider what performances and products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
[pic] Height vs. Historical Period - Student Assessment Task
Is farming the root of all evil?1
Academics have claimed that moving away from a hunter-gatherer lifestyle was 'the worst mistake in history'. But are they right?
Adapted from an article by Sanjida O'Connell
Could it be that rather than being a boon to mankind, the invention of agriculture was, in the words of one academic, "the worst mistake in human history"?
Dr. Jay Stock, an evolutionary anthropologist at the Leverhulme Centre for Human Evolutionary Studies in Cambridge, believes that farming has played a powerful role in distorting human development. This idea first came to prominence through Professor Jared Diamond from the University of California in Los Angeles in his 1997 book, Guns, Germs and Steel. According to Diamond, before agriculture evolved about 12,000 years ago hunter-gatherer societies were at times subject to starvation. With the advent of agriculture, humans became malnourished and disease-ridden compared to hunter-gatherers. Hunter-gatherers, for example, ate a wide variety of foods, around 60-70 kinds a year. But once humans switched to agriculture, we became dependent on a small number of crops that do not have all the nutrients essential for a healthy life and are also vulnerable to disease and drought. Today, wheat, rice and corn, provide the bulk of calories for the world's population. Further, agriculture allowed food to be stockpiled and freed some people to live in cities, invent more and better weapons, to be soldiers and go to war, and to create political, gender and class inequalities. The close proximity of livestock also enabled cross-over diseases such as avian flu and H1N1. Diamond is supported by Tom Standage in his book, An Edible History of Humanity, in which he argues that agriculture is a "profoundly unnatural activity".
Neolithic Hunter-Gatherers
Modern Homo-Sapiens-Sapiens
Agrarian Nile-Valley Egyptians
|Year Number* |Height (inches)|
|013 |66.0 |
|212 |66.8 |
|388 |67.6 |
|572 |68.4 |
|801 |69.2 |
|998 |70.4 |
*years after 11 000 BC
|Year Number* |Height (inches) |
|0007 |70.4 |
|0603 |69.3 |
|1615 |68.2 |
|4200 |63.7 |
|6397 |59.9 |
|7990 |58.2 |
*years after 10 000 BC
|Year Number* |Height (inches)|
|0085 |58.0 |
|0719 |58.7 |
|1485 |59.8 |
|2324 |60.4 |
| | |
| | |
*years after 2000 BC
Height (in inches)
Time (years after 11 000 BC)
Height (in inches)
Time (years after 10 000 BC)
Height (in inches)
Time (years after 2000 BC)
Era 2
[pic]
Era 1
[pic]
Era 3
[pic]
Implementation note:
Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.
|x |y |
|-2 |0.25 |
|-1 |0.5 |
|0 |1 |
|1 |2 |
|2 |4 |
|3 |8 |
|x |y |
|-2 |8 |
|-1 |5 |
|0 |2 |
|1 |-1 |
|2 |-4 |
|3 |-7 |
|x |y |
|1 |1 |
|2 |4 |
|3 |9 |
|4 |16 |
|5 |25 |
|6 |36 |
|x |y |
|1 |8 |
|2 |10 |
|3 |12 |
|4 |14 |
|5 |16 |
|6 |18 |
[pic]
[pic]
[pic]
[pic]
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.
Y=____
Y=____
Y=____
Y=____
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