Course Profile Mathematics, Locally
Developed, Grade 10, Catholic
Unit 3: Explorations in Two Dimensions
Time: 28 hours
Activity 1 | Activity 2
| Activity 3 | Activity
4 |
Activity 5 | Activity 6
| Activity 7
Unit Description
In this unit, students explore measurement aspects of two-dimensional figures in applications related to the architectural and design features of a home. The activities in this unit address applications of perimeter and area including the determination of optimal values for two-dimensional figures. The Pythagorean theorem is investigated through concrete materials and realistic examples. Students develop their skills in estimation and calculation by exploring some properties of triangles and circles. Their expertise in solving practical problems and their development of literacy skills are fostered through the activities.
Strand(s) and Expectations
Ontario Catholic School Graduate
Expectations: CGE 2c, 2d, 3c, 3e, 4a, 4f, 5a, 7i.
Strand(s): Number Sense, Measurement and
Geometry, Patterns and Relationships
Overall Expectations: NSV.01, NSV.02,
PRV.01, MGV.01, MGV.02, MGV.03, MGV.04.
Specific Expectations: NS1.01, NS1.02,
NS1.04, NS1.05, NS1.07, NS1.08, NS2.02, NS2.03, NS2.05, NS2.06, PR1.03, PR2.01,
MG1.01, MG1.02, MG1.03, MG2.01, MG2.02, MG2.03, MG2.04, MG2.05, MG2.06, MG3.01,
MG3.02, MG3.03, MG 4.01, MG4.02, MG4.03, MG4.05, MG4.06.
Activity Titles (Time + Sequence)
|
Activity 1 |
Setting the Scale [measuring angles, illustrating angles and triangles, scale drawings] Follow-up Activity |
4.5 hours 0.5 hours |
|
Activity 2 |
Designing a Home [perimeter and area of triangles, rectangles and squares; estimation] Follow-up Activities |
2.75 hours 1.75 hours |
|
Activity 3 |
What is a Pi? [circumference and area of circles; scale; estimation] Follow-up Activities |
2.25 hours 0.75 hours |
|
Activity 4 |
Designing Ways [perimeter and area of composite figures] Follow-up Activities |
3 hours 1 hour |
|
Activity 5 |
Optimizing the Design [optimal values in measurement of two-dimensional figures; problem-solving] Follow-up Activity |
4.5 hours 1.5 hours |
|
Activity 6 |
A Tool from Ancient Times [Pythagorean theorem] Follow-up Activity |
1.25 hours 1.25 hours |
|
Activity 7 |
Summative Assessment: Making a Decision |
3 hours |
Prior Knowledge Required
Number Sense
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement and Geometry
· measure lengths accurately
· illustrate the meanings of the concept of perimeter, and area
· use the formulas for the area of rectangles, squares, triangles and circles
· determine the relationship between linear units
Unit Planning Notes
· At times in this unit, students are required to write a “technical manual”. Within this manual, students record the formulas for perimeter, area and the Pythagorean theorem. They, therefore, can reference the manual during subsequent activities when the need for the formulas arises. As a formula is entered into the manual, students record the steps that must ensue when the formula is being applied. This process serves to reinforce students’ understanding of the mathematical concepts and to develop writing proficiency necessary for the Grade 10 Test of Reading and Writing Skills.
· In the first activity, students begin the development of an individual portfolio related to measurement aspects. The teacher may wish to have some students work in groups of two to facilitate the acquisition of skills. In the portfolio, students apply the formulas recorded in the technical manual.
· Throughout this unit, opportunities arise for the teacher to incorporate situations relevant to other disciplines and with practical significance to activities beyond the classroom. The teacher may therefore wish to link some of the activities to career possibilities for students.
· Prior learning skills should be diagnosed throughout the unit. Remediation activities should be provided to compensate for deficiencies.
· The activity, “Optimizing the Design” enables students to gain familiarity with using a spreadsheet. However, the mathematical activities can be easily adapted to use with grid paper or unicubes.
Teaching/Learning Strategies
· When forming peer groups, consideration should be given to balancing the strengths and weaknesses of members. The assistance of peer tutors during some activities will enable students to successfully complete some tasks with minimal frustration.
· The teacher may wish to change the composition of groups for differing activities in this unit. A change in group membership may allow some students to extend their skill development. Students would be given the opportunity to support and appreciate the talents of all class members.
· The suggested timelines may be modified to allow students to further develop their skills in a particular area or to expand their knowledge of concepts. The timing of activities should be sequenced to enable students to develop their confidence in doing mathematics.
· As some students may have difficulty in sequencing tasks, the activities in this unit have been structured to allow frequent teacher direction or intervention. The teacher may wish to elaborate on specific tasks or, to combine activities if students are able to independently follow instructions.
· Since Imperial Units are used in the building trades, the teacher may wish to provide a summary of the components of Imperial Units. As students will be scaling their “home” in Metric Units, the teacher should provide examples of converting dimensions from Metric to Imperial Units if the cost of supplies in the catalogues or flyers is measured in Imperial Units.
· Mental mathematics activities will serve to enhance the progress of the activities and student proficiency in arithmetic operations.
· Throughout the unit, students are expected to practise skills independently. Although assistance may be required initially in recording details in the technical manual, students are expected to gain proficiency in writing the steps of the procedures.
Assessment and Evaluation
· Although sample generic rubrics have been provided in this profile, they should be modified to suit the need of the students within the classroom.
· Prior to an activity, students should be aware of the method and criteria upon which they will be assessed.
· Opportunities should be provided for students to self-evaluate and to identify areas and methods for improvement.
Resources
Internet Websites
http://archives.math.utk.edu/
links to teaching materials and software; searchable database
http://forum.swarthmore.edu/
provides resources organized by mathematics subject area (K-12 and advanced);
key issues in education including assessment issues
http://www.learner.org/
supplies information on reform initiatives including trends in mathematics
education
http://www.mathgoodies.com/
links to interactive mathematics lessons, activities, worksheets and puzzles
http://www.mth.msu.edu/cmp/
presents teaching materials and suggestions for assessment
Licensed spreadsheet software
ClarisWorks
Microsoft Works
Corel WordPerfect Suite
Activity 3.1: Setting the Scale
Time: 5 hours
Description
In this activity, which introduces the theme of home design, students review the correct mathematical terminology for shapes, angles, and lines. They develop practical experience in determining an appropriate scale for illustrations and apply that scale in problem situations. The explorations in this and subsequent activities are to be compiled by the students into a portfolio format. Opportunities are provided for students to reflect upon their learning through written observations on mathematical aspects of home design.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE3c - thinks reflectively and creatively to evaluate situations and solve problems;
CGE3e - adopts a holistic approach to life by integrating learning from various subject areas and experience;
CGE7i - respects the environment and uses resources wisely.
Strand(s): Number Sense,
Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
NSV.02 - consolidate the meaning and use of proportionality through applications drawn from student experiences and broader contexts;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects;
MGV.04 - demonstrate an understanding of the properties and sides and angles in triangles and parallel lines through investigations using concrete materials and appropriate technology.
Specific Expectations
NS1.01 - use pencil and paper computational methods effectively to evaluate expressions involving fractions, decimals, and exponents as they arise in problems throughout the course;
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.07 - judge the reasonableness of answers to problems by considering likely results;
NS2.02 - demonstrate an understanding of and apply unit rate in problem solving situations;
NS2.03 - solve problems involving ratio, proportion, and scale drawn from familiar applications;
NS2.05 - solve problems involving scale, proportionality, and similar figures drawn from familiar applications;
MG4.01 - illustrate the meanings of key terms associated with angles and triangles by constructing diagrams;
MG4.02 - estimate the measures of angles and line segments;
MG4.03 - determine the measures of angles and line segments using appropriate tools;
MG4.05 - solve simple geometric problems;
MG4.06 - communicate solutions to problems and the results of investigations, using appropriate terminology, symbols and form.
Planning Notes
· In this unit, students select a photograph or sketch of the exterior of a home from a magazine or newspaper. Rectangles, squares, triangles, and circles should be clearly discernible in the architectural features of the home. Although the teacher may request that students retrieve their own model, a wide selection of sample designs should be available if a student’s choice lacks relevant features.
· During this activity, students are requested to outline five geometric figures on their “home”. The shape will then be more clearly discernible by students when they are measuring the figure and will enable the teacher to identify the original figure when the completed portfolio is assessed.
· The teacher may wish to incorporate a study of the properties of parallel lines within this unit. Students could consider the methods used by builders to obtain parallel beams: measuring the distance between the beams or using a carpenter’s level.
· A large illustration of a home’s exterior should be used by the teacher for instructional purposes.
· Students require scientific calculators, metric rulers, protractors and 1 mm graphing paper.
· A set of pre-cut two-dimensional triangles are required by each student for the investigation into the types of angles and triangles. The figures should be such that they represent the spectrum of right, acute, and obtuse angles and scalene, obtuse, isosceles, right, and equilateral triangles.
· Technological resources such as The Geometer’s Sketchpad™ may be used to enable students to differentiate the variety of angles in triangles and the types of triangles.
· The initial activity in which student measure the doors and window of the classroom and determine an appropriate scale of conversion for the teacher’s model home is expected to consume approximately one hour.
Prior Knowledge Required
Number Sense and Algebra
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement
· measure lengths accurately
· determine the relationship between linear units
Teaching/Learning Strategies
Teacher Facilitation: The teacher poses the following scenario to the class:
As the owner of a construction company, you have received a contract to build a home for a family. You will be involved in aspects of the design and decorating of the home. Your client would like to receive a written description of the exterior or facade of the home including its interesting geometric properties. What is the shape of doors, windows, roof? Are there any dormers or arches?
The teacher informs the class that prior to writing the client, they need to insure that they know the correct terminology. Students fold a paper in half and describe the type of angle formed. They are advised to use the paper to determine if a series of angles are less than 90, equal to 90 or more than 90 degrees.
Student Activity
Students are then each given a set of triangles. Within each set are scalene, equilateral, isosceles and obtuse triangles. Using the folded piece of paper, students determine whether each angle of the triangle is less than 90 degrees, equal to 90 degrees, or more than 90 degrees. They record the designation on the angle.
Teacher Facilitation: The class reconvenes and students present their findings. During this presentation, the teacher develops a list of terminology, such as acute angle, right angle, obtuse angle, which students are expected to use. Throughout this activity, the teacher adds appropriate terms. Each term is accompanied by a diagram or written definition. Students should be provided with a copy of the terminology when it is complete.
The teacher has students estimate which triangles have equal and unequal sides. Student volunteers measure the angles and sides of the triangles. Students and the teacher categorize angles and triangles by their respective names. Students are led to realize that the sum of all angles in a triangle equals 180 degrees. They recognize that they need only measure two angles in a triangle. The measure of the third angle may be obtained by subtracting the sum of the measured two angles from 180 degrees.
The teacher focusses discussion on the architectural features of the displayed home, such as dormers or bay windows, using correct mathematical terminology for shapes, angles, and lines. The class considers appropriate building materials for the home given geographic, monetary, and environmental considerations.
Student Activity
Students write their impressions of interesting aspects of their chosen homes’ design. Descriptions in this first portfolio submission are to be phrased in appropriate mathematical language.
|
Follow-up Activity |
|
Time: 0.5 hours |
|
Students are informed that they need to write a technical manual to assist students hired for the summer. They must make a diagram of the types of angles and triangles, indicating the distinguishing properties that serve to define the angles and triangles. |
Teacher Facilitation: Elaborating on the architectural features of the displayed home, the class relates the occupations of individuals who would be hired for the construction. The teacher recounts various circumstances in which these individuals use mathematical principles on their jobs. The teacher initiates discussion to determine a method for calculating the amount of wood required to frame a triangular, square, or rectangular window or door. The class recognizes that the dimensions of the figure would need to be measured.
The teacher informs the students that actual measurements are to be used in calculating the cost of supplies. Students brainstorm on the method of determining a proper scale for converting the measurements of their model home to that of an actual sized home. A short review of metric conversions follows or the teacher may wish to develop the relationship of Imperial Units to Metric Units, given the importance of Imperial Units in the building trades.
Student Activity
In groups of three or four, students measure the windows and doors of the classroom. Each group then uses the dimensions of the length or width of the teacher’s model home to determine a scale for conversion.
Teacher Facilitation: The teacher may wish to review the method of using a protractor. The teacher reviews similar triangles in relation to a scale drawing of a triangular figure and its actual size.
Student Activity
Students reinforce their understanding of scale drawings and metric conversions through completing the following activity.
1. Outline with a pen, highlighter, or coloured pencil, 5 geometric figures on your home. Your choices must contain at least 1 of each of the following: circle (a semi-circle is acceptable), square, rectangle, triangle. Label each figure from A to E.
2. On a separate piece of graphing paper, sketch each figure (to the size portrayed in the picture), except for the circle. Leave a blank page for the circle. Your diagrams should be exact. Measure angles and lines carefully.
3. Record the dimension of each line segment and angle.
4. Label each figure according to its type of shape. Be specific. Is your triangle equilateral, isosceles, scalene or obtuse?
5. Calculate the actual dimensions of each side of the figures. Use the measurements of a classroom door or window to develop the scale for your picture. Display your calculations.
Extension
· Students use chart paper that is measured in grids. Using the picture of a home, they sketch the home onto the chart paper using an appropriate scale of measurement.
Accommodations
· The writing utensil used by learning disabled and low vision students, for this and subsequent activities requiring the use of grid paper, should be a different colour from the grid lines.
· The right angle in a triangle should be identified with a consistent marking for blind or low vision students.
· Descriptions of the exterior of the home should be supported with synonyms and examples to facilitate understanding by deaf and hard of hearing students. Specific terms, such as dormer or fascia, may be compiled in a glossary for quick reference.
· Measuring instruments for use by low vision students can be modified by darkening divisions on the instruments, at intervals of 2 cm, to aid in determining measurements. Students should receive guidance in how to use the instruments.
· The teacher may wish to colour code the triangles to provide an easy method of designation for class discussions.
Assessment/Evaluation Techniques
· The teacher circulates among the groups to ensure that the scale conversion is understood. Some students are assessed, using the Observational Checklist, Appendix H, on their strategies in determining the appropriate steps for scale conversion.
· The learning skills areas of independence and initiative are assessed using the Learning Skills Checklist, Appendix J. The teacher develops anecdotal notes on the level of support that students require to perform the scaling tasks and, on the students’ persistence in completing the procedural steps.
· The teacher assesses the portfolio and technical manual of some students using Appendix I, the Written Report Rubric. At this juncture, assessment is appropriate in the categories of Knowledge, Written Communication and Application.
Resources
New home sections of newspaper or magazines related to new homes
Appendices
Appendix H – Observational Checklist
Appendix I – Written Report Checklist
Appendix J – Learning Skills Rubric
Activity 3.2: Designing a Home
Time: 4.5 hours
Description
Throughout this activity which focusses on the architectural or decorative features of homes, students develop their understanding of perimeter and area. The problems in this activity enable students to develop their expertise in basic arithmetic operations, with and without the use of a calculator. Students make additional contributions to the portfolio and technical manual.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE2c - presents information and ideas clearly and honestly and with sensitivity to others;
CGE2d - writes and speaks fluently one or both of Canada’s official languages;
CGE3e - adopts a holistic approach to life by integrating learning from various subject areas and experience;
CGE7i - respects the environment and uses resources wisely.
Strand(s): Number Sense,
Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
NSV.02 - consolidate the meaning and use of proportionality through applications drawn from student experiences and broader contexts;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects.
Specific Expectations
NS1.01 - use pencil/paper computational methods effectively to evaluate expressions involving fractions, decimals, and exponents as they arise in problems throughout the course;
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.04 - use estimation and mental computation to approximate and/or calculate answers to numerical problems as they arise throughout the course;
NS1.07 - judge the reasonableness of answers to problems by considering likely results;
MG1.01 - solve simple problems using the formulas for the perimeter and area of triangles, rectangles, squares and circles;
MG1.03 - substitute into and evaluate measurement formulas as the need arises in problem solving.
Planning Notes
· In the “technical manual”, students detail the method of calculating perimeter and area. As this manual is to serve as a reference in future activities, the teacher is advised to check the manual of as many students as possible, at each stage, to ensure that the formulas have been entered accurately.
· The concepts developed in these activities are to be presented in real-life situations. For investigations related to monetary matters, such as determining the cost of wood trim, students should consult newspaper ads or hardware catalogues to obtain the cost per metre.
· When the cost of articles is provided in inches or feet, the teacher instructs the students to convert the dimensions of the figures in the portfolio from Metric Units to Imperial Units.
· When developing the formulas for the perimeter of squares and rectangles, the teacher should expose students to and explain the steps in using the formula, P = 2(l+w).
Prior Knowledge Required
Number Sense and Algebra
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement and Geometry
· measure lengths accurately
· use the formulas for the area of rectangles, squares, triangles and circles
· determine the relationship between linear units
Teaching/Learning Strategies
Teacher Facilitation: The teacher reminds the class of the following scenario:
As the owner of a construction company, you have received a contract to build a home for a family. You will be involved in aspects of the design and decorating of the home. The teacher again focusses discussion on the architectural features of the displayed home, such as dormers or bay windows, using correct mathematical terminology for shapes, angles and lines
The teacher initiates discussion to determine a method for calculating the amount of wood required to frame a triangular, square or rectangular window or door. With reference to the model home, the formulas for the perimeter of rectangles, squares and triangles are developed. During this development, students explore situations in which the formula for perimeter would be used during the design and construction process.
The teacher informs the students that actual measurements are to be used in calculating the cost of supplies. Students brainstorm on the method of determining a proper scale for converting the measurements of their model home to that of an actual sized home. A short review of metric and imperial conversions, with applications to framing a door or window on the teacher’s model, serves to prepare students for the subsequent activity.
Student Activity
Students refer to their home portfolio and are required to complete the following activity.
1. Determine the perimeter of the original figure. Display your calculations.
2. Using a newspaper or catalogue, calculate the cost of purchasing wood to frame the door or window. Display your calculations. Will you frame all sides?
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Follow-up Activity |
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Time: 0.75 hours |
|
Students are informed that they need to record the method of determining the actual measurements of a figure from a scale drawing. They are also to record the method to be used in calculating the perimeter of squares, rectangles, and triangles. |
Teacher Facilitation: To introduce the concept of area, the teacher informs students that their client has decided that the front and back door of the home, and door leading to the garage, will be constructed of wood and then painted. The students must determine the amount of paint to be used. Discussion occurs to determine the area of a rectangle. During this process, the area of a square should be developed.
The teacher leads students in determining the amount of paint needed for the three doors of the model home. Depending on the ability of the class, the teacher may instruct students to consider a door without windows or to discuss painting a door to which windows are to be inserted. The latter choice would lead readily into the next activity in which students calculate the area of a triangular window. The teacher models the steps in determining the amount of paint. During this discussion, the teacher may wish to have students elaborate on practical considerations such as the number of coats of paint, types of paint, colour scheme. The teacher then informs students that they need to calculate the amount of paint needed for a triangular segment above a door or window.
Student Activity
To develop the formula for the area of a triangle, students fold a square piece of paper in half
|
|
along the diagonal. Students cut the
square piece of paper along the diagonal. Through discussion, it is determined
that the area of the triangle is half the area of the square or, half the
length times the width, ½ (l x w).
Using a rectangular piece of paper, students fold the paper in half along the longer side. Students then, using pencil or pen, join the top point of the centre fold to each of the lower corners. They then cut along
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|
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the lines so formed. Students place each of the segments which were removed on top of the remaining portions of the triangle. They again determine that the area of the triangle is half the length times the width (or half the base times the height or altitude), ½ (b x h)
Teacher Facilitation: To consolidate the study of the area of a triangle, the teacher assists the class in discerning that any side of the triangle can be considered its base.
Student Activity
The class is divided into three groups and required to measure the base of the triangle that was used in the previous activity. However, each group is assigned a different side as the base. Students calculate the height of the triangle and then the area of the triangle.
Teacher Facilitation: All groups present their determination of the area of the triangle and are led to realize that any side of the triangle may be considered its base. The teacher may wish to explore with students the method of calculating the area of a triangle with an external altitude.
Student Activity
In their portfolios, students calculate the actual area for all figures of their model home, except circular shapes. They calculate the cost of painting the figures and the amount of paint needed, if doors, or the cost of inserting glass into the figures, if windows. For these calculations, students should obtain the cost of cans of paint or glass from newspaper flyers or store catalogues.
Students record the method of calculating the area of squares, rectangles and triangles in the technical manual.
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Follow-up Activity |
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Time: 1 hour |
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Students should be given the opportunity to practise the calculations for area and perimeter with questions ranging from simple calculations to the more complex as indicated below. Similar questions may be obtained from the textbook references or the teacher may wish to develop questions related to the school or home environment. |
|
Find the perimeter of a garden if it
is 8 m long and 5.8 m wide. |
|
A triangular lot has a base of
length 17 m. If the height of the lot is 8 m, what is the area of the lot? |
|
You wish to paint two walls of the
gymnasium. One wall is 6.5 m by 7.4 m. The other wall is 9.3 m by 7.4 m. The
store has six 4 litre cans of paint in the colour which you need. Is there
enough paint for one coat if a litre of paint usually covers 9.5m2? The teacher should circulate while students are completing the questions to provide assistance where needed. |
Extension
· Students may wish to scan the sketch of the home into computer software and then make changes in the structural design of the home. They would present the calculations for the dimensions of the original home and the modifications. Students would calculate the cost of framing the figures and of painting them (perimeter and area). They would determine which design was more cost efficient.
Accommodations
· The use of precut shapes will enable blind or low vision students to complete the home portfolio more readily.
· The teacher may wish to provide illustrations for some questions as an aid for ESL students or for those students experiencing difficulty with the mathematical content.
Assessment/Evaluation Techniques
· The student portfolio should be marked in stages, for accuracy, to ensure that students are grasping the concepts and are making acceptable progress. Peer tutoring may encourage students whose work habits are lagging.
· The teacher assesses the portfolio and technical manual of some students using Appendix I, the Written Report Rubric. Assessment is appropriate in the four categories of Knowledge and Understanding, Thinking/Inquiry and Problem Solving, Written Communication and Application.
· As students complete the problems during the last Follow-up activity, the teacher should ask individual students to explain why they chose to use a particular formula. This method would allow teachers to assess applications. By noticing how students solve problems requiring them to differentiate between area and perimeter, deficiencies could also be noted and remediation occur.
Appendices
Appendix I – Written Report Rubric
Activity 3.3: What is a Pi?
Time: 3 hours
Description
This activity enables students to measure circular objects and diagrams of circular objects while developing an understanding of the meaning of π. Through this understanding, students realize that the ratio of circumference to diameter is always a constant number. Students extend their knowledge of perimeter and area by developing the formulas for the circumference and area of circular figures.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE2c - presents information and ideas clearly and honestly and with sensitivity to others;
CGE2d - writes and speaks fluently one or both of Canada’s official languages;
CGE4a - demonstrates a confident and positive sense of self and respect for the dignity and welfare of others;
CGE5a - works effectively as an interdependent team member.
Strand(s): Number Sense,
Patterns and Relationships, Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
PRV.01 - use patterning strategies to solve simple problems that arise in activities throughout the course;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects.
Specific Expectations
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.04 - use estimation and mental computation to approximate and/or calculate answers to numerical problems as they arise throughout the course;
NS1.07 - judge the reasonableness of answers to problems by considering likely results;
NS2.03 - solve problems involving ratio, proportion, and scale, drawn from familiar applications;
PR1.03 - identify and extend simple patterns within problem solving situations;
PR2.01 - formulate an hypothesis about a relationship between two variables and express the relationship in words and/or symbols;
MG1.01 - solve simple problems using the formulas for the perimeter and area of triangles, rectangles, squares and circles.
Planning Notes
· The formula for the area of a circle is introduced prior to the formula for circumference. However, the teacher may wish to reverse the order of introduction and commence with the circumference of a circle.
· A variety of circular objects are to be measured by students in the second part of the activity. These circular objects should vary in size to facilitate student’s realization that the ratio of circumference divided by diameter (or radius) is not dependent upon the size of the object.
· One group of students requires actual balls such as golf, baseball, soccer and basketballs.
· The teacher provides measuring or masking tape to the students.
· Students require compasses when adding the circular figure to their portfolio.
Prior Knowledge Required
Number Sense and Algebra
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement and Geometry
· measure lengths accurately
· illustrate the meanings of the concept of perimeter and area
· use the formulas for the area of rectangles, squares, triangles and circles
Teaching/Learning Strategies
Teacher Facilitation: The teacher and students discuss modifications to the model home which may enhance its aesthetic appearance including the addition of a circular flower garden or semicircular window.
From this discussion, students are led to realize that they need to
calculate the area and circumference of a circle. The teacher reviews the
terminology associated with circles (radius, diameter, circumference) and then
provides each student with two diagrams. Displayed in the first diagram, as
shown in Worksheet 3.1a (Appendix E), is a circle embedded in a square and the
second, illustrates a square with dimensions equal to the first. The square
containing the embedded circle is to be folded in half, vertically and
horizontally, to form the four sections as indicated. Each student is also
given page 1 of Worksheet 3.1b (Appendix E), entitled “Finding the Area of a
Circle”. The teacher directs the students to complete the activities listed on
page 1. The teacher circulates among the students, ensuring that they are
progressing successfully in the activity.
Student Activity
Students, individually, complete Worksheets 3.1a and b (Appendix E) which lead them to develop the area of a circle while cultivating a representation for π (pi).
Teacher Facilitation: When circulating amongst the students or when directing comments to the entire class, the teacher ensures that students realize that the radius is any line drawn from the centre of the circle to any point on the circle’s edge. To enable students to solidly grasp this concept, students could be required to fold the circle in half three different ways and then mark the point at which the folds intersect. Students would concur, by measuring the six lines from the intersection point (the centre of the circle), that all radii are equal.
When the class has completed Worksheet 3.1b, students explain that from their investigation, they have deduced that any circle embedded in a square will fill more than 3 segments of the square but not 4 segments. The teacher then introduces the mathematical formula for the area of a circle. Students are encouraged to refer to their calculators to establish an accurate representation of π (pi).
The teacher extends the exploration of the circle by dividing students into groups. Groups are given differing sets of circular figures or objects. The teacher requests that students measure the circumference of each circle or circular object. The teacher may provide students with the instruments to conduct the measurement (measuring tape) or may brainstorm with groups of students to determine possibilities for obtaining the measurements (cutting a piece of masking tape the circumference of the ball and determining the length of the tape using a ruler measured in cm).
Student Activity
Students are in groups and each group is required to measure one of the following collections:
· circles formed on paper. The centre of each circle is clearly indicated.
· flat, circular objects (such as the tops of margarine containers, a loonie).
· different spherical balls.
The groups record their results on chart paper of which an example is presented below for the group measuring the 3 dimensional objects. For this group, the diameter of each ball is provided but the other groups are required to determine the circumference and diameter of each figure. The teacher indicates to the group measuring the spherical balls that one usually discusses a cross-section of the ball when one is discussing circumference.
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Type of Ball |
Circumference |
diameter |
C ÷ d |
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Golf |
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4.2 cm |
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Baseball |
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7.4 cm |
|
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Soccer |
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22.28 cm |
|
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Basketball |
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23.86 cm |
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Students ascertain the missing values for each cell in the table. Students are required to complete the cells, C¸ d, with entries correct to three decimal places. In so doing, they are to notice any pattern that develops. When completed, the charts are displayed for the class to view. Each group relates to the class the method used to obtain the measurements. They indicate any pattern that emerged in their measurements of the entries for circumference divided by diameter.
Teacher Facilitation: The teacher relates the diameter and circumference of a circle by recalling the mathematical symbol, π (pi), from the previous segment of this activity. The teacher assists the students in developing the formula for the circumference of a circle, using the diameter. The class then discusses how the formula for circumference can be modified to include the radius. If any group had entries in the columns, C ÷ d, which deviated substantially from 3.142, the teacher may request that students in those groups repeat a portion of the activity.
Student Activity
The students recalculate the
circumference of the figures or objects on their chart, using the formula, C
= π d.
Students refer to their home portfolio and determine the diameter or radius of circular figures. Prior to obtaining the circumference and area of these circular figures, they convert radius or diameter to the actual measurements.
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Follow-up Activity |
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Time: 0.75 hours |
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In their technical manual, students include the method of calculating the circumference and area of circular figures or objects. In their portfolio, student complete the page reserved for the circular figure. They use compasses to construct the figure and use the previously determined scale to obtain the true dimensions of the figure. They are to calculate the circumference and area of the actual figure. |
Extension
Students compile the measurements for the circumference and/or area of circular objects used in the field of sports. Under proper supervision, students could measure the circumference of the basketball nets in the school gym or the holes on a golf course. They could relate these dimensions to the dimensions of the balls used in the same sport and suggest reasons for the differential variation between the two items.
Accommodations
· Precut diagrams will assist blind and low vision students in readily completing portions of this activity.
Assessment/Evaluation Techniques
· The teacher continues to assesses the portfolio and technical manual of some students using Appendix I, the Written Report Rubric. Assessment is appropriate in the four categories of Knowledge and Understanding, Thinking/Inquiry and Problem Solving, Written Communication, and Application.
· Using Appendix H, the Observational Checklist, the teacher assesses the skills of group effectiveness and group contributions. The teacher should ensure that students are informed of their skills and of ways to enhance their skills as some subsequent activities are also to be completed through group interaction.
· A formative paper and pencil quiz on the fundamentals of perimeter and area would be written to evaluate the categories of Knowledge and Understanding, Thinking/Inquiry and Problem Solving, and Applications. The results of the quiz would identify, to the teacher, the students who were experiencing difficulty. The teacher could thus monitor the students’ progress and deliver remedial assistance.
Resources
Circular objects and figures
Internet Web sites detailing activities related to Pi or the circumference or area of the circle:
http://archives.math.utk.edu/
http://www.learner.org/
http://www.mathgoodies.com/
Adomeit, J. et al. The Learning
Equation Ontario Mathematics 7 Teacher’s Manual, ITP Nelson, 1999,
pg. 98.
Appendices
Appendix H – Observational Checklist
Appendix I – Written Report Rubric
Appendix E – Worksheet 3.1
Activity 3.4: Designing Ways
Time: 4 hours
Description
Students explore the perimeter and area of composite figures by examining or creating the floor plans of homes. They are led to discover that although composite figures may be separated into two-dimensional figures in more than one way, the area and perimeter of the composite figures remain constant. In this activity, through concrete examples, they obtain practice in selecting the appropriate formula to use for area and perimeter.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE4f - applies effective communication, decision-making, problem-solving, time and resource management skills;
CGE7i - respects the environment and uses resources wisely.
Strand(s): Number Sense,
Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects.
Specific Expectations
NS1.01 - use pencil and paper computational methods effectively to evaluate expressions involving fractions, decimals, and exponents as they arise in problems throughout the course;
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.04 - use estimation and mental computation to approximate and/or calculate answers to numerical problems as they arise throughout the course;
NS1.05 - apply percents in solving problems;
NS1.07 - judge the reasonableness of answers to problems by considering likely results;
MG1.02 - calculate the perimeter and area of composite figures;
MG1.03 - substitute into and evaluate measurement formulas as the need arises in problem solving.
Planning Notes
· A variety of floor plans may be obtained from newspapers, magazines or the Internet and shared with the class.
· This activity lends itself to incorporating patterning elements. The teacher may wish to revisit sections of the patterning unit within this activity.
· Cross-curricular connections with computer science and technological (drafting) departments may be explored.
· The teacher may wish to provide students with the examples drawn on grid paper, as an introduction. As it may then be easier for students to recognize the shapes comprising the total, students would gain confidence in performing the task.
· Some students may require the use of tangrams to determine the shapes comprising the more complex composite figures or to have the shapes displayed on grid paper.
· The teacher may wish to display the two introductory examples on overheads. By so doing, the teacher could then review the segmentation of the composite figures both to the whole class and small groups of students.
· Each student receives a copy of the second diagram as it is referred to in the worksheet.
· Students are to be placed in groups of two or three. The teacher may wish to form a group of students who, from the initial instruction, seem to grasp the activity more readily. These students could be presented with more challenging tasks to complete.
· Students require scientific calculators.
Prior Knowledge Required
Number Sense
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement and Geometry
· use the formulas for the area of rectangles, squares, triangles and circles
· use the formulas for the perimeter of rectangles, squares and triangles and, the circumference of circles
Teaching/Learning Strategies
Teacher Facilitation: The teacher introduces the following scenario to the class:
Your business is booming! You have now obtained more contracts to design and construct homes. You are working on a number of projects including the following assignment.
A rectangular shaped floor plan has been chosen by a client.
Your client wishes to carpet the bedroom and living room. The bathroom floor will be tiled and the hallway will be hardwood flooring. Calculate the amount of carpeting, tiles and hardwood that is needed.
The teacher assists students in determining the missing measurements before completing the calculations.
The teacher then informs the students that their client has changed his/her mind about the layout of the house.
The following diagram illustrates the new layout. How does the amount of flooring change?
Under the teacher’s direction, the students calculate the area of the living room by first determining the dimensions of the rectangle and triangle which comprise the living room. The teacher illustrates the decomposition by sketching the two components separately.
Students may need considerable support and reiteration for this activity. Some students may require tangrams for the more challenging problems. The students would model the diagrams using geometrical shapes.
Student Activity
Working in pairs, students investigate the perimeter and area of composite figures. Worksheet 3.2 (Appendix F) illustrates some problems with real-world applicability. Students present the diagrams for question #3 of the worksheet to the whole class.
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Follow-up Activity |
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Time: 1 hour |
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Students work on teacher-compiled problems to clarify their understanding. These problems may vary among different groups. Some students may require questions with the diagrams clearly outlined on grid paper whereas other students may enjoy the challenge of determining the shapes necessary for calculating the area of the composite figures. In the technical journal, students record the method for determining the area or perimeter or composite figures. The method is illustrated with a diagram. |
Extension
Students retrieve a floor plan from a newspaper, magazine, or the Internet. They determine the amount and type of flooring needed for rooms within the home and the cost of such flooring. They then redesign the home by altering some of the dimensions of the rooms. They determine the flooring and the cost of such flooring (carpet, tile, or hardwood) for the newly designed habitat. Students could present their designs to the class and inform the class of the reasons for their choices.
Students may wish to construct their own floor plans using various computer programs (Wordperfect, Word, The Geometer’s Sketchpad™). The floor plans would indicate the purpose of the room and its dimensions. The area of each room and total area and perimeter would be indicated.
Accommodations
· Students who experience difficulty in spatial perception would be well served by initially having all diagrams placed on grid paper. The students outline the individual shapes and redraw them separately on grid paper. The area of each shape is then calculated.
· Students may require practice before the class presentation.
Assessment/Evaluation Techniques
· Individual student problem solving may be assessed during the discussion periods, using the Observational Rubric, Appendix H.
· Assessment in the Learning Skills areas of teamwork and initiative (Appendix J) is possible as students complete Worksheet 3.2 or the teacher compiled questions in the Follow-up Activity.
· Student worksheets may be assessed using a modification of the Written Work Rubric, Appendix I.
· The teacher assesses some of the technical manuals using Appendix I, the Written Report Rubric, in the four categories of Knowledge and Understanding, Thinking/Inquiry and Problem Solving, Written Communication and Application.
Resources
Internet Web sites
http://archives.math.utk.edu/
http://www.learner.org/
Appendices
Appendix F – Worksheet 3.2
Appendix H – Observational Rubric
Appendix I – Written Report Rubric
Appendix J – Learning Skills Rubric
Activity 3.5: Optimizing the Design
Time: 6 hours
Description
This activity affords students the opportunity to explore the variety of rectangles which can be constructed for a given area or perimeter. In the process, students will gain practice in determining the factors of a number. Technology, concrete materials and diagrams will be employed to ascertain the maximum area for a given perimeter and minimum perimeter for a given area.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE3c - thinks reflectively and creatively to evaluate situations and solve problems;
CGE5a - works effectively as an interdependent team member.
Strand(s): Number Sense,
Patterns and Relationships, Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
PRV.01 - use patterning strategies to solve simple problems that arise in activities throughout the course;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects;
MGV.02 - determine the optimal values of various measurements, through investigations using concrete materials, diagrams, and calculators or computer software.
Specific Expectations
NS1.08 - judge the reasonableness of answers produced by a calculator or computer, using mental mathematics and estimation;
PR1.03 - identify and extend simple patterns within problem solving situations;
MG1.01 - solve simple problems using the formulas for the perimeter and area of triangles, rectangles, squares and circles;
MG2.01 - construct a variety of rectangles for a given perimeter and determine the maximum area for a given perimeter;
MG2.02 - construct a variety of rectangles for a given area and determine the minimum perimeter for a given volume;
MG2.04 - describe applications in which it would be important to know the maximum area for a given perimeter or the maximum volume for a given surface area;
MG2.05 - describe applications in which it would be important to know the minimum perimeter for a given area;
MG2.06 - solve simple problems involving optimal values.
Planning Notes
· The activity commences with an exploration of the dimensions of rectangles for a given area. The teacher may wish to commence instead with an exploration of the dimensions of rectangles for a given perimeter.
· The exercises in this activity can be completed without the use of technology. The activity would then be modified by permitting students to use geoboards to modify the area or perimeter of the rectangle with ease.
· The accompanying worksheet has been written for use with Corel Quattro Pro 9. The worksheet may be modified for use with other versions of the program or for use with Excel.
· The investigations using computer technology are expected to consume 2.5 hours.
· The teacher may wish to have a template loaded onto the computers to enable students to obtain the calculations for length, width, perimeter, or area directly without student modification of the spreadsheet.
· A computer lab is required for this activity. If students have not had exposure to menu bars and graphics in computer programs, students should work in pairs to facilitate completion and comprehension of the tasks.
· Prior to the initial visit to the computer lab, the teacher may wish to display the menu bars for Quattro Pro on an overhead. The teacher would assist the class in reading the preliminary parts of the accompanying worksheet and determining the appropriate toolbar and tabs to use.
· The teacher ensures that students understand that the cells of a spreadsheet are identified by row and column (A1).
· Worksheets 3.3a and 3.3b (Appendix G) detail the instructions for using the spreadsheet with uppercase letters. However, Quattro Pro is not case sensitive.
· The teacher may wish to allow students some time to explore functions, such as shading, in the Quattro Pro worksheet.
Prior Knowledge Required
Number Sense
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
Measurement and Geometry
· use the formulas for the area of rectangles, squares, triangles and circles
· determine the relationship between linear units
Teaching/Learning Strategies
Teacher Facilitation: The teacher introduces the activity by relating the following scenario to the students. A client has decided that the master bedroom of the house should have a walk-in closet. Upon examining the floor plan, you determine that you could section about 12 m2 of the bedroom for the walk-in. The client would like the baseboards that are in the bedroom to be used in the closet. The cost of the baseboards should then be minimized. What dimensions should you have for the closet?
The class systematically develops a list of all of the factors of 12 and uses the list to determine the dimensions of the rectangles. To aid students in determining the factors of larger numbers, the teacher leads students in factoring a number such as 48. Using the formula for area, the teacher insures that students understand how to isolate the l or w when the A is known. The teacher may wish to write the spreadsheet columns on the blackboard first and have students complete a calculation by hand.
Student Activity
Using technology or grid paper, students explore a variety of rectangles for a stated area. In the process, they determine the minimum perimeter for the stated area. If students have access to computers and Quattro Pro 9, they perform the sequence of operations outlined in Worksheet 3.3, Appendix G.
Teacher Facilitation: The teacher and students discuss the results of their investigation. From their investigations, students should realize that the rectangle with minimum perimeter has dimensions that are close together. The teacher leads the students to realize that a rectangle with dimensions, 3 m by 4 m, can be rotated to be a rectangle with dimensions, 4 m by 3 m. The class continues this investigation by brainstorming other scenarios when it is desirable to have the maximum area but minimum perimeter (Building a dog run, fencing a playground, keeping the cost of the exterior foundation of a home to a minimum).
Students also relate why it is often not desirable to have a rectangle with dimensions 1 m by ____ m. The students are led to realize that some rectangles are more practical for a closet and that the dimensions of the closet affect the bedroom layout and vice versa.
Prior to introducing the spreadsheet activity in which students construct rectangles for a given perimeter, the teacher has the class consider the situation in which they are to fence an area of the school’s athletic field for the barbeque section, during the school’s carnival. They have a limited amount of rope to use as fencing. How could they determine the dimensions of the barbeque section with the maximum area, given the restriction on the amount of rope to be used? The teacher changes the directions in Worksheet 3.3b to enable the calculation of length, width and area from the perimeter.
Student Activity
As in the previous activity, students explore a variety of rectangles for a stated perimeter. They determine the maximum area for the given perimeter. Students provide examples of when it is important for homeowners to determine the shape of a rectangle which would provide maximum area for a given perimeter.
Extension
· Students could create a word problem about area or perimeter and optimal values.
· The teacher could provide students with alternate questions that are more challenging. As an example, the previous question could be altered such that Jason would only place the rope on three sides of the garden.
· Students determine if they could form a triangle in the spreadsheet and how they would calculate the area. They may wish to attempt to write directions for the program.
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Follow-up Activity |
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Time: 1.5 hour |
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Students complete simple problems in which they relate the optimal values of area and perimeter to home construction. They perform the operations in steps and verify their answers by sketching the rectangle and calculating both the perimeter and area from the dimensions. These problems would be of the form: |
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Jason intends to plant a rectangular flower garden near his front window. He has 10 m of rope to form the sides. What is the rectangle of greatest area that he could form? |
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In their journals, students describe a situation when it might be beneficial to know the maximum area for a given perimeter or minimum perimeter for a given area. |
Accommodations
· Students could complete this activity by using two-dimensional squares. The use of such manipulatives may assist low vision or blind students, in particular, in determining the optimal values.
· The instructions for changing the cells in the spreadsheet may at first appear daunting for learning disabled students. Assistance from peer tutors or educational assistants would alleviate some of the anxiety. The teacher may instead wish to modify the assignment and have the cells in the spreadsheet preset prior to the beginning of the class.
Assesssment/Evaluation Techniques
· All students complete the Self-Assessment, Appendix M, to assess the Learning Skills of On-Task Behaviour, Co-operation and Teamwork.
· The teacher assesses some students on the Skills of Group Effectiveness, Group Contributions, and Initiative, using Appendix H, the Observational Checklist.
· The teacher confers with the students for whom the Observational Checklist was used, to discuss and compare the teacher’s assessment with the student’s self-assessment.
· The teacher assesses knowledge and application through an analysis of the problems in the follow-up activity. The teacher develops a modification of Appendix I, the Written Report Rubric.
· The teacher modifies Appendix I to assess the written communication displayed by students in developing a problem for the journal.
Resources
Licensed spreadsheet software
ClarisWorks
Microsoft Works
Corel WordPerfect Suite
Alexander, R., B. Canton, P. Harrison, R. McLeish, N. Nielson, and M. Sinclair. Mathematics 9. Addison-Wesley, 1999, pg. 445 - 450.
Allison, P., et al. The Learning Equation Mathematics 9 Student Refresher, ITP Nelson, 1998, pg. 72-73.
Knill, G., R. Baxter, D. Dottori, G. Fawcett, M.L. Forest, M. Hamilton, S. Pasko, H. Traini, and M. Webb. Mathpower 9, McGraw-Hill Ryerson, 1999, pg. 460 - 463.
Appendices
Appendix G – Worksheet 3.3
Appendix H – Observational Checklist
Appendix I – Written Report Rubric
Appendix M – Self-Assessment Checklist
Activity 3.6: A Tool from Ancient Times
Time: 2.5 hours
Description
The 3-4-5 triangle and its use by ancient Egyptian surveyors serves as a context for an investigation of the Pythagorean theorem. Students develop their own instrument for determining a right angle and solve practical problems using the Pythagorean theorem.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE4a - demonstrates a confident and positive sense of self and respect for the dignity and welfare of others;
CGE4f - applies effective communication, decision-making, problem-solving, time and resource management skills.
Strand(s): Number Sense,
Patterns and Relationships, Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
PRV.01 - use patterning strategies to solve simple problems that arise in activities throughout the course;
MGV.03 - solve simple problems involving the Pythagorean theorem.
Specific Expectations
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.08 - judge the reasonableness of answers produced by a calculator or computer, using mental mathematics and estimation;
PR1.03 - identify and extend simple patterns within problem solving situations;
MG1.01 - solve simple problems using the formulas for the perimeter and area of triangles, rectangles, squares and circles;
MG3.01 - illustrate the Pythagorean theorem, using concrete materials;
MG3.02 - use the Pythagorean theorem to construct right angles in practical situations;
MG3.03 - solve simple problems drawn from familiar applications, using the Pythagorean theorem.
Planning Notes
· The teacher may wish to review the vocabulary of right-angled triangles prior to commencing the activity.
· The teacher reviews the methods of squaring and taking the square root of a number prior to introducing this activity.
· Students construct a surveyor’s rope as a tool for understanding the Pythagorean theorem. For this construction, they will need heavy rope (such as for wrapping a parcel), masking tape and a ruler measured in centimetres.
· This activity may be interspersed with the first activity on home design. However, as some students frequently have difficulty in retaining large segments of new material in a short span of time, it has been placed at the end of the development of two-dimensional measurements concepts. This activity can then be used to review some of the principles of perimeter and area of composite figures.
· Students may work in pairs or individually.
Prior Knowledge Required
Number Sense
· apply strategies for mental mathematics and estimation
· use a scientific calculator effectively
· judge the reasonableness of answers
Patterns and Relationships
· understand the concept of a variable
Measurement and Geometry
· measure lengths accurately
· determine the relationship between linear units
Teaching/Learning Strategies
Teacher Facilitation: The teacher displays a 3-4-5 triangle while describing to the class the importance of the triangle for surveying, construction and in engineering applications. The teacher relates the following information to the class to set the context for the exploration.
In ancient times, each year, the Nile River flooded farmers’ fields that were situated along the river’s banks. Subsequently, Egyptian surveyors would be required to section the farmers’ lands into squares or rectangles. For this purpose, the surveyors carried a rope, with knots tied at equal intervals, and what was known as a cubit rod. The surveyors would use the piece of rope to form right angles and use the rod, that ranged between 43 to 53 cm in length, to measure the length and width of each parcel of land. In this activity, you will develop a surveyor’s rope to investigate some of the properties of right-angled triangles.
The teacher relates the method used by the ancient surveyors to the 3-4-5 triangle used in the present day construction industry.
Student Activity
Individually or in pairs, students are provided with a 60 cm piece of rope, masking tape and a form with columns and rows such as the example below. The shaded regions indicate information that is to be already supplied on the form.
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Base |
Height |
Hypotenuse |
Base Squared |
Height Squared |
Hypotenuse Squared |
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9 |
12 |
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Students are to divide the strip of rope into 12 equal segments of length 5 cm, marking each segment with masking tape. Students formulate various methods to perform this task other than measuring 12 times.
Students form a right-angled triangle with their piece of rope. Their construction is verified with objects in the classroom such as straight-edged books or corners of the room. The students complete the first row of the chart by counting the number of “knots” or segments of masking tape along the length of each side of the triangle.
Teacher Facilitation: The teacher circulates among the groups to assist and encourage students.
Student Activity
Students now subdivide the strip of rope into 24 equal segments. Students again form a right-angled triangle with their piece of rope. The students complete the second row of the chart by again counting the number of “knots” or segments of masking tape along the length of each side of the triangle. Students should look for a pattern in the first two rows and use the pattern to complete the last row.
Teacher Facilitation: The teacher, building on the pattern, introduces the Pythagorean theorem. During this segment, the teacher provides examples in which students solve problems to determine the length of the hypotenuse and, problems in which the length of the hypotenuse and another side is provided, and students determine the length of the remaining side of the triangle. The teacher encourages the students to suggest practical activities in which the Pythagorean theorem would serve to simplify problems.
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Follow-up
Activity |
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Time: 1.25 hour |
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Students complete simple contextual problems using the Pythagorean Theorem. The teacher provides diagramatic representations for some of the problems and has students draw the diagrams for other problems. The problems proceed from those seeking the measure of the hypotenuse to those seeking a measure of the height or base of the diagram. The problems may be of the form: |
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A client’s garden is 18 m by 22 m.
The client wishes to have a diagonal path going from one end of the garden to
the other. |
|
|
a) What
would be the length of the path? |
|
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b) In
a catalogue or newspaper ad, find the cost of some interlocking bricks to
place along the path. |
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What would be the cost of laying the
bricks on the path? |
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Students record the Pythagorean theorem in the technical manual. |
Extension
· Students research in the school library or through the Internet to compile a list and description of instruments used in ancient times. Students would describe the evolution of the instrument to its present day stage.
· Students use the Pythagorean theorem to determine the measure of a length in a composite figure.
· Students write their own problems for which the Pythagorean theorem is required for determining a solution.
· Students are given the following assignment. An ad for a television indicates that the measurement of the television is 17". This measurement corresponds to the diagonal of the television, not its length or width. Investigate the possible dimensions of this television. Find ads for larger televisions. Determine the possible dimensions of these televisions.
Accommodations
· A paper clip should be provided to blind or low vision students to serve as a marker on the rope for the vertex of the right angle.
· The teacher may wish to provide diagrams for all questions.
Assessment/Evaluation Techniques
· The teacher assesses Knowledge and Understanding by conferencing with students about their findings as students work through the rope investigation.
· The Learning Skills of Independence and Initiative can be assessed using the rubric in Appendix J.
· A paper and pencil quiz on the Pythagorean theorem would evaluate students in the categories of Knowledge and Problem Solving. The teacher may wish to use a modification of Appendix I.
Resources
Rope, masking tape, paper clips
Internet web sites providing information related to the Pythagorean theorem:
http://forum.swarthmore.edu/
http://www.learner.org/
http://www.mathgoodies.com/
Appendices
Appendix I – Written Report Rubric
Appendix J – Learning Skills Checklist
Activity 3.7: Summative Assessment: Making a Decision
Time: 3 hours
Description
In this activity, students determine if a plot of land is suitable for a client’s restaurant. In the process, they calculate the area and perimeter of two-dimensional figures and use the Pythagorean theorem. They explore the variety of rectangles which can be constructed for a given area and determine the dimensions of the rectangle which will have a minimum perimeter for the given area.
Strand(s) and Expectations
Ontario Catholic School Graduation Expectations
CGE3c - thinks reflectively and creatively to evaluate situations and solve problems;
CGE5a - works effectively as an interdependent team member;
CGE7i - respects the environment and uses resources wisely.
Strand(s): Number Sense,
Measurement and Geometry
Overall Expectations
NSV.01 - use a variety of methods for calculations when solving problems;
NSV.02 - consolidate the meaning and use of proportionality through applications drawn from student experiences and broader contexts;
MGV.01 - solve problems involving the measurement of two-dimensional figures and three-dimensional objects;
MGV.02 - determine the optimal values of various measurements, through investigations using concrete materials, diagrams, and calculators or computer software;
MGV.03 - solve simple problems involving the Pythagorean theorem.
Specific Expectations
NS1.01 - use pencil and paper computational methods effectively to evaluate expressions involving fractions, decimals, and exponents as they arise in problems throughout the course;
NS1.02 - use a scientific calculator effectively to evaluate expressions involving fractions, decimals, percent, exponents and square roots, as they arise in problems throughout the course;
NS1.04 - use estimation and mental computation to approximate and/or calculate answers to numerical problems as they arise throughout the course;
NS1.07 - judge the reasonableness of answers to problems by considering likely results;
NS1.08 - judge the reasonableness of answers produced by a calculator or computer, using mental mathematics and estimation;
NS2.03 - solve problems involving ratio, proportion and scale, drawn from familiar applications;
NS2.05 - solve problems involving scale, proportionality, and similar figures drawn from familiar applications;
MG1.01 - solve simple problems using the formulas for the perimeter and area of triangles, rectangles, squares and circles;
MG1.02 - calculate the perimeter and area of composite figures;
MG1.03 - substitute into and evaluate measurement formulas as the need arises in problem solving;
MG2.06 - solve simple problems involving optimal values;
MG3.01 - illustrate the Pythagorean theorem, using concrete materials;
MG3.03 - solve simple problems drawn from familiar applications, using the Pythagorean theorem;
MG4.05 - solve simple geometric problems;
MG4.06 - communicate solutions to problems and the results of investigations, using appropriate terminology, symbols and form.
Planning Notes
· Students will need graph paper, compasses (or circular objects) and rulers.
· This activity is group oriented. The teacher should be aware of students’ strengths and weaknesses when forming the groups.
· As this is a group activity, students should be aware that all members of the group will receive the same assessment. The teacher monitors the groups closely to encourage all students to provide input into the activity.
· The teacher may request that all students submit a written project although it will be the same project for all members of the group.
· Students are encouraged to refer to their technical manuals during the activity.
· Students are encouraged to prepare a rough draft of the lot and then to redraw it as a good copy for the assessment.
Prior Knowledge Required
Number Sense
· judge the reasonableness of answers
· determine the dimensions of a figure in another scale
Measurement
· measure lengths accurately
Teaching/Learning Strategies
Teacher Facilitation: The teacher introduces the activity to the students with the following scenario:
A client who is hoping to open a restaurant has found a lot in the area in which he/she would like the restaurant to be located. You have been asked to determine if the lot is suitable for the client’s purposes. If the lot is suitable, you are to draw a plan for the layout of the facility. Your client has faxed the sketch of the lot and other relevant information to you.
Students are clustered in groups of two or three individuals and given Appendix K, Making a Decision. The teacher and students brainstorm methods of organizing the project. The teacher explains the instructions to the students and responds to questions. Students are also given Appendix L, Written Report Checklist and discuss with the teacher the method and criteria of evaluation, prior to commencing the activity.
Student Activity
Students, working in groups, complete the activity outlined in Appendix K.
Extensions
· Students may wish to determine and to sketch a walkway around the patio and fountain.
· An alternate shape for the restaurant could be determined. Although the area would remain as 100 m2, students may wish to form a composite shape for the restaurant. Students would complete the activity as designed but change the shape of the restaurant.
Accommodations
· Tactile learners should be encouraged to cut shapes of the required figures. The shapes may then be moved to various locations on the lot.
Assessment/Evaluation Techniques
· This activity provides teachers with the opportunity to assess Communication by listening to the group discussions.
· Some students could be assessed on their skills in finding and using resource materials using Appendix H, the Observational Checklist.
· Students are assessed on all four categories of learning using the Written Report Rubric, Appendix L.
· If not done so previously, the teacher should assess the completed portfolio and technical manual for all categories of learning, using a modification of Appendix I, the Written Report Rubric.
Resources
Graphing paper, compasses, rulers
Appendices
Appendix H – Observational Rubric
Appendix I – Written Report Rubric
Appendix J – Learning Skills Rubric
Appendix K – Instructions
Appendix L – Written Report Rubric