Grade 8: Content and Reporting Targets
Across the strands and the terms
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters.
Term 1 – Content Targets
Number Sense and Numeration*
• Integers
• Powers and Square Roots
Geometry and Spatial Sense*
• Pythagorean Relationship
Patterning and Algebra*
• Algebraic Expressions
• Solving Equations
• Patterning
• Writing nth Terms
Data Management and Probability
• Experimental vs. Theoretical Probability
Term 2 – Content Targets
Number Sense and Numeration*
• Fractions
• Order of Operations
• Unit Rate
Measurement*
• Circles
Geometry and Spatial Sense*
• Angle Properties
• Construction of a Circle
Data Management and Probability*
• Complex Probabilities
• Best Measure of Central Tendency
• Census vs. Sample
Term 3 – Content Targets
Number Sense and Numeration*
• Order of Operations with Exponents and
Fractions
• Unit Rates and Percents
• Mental Math Skill
Measurement*
• Triangular Prisms
• Valuing Measurement
Geometry and Spatial Sense
• Connect the Pythagorean Relationship to
3-D figures.
Patterning and Algebra*
• Review and Extend Solving Equations in
Contexts
• Inequalities
Data Management and Probability*
• Comparative Bar Graph
• Bar Graph vs. Histogram
Rationale
Connections between:
- integer size/area of squares
- integer sign/colour of integer tile
- square roots/measurements in right
triangles
- scientific notation/powers
- Pythagorean relationship/data
management through inquiry
- equation solving/applications of
Pythagorean Relationship
- algebraic expressions/generalizations of
patterns
- different algebraic representations of a
pattern/the values generated by
substitution into those representations
- statements/algebraic expressions/
equations
- algebraic expressions/unknowns in
equations
Leading to:
- connection between powers/
measurement units (Terms 2 and 3)
- applications of algebraic expressions
and equations (Terms 2 and 3)
- solving equations requiring collection of
like terms (Grade 9)
- using both theoretical and experimental
means of finding patterns (Terms 2 and 3)
Connections between:
- integers/order of operations
- unit rate problems/Term 1 algebra
- constructing circles/discovering
relationships between circle
measurements
- angle properties/data management
- angle properties/Term 1 algebra
- theoretical and experimental probability/
complex probabilities
- best measure of central tendency and
data for developing circle formulas
Leading to:
- combining fractions with order of
operations (Term 3)
- connecting unit rates with percents and
fractions (Term 3)
- combining perimeter/area of irregular
shapes with circles (Grade 9)
- connecting circles to volume of a
cylinder (Grade 9)
- understanding the effect of outlier data
points (Grade 9)
- extending probability/statistics
(Grade 12)
Connections between:
- order of operations/fractions, integers,
powers
- fractions/unit rates/percent
- Natural/Whole/Integer/Fractional/
Rational/Irrational sets of numbers
(combining Natural, Whole, Integer, and
Fractional numbers)
- volume of triangular prism and Grades 6
and 7 concept of Volume = area of base
× height
- inequalities and patterning/problem
solving
- solving equations/Pythagorean
relationship/triangular prisms
- solving equations/unit rates
- comparative bar graphs/histograms
- measures of central tendency (Term 1)/
dispersion shown in a histogram
- data from Term 1 and 2 investigations/
associated concepts/histograms
Leading to:
- combining rational numbers (Grade 9)/
irrational numbers (Grade 11 University
destination)
- volume of a cylinder understood as area
of base × height (Grade 9)
* Strands for reporting purposes
See Appendix for the clusters of curriculum expectations attached to each of the content targets.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 1
Appendix: Curriculum Expectation Clusters
Term 1
Grade 8: Number Sense and Numeration
Term 2
Term 3
Across the stands and the terms:
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters
8m6 • use a calculator to solve number questions that are beyond the proficiency expectations for operations using pencil and paper;
8m7 • justify the choice of method for calculations: estimation, mental computation, concrete materials, pencil and paper, algorithms
(rules for calculations), or calculators;
8m8 • solve and explain multi-step problems involving fractions, decimals, integers, percents, and rational numbers;
8m9 • use mathematical language to explain the process used and the conclusions reached in problem solving;
8m14 – explain numerical information in their own words and respond to numerical information in a variety of media;
8m28 – use estimation to justify or assess the reasonableness of calculations;
8m30 – *ask “what if” questions; pose problems involving fractions, decimals, integers, percents, and rational numbers; and
investigate solutions;
8m31 – explain the process used and any conclusions reached in problem solving and investigations;
8m32 – reflect on learning experiences and interpret and evaluate mathematical issues using appropriate mathematical language (e.g.,
in a math journal);
8m33 – solve problems that involve converting between fractions, decimals, percents, [unit rates, and ratios (e.g., that show the
conversion of 1/3 to decimal form)].
Integers
8m1 • compare, order, and represent
fractions, decimals, integers, and square
roots;
8m5 • demonstrate an understanding of the
rules applied in the multiplication and
division of integers;
8m11 – compare and order fractions,
decimals, and integers;
8m21 – *discover the rules for the
multiplication and division of integers
through patterning (e.g., 3 × [–2] can be
represented by 3 groups of 2 blue disks);
8m22 – add and subtract integers, with and
without the use of manipulatives;
8m23 – multiply and divide integers.
Powers and Square Roots
8m10 – represent whole numbers in
expanded form using powers and scientific
notation (e.g., 347 = 3 × 102 + 4 × 10 + 7,
356 = 3.56 × 102);
8m17 – express repeated multiplication as
powers;
8m24 – understand that the square roots of
non-perfect squares are approximations;
8m25 – estimate the square roots of whole
numbers without a calculator;
8m26 – find the approximate values of
square roots of whole numbers using a
calculator;
8m27 – use trial and error to estimate the
square root of a non-perfect square.
Fractions
8m2 • demonstrate proficiency in
operations with fractions;
8m13 – represent composite numbers as
products of prime factors
(e.g., 18 = 2 × 3 × 3);
8m15 – demonstrate an understanding of
operations with fractions;
8m18 – add, subtract, multiply, and divide
simple fractions.
Order of Operations with Integers
and Decimals
8m4 • understand and apply the order of
operations with brackets for integers;
8m16 – perform multi-step calculations
involving whole numbers and decimals in
real-life situations, using calculators.
Unit Rates
8m29 – demonstrate an understanding of
and apply unit rates in problem-solving
situations.
Order of Operations with Exponents
and Fractions
8m3 • understand and apply the order of
operations with brackets and exponents in
evaluating expressions that involve
fractions;
8m19 – understand the order of operations
with brackets and exponents and apply the
order of operations in evaluating
expressions that involve fractions;
8m20 – apply the order of operations (up
to three operations) in evaluating
expressions that involve fractions.
Unit Rates and Percents
8m34 – apply percents in solving problems
involving discounts, sales tax,
commission, and simple interest.
Mental Math Skill
8m12 – mentally divide numbers by 0.1,
0.01, and 0.001.
* Expectations require that students be given the opportunity to learn through inquiry. Learning through problem
solving is recommended for most other curriculum expectations.
Overall curriculum expectations are designated by the • after the number.
Specific curriculum expectations are designated by the – after the number.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 2
Appendix: Curriculum Expectation Clusters
Term 1
Grade 8: Measurement
Term 2
Term 3
Across the stands and the terms:
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters
8m35 • demonstrate a verbal and written understanding of and ability to apply accurate measurement and estimation strategies that
relate to their environment;
8m36 • identify relationships between and among measurement concepts (linear, square, cubic, temporal, monetary);
8m39 – use listening, reading, and viewing skills to interpret and evaluate the use of measurement formulas;
8m40 – explain the relationships between various units of measurement;
8m42 – make increasingly more informed and accurate measurement estimations based on an understanding of formulas and the results
of investigations;
8m43 – ask questions to clarify and extend their knowledge of linear measurement, area, volume, capacity, and mass, using appropriate
measurement vocabulary.
Circles
8m37 • solve problems related to the
calculation of the radius, diameter, and
circumference of a circle;
8m44 – measure the radius, diameter, and
circumference of a circle using concrete
materials;
8m45 – *recognize that there is a constant
relationship between the radius, diameter,
and circumference of a circle, and
approximate its value through
investigation;
8m46 – *develop the formula for finding
the circumference and the formula for
finding the area of a circle;
8m47 – estimate and calculate the radius,
diameter, circumference, and the area of a
circle, using a formula in a problemsolving context;
8m48 – draw a circle given its area and/or
circumference;
8m49 – define radius, diameter, and
circumference and explain the
relationships between them.
Triangular Prisms
8m38 • apply volume and area formulas to
problem-solving situations involving
triangular prisms;
8m50 – *develop the formula for finding
the surface area of a triangular prism using
nets;
8m51 – *develop the formula for finding
the volume of a triangular prism;
8m52 – understand the relationship
between the dimensions and the volume of
a triangular prism;
8m53 – calculate the surface area and the
volume of a triangular prism, using a
formula in a problem-solving context;
8m54 – sketch a triangular prism given its
volume.
Valuing Measurement
8m41 – research, describe, and report on
uses of measurement in projects at home,
in the workplace, and in the community
that require precise measurements.
* Expectations require that students be given the opportunity to learn through inquiry. Learning through problem
solving is also recommended for most other curriculum expectations.
Overall curriculum expectations are designated by the • after the number.
Specific curriculum expectations are designated by the – after the number.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 3
Appendix: Curriculum Expectation Clusters
Grade 8: Geometry and Spatial Sense
Across the stands and the terms:
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters
8m60 • use mathematical language effectively to describe geometric concepts, reasoning, and investigations.
Term 1
Term 2
Term 3
Pythagorean Relationship
8m65 – *investigate the Pythagorean
relationship using area models and
diagrams;
8m70 – apply the Pythagorean relationship
to numerical problems involving area and
right triangles;
8m73 – explain the Pythagorean
relationship.
Angle Properties
8m55 • identify, describe, compare, and
classify geometric figures;
8m57 • identify and investigate the
relationships of angles;
8m58 • construct and solve problems
involving lines and angles;
8m63 – identify the angle properties of
intersecting, parallel, and perpendicular
lines by direct measurement: interior,
corresponding, opposite, alternate,
supplementary, complementary;
8m64 – *explore the relationship to each
other of the internal angles in a triangle
(they add up to 180°) using a variety of
methods;
8m66 – solve angle measurement
problems involving properties of
intersecting line segments, parallel lines,
and transversals;
8m67 – create and solve angle
measurement problems for triangles;
8m68 – construct line segments and angles
using a variety of methods (e.g., paper
folding, ruler and compass);
8m71 – describe the relationship between
pairs of angles within parallel lines and
transversals;
8m72 – explain why the sum of the angles
of a triangle is 180º.
Connect the Pythagorean
Relationship to 3-D figures
8m56 • identify, draw, and represent threedimensional geometric figures;
8m61 – recognize three-dimensional
figures from their top, side, and front
views;
8m62 – sketch and build representations of
three-dimensional figures (e.g., nets,
skeletons) from front, top, and side views.
Construct a Circle
8m59 • *investigate geometric
mathematical theories to solve problems;
8m69 – construct a circle given its centre
and radius or centre and a point on the
circle or three points on the circle.
* Expectations require that students be given the opportunity to learn through inquiry. Learning through problem
solving is also recommended for most other curriculum expectations.
Overall curriculum expectations are designated by the • after the number.
Specific curriculum expectations are designated by the – after the number.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 4
Appendix: Curriculum Expectation Clusters
Grade 8: Patterning and Algebra
Across the stands and the terms:
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters.
Processes involving problem-solving communication technology and reasoning are embedded in the expectations below.
Term 1
Term 2
Term 3
Review and Extend Solving
Equations in Contexts
8m84 – write statements to interpret
simple equations;
8m87 – translate complex statements into
algebraic expressions or equations;
8m88 – solve and verify first-degree
equations with one variable, using various
techniques involving whole numbers and
decimals.
Algebraic Expressions
8m74 • identify the relationships between
whole numbers and variables;
8m76 • evaluate algebraic expressions;
8m86 – evaluate simple algebraic
expressions, with up to three terms, by
substituting fractions and decimals for the
variables.
Solving Equations by Inspection
and Systematic Trial
8m77 • identify, create, and solve simple
algebraic equations;
8m82 – use the concept of variable to
write equations and algebraic expressions;
8m89 – create problems giving rise to
first-degree equations with one variable
and solve them by inspection or by
systematic trial;
8m90 – interpret the solution of a given
equation as a specific number value that
makes the equation true.
Inequalities
8m83 – *investigate inequalities and test
whether they are true or false by
substituting whole number values for the
variables (e.g., in 4x ≥ 18, find the whole
number values for x).
Patterning
8m75 • identify, create, and discuss
patterns in algebraic terms;
8m78 • apply and defend patterning
strategies in problem-solving situations;
8m79 – describe and justify a rule in a
pattern;
8m85 – present solutions to patterning
problems and explain the thinking behind
the solution process.
Writing nth Terms
8m80 – write an algebraic expression for
the nth term of a numeric sequence;
8m81 – find patterns and describe them
using words and algebraic expressions;
8m82 – use the concept of variable to
write equations and algebraic expressions.
* Expectations require that students be given the opportunity to learn through inquiry. Learning through problem
solving is also recommended for most other curriculum expectations.
Overall curriculum expectations are designated by the • after the number.
Specific curriculum expectations are designated by the – after the number.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 5
Appendix: Curriculum Expectation Clusters
Grade 8: Data Management and Probability
Across the strands and the terms.
Problem Solving, Communication, Technology, and Reasoning - expectations to be applied to any/all content clusters.
8m91 • systematically collect, organize, and analyse primary data;
8m92 • use computer applications to examine and interpret data in a variety of ways;
8m93 • interpret displays of data and present the information using mathematical terms;
8m94 • evaluate data and draw conclusions from the analysis of data;
8m95 • identify probability situations and apply a knowledge of probability;
8m99 – read a database or spreadsheet and identify its structure;
8m100 – manipulate and present data using spreadsheets, and use the quantitative data to solve problems;
8m101 – search databases for information and use the quantitative data to solve problems;
8m102 – know that a pattern on a graph may indicate a trend;
8m104 – discuss trends in graphs to clarify understanding and draw conclusions about the data;
8m105 – discuss the quantitative information presented on tally charts, stem-and-leaf plots, frequency tables, and/or graphs;
8m106 – explain the choice of intervals used in constructing bar graphs or the choice of symbols in pictographs;
8m112 – make inferences and convincing arguments that are based on data analysis;
8m113 – evaluate arguments that are based on data analysis;
8m114 – determine trends and patterns by making inferences from graphs;
8m115 – explore with technology to find the best presentation of data.
Term 1
Term 2
Term 3
Experimental vs. Theoretical
Probability
8m96 • appreciate the power of using a
probability model by comparing
experimental results with theoretical
results;
8m117 – identify 0 to 1 as a range from
“never happens” (impossibility) to “always
happens” (certainty) when investigating
probability;
8m118 – list the possible outcomes of
simple experiments by using tree
diagrams, modelling, and lists;
8m119 – identify the favourable outcomes
among the total number of possible
outcomes and state the associated
probability (e.g., of getting chosen in a
random draw);
8m121 – compare predicted and
experimental results.
Complex Probabilities
8m116 – use probability to describe
everyday events;
8m120 – use definitions of probability to
calculate complex probabilities from tree
diagrams and lists (e.g., for tossing a coin
and rolling a die at the same time);
8m122 – apply a knowledge of probability
in sports and games, weather predictions,
and political polling.
Best Measure of Central Tendency
8m103 – understand and apply the concept
of the best measure of central tendency;
8m109 – determine the effect on a measure
of central tendency of adding or removing
a value (e.g., what happens to the mean
when you add or delete a very low or very
high data entry).
Comparative Bar Graph
8m107 – assess bias in data-collection
methods;
8m108 – read and report information about
data presented on line graphs, comparative
bar graphs, pictographs, and circle graphs,
and use the information to solve problems.
Bar Graph vs. Histogram
8m110 – understand the difference
between a bar graph and a histogram;
8m111 – construct line graphs,
comparative bar graphs, circle graphs, and
histograms, with and without the help of
technology, and use the information to
solve problems.
Census vs. Sample
8m97 – collect primary data using both a
whole population (census) and a sample of
classmates, organize the data on tally
charts and stem-and-leaf plots, and display
the data on frequency tables;
8m98 – understand the relationship
between a census and a sample.
* Expectations require that students be given the opportunity to learn through inquiry. Learning through problem
solving is also recommended for most other curriculum expectations.
Overall curriculum expectations are designated by the • after the number.
Specific curriculum expectations are designated by the – after the number.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 6
Grade 8: Term 1 Content Flow
Introduction
Powers and
Square Roots
Days 1-6
Establish the
importance of
problem solving,
communication,
cooperative
learning skills
Applied in context
Pythagorean
Relationship
Days 7-10
Days 11-16
(Number Sense
and Numeration
strand)
(Geometry and
Spatial strand)
Exemplar task
provides a
segue
An
experiment
provides a
segue
Revisits
Sets the stage by
requiring application
of Grade 7 skills and
concepts (inspection
and systematic trials)
Writing nth
Terms
Days ?- ?
(Patterning
and Algebra
strand)
Experimental
vs. Theoretical
Probability
Days 17-23
(Data
Management
and Probability
strand)
An experiment
provides a segue
Provides examples for
Patterning
Solving
Equations
Days ?-?
(Patterning
and Algebra
strand)
Days ?-?
(Patterning
and Algebra
strand)
Integers
Days 24- ?
(Number Sense
and Numeration
strand)
Applied in context for practice
Other sequences are possible.
Suggestions for development of further Term 1 lessons are included on page 8.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 7
Developing Lessons Targeting Term 1 Curriculum Clusters –
Integers, Solving Equations, Patterning, Writing nth Terms
Suggestions
•
These curriculum clusters are supported by comprehensive content-based packages – Integers,
Solving Equations and Using Variables as Placeholders, and Patterning to Algebraic Modelling. It is
recommended that groups of teachers collaboratively develop lessons using the contents of these
packages as a starting point.
•
Each package includes:
− scope and sequence across grades
− suggested instructional strategies for each of Grade 7, Grade 8, and Grade 9 Applied
− suggestions for helping students develop understanding in the areas where experience shows that
some students may struggle
− cross-strand connections
− sample questions addressing key expectations based on the four mathematical process areas
identified and supported in this project
− sample Developing Proficiency tests based on key expectations
− Extend Your Thinking questions that ask for multiple solutions
− Is This Always True? Questions to help student deepen their understanding of key concepts.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 8
Interpreting the Lesson Outline Template
Download the template at www.curriculum.org/occ/tips/downloads.shtml
Lesson Outline: Days 5 - 9
Grade 7
Sequence of Lessons Addressing a Theme
BIG PICTURE
Grade Level
Students will:
• explore and generalize patterns;
• develop an understanding of variables;
• investigate and compare different representations of patterns.
Lessons are planned to help students develop and demonstrate the skills and knowledge
detailed in the curriculum expectations.
•
•
•
To help students value and remember the mathematics they learn, each lesson is connected to and focussed on important
mathematics as described in the Big Picture.
Since students need to be active to develop understanding of these larger ideas, each point begins with a verb.
Sample starter verbs: represent, relate, investigate, generate, explore, develop, design, create, connect, apply
Day
5
Lesson Title
Toothpick Patterns
•
•
•
Description
Review patterning concepts
Build a growing pattern
Explore multiple representations
Expectations
7m70, 7m72
7m66, 7m71
CGE 3c, 4f
6
Patterns with Tiles
•
•
Build a pattern
Introduce the nth term
7
Pattern Practice
•
Continued development of patterning skills
A brief descriptive lesson title
8
9
Pattern Exchange
Performance Task
•
•
•
•
•
CGE 4b
7m67, 7m71,
List curriculum7m75
expectations
(and CGEs)
CGE 2c, 5e
7m69, 7m75
Class sharing of work from previous day. by code
Performance Task - individual
Two or three points to describe the content of this lesson.
Points begin with a verb.
Individual lesson plans elaborate on how objectives are met.
CGE 2c, 5e
7m66, 7m67,
7m73, 7m75
CGE 5g
NOTES
a) While planning lessons, teachers must judge whether or not pre-made activities support development of big ideas and provide
opportunities for students to understand and communicate connections to the “Big Picture.”
b) Ontario Catholic School Graduation Expectations (CGEs) are included for use by teachers in Catholic schools.
c) Consider auditory, kinesthetic, and visual learners in the planning process and create lessons that allow students to learn in different
ways.
d) The number of lessons in a group will vary.
e) Schools vary in the amount of time allocated to the mathematics program. The time clock/circle on completed Grade 7 and 8
lessons suggests the fractions of the class to spend on the Minds On, Action!, and Consolidate/Debrief portions of the class. Grade
9 Applied lessons are based on 75-minute classes.
f)
Although some assessment is suggested during each lesson, the assessment is often meant to inform adjustments the teacher will
make to later parts of the lesson or to future lessons. A variety of more formal assessments of student achievement could be added.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 9
Interpreting the Lesson Planning Template
Download the template at www.curriculum.org/occ/tips/downloads.shtml
Grade Level
Day #: Lesson Title
Day 1: Encouraging Others
Grade 8
Same two or three objectives
listed in the lesson outline
Description
the social skill of encouraging others.
• Identify strategies involving estimation problems.
• Set the stage for using estimation as a problem-solving strategy.
• Practise
Time colour-coded to the three
parts of the day’s lesson
Minds On ...
Materials
used in the
lesson
Suggested student grouping Æ teaching/learning strategy for the activity.
Assessment
Opportunities
Whole Group Æ Brainstorm
Explain why it is important to encourage others. Explicitly teach the social
skill, “Encouraging Others,” through a group brainstorm. Create an anchor
chart using the criteria: What does it look like? What does it sound like?
• Mentally engages students at start of class
• Makes connections between different math strands, previous lessons or groups of
lessons, students’ interests, jobs, etc.
• Introduces a problem or a motivating activity - orients students to an activity or materials.
Action!
Think/Pair/Share Æ Gather Data
Use an overhead of the Think/Pair/Share process (TIP 2.1) and student copies
• Students do
of BLM 1.1. Students gather data.
mathematics: Learning Skill/Observation/Mental Note: Circulate, observing social skills
reflecting,
and listening to students.
discussing,
Indicates an assessment opportunity
- what is assessed/strategy/scoring tool
observing,
investigating,
Share with students some of the positive words and actions observed during the
exploring,
activity and invite students to make additions to the anchor chart on
creating,
Encouraging Others.
listening,
Indicates suggested
reasoning,
assessment
making
Whole Class Æ Sharing
connections,
Based on ‘teachable topics’ during the Think/Pair/Share Activity, e.g., a
demonstrating
particularly effective phrase/statement expressed by a student, clarification of
understanding,
the cooperative learning strategy, an interesting result on BLM 1.1, ask
discovering,
hypothesizing representatives of groups to share their results or report on their process.
• Teachers listen,
observe, respond
Consolidate
Debrief
• “Pulls out´ the
math of the
activities and
investigations
• Prepares
students for
Home/Further
Classroom
Consolidation
Materials
• BLM 1.1
• birdseed
Whole Class Æ Discussion
Use the posters Inquiry Model Flow Chart, Problem-Solving Strategies, and
Understanding the Problem. Discuss how these posters will be of assistance
over the next few days as well as during the whole math program. Point out
that when students encourage others, it makes it safe for them to try new things
and contribute to group activities.
Home Activity or Further Classroom Consolidation
Interview one or more adults about estimation using the following guiding
questions and record your responses in a math journal. Summarize what you
notice about the responses. You may be asked to share this math journal entry
with the class.
Social Skill
Practice
Reflection
“Learning is
socially
constructed; we
seldom learn
isolated from
others.”
- Bennett & Rolheiser
Consider using
stickers as a
recognition for
examples of the
social skill being
applied by a group.
Solving Fermi
problems is a way
to collect
diagnostic
assessment data
about social skills,
academic
understandings,
and attitudes
towards
mathematics
(see TIP 1.2).
Tips for the Teacher
These include:
- instructional hints
- explanations
- background
- references to resources
- sample responses to
questions/tasks
Meaningful and appropriate follow-up to the lesson.
Focus for the follow-up activity
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 10
Lesson Outline – Days 1 - 6
Grade 8
BIG PICTURE
Students will:
• develop teamwork skills through cooperative learning;
• take risks when carrying out an investigation and demonstrate perseverance;
• apply a variety of problem-solving strategies;
• apply a number of estimation strategies during problem solving;
• justify their solutions and choice of strategies;
• make connections between prior and new knowledge to draw conclusions;
• represent their thinking in a variety of ways, reflect on their learning, and communicate effectively.
Day
Lesson Title
Description
Expectations
8m7, 8m28
1
Encouraging Others • Practise the social skill of encouraging others.
• Identify strategies involving estimation problems.
CGE 5a
• Set the stage for using estimation as a problem-solving
strategy.
2
Solving a Fermi
• Find a solution to a problem involving estimation.
8m6, 8m31,
Birdseed Problem
8m112
3
Taking Turns
• Practise
CGE 3c
8m6, 8m9,
8m32
4
Paraphrasing and
Summarizing
• Practise
CGE 5a
8m6, 8m9,
8m32, 8m112
5
Including All
Participants and
Recording
Mathematics
• Practise
CGE 2a
8m6, 8m9,
8m14, 8m31
6
Disagreeing in an
Agreeable Way
While Analysing
Good Math
Records
TIPS: Section 3 – Grade 8
the social skill of taking turns.
• Find a solution to a problem involving estimation.
the social skill of active listening and
paraphrasing.
• Practise developing good problem-solving strategies.
the social skill of including all participants.
• Develop a method for effective recording of mathematics
learning.
• Find a solution to a problem involving estimation.
• Practise the social skill of disagreeing in an agreeable
way.
• Examine math recordings, suggest how to improve them,
and articulate what good writing looks like in
mathematics.
• Create a concept map to help consolidate their thinking
over the last few days.
© Queen’s Printer for Ontario, 2003
CGE 5a
8m6, 8m9,
8m14, 8m31,
8m32, 8m35,
8m39
CGE 5e
Page 11
Day 1: Encouraging Others
Grade 8
Materials
• BLM 1.1
• birdseed
Description
the social skill of encouraging others.
• Identify strategies involving estimation problems.
• Set the stage for using estimation as a problem-solving strategy.
• Practise
Assessment
Opportunities
Minds On ...
Action!
Consolidate
Debrief
Social Skill
Practice
Reflection
Whole Group Æ Brainstorm
Explain why it is important to encourage others. Explicitly teach the social
skill, Encouraging Others, through a group brainstorm. Create an anchor chart
using the criteria: What does encouragement look like? What does it sound
like? (TIP 2).
Think/Pair/Share Æ Gather Data
Use an overhead of the Think/Pair/Share process (TIP 8) and student copies of
BLM 1.1. Students gather data.
Learning Skill/Observation/Mental Note: Circulate, observing social skills
and listening to students.
Share with students some of the positive words and actions observed during the
activity and invite students to make additions to the anchor chart on
Encouraging Others.
Whole Class Æ Sharing
Based on ‘teachable topics’ during the Think/Pair/Share Activity, e.g., a
particularly effective phrase/statement expressed by a student, clarification of
the cooperative learning strategy, an interesting result on BLM 1.1, ask
representatives of groups to share their results or report on their process.
Whole Class Æ Setting Stage
Explain that during the first week, the class will solve a number of different
kinds of Fermi problems to sharpen their problem-solving and estimation skills
(Poster: Teaching Through Problem Solving). Discuss the concept of Fermi
problems (TIP 3).
Show a large bag of birdseed and ask, How many seeds do you think are in the
bag? Tell the class that tomorrow they will work on solving this problem.
“Learning is
socially
constructed; we
seldom learn
isolated from
others.”
- Bennett & Rolheiser
Consider using
stickers as
recognition for
examples of the
social skill being
applied by a group.
Solving Fermi
problems is a way
to collect
diagnostic
assessment data
about social skills,
academic
understandings,
and attitudes
towards
mathematics
(TIP 3).
Whole Class Æ Discussion
Use the posters Teaching Through Problem Solving, Problem-Solving
Strategies, and Understand the Problem. Discuss how these posters will be of
assistance over the next few days as well as during the whole math program.
Point out that when students encourage others, it makes it safe for them to try
new things and contribute to group activities.
Home Activity or Further Classroom Consolidation
Interview one or more adults about estimation using the following guiding
questions and record your responses in a math journal. Summarize what you
notice about the responses. You may be asked to share this math journal entry
with the class.
• When do you use estimation and how often?
• When are accurate calculations necessary?
Answer the following questions in the math journal to assess your growth in
social skills.
• The social skill focus of this activity was ___________.
• Something I said or did to demonstrate the social skill was _________.
• I helped the group work in a positive way by ________.
• An area I want to work on is _____________.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 12
1.1: What Do We Have in Common?
Name:
Date:
Fill in the following table with your partner.
Name:
Which TV shows do you like to
watch?
What kind of music do
you like to listen to?
What do you enjoy doing in your
spare time?
What movies have you seen
lately?
What sports do you like to
watch or participate in?
Do you have brothers and sisters?
What are some activities you
participate in during the summer?
What school subjects
do you enjoy?
What did you find the most
interesting in your comparison?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 13
Day 2: Solving a Fermi Birdseed Problem
Description
• Find a solution to a problem involving estimation.
Minds On…
Action!
Consolidate
Debrief
Application
Exploration
Grade 8
Materials
• large bag of
birdseed
• chart paper
• Impact Math –
Number Sense
Assessment
Opportunities
Whole Class Æ Introduce the Problem
Learning Skills/Journal/Anecdotal: Ask selected students to share their math
journal responses.
Read one of the books listed as a jumping-off point for the lesson. This makes
an important link between mathematical and language literacy. A King/Rajah
starts by placing one grain of rice on a chessboard and then doubles the value
for each consecutive square, i.e., 1, 2, 4, 8, 16, until the last square. Stop the
story before it gets too far along in order to not give away the strategy used for
measuring the rice, i.e., volume or mass. Introduce the problem: How many
seeds do you think are in this large bag of birdseed?
Think/Pair/Share Æ Guided Cooperative Problem Solving
Guide students through the Think/Pair/Share process as it pertains to the
birdseed problem. (TIP 8) Tell students that you are looking for a variety of
creative strategies for solving the problem.
Curriculum Expectations/Question & Answer/Anecdotal: Circulate to look
for evidence of the social skill, strategies used, and students who are having
difficulty. If there are a number of students who need help:
• scaffold through questioning: How can we measure the birdseed? How can
these measurements help us solve the problem? What tools might help us?
How? How might a smaller container help? How would a scale help?
• model your thinking process through a think aloud (See Section 2 – Research,
Scaffolding) pausing to allow students to contribute their own ideas and
strategies as the group solves the problem together.
Whole Class Æ Sharing Strategies
Share some of the positive words and actions observed during the activity and
invite students to make additions to the anchor chart on Encouraging Others
(TIP 2). Pose the question: How did you decide on your estimate?
Select one person from three or four groups to share their group strategies and
estimations. Pick groups with different approaches to help students realize
there are many ways to solve this problem. Have the class compare group
estimations and decide on reasons why they may vary and whether some are
more valid.
Clarify, summarize, and record student responses on chart paper.
Stories to set a
context: The King’s
Chessboard by
David Birch, ISBN
0140548807 or
The Rajah’s Rice
by David Barry,
ISBN 0716765683
An on-line version
of the King’s
Chessboard –
http://www.2.bc.ed
u/~grout/chessboa
rd/html/pg01.htm
Set ground rules
for sharing:
• Everyone has a
perspective that
should be
considered.
• We need
everyone’s ideas
for the best
result.
• We could miss
an important
point/
perspective if we
do not share our
thinking.
• Suggestions of
all students will
be listened to
and used.
Possible strategies
include:
• Using mass
comparison
• Using volume
comparisons
• Counting seeds
in a small sample
• Using measuring
cups
• Using measuring
tapes
Home Activity or Further Classroom Consolidation
Discuss the concept of a Fermi problem with a friend or someone at home, then
generate an example of a Fermi problem that deals with something from home.
Write up your problem and its solution in your math journal.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 14
Day 3: Taking Turns
Grade 8
Materials
• BLM 3.1, 3.2
• phone books
• calculators
Description
• Practise the social skill of taking turns.
• Find a solution to a problem involving estimation.
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Reflection
Whole Class Æ Sharing
Curriculum Expectations/Observation/Anecdotal: Listen to students and
provide immediate feedback as Inside Outside Circle (TIP 13) are used to have
students share the Fermi problem they developed.
Rotate the circles and have students share their Fermi problem with a new
partner. Review with the class any elements that have been misunderstood.
Whole Class Æ Brainstorm
Brainstorm to create an anchor chart for Taking Turns (TIP 2).
Groups of 4 Æ Cooperative Group Problem Solving
Use BLM 3.1 for a Placemat cooperative activity (TIP 9) Ask each group: Why
was it difficult to solve the birdseed problem? What information did you need
to know? Based on the class list of questions, each group generates a list of
questions to guide them in making their estimates. Model for the class how to
create one or two of these questions.
Whole Class Æ Discussion
One person from each group is selected at random to share one question from
the Placemat activity. Record the questions on chart paper or on a transparency.
Using a transparency of BLM 3.2, the class orders the questions from the class
chart from broadest at the top to more narrow information at the bottom so that
the combined answers give an appropriate estimation. How did you decide on
your questions?
Groups of 4 Æ Cooperative Problem Solving
Learning Skills & Curriculum Expectations/Observation/Anecdotal:
Circulate while groups are working.
Using a Placemat activity, students choose a strategy to solve the problem,
How many names are there in the phone book? After solving the problem,
students use the Ranking Ladder (BLM 3.2) to sequence the questions they
used to arrive at an accurate estimation.
Whole Class Æ Discussion
Curriculum Expectations/Exhibition/Checklist: Select one person from two
or three of the groups to present their problem-solving strategies to the class.
Choose groups with different methods for solving the problem. Encourage
students to show how each strategy follows the estimation model. Record
strategies on a transparency or chart paper.
Summarize strategies with the class, modelling the selection of important
information. Tell the class that they will build the summarizing skills used
today during next class. Reaffirm how estimation skills improve with practice.
Inside/Outside
Circles help
develop a positive
classroom climate
and a community
of learners.
During cooperative
learning, use a 2colour disk as a
barometer.
Show the white
side when the
group is
demonstrating the
social skill. Show
the red side when
they are not using
the social skill.
Fermi solved his
legendary
problems by
developing a
series of questions
and estimating the
answers.
Give examples of
some of the
positive things
(Taking Turns and
Encouraging
Others) and add
them to the class
anchor charts.
See TIP 15,
Questioning, for
suggestions on
how to elicit
mathematical
thinking.
Home Activity or Further Classroom Consolidation
In your math journal, identify a situation where estimation is needed, then
describe a strategy that could be used to establish a reasonably accurate
estimate.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 15
3.1: Placemat
Names:
Date:
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 16
3.2: Ranking Ladder
Name:
Date:
Use the ranking ladder to organize the questions you used to arrive at an accurate estimation.
List the first question you would ask yourself to solve the problem at the top of the ladder.
List the last question at the bottom of the ladder, and use the middle rungs to put the other
questions in order.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 17
Day 4: Paraphrasing and Summarizing
Grade 8
Description
the social skill of active listening and paraphrasing.
• Practise developing good problem-solving strategies.
Materials
• BLM 3.2
• Practise
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Reflection
Whole Group Æ Active Listening and Summarizing
Explain the meaning of Active Listening and Paraphrasing and through
brainstorming develop a class anchor chart for this social skill (TIP 2).
Groups of 3 Æ Cooperative Problem Solving
Introduce the Fermi problem: How many times does the wheel of your bicycle
turn on a trip from the school to the Sky Dome in Toronto? (Change the
destination as needed.) Have the class think of some questions the interviewer
may ask:
• What strategies will you use?
• What questions do you need answered to estimate the solution?
• What information do you need to know?
• What confuses you?
In small groups students quietly think for a few minutes before starting the
interviews.
Whole Class Æ Sharing Session
Select one or two groups who were successful in paraphrasing and have them
model for the class. On the anchor chart, make any additions that emerge from
the sharing.
Whole Group Æ Discussion
Guide a class reflection on problem-solving steps and strategies that students
effectively used to solve Fermi problems. Students brainstorm criteria for good
problem solving using the focus question: What does a good problem solver
do? Students briefly discuss with a partner, then draw out and record their
ideas. Refer to classroom posters on Problem Solving.
Groups of 4 Æ Cooperative Problem Solving
Each group appoints a recorder and tracks the steps and strategies the group
follows to solve the Fermi problem. Another member tracks the questions
asked to arrive at a reasonable estimate. Each group records its hierarchy of
questions on BLM 3.2.
Curriculum Expectations/Performance Task/Anecdotal: Circulate and look
for groups to share their problem solving process, strategies and questions.
Learning Skills/Observation/Tracking Sheets: Encourage the groups as
necessary using prompting questions (TIP 17 Learning Skills Tracking Sheet).
Whole Class Æ Sharing
Select one student from two or three groups to present their problem solving
strategies, and ranking ladder questions. Choose groups with different methods
for solving the problem, demonstrating that there are many good ways to solve
the same problem. Use think aloud to model, paraphrase, and record each
group’s ideas on chart paper or overhead.
Learning Skills/Exhibition/Mental Note: Assess students as they present
their strategies.
Whole Class Æ Consolidate
Debrief the steps for being a problem solver. Compare all the strategies used
throughout Lessons 1-4 to the ones listed on the poster, Problem-Solving
Strategies. Use the strategy, Logical Reasoning, when developing and ordering
questions.
The interview
process provides
an opportunity to
apply the skill of
active listening and
paraphrasing to
collectively
determine an
appropriate
strategy for solving
the problem.
Provide road or
computer maps.
Prompting
questions:
How do you make
sense of the
problem?
How do you get
started?
How do you know
what to do?
How do you
organize and
communicate your
thinking?
How do you pick a
strategy?
How do you solve
problems?
Home Activity or Further Classroom Consolidation
Reflect on the steps and strategies you use to problem solve and write about
them in your math journal.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 18
Day 5: Including All Participants and Recording Mathematics
Grade 8
Description
the social skill of including all participants.
• Develop a method for effective recording of mathematics learning.
• Find a solution to a problem involving estimation.
Materials
• BLM 1.2, 3.2,
• Practise
5.1
• Math posters
• colour markers
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Application
Concept Practice
Whole Class Æ Reflection
Introduce the social skill: Including All Participants. Students discuss why the
skill is important and what it looks like and sounds like (TIP 2).
Learning Skill/Self-Assessment/Anecdotal: Using journal question at the end
of Day 1, have students self-assess their social skills development to date.
Groups of 4 Æ Graffiti Board
Use the following questions to help students begin their graffiti board:
• Why record in math?
• What should the written explanation of your records include?
• In what other ways besides words can you organize and show your thinking?
• If you were trying to understand someone else’s thinking, what information
and organizational formats (diagrams, tables, charts, etc.) would help you?
Circulate and use prompting questions, as necessary (TIP 15).
Whole Class Æ Sharing
Students consolidate their thinking and develop a list of criteria for good math
records. Create and post a class anchor chart listing the criteria for good math
records.
Whole Class Æ Introduce the Fermi problem
Introduce today’s problem: How many hours do students in Grade 7 and 8 in
Ontario talk on the telephone in one year?
Groups of 4 Æ Solve the Problem
Students discuss and record their questions on the Ranking Ladder (BLM 3.2).
When they have found a satisfactory solution, the group discusses and creates
their best record using the criteria developed during the Graffiti Board exercise.
Provide markers and chart paper. Use BLM 5.1 on a transparency to guide
students’ thinking.
Learning Skills/Observation/Checklist: During the problem-solving process,
look for students who are recording their series of questions in sequence and
groups that are using different strategies. Call on these groups during
consolidation.
Whole Class Æ Discussion
Curriculum Expectations/Performance Task/Rating Scale: Select groups to
display their recordings and explain their estimation/problem-solving process.
Clarify, if necessary, having students turn to a partner to paraphrase what was
explained. Guide the discussion, as necessary.
Ask students to reflect on whether they are becoming more accomplished
estimators. Discuss estimation strategies that you have observed throughout the
class. Tell students they will discuss records in more detail during the next
class.
Note: Display the
Math posters
prominently:
Teaching Through
Problem Solving,
Representations
Make Our Thinking
Visible,
Understand the
Problem, and
Problem-Solving
Strategies.
Link to writing for
different
audiences.
Display the
teacher-made
charts of student
strategies for
Fermi problems
Remind students
to use the social
skills they have
learned to date.
Be sure that
students note that
good records
should include one
or more
representations of
thinking: diagrams,
words, numbers or
symbols, tables,
etc.
Home Activity or Further Classroom Consolidation
Explain to someone how you would solve today’s Fermi problem.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 19
5.1: Thinking to Solve Problems
Name:
Date:
What do you predict? Why?
What question will you use to begin estimating?
How will you decide how many students there are in Grades 7 and 8?
What surprises you? Why?
What do you find interesting? Explain.
Describe any trends you see in the data?
Why do you think these trends are happening?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 20
Grade 8
Day 6: Disagreeing in an Agreeable Way While Analysing Good Math Records
Description
the social skill of disagreeing in an agreeable way.
• Examine math records, suggest how to improve them, and articulate what
good writing looks like in mathematics.
• Create a concept map to help consolidate their thinking over the last few days.
• Practise
Materials
• BLM 6.1
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Reflection
Whole Class Æ Discovery
Present both positive and negative examples of the social skill shown on BLM
6.1. With a partner, students compare the two scenarios. When the class agrees
on the social skill, help them decide on an appropriate name for it. Discuss why
this skill is important to their learning and cooperative group work. Create an
anchor chart for Disagreeing in an Agreeable Way (TIP 2).
Explain that today the class will analyse some examples from the previous
day’s student records, specifically looking for evidence of good mathematics
communication. Select two examples of group records from Day 5’s work
(remove student names). Remind them that students worked hard to make these
the best records possible. It is important to respect their effort by noting the
strengths of the recordings, making positive suggestions for improvement.
Remind students of all the positive social skills they have developed to date.
Think/Pair/Share Æ Peer Assessment
Students examine the examples and jot down at least three things that
demonstrate the criteria the class established and one or two positive ways the
authors can improve their records. They pair and share their findings. Look for
students who have found evidence of the established criteria and for examples
of students disagreeing positively. Use prompting questions to encourage
students, as needed.
Students self-assess their group work, using the questions from Day 1.
Whole Class Æ Sharing Session
Select one person from each group to make thoughtful positive comments and
suggestions for improvement. Record each of the group’s suggestions on chart
paper and summarize their findings. Point out any evidence students may have
missed of representations, thinking, strategies, and noticing patterns.
Whole Class Æ Brainstorm
Ask what students have done during the first five math classes. Record
responses on chart paper or transparency (social skills, cooperative group work,
Fermi problems, estimation, setting the criteria for good mathematical
recordings). Discuss a concept map (poster) with students.
Groups of 4 Æ Concept Map Activity
Students make a concept map to summarize what they have learned so far in
math class. Remind them to use the social skills they have learned.
Learning Skills/Observation/Checklist: Circulate and observe, noting the
symbols and other features students use on their concept maps. Listen for good
use of social skills. Assess each student’s contribution to the group as they
work on their concept maps.
Whole Class Æ Sharing
Learning Skills/Exhibition/Checklist: Post concept maps. Students name
some symbols that help them to remember the past week.
Choose one student from each group to tell what the Fermi problems/
estimating taught them.
Be sensitive to the
fact that in some
cultures it is
considered
disrespectful to
maintain direct eye
contact with
another person.
Post TIP 2 or place
it on a
transparency.
Social skills should
be left posted to
remind students of
the expectations
when working in
groups.
Students can be
more successful
making concept
maps if they have
time to talk about
and process their
memories.
Concept maps
may be collected
and commented
on for group work
and effort.
Home Activity or Further Classroom Consolidation
Reflect on your group records in your math journal looking for strengths and
improvements. Write a letter to the teacher to explain what you learned this
week in math class and your goals for the term. Explain how you learn
mathematics best.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 21
6.1: What is the Social Skill?
Name:
Date:
Look at the examples below and decide which social skill is being demonstrated by the positive
examples:
Positive Examples
Negative Examples
Looks like …
♦ Eye contact with a slight shake of the head
♦ Listening to someone’s entire idea before
speaking
♦ Smiling at the speaker
♦ Puzzled or questioning look
Sounds like …
♦ I understand what you are thinking but
have you ever considered ….?
♦ Your idea is important but have you
thought about …?
♦ I think I understand what you are saying
but have you thought about …?
♦ Calm, quiet, controlled voices
Looks like …
Listener interrupts the speaker
Shaking the head rapidly back and forth
♦ Impatiently challenging the speaker
♦ Rapidly tapping the fingers
♦ Angry challenging look
♦
♦
Sounds like …
No way! I disagree, my idea is much better
than that.
♦ So what. Who cares? I have a different
idea.
♦ I totally disagree with everything you just
said.
♦ Loud, angry, or aggressive voices
♦
Look at the examples below and decide whether each is a positive or negative example for the
social skill. Discuss why this skill is important for successful learning and for getting along in
your teams:
Examples
1. That’s what you think…My idea is much better!
2. Something else to consider is _______________, which is a little different than your idea.
3. My idea is fine. I’m not changing anything.
4. Is there anything we can add to _______________’s idea?
5. You think only your ideas are important. What about mine?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 22
Lesson Outline: Days 7 - 10
Grade 8
BIG PICTURE
Students will:
• appreciate that numbers can appear in numerous written and numerical forms;
• represent whole numbers in expanded form using powers and scientific notation;
• represent whole numbers using words and expanded notation;
• apply rules for multiplying and dividing by powers of 10 to mentally solve problems;
• develop rules for multiplying and dividing by powers of 10;
• appreciate the need to find square roots;
• use calculators to estimate the square root of a number.
Day
Lesson Title
Description
7
The Value of Place • Review place value and correct reading of large and
Value
small numbers from 0.001 to 999 999 999.
• Represent whole numbers using word form, expanded
form, and expanded form using powers and scientific
notation.
8
Powering Up with
• Observe patterns for multiplying and dividing by
Powers of 10
powers of 10. Develop a set of rules for multiplying
and dividing by powers of 10.
Expectations
8m10, 8m11
CGE 5a
8m12, 8m32,
8m36
9
Making Sense of
Squares
•
Review area of squares.
CGE 5a
8m91, 8m79
10
Finding the Root of
the Problem
•
Determine square roots of perfect and non-perfect
squares.
CGE 4b
8m24, 8m25,
8m26, 8m27
CGE 4b, 5a
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 23
Day 7: The Value of Place Value
Grade 8
Description
place value and correct reading of large and small numbers from
0.001 to 999 999 999.
• Represent whole numbers using word form, expanded form, and expanded
form using powers and scientific notation.
• Review
Materials
• place value mats
• centicubes
• BLM 7.1, 7.2
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Concept Practice
Whole Class Æ Connection to Jobs
Pose these questions: What jobs involve the use of very large numbers? What
is being measured by these very large numbers?
Have students volunteer their ideas and record the answers on the board.
Whole Class Æ Applying Concepts
Place a transparency of BLM 7.1 on the overhead projector and hand out
student copies. Prompt students to name the columns with place values from
hundred millions on the left to thousandths on the right of the decimal column.
Fill in BLM 7.1 on the transparency and ensure each student has it completed
correctly.
Write a number on BLM 7.1 and say the number correctly as it is being written
down, e.g., 2.47 - two and forty-seven hundredths.
Curriculum Expectations/Question & Answer/Mental Note: Repeat for
more numbers, prompting different students to correctly read the new number.
Write one of the numbers from the chart on the board and ask, In how many
different forms can you represent the number 574?
Form of the number Æ Representations
Standard form Æ 574
Word form
Æ five hundred seventy-four
Expanded form Æ 5 × 100 + 7 × 10 + 4 × 1
Students will be familiar with these three forms from previous grades. To
introduce another form which expresses the expanded form with powers, ask:
How can we represent 100 as a power of base 10? 100 = 10 × 10 = 102
Represent as a power of base 10: 100 000 = 10 × 10 × 10 × 10 × 10 = 105
Ask: How do you determine the exponent of the base 10?
The final form of 574 in expanded form with powers Æ 5 × 102 + 7 × 101 + 4,
and in scientific notation 5.74 × 102
Whole Class Æ Demonstrate Understanding
To reinforce understanding of the different forms, complete two or three
exercises with students.
Individual Æ Practise
Students complete BLM 7.2 individually.
Among the
possibilities are
jobs involving
money, cell or
bacteria counts,
outer space, and
astronomy.
Home Activity and Further Classroom Consolidation
Order all the numbers on worksheet 7.2 from smallest to largest.
In your math journal, under the heading Using Large Numbers, describe a
context where large numbers are used, where you obtained this information,
and express a number used in this situation in four different ways.
In scientific
notation, a number
looks like a
number with one
non-zero digit to
the left of the
decimal times a
power of 10, e.g.,
1.23 × 10, 9.6 ×
3
4
10 , 5.001 × 10
Explain in your math journal what you think the exponent of base 10 would be
1
1
for the number 1 or 100
= 0.01 or 10000
= 0.0001.
Pose this question
for students who
need a challenge.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Students may
need to use place
value mats and
base 10 blocks.
When reading
numbers aloud it is
important to
remember ‘and’ is
used to express a
decimal point. e.g.,
sixteen and eight
tenths - 16.8,
fourteen and nine
thousandths 14.009
No ‘and’ is used in
one thousand forty
– 1040
Students are not
expected to work
with zero or
negative
exponents until
Grade 9.
Page 24
7.1: Place Value Chart
Name:
Date:
Sample Numbers
Place Value
Hundred millions
Ten thousands
3
5
2
9
Units
.
Decimal
6
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 25
7.2: Place Value and Representing Numbers
Name:
Date:
Complete the charts.
Standard Form
894
87.65
1 000 326
five hundred
million and four
tenths
forty-seven and six
tenths
seventy-eight
million
Word Form
Expanded Form
Expanded Form with Powers
Scientific Notation
Standard Form
Word Form
Expanded Form
Expanded Form with Powers
Scientific Notation
Standard Form
Word Form
seven thousandths
Expanded Form
6 × 102 + 8
Expanded Form with Powers
Scientific Notation
TIPS: Section 3 – Grade 8
6.054 × 103
© Queen’s Printer for Ontario, 2003
Page 26
Day 8: Powering Up with Powers of 10
Description
• Observe patterns for multiplying and dividing by powers of 10.
• Develop a set of rules for multiplying and dividing by powers of 10.
Minds On…
Action!
Consolidate
Debrief
Application
Reflection
Concept Practice
Skill Drill
Grade 8
Materials
• place value
mats
• centicubes,
algeblocks
• BLM 8.1, 8.2
Assessment
Opportunities
Whole Class Æ Review Concepts
While taking up student responses to BLM 7.2, assess whether students require
further opportunities to learn and practise or whether they are ready for a quiz.
Discuss how to get 10% of a number mentally and review metric units (metric
staircase).
Curriculum Expectations/Question & Answer/Mental Note: Ask students to
give the answers to simple questions, without using a calculator,
e.g., 246 × 100, 246 ÷ 100, 246 ÷ 0.01.
Explain why it would be useful to develop rules for multiplying and dividing
by powers of 10. Multiplying and dividing by powers of 10 are often part of
calculations related to the metric system and finance.
Think/Pair/Share Æ Building Algorithmic Skills
Students individually complete Part A of BLM 8.1. In pairs, they check results
and look for patterns to complete the rules in Part B. Students test the rules on
new examples and then check their answers with calculators. Have two or more
groups write their rules on chart paper.
Whole Class Æ Summarizing
As a class, agree on the best wording for the rules in Part B, BLM 8.1.
The rules can be summarized as follows:
1. When multiplying by 10, 100, 1000, etc., the number gets larger, so move
the decimal point the same number of places as there are zeros in the
power to the right.
2. When multiplying by 0.1, 0.01, 0.001, etc. the number gets smaller, so
move the decimal point the same number of places as there are digits to the
right of the decimal point to the left.
3. When dividing by 10, 100, 1000, etc., the number gets smaller, so move
the decimal point the same number of places as there are zeros in the
power to the left.
4. When dividing by 0.1, 0.01, 0.001, etc., the number gets larger, so move
the decimal point the same number of places as there are digits to the right
of the decimal point to the right.
Assign appropriate concept practice exercises from textbook – look for context
questions.
Home Activity or Further Classroom Consolidation
Complete the exercises assigned from your textbook.
Answer one of these questions in your math journal:
• Taxes on purchases in Ontario are 15% (7% GST and 8% PST). To do a
quick calculation of the tax owing on a purchase, you can mentally take 10%
of the total and then half of that and add them together. Explain what the tax
would be on a purchase of $186.00 using this method.
• In Grade 6, you converted one metric unit to another, e.g., metres to
centimetres or grams to kilograms. Explain how to change from one metric
unit to another, e.g., 80 m to cm, without using a calculator.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Negative
exponents are not
introduced until
Grade 9.
Students who are
having difficulty
may use place
value charts to see
the direction the
decimal moves.
Rules may have to
be modified until
they are accurate.
Demonstrate how
division by a
number less than 1
produces an
answer greater
than the dividend.
The math journal
entry can be
assessed for
Curriculum
Expectation/
Journal/Rubric.
[See 8.2
Assessment Tool]
Page 27
8.1: Finding a Pattern
Multiplying and Dividing using Powers of 10
Name:
Date:
A) Complete the chart.
Number
1. a) 35.2
Instruction
Multiply by 10
b) 35.2
Multiply by 100
c) 35.2
Multiply by 1 000
2. a) 35.2
Multiply by 0.01
c) 35.2
Multiply by 0.001
352
Divide by 10
b) 35.2
Divide by 100
c) 35.2
Divide by 1 000
4. a) 35.2
35.2 × 10
= 35.2 × 101
35.2 × 100
= 35.2 × 102
Result
Multiply by 0.1
b) 35.2
3. a) 35.2
Calculation
Divide by 0.1
b) 35.2
Divide by 0.01
c) 35.2
Divide by 0.001
B) Look for patterns and complete the rules:
1. When multiplying by 10, 100, 1 000, etc., the number gets ______________ so move the
decimal point __________________________________ to the ______________ .
2. When multiplying by 0.1, 0.01, 0.001, etc., the number gets ______________ so move the
decimal point __________________________________ to the ______________ .
3. When dividing by 10, 100, 1 000, etc., the number gets ______________ so move the
decimal point __________________________________ to the ______________ .
4. When dividing by 0.1, 0.01, 0.001, etc., the number gets ______________ so move the
decimal point __________________________________ to the ______________ .
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 28
8.2 Assessment Tool: Journal Entry
Name:
Date:
Journal Entry Topic:
Mathematical
Process
(Category)
Criteria
Below
Level 1
Level 1
Level 2
Level 3
Level 4
Making
Connections
(Understanding of
Concepts)
- metric
conversion
Depth of
understanding
- little or no
evidence
- superficial
depth
- moderate
depth
- substantial
- insightful
Communicating
(Communication)
- explains metric
conversion
Clarity
- unclearly
- with limited
clarity
- with some
clarity
- clearly
- precisely
- uses
mathematical
language,
symbols, forms,
and conventions
Use of
conventions
(accurately,
effectively,
and fluently)
- demonstrates
an
undeveloped
use of
conventions
- demonstrates
minimal skill
in the use of
conventions
- demonstrates
moderate skill
in the use of
conventions
- demonstrates
considerable
skill in the use
of conventions
- demonstrates
a high degree
of skill in the
use of
conventions
Comments:
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 29
Day 9: Making Sense of Squares
Grade 8
Materials
• 5 × 5 geoboards
• overhead
geoboard
• BLM 9.1, 9.2,
9.3
• Ministry
Exemplar Task
(2002)
Assessment
Opportunities
Description
• Review area of squares.
Minds On…
Action!
Consolidate
Debrief
Whole Class Æ Orienting Students to an Activity
Look into a box or an envelope and say, I’m looking at a quadrilateral (foursided polygon). I think it is a square. How do I know if it really is a square?
(See Grade 8 Exemplar.)
Use on a transparency, of 3 × 3 dot grid, BLM 9.1. How many different-sized
squares can be drawn? Have students draw the different squares on the
transparency. Demonstrate the overlapping nature of 2 × 2 squares on a 4 × 4
dot grid.
Pairs Æ Shared Exploration
Working in pairs, students determine all the different-sized squares they can
construct on a 5 × 5 geoboard. Students record their findings on BLM 9.2.
Students who finish early can explore the total number of squares that can be
generated on a 5 × 5 grid. This includes counting all squares with the same
area. For example, there are sixteen 1 × 1 squares. Students who require
scaffolding could work with a 4 × 4 grid first, then move
to a 5 × 5 grid.
Challenge students to confirm that the shapes constructed
using diagonal sides are squares. Further challenge them to
make the confirmation in several ways (TIP 4).
This lesson is
based on one of
the Grade 8
Ministry Exemplar
Tasks (2002)
Some students
may not think of
creating squares
using diagonal
sides.
Whole Class Æ Summarizing
Learning Skill (class participation, initiative)/Presentation/Checklist: Have
some students use the overhead BLM 9.2 to show the different-sized squares
generated on the 5 × 5 geoboard.
Individual Æ Making Connections
Students begin BLM 9.3. This exercise helps increase student familiarity with
perfect square numbers. Ensure that all squares are counted and the connection
to perfect square numbers is made.
Application
Concept Practice
Skill Drill
Home Activity or Further Classroom Consolidation
Complete worksheet 9.3.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 30
9.1: Overhead Grid Dot Paper – Teacher
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TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 31
9.2: 5 × 5 Grids
Name:
Date:
Use the grids below to record the results of your work.
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TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 32
9.3: Mmmmmm Synonym Squares
Name:
Date:
1. How many squares in total can you find in each of the following?
a)
b)
c)
d)
2. Describe at least one pattern you observe to define the relationship between the number of
the term and the total number of squares.
3. Explain how you would find the total number of squares for a 10 × 10 grid?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 33
9.3: Mmmmmm Synonym Squares (continued)
(Answers)
1.
a)
1
b)
4 small + 1 larger = 5 squares
c)
9 small + 4 medium + 1 large = 14 squares.
Encourage students to trace the different-sized squares in their
notes.
d)
16 + 9 + 4 + 1 = 30.
There are 16 1 ×1 squares, plus 9 2 x 2 squares,
plus 4 3 × 3 squares, plus 1 4 × 4 square in a 4 × 4 grid.
2. The total number of squares for a term is the sum of the perfect square numbers
12 + 22 + 32 + …up to and including the term number.
3. There will be:
102 1 × 1 squares, plus 92 2 × 2 squares, plus 82 3 × 3 squares, plus… 12 10 × 10 square or
102 + 92 + 82 + … + 12 = 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 385 squares.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 34
Day 10: Finding the Root of the Problem
Grade 8
Materials
• 5 × 5 geoboards
• BLM 9.3, 10.1
• calculators
Description
• Determine square roots of perfect and non-perfect squares.
Assessment
Opportunities
Minds On…
Action!
Whole Class Æ Connect to Previous Lesson
Emphasize the patterns between perfect square numbers and square shapes as
BLM 9.3 is taken up.
Students write the list of perfect square numbers from 1 to 225 (12 to 152).
They may need their calculators for 132, 142, and 152.
Learning Skills/Observation/Checklist: Have selected students share what
they found in the exploration on BLM 9.3.
Curriculum Expectations/Demonstration/Marking Scheme: Collect and
mark BLM 9.3.
Pairs Æ Shared Exploration
Revisit the different squares generated on the 5 × 5 geoboard. Challenge
students to calculate the area of as many squares as they can. Students record
their findings on additional copies of BLM 9.2.
Students who finish early can find four different ways of
calculating the area of the square shown (TIP 5):
students can use other areas or measurements to
determine the area of the square.
Consolidate
Debrief
Whole Class Æ Demonstrate Understanding, Extend Thinking
Have some students share how they determined the area of the different-sized
squares generated on the 5 × 5 geoboard, using a transparency.
Challenge students to find the perimeter of as many of the squares as they can.
Students may find it more difficult to find the perimeter of squares that do not
have an area that is represented by a perfect square. This establishes the need to
learn about square roots. Define the square root of a number. Start with perfect
square numbers and lead to non-perfect square numbers.
Try not to introduce the term square root too soon. Students look for a number
that, when multiplied by itself, gives them ten.
Challenge: Q includes the set of fractions like
1
2
, 32 , 52 , 98
56 , etc.
Q
includes
numbers like 2 , 3 , 5 , 6 , 7 , 8 , 10 , K , and π. Compare the decimal
forms of numbers in Q and numbers in Q .
Discuss the Q and Q notations in connection to symbols like ≠ and to negative
prefixes.
Pairs Æ Introduce Concept of Irrational Numbers
Students play Root Magnet, using BLM 10.1 and determine which target
numbers are not perfect squares.
Application
Reflection
Concept Practice
Familiarity with
perfect square
numbers
significantly helps
when exploring
other applications
that include the
Pythagorean
relationship.
The square root of
a non-perfect
square whole
number is an
Irrational Number.
It can never be
expressed as a
fraction, and is a
decimal that never
ends or repeats
(like pi,
3.14159265…).
Q (for quotients)
represents the
rational numbers.
Q represents the
irrational numbers.
Home Activity or Further Classroom Consolidation
In your math journal describe the steps you would use to approximate the
square root of non-perfect squares.
For more practice with perfect squares, non-prefect squares, and square roots,
complete the assigned exercises from your math textbook.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 35
10.1: Root Magnet
Names:
Date:
Getting Ready
Work in pairs. You need a calculator for this activity.
How to Play
Each player selects three target numbers from 1 to 1000 and enters them on their opponent’s
score sheet. Each player then estimates to find a number that must be multiplied by itself to get
the target number. Make an estimate for all three target numbers. To score, multiply the
estimate by itself then subtract from the target number. The score is the total of all the
differences. The player with the lowest score is declared the Root Magnet!
Example:
Target
12
100
2
Player A
Estimate
6
10
1.3
Estimate2
36
100
1.69
Total
Score
24
0
0.31
24.31
Let’s Play
Round One
Target
Estimate
Player:
Round Two
Target
Estimate
Player:
Round Three
Target
Estimate
Player:
TIPS: Section 3 – Grade 8
Estimate2
Score
Total
Estimate2
Total
Estimate
Player:
Score
Total
Estimate2
Target
Target
Target
Score
Total
Estimate
Player:
Score
Estimate2
Estimate2
Score
Total
Estimate
Player:
© Queen’s Printer for Ontario, 2003
Estimate2
Score
Total
Page 36
Lesson Outline: Days 11 - 16
Grade 8
BIG PICTURE
Students will:
• explore a real-life problem to appreciate the need to learn more mathematics, specifically the Pythagorean
relationship;
• represent right-angled triangles in different orientations on a geoboard;
• investigate the relationship between the areas of squares constructed along the sides of a right-angled
triangle;
• test a conjecture as to whether or not the Pythagorean relationship applies to triangles other than right-angled
triangles;
• consolidate understanding of the Pythagorean relationship.
Day Lesson Title
Description
Expectations
11
Will it Fit?
• Set the stage for connecting the Pythagorean
8m9, 8m59
relationship to problem solving.
CGE 2c
12
Geoboards and the
• Develop the Pythagorean relationship.
8m26, 8m65, 8m73,
Pythagorean Relationship
8m91, 8m94,
13
14
15
16
Investigating the
Pythagorean Relationship
using The Geometer’s
Sketchpad ®
Applying the
Pythagorean Relationship
Bringing It Together
What’s the Area?
TIPS: Section 3 – Grade 8
• Use
®
The Geometer’s Sketchpad to
investigate the Pythagorean relationship.
• Apply the Pythagorean relationship.
• Apply
the Pythagorean relationship.
• Apply
knowledge of various concepts to
solve a variety of problems in small groups
to help consolidate learning.
• Apply
knowledge of the Pythagorean
relationship, square roots, perfect squares,
and geometric properties to solve an area
problem in a variety of ways.
© Queen’s Printer for Ontario, 2003
CGE 5a, 4b
8m31, 8m59, 8m64,
8m70
CGE 5d
8m31, 8m59, 8m70,
8m60
CGE 2c, 2d
Expectations cited in
prior lessons
CGE 5e, 5a
8m59, 8m65, 8m70
CGE 3c
Page 37
Day 11: Will it Fit?
Grade 8
Description
the stage for connecting the Pythagorean relationship to problem solving.
• Set
Materials
• chart paper or
mural paper
• markers
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Exploration
Reflection
Whole Group Æ Introducing a Problem
Begin the lesson with a problem that engages prior knowledge and establishes a
practical motivation for learning the Pythagorean relationship:
You have a piece of plywood with dimensions that are 1.2 m × 2.4 m. You
would like to pass it through a window that is 1 m by ¾ m. Can you pass the
plywood through the window? Explain your reasoning.
To enhance student interest, consider entering the room in role as a customer at
Pat the Builder’s Build-all. Approach a student in the room as if he or she is an
employee and present them with your dilemma. You really don’t want to go to
all the trouble of having to return the piece of plywood, so you need a
definitive answer for whether the plywood can pass through the window.
Small Groups Æ Exploring
Students work in groups to think creatively and suggest a response to the
problem. The focus is to motivate students in learning how to compute the
length of the hypotenuse of a triangle.
Curriculum Expectations/Observation/Question & Answer/Checkbric:
Students demonstrate the strategies they choose to solve the problem and
justify and communicate their reasoning.
Record the solution options on chart paper or mural paper to be shared during
the final portion of the lesson. Students may suggest scale models or other
viable solutions to this problem. Evaluate their plans on practicality as well as
the potential for the plan to give a definitive “yes” or “no” answer.
Bring in visuals –
possibly a scale
model or a picture
to help focus
student attention
on the problem.
A checkbric
identifies the
criteria by which
work will be
assessed, but
does not contain
descriptors of
levels. See 16.2
Assessment Tool
for an example.
Whole Class Æ Presentation
After each small group presentation of their strategy, guide the class in
concluding that mathematics can provide an efficient, accurate solution to this
life problem, and that a discovery of this mathematical relationship will be the
focus over the next few days. Use a graphic organizer to illustrate yes/no
examples of right triangles in various orientations.
Home Activity or Further Classroom Consolidation
Interview someone at home. Have they ever had trouble fitting an item through
a doorway or window? Find out the details. What happened? Was mathematics
involved in any part of the process, or should it have been? Come prepared to
share your findings.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 38
Day 12: Geoboards and the Pythagorean Relationship
Grade 8
Materials
• geoboards
• BLM 12.1, 12.2,
12.3
Description
• Develop the Pythagorean relationship.
Assessment
Opportunities
Minds On…
Action!
Consolidate
Debrief
Whole Class Æ Connecting to previous lesson and orientating
students to an activity
Ask students to recall the problem posed during the previous class and to share
their findings. If the connection is not made to putting the piece of plywood on
a diagonal, introduce the idea of diagonal by drawing a rectangle to represent a
window and making a right-angled triangle using one corner.
Pairs Æ Exploration
Give students BLM 12.2 and go over the instructions with them. Mixed-ability
student pairs pick a card from BLM 12.1. Students do their constructions or
take their measurements and enter the data, select another card, and repeat the
process. In the end, there should be a minimum of fifteen entries in the table,
with more than one pair working on each entry.
Curriculum Expectations/Question & Answer/Mental Note: Circulate
among the groups, observing student strategies for calculating the area of the
square on the diagonal side (TIP 5).
Use probing questions to prompt student thinking as needed:
• What are the largest and smallest right-angled triangles that can be
constructed on a 5 × 5 geoboard?
• Which of the geoboard triangles have two sides that are the same length?
• Are you able to make a right-angled triangle with three equal sides? Explain.
• Where is the largest square in relation to the triangle’s 90-degree angle?
Challenge some students to confirm that the shapes on the diagonals are indeed
squares (TIP 4).
Whole Class Æ Making Connections and Summarizing
Students examine the data, as displayed on BLM 12.2, and identify patterns.
Provide sufficient time before accepting any student answers so that each
student has an opportunity to participate in the thought process. Be prepared to
discuss any rows on BLM 12.2 where the answers do not fit the pattern due to
student error. (For each row, Value of Column 3 + Value of Column 4 should =
Value of Column 6.) Guide students to describe the Pythagorean relationship in
words. Debriefing should include:
2
2
2
• the articulation of the a + b = c relationship which refers to the relationship
of the areas of squares built on the sides of a right-angled triangle
• the convention of labelling the sides containing the right angle “a” and “b”
and the hypotenuse “c” but reinforcing that any variable can be used to
indicate the sides of the triangle.
Home Activity or Further Classroom Consolidation
You are a mathematical advice column writer responding to this letter:
Application
Concept Practice
Reflection
“Dear Math Maniac,
I pride myself in being quite a well-informed Grade 8 math student, but when I was
watching the Wizard of Oz as I was babysitting, I heard the Scarecrow make a
mathematical statement that confused me. It was: Once the Scarecrow received his
“brains” he immediately tried to impress his friends by reciting the following
mathematical equation, “The sum of the square roots of any two sides on an
isosceles triangle is equal to the square root of the remaining side.” Was the
Scarecrow correct? What did he mean?
Provide a large
sheet of paper in
the centre of a
table as a
Placemat.
Students write
their conjectures in
their own space on
this Placemat
without talking.
Once each student
has written a
conjecture, the
group decides on
the best conjecture
and how to word it.
Source:
http://www.geocitie
s.com/Hollywood/
Hills/6396/ozmath.
htm
Complete worksheet 12.3.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 39
12.1: Pairs Investigation Cards
Triangle 1
Triangle 2
Triangle 3
Side a is 1 cm
Side b is 1 cm
Side a is 2 cm
Side b is 2 cm
Side a is 3 cm
Side b is 4 cm
Triangle 4
Triangle 5
Side a is 6 cm
Side b is 8 cm
Side a is 5 cm
Side b is 12 cm
Triangle 7
Triangle 8
Triangle 9
Side a is 1 cm
Side b is 3 cm
Side a is 1 cm
Side b is 4 cm
Side a is 2 cm
Side b is 3 cm
Triangle 10
Triangle 11
Triangle 12
Side a is 4 cm
Side b is 5 cm
Side a is 3 cm
Side b is 5 cm
Side a is 3 cm
Side b is 6 cm
Triangle 13
Triangle 14
Triangle 15
Side a is 5 cm
Side b is 6cm
Side a is 4 cm
Side b is 7 cm
Side a is 2 cm
Side b is 5 cm
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Triangle 6
Side a is 3 cm
Side b is 3 cm
Page 40
12.2: Squares on Sides of a Right-angled Triangle
Names:
Date:
1. Using a geoboard and coloured elastics or square dot paper, or both, construct the right
triangle described on your card. Remember the two sides of the triangle are at 90° to each
other. Construct the third side (the hypotenuse).
2. Construct a square on each side of the triangle, using each side length.
3. Complete the row that corresponds to your triangle number on the chart (first 5 columns
only).
4. Use the area of the square on the hypotenuse to determine the length of side “c” (column 6).
Check with a ruler.
5. Add this data to the class chart.
Triangle
#
1
2
Length of
Side “a”
Length of
Side “b”
TIPS: Section 3 – Grade 8
3
Area of
Square
on Side
“a”
4
Area of
Square
on Side
“b”
5
Length of
hypotenuse
“c”
© Queen’s Printer for Ontario, 2003
6
Area of
Square on
hypotenuse
“c”
Page 41
12.2: Squares on Sides of a Right-angled Triangle (continued)
Class Chart
Triangle
#
1
2
Length of
Side “a”
Length of
Side “b”
3
Area of
Square
on Side
“a”
4
Area of
Square
on Side
“b”
5
Length of
hypotenuse
“c”
6
Area of
Square on
hypotenuse
“c”
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 42
12.3: Social Triangles
Name:
Date:
Each square represents 0.5 m x 0.5 m.
1. You live at the location of rectangle #1. The other numbered rectangles represent the
homes of your friends. The scale is 500 m per unit length. Calculate the distance between
your home and that of each of your friend’s. Show your work in good form.
2. A new friend lives exactly five kilometres away from your home.
a) Show all possible locations for this friend’s home.
How is this set of possible locations shown on your grid?
b) Use an X to mark which of these locations is on a grid point.
How did you determine the locations?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 43
Day 13: Exploring the Pythagorean Relationship using
The Geometer’s Sketchpad®
Grade 8
Materials
• carpenter’s
triangle
• The Geometer’s
Sketchpad 4 ®
• BLM 13.1
• GSP file:
Pythagorean
Relationship
Assessment
Opportunities
Description
®
• Use The Geometer’s Sketchpad to investigate the Pythagorean relationship.
• Apply the Pythagorean relationship.
Minds On…
Whole Class Æ Connections to Careers and Posing a Question
Curriculum Expectation and Learning Skills/Portfolio/Marking Scheme
and Checklist: Collect student work on BLM 12.3 for assessment.
Use a carpenter’s triangle as an example of a tool. Explain that it is based on
the 3:4:5 Pythagorean Triple (a set of 3 whole numbers which are the lengths of
the sides of a right-angled triangle) used by carpenters to ensure that their walls
are “square.”
Using students’ entries on BLM 12.2, identify Pythagorean Triples and start a
cumulative class list to be augmented as the lessons continue.
Draw student attention to the 3:4:5 and 6:8:10 triples. Might there be a
relationship between these two? (multiples) Can we use this to generate other
triples?
Extension: Find another Pythagorean Triple besides the 3:4:5, 6:8:10,
and 5:12:13 encountered on BLM 12.2, e.g., 8:15:17
Hypothesize whether or not the Pythagorean relationship is true for all types of
triangles. How could we confirm or refute your hypothesis? What tool could
we use to test the hypothesis?
Action!
Pairs Æ Guided Exploration
Mixed-ability pairs use The Geometer’s Sketchpad® and follow instructions on
BLM 13.1 to discover that the Pythagorean relationship is unique to rightangled triangles.
Learning Skills (Co-operation with others)/Question & Answer/Checklist:
Observe students as they work through the activity.
Students who finish early can develop and explore “what if” questions they can
pose in relation to the Pythagorean relationship.
Consolidate
Debrief
Whole Class Æ Demonstrating Understanding
Use GSP file: The Pythagorean Relationship (p. 47) to consolidate student
understanding and to introduce them to different visual proofs.
Say, “Now that we understand what the Pythagorean relationship is and know
that it is unique to right-angled triangles, we want to see where it applies to real
life situations.” Return to the original problem from Lesson 11 and assign it.
Application
Concept Practice
The 3:4:5
Pythagorean Triple
was also used by
the Egyptians in
the building of
pyramids.
Home Activity or Further Classroom Consolidation
Solve the window and plywood problem from Day 11.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 44
13.1: The Pythagorean Relationship – Investigation
Construct a Right Triangle using the following steps
•
•
•
•
•
•
•
Select GRID from the overhead menu and click on “snap to grid.”
Select the POINT TOOL from the side menu to construct a point where the x-axis and y-axis
intersect on the grid.
Create another point at (0, 4) which is four steps up the vertical or y-axis.
Create a third point at (3, 0), three steps along the horizontal or x-axis.
Highlight the three points you have just constructed by holding down the shift button while
clicking on each with the SELECT TOOL.
Click on CONSTRUCT and then select “Segment.”
You should now have a right-angled triangle. Use the LABELLING TOOL to name each
vertex A, B and C respectively.
Construct a Square on each of the three sides of the triangle
•
•
•
•
•
•
•
•
•
•
Double click on a vertex (look for a “bulls eye”- this marks that point as the centre of rotation).
Select an adjacent side and the point at the end of that segment
Click on TRANSFORM, select “Rotate.”
Choose 90 degrees (or –90 degrees, depending on which side was selected).
If there is no point at the end of this line segment – select the segment, click on
CONSTRUCT, select “Point on Object”. Then, drag this point completely to the end of the
segment.
Double click on this newly created point (to mark it as the centre of rotation).
Select the segment.
Click on TRANSFORM, select “Rotate.”
Choose 90 degrees (or –90).
Repeat until a square is constructed on each of the 3 sides.
Measure the area of each square
•
•
•
•
•
•
Holding the SHIFT key down, point and click on each of the 4 vertices in clockwise or counter
clockwise order.
Click on CONSTRUCT.
Select “Polygon Interior”. Colour each square’s interior differently if you choose.
Click on “Measure.”
Select “Area.”
Repeat for each of the 3 squares.
Use the Geometer’s Sketchpad Calculator
•
•
•
•
•
•
Click on “Measure.”
Select “Calculate”
Highlight the area of the smallest square.
Click the “Add” button on the calculator.
Highlight the area of the next smallest square.
Click on “OK” (there is no = sign).
Look for a relationship between the values.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 45
13.1: The Pythagorean Relationship – Investigation (continued)
Experiment
•
•
Click and drag point any vertex other than the 90 degree one.
Examine the area values as they change. What changes? What stays the same?
• Ask some “what if” questions here and experiment.
• Talk with your partner to further clarify any relationships you notice. Consider how these
relationships might be important in mathematics. Each student writes a journal entry as a
personal interpretation of the relationship.
Journal
Students use the following prompts to write and reflect upon their learning:
“After investigating the squares on the sides of right angle triangles using Geometer’s
Sketchpad, my partner ________ and I discovered that…
We experimented with … and found that …
We also developed the following “what if” questions.
What if?
Encourage students to develop and explore “what if” questions:
“What if the triangle is not a right angled triangle? Will the relationship still hold true?”
Students can explore this question quickly and easily with GSP.
“What if a semi-circle or some other geometric figure is built on each side of the right triangle,
will the relationship still exist?
Encourage students to investigate this on GSP.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 46
The Pythagorean Relationship (GSP file)
Download this file at www.curriculum.org/occ/tips/downloads.shtml
The Pythago rean Relatio nship
Given: Righ t Trian gle ABC, ∠C = 90 ° .
Sq uare s will b e drawn on the th re e sid es.
Click on the link belo w to sho w th e sq uare s.
B
Sh ow Sq uare s of the Sid e
A
C
The Pythag orean Relationship
A rig ht an gled trian gle is sh own at righ t
with a right an gle at C.
Sh ow Pythagore an Th eo re
Re se t
drag this point
The Pythago rean Relationship
A rig ht an gled triang le is sh own at righ t
with a right an gle at A.
Fo llo w th e ste ps b elow.
1)
Sh ow Sq uare s of Side
2)
Sh ow Altitude
3)
4)
Sh ow Qu ad rilatera
Sh ow Area Measu re men
TIPS: Section 3 – Grade 8
A
B
C
© Queen’s Printer for Ontario, 2003
Page 47
Day 14: Applying the Pythagorean Relationship
Grade 8
Description
the Pythagorean relationship.
Materials
• transparencies and
• Apply
markers
• BLM 14.1
Assessment
Opportunities
Minds On…
Whole Class Æ Connecting to Previous Lessons and Introducing
Problems
Revisit the problem of putting a 1.2 m × 2.4 m piece of plywood through a
1 m × 34 m window. Ask students to indicate whether or not they thought the
plywood will fit through the window. Have volunteers present each side of the
argument (if there is a difference of opinion). Discuss and determine the correct
answer. (Yes, the plywood will just fit, as long as it is not too thick. The
diagonal of the window is 1.25 m. Some students may recognize that 3: 4 : 5 is
a multiple of 34 : 1 : 1.25.) Use this as a point of departure to lead into other
practical problems that can be solved using the Pythagorean relationship.
Curriculum Expectations/Performance Task/Marking Scheme: Collect and
assess students’ follow-up activity responses.
Action!
Small Groups Æ Developing Understanding
Student groups rotate through four different problem centres over the course of
the lesson, solving the problems on BLM 14.1. Each group is given the task of
writing up a full solution to one specific problem on overhead transparencies
for discussion.
Consolidate
Debrief
Whole Class Æ Presentations
Solutions are presented and discussed. Clarity of communication and effective
use of mathematical terminology are highlighted.
Encourage students to take careful notes during the presentations to use with
the consolidation activity and assessment.
Curriculum Expectations/Learning Skills/Presentation/Rubric/Checklist:
Assess student presentations for understanding of concepts, communication,
application of procedures, and problem-solving skills.
Application
Concept Practice
If there are four
stations and 7 or 8
groups in the
class, two groups
can work
independently at
each station.
Have two versions
of a written
solution on an
overhead
transparency to
compare at the
close of the
lesson.
Home Activity or Further Classroom Consolidation
Write complete solutions for all four problems on worksheet 14.1. These
solutions will be collected next class and assessed. Critically look at your work
to ensure that it is your best.
Create and record a practical problem that would require the use of the
Pythagorean relationship in its solution. Solve the problem on a separate sheet
of paper.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 48
14.1: Carousel of Pythagorean Problems
Station 1
Station 2
Find the distance from A to B given the
smaller square has a perimeter of 4 cm
and the larger square has an area of 16
cm2.
A ladder leans against a brick wall that
is 8 m high. The base of the ladder is
2 m away from the base of the wall and
3
of the way up the
the ladder extends
4
wall.
How long is the ladder?
Station 4
Station 3
A rocket is launched into the sky on a
windy day. The rocket has a vertical
velocity of 15 m/s. There is a strong
wind blowing east to west at 35 m/s.
The distance between the bases in a
baseball diamond is 27.4 metres. You
picked up a ground ball at first base and
you see the other team's player running
towards third base.
How far from the start point is the rocket
after 60 seconds?
TIPS: Section 3 – Grade 8
How far do you have to throw the ball to
get it from first base to third base?
© Queen’s Printer for Ontario, 2003
Page 49
Day 15: Bringing It Together
Grade 8
Description
• Apply knowledge of various concepts to solve a variety of problems in small
groups to help consolidate learning.
Materials
• scientific
calculator
• BLM 15.1
Assessment
Opportunities
Minds On…
Whole ClassÆ Brainstorm
Curriculum Expectations/Portfolio/Marking Scheme & Learning Skills/
Observation/Checklist: Collect completed solutions for worksheet 14.1 and
assess students’ responses.
Individual Æ Reflection
Ask students to brainstorm all the different mathematical concepts they have
learned during this term. Make a list on the board or chart paper. Each student
sorts these concepts in the table provided in question 1 on BLM 15.1.
Curriculum Expectations/Anecdotal/Mental Note: Collect and read the
students’ work to determine which students need assistance.
Advise students that they will be revisiting concepts done previously through
various activities and will be preparing for another assessment piece on the
Pythagorean relationship.
Action!
Pairs Æ Conferencing
Randomly number students 1 and 2. They are to find a partner with a different
number than themselves and share the practical problem they created. They
discuss and check it to ensure that their solutions are correct. If a student had
difficulty creating a problem, the partners could create one together.
Pairs Æ Problem Solving
Students work with another partner with whom to share their problem. Students
solve their partner’s problem and the partner checks the solution. Each student
is responsible for assisting classmates by providing hints and explanations of
the solution. Repeat the process with as many partners as the class period
allows.
Consolidate
Debrief
Whole Group Æ Discussion
Invite comments about what students found out in trying to write, solve, and
share the creation of problems that apply a particular mathematics relationship.
Individual Æ Self-Assessment
Learning Skills/Worksheet/Conference: Students complete questions 2
through 4 on BLM 15.1. Conference with students as they are working on their
self-assessments and working through problems individually or in small
groups.
Provide a series of practice questions from the textbook or supplementary
resource(s) to help reinforce the skills.
Concept Practice
Circulate as
students are
working to note
which students
may have had
difficulty with the
task.
Take note of any
especially
interesting
problems that
students can share
with the whole
class.
Home Activity or Further Classroom Consolidation
Complete the questions assigned from your textbook as a review of the
Pythagorean relationship.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 50
15.1: What Have I Learned?
Name:
Date:
1. Classify all the mathematical concepts and skills added to the class list
into the following categories. This will help you to focus on concepts you
had difficulty with when reviewing for future activities and assessments.
Do not yet understand
thoroughly
Consolidating
Mastered
2. What do I need to review?
3. How will I improve my understanding of the concepts listed above?
4. Problems I can redo or practise to deepen my understanding.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 51
Day 16: What’s the Area?
Grade 8
Description
• Apply knowledge of the Pythagorean relationship, square roots, perfect
squares, and geometric properties to solve an area problem in a variety of
ways.
Minds On…
Action!
Consolidate
Debrief
Reflection
Materials
• scientific
calculator
• BLM 16.1
• 16.2 Assessment
Tool
• geoboards, dot
paper
• The Geometer’s
Sketchpad®
Assessment
Opportunities
Whole Class Æ Introduction
Address any problems from the consolidation activity. Have students share
their solutions on the board, overhead transparency, or chart paper.
Introduce the problem on BLM 16.1
Groups of 4 Æ Brainstorm
Students discuss the problem and brainstorm various ways in which they can
find the area. They discuss different approaches using all materials and tools
available, but do not take notes.
Learning Skills/Observation/Mental Note: Circulate, listen to conversations,
and note contributions of students.
Individual Æ Problem Solving
Students work independently to solve the problem in as many ways as they can,
using the ideas generated in their brainstorming group session. Remind
students that they can use all materials that they suggested in their groups.
The time in groups
should be brief in
order to allow time
for students to
individually
complete the
performance task.
See the Grade 8
Exemplars 2002
for scored samples
of student work.
Curriculum Expectations/Performance Task/Checkbric/Marking Scheme:
Collect student work and assess using 16.2 Assessment Tool
Comment on the students’ strengths and next steps that they can take to
improve performance.
Home Activity or Further Classroom Consolidation
Reflect on your ability to solve the problem presented in the assessment task
and answer the following questions in your math journal:
• Did you find it easy or difficult to solve the area problem? Explain why.
• What tools helped you to solve the area problem?
• Select one of your solution methods and describe how you thought of using it
to solve the area problem.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
See Section 2 –
Developing
Perimeter and
Area for samples
of different ways
students could
approach the
problem.
When returning
graded work to
students, consider
photocopying
samples of Level 3
and 4 responses
with student
names removed.
Select and
discuss, with the
class, samples that
show a variety of
strategies.
Page 52
16.1: Area of Rectangle ABCD
Name:
Date:
Find different ways to determine the area of rectangle ABCD.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 53
16.2 Assessment Tool: Ways to Determine the Area
Name:
Date:
Mathematical
Process
(Category)
Making
Connections
(Understanding of
Concepts)
Communicating
(Communication)
Below
Level 1
Criteria
Level 1
Level 2
Level 3
Level 4
Appropriateness of
strategies selected
Completeness of
suitable strategies
Clarity of
explanation
Use of conventions
(accurate, effective,
and fluent)
Accuracy of
computations
Correctness of
recalled facts, e.g.,
area of the triangle
Knowing Facts
and Procedures
(Application)
is
1
the area of the
2
rectangle;
Pythagorean
relationship; area of
the triangle is
Use a marking scheme
1
the
2
area of the
parallelogram that
can be formed;
formulas for areas
Comments:
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 54
Lesson Outline: Days 17 - 23
Grade 8
BIG PICTURE
Students will:
• apply their knowledge of the Pythagorean relationship;
• investigate side length combinations for triangles;
• determine and compare the theoretical and experimental probability of events;
• make predictions based on probability;
• analyse “fairness “ in games of chance;
• review addition and subtraction of integers using concrete materials and drawings.
Day Lesson Title
Description
17
Rolling Number Cubes • Discover how three side lengths must be related to
for Pythagoras
create a triangle.
• Apply the Pythagorean relationship to determine if
a triangle is a right-angled triangle.
18
Experimental and
• Express probability using multiple representations.
Theoretical Probability: • Introduce concepts of theoretical and experimental
Part 1
probability.
19
20
21
22
23
Experimental and
Theoretical Probability:
Part 2
Theoretical and
Experimental
Probability of Events:
Part 3
Checkpoint
Revisiting Rolling
Number Cubes for
Pythagoras
Investigating
Probability Using
Integers
• Compare
theoretical and experimental probability.
Expectations
8m65, 8m73, 8m91,
8m94
CGE 5a
8m95, 8m96, 8m116,
8m117, 8m118,
8m120, 8m121
CGE 2c
8m95, 8m96, 8m121
• Analyse
CGE 3c
8m118, 8m119,
8m121
• Consolidate
CGE 4b
Revisit expectations
listed above
• Investigate
CGE 2b
8m118, 8m119,
8m121
a game of chance to demonstrate
understanding of theoretical and experimental
probability.
concepts of theoretical and
experimental probability.
order of outcomes on theoretical and
experimental probability.
• Review
• Link
addition and subtraction of integers.
probability to the study of integers.
CGE 2c
8m22, 8m91, 8m94,
8m118, 8m119,
8m121
CGE 5e
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 55
Day 17: Rolling Number Cubes for Pythagoras
Grade 8
Materials
• number cubes
• straws/scissors
• ruler/compasses
• The Geometer’s
Sketchpad®
• BLM 17.1
Assessment
Opportunities
Description
• Discover how three side lengths must be related to create a triangle.
• Apply the Pythagorean relationship to determine if a triangle is a right-angled
triangle.
Minds On…
Action!
Consolidate
Debrief
Reflection
Concept Practice
Whole Class Æ Connecting to Previous Lessons and Posing
Questions
Review how to construct a triangle of side lengths 2 units, 4 units, and 5 units
using ruler and compasses or using three transparencies with lengths 2 cm, 4
cm, and 5 cm or using three straws cut to the given lengths.
Pose the following questions and have students write down their hypotheses,
explaining their reasoning. Do not confirm or deny their hypotheses at this
time.
• If I roll three standard number cubes can the three numbers that appear
always be the side lengths of a triangle?
• Are we able to construct a right-angled triangle using the three lengths rolled?
Small Groups Æ Investigation
Curriculum Expectations/Observation/Mental Note: Observe students as
they work and assist groups who have trouble creating their triangles properly.
Students conduct an experiment and fill in the chart on BLM 17.1. They can
use strings or straws to form the triangles or use a ruler and compasses. Each
student completes the sheet and keeps it for a later activity. Students record
how many sets of three numbers formed a triangle and how many sets formed a
right-angled triangle.
Whole Class Æ Summarizing
Several students describe to the class what they found when trying to construct
triangles. Help them come to the conclusion that the sum of the two shorter
sides must be greater than the length of the longest side.
Ask: What relationship needs to exist among the three numbers rolled, so that a
triangle can be constructed?
Discuss the following questions:
1. Out of 30 trials, how many triangles were formed? (Since each group may
have a different answer, discuss why this happens.)
2. How can you determine if a triangle that was formed was a right-angled
triangle? How many right-angled triangles did your group get?
Students write conclusions to these questions in their notebooks or math
journals. Tell them that these will be collected and assessed.
Lead a discussion on how Pythagorean Triples connect to the experiment.
Home Activity or Further Classroom Consolidation
In your math journal, answer the following questions:
• What does it take to form a triangle given three lengths? Why?
• What is a Pythagorean Triple?
• Make a table listing all possible number cube combinations that will form a
triangle. Identify the right-angled triangles.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
This activity is part
of the Grade 8
Math Exemplar,
2002.
This investigation
could be done as a
class
demonstration
using The
Geometer’s
®
Sketchpad .
Although order
does not matter for
constructing a
triangle, in a later
lesson students
return to their math
journal entry to
determine the
probability of
forming a
particular type of
triangle. At that
time, it will be
important to notice
that a roll of 2, 2,
and 2 occurs less
frequently than a
roll of 2, 3, and 4.
Page 56
17.1: Rolling Number Cubes for Pythagoras
Experimental Results
Name:
Date:
1. Roll three number cubes 30 times. The largest number should be side c. The other numbers
are the lengths of sides a and b.
2. Construct a triangle, using one of the methods you know, and record if the roll will form a
triangle by writing Yes or No.
Length of
Side a
TIPS: Section 3 – Grade 8
Length of
Side b
Length of
Side c
Triangle
formed
Yes or No
© Queen’s Printer for Ontario, 2003
Right triangle
Yes or No
Page 57
Day 18: Experimental and Theoretical Probability: Part 1
Grade 8
Materials
• coins
• BLM 18.1, 18.2
Description
• Express probability with multiple representations.
• Introduce concepts of theoretical and experimental probability.
Assessment
Opportunities
Minds On…
Action!
Whole Class Æ Guided
Curriculum Expectations/Journal/Rubric: Collect and assess math journal
entries from the follow-up activity.
Review the meaning and terminology of the vocabulary associated with
probability situations using BLM 18.1. Students brainstorm, write, and share
their own statements, using correct terminology. Students should be prepared to
offer reasoning for their decisions. In discussion, focus on those events which
students identify as “maybe” to decide whether these events are likely or
unlikely to occur.
Pairs Æ Demonstrating Concepts
Working in pairs, students toss one coin and state the number of possible
outcomes. Each pair tosses two coins and suggests possible outcomes.
Demonstrate how a tree diagram can be used to organize outcomes. Focus
students’ attention on the representation of choices by branches in the tree.
Each pair of students creates a tree diagram for tossing three coins. As an
example, when tossing three coins, we wish to see 1 head and 2 tails. What is
the probability of this occurring? Explain how a preference (or what we want to
occur) is considered to be a favourable outcome; how probability is considered
to be the ratio of the number of favourable outcomes to the total number of
possible outcomes.
P=
Consolidate
Debrief
Reflection
Concept Practice
Skill Drill
Number of favourable outcomes
Number of possible outcomes
Pairs Æ Investigation
Each pair tosses two coins twenty times (20 is the sample size) and records
their outcomes. They compare their experimental results to the theoretical
results of 1 out of 4 for two heads or for two tails and 2 out of 4 for one head
and one tail. They discuss how changing sample size (to more or fewer than
20) would affect their results.
Whole Class Æ Student Presentation
One student from each pair presents their results for tossing two coins twenty
times. Discuss the effect of sample size on experimental outcomes. Discuss
what a probability of 0 and a probability of 1 would mean in the context of coin
tosses.
Home Activity or Further Classroom Consolidation
Complete worksheet 18.2. Devise your own simulations using spinners,
combination of coins and spinners, etc.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Words for the
Word Wall: certain
or sure,
impossible, likely
or probable,
unlikely or
improbable,
maybe, uncertain
or unsure, equally
likely and equally
unlikely.
Probability is
always a number
between zero and
one. Zero would
indicate an
impossible event.
One would indicate
a certainty.
Theoretical
Outcomes: all
outcomes that
could happen.
Experimental
Outcomes: all
outcomes that
occur when we do
an experiment.
Theoretical
outcomes can be
used to predict the
experimental
outcomes.
A comparison of
theoretical
outcomes with
experimental
results should
allow students to
draw the
conclusion that the
experimental
results will usually
be close to the
theoretical
outcomes, but it
may depend on a
variety of factors,
sample size, etc.
See Answers to
BLM 18.2.
Page 58
18.1: Talking Mathematically
Name:
Date:
Read each statement carefully. Choose from the terms to describe each event and record your
answer in the space provided:
• certain or sure
• impossible
• likely or probable
• unlikely or improbable
• maybe
• uncertain or unsure
Consider pairs of statements and determine which of them would be:
• equally likely
• equally unlikely
1. Tomorrow is Saturday.
2. I will be in Australia this afternoon.
3. It will not get dark tonight.
4. I will have pizza for supper.
5. I will be in school tomorrow.
6. It will snow in July.
7. The teacher will write on the board today.
8. January will be cold in Ontario.
9. My dog will bark.
10. I will get Level 4 on my science fair project.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 59
18.2: Investigating Probability
Name:
Date:
Solve the following problems in your notebook:
1. Keisha’s basketball team must decide on a new uniform. The team has a choice of black
shorts or gold shorts and a black, white, or gold shirt.
Use a tree diagram to show the team’s uniform choices.
a)
b)
c)
d)
What is the probability the uniform will have black shorts?
What is the probability the shirt will not be gold?
What is the probability the uniform will have the same-coloured shorts and shirt?
What is the probability the uniform will have different-coloured shorts and shirt?
2. Brit goes out for lunch to the local sub shop. He can choose white or whole wheat bread for
his sub. The filling for Brit’s submarine sandwich can be turkey, ham, veggies, roast beef, or
salami.
Use a tree diagram to show all Brit’s possible sandwich choices.
a) His choice of a single topping includes tomatoes, cheese, or mixed veggies. How does
this affect his possible sub choices?
b) If each possibility has an equal chance of selection, what is the probability that Brit will
choose a whole wheat turkey sub topped with tomatoes?
c) What is the probability of choosing a veggie sub topped with cheese?
d) What is the probability of choosing a meat sub topped with mixed veggies?
e) What is the probability of choosing any meat sub topped with mixed veggies on white
bread?
3. The faces of a cube are labelled 1, 2, 3, 4, 5, and 6. The cube is rolled once.
a)
b)
c)
d)
e)
f)
What is the probability that the number on the top of the cube will be odd?
What is the probability that the number on the top of the cube will be greater that 5?
What is the probability that the number on the top of the cube will be a multiple of 3?
What is the probability that the number on the top of the cube will be less than 1?
What is the probability that the number on the top of the cube will be a factor of 36?
What is the probability that the number on the top of the cube will be a multiple of 3
and 6?
TIPS: Section 3 – Grade 8
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Page 60
18.2: Investigating Probability
(Answers)
Question 1
a) The probability the uniform will have black shorts is
b) The probability the shirt will not be gold is
3
1
or .
6
2
4
2
or .
6
3
2
1
or .
6
3
4
2
d) The probability the uniform will have different-coloured shorts and shirt is
or .
6
3
Question 2
a) Brit has the choice of 2 breads and 5 fillings. So, he has the choice of 2 x 5 = 10
sandwiches. This can be shown using a tree diagram that first has 2 branches (one for each
of the bread types) and then 5 branches at the end of the first branches (one for each of the
fillings). This will give 10 ends to the tree. You can add 3 branches at the end of each
branch to indicate each of 3 topping choices. This gives 30 possible outcomes.
c) The probability the uniform will have the same-coloured shorts and shirt is
b) Only one of these outcomes is a whole-wheat turkey sandwich topped with tomatoes. So the
1
probability that he chooses this sandwich is
. It is only one of 30 possible sandwiches.
30
2
1
c) The probability of choosing any veggie sub topped with cheese is
or
. The student
30
15
must remember to use both the whole wheat and white bread possibility in this answer.
8
4
or
. The
d) The probability of choosing a meat sub topped with mixed veggies is
30
15
student must remember to use all possible meat selections in this answer.
4
e) The probability of choosing any meat sub topped with mixed veggies on white bread is
30
2
or
.
15
Question 3
a) There are 3 odd numbers, so the probability is
3
1
or .
6
2
1
.
6
2
1
c) There are two multiples of 3, i.e., 3 and 6, so the probability is
or .
6
3
d) There is no number less than one, so the probability is zero.
b) There is only one number greater than 5, so the probability is
5
.
6
1
There is only one number that is a multiple of both 3 and 6, i.e., 6, so the probability is .
6
e) There are 5 numbers that are factors of 36, i.e., 1, 2, 3, 4, and 6, so the probability is
f)
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 61
Day 19: Theoretical and Experimental Probability: Part 2
Description
• Compare theoretical and experimental probability.
Minds On…
Grade 8
Materials
• BLM 19.1, 19.2
• differentcoloured number
cubes
• coloured disks or
paper squares
Assessment
Opportunities
Pairs Æ Motivating Activity
Curriculum/Journal/Rubric: Collect and assess BLM 18.2 and simulation
explanations from the follow-up activity.
One student chooses a number and records the number of times he/she predicts
the number cube would have to be rolled in order for this number to appear.
The other student rolls the number cube until the partner’s number comes up.
Students switch responsibilities and repeat the activity.
Discuss the probability of an event using one numbered cube, e.g., P (rolling
a 4) = 61
• Is
there a number that occurs more frequently? (No).
did your results compare to your predictions?
• Did your results surprise you?
• Which sum is the most frequent when 2 number cubes are rolled? (7, six
combinations)
• How
Action!
Consolidate
Debrief
Whole Class Æ Connecting Concepts
Demonstrate the activity (BLM 19.1).
Pairs Æ Hypothesizing and Exploring
Using investigation techniques and BLM 19.1, students predict, record, and
analyse their results, using two number cubes. Students switch roles and
continue the experiment until all squares on the board have at least one marker
on them. Demonstrate how to fill in the recording chart (BLM 19.2). In their
pairs, students complete the recording charts.
Whole Class Æ Communicating Understanding
Lead a whole class discussion to find the theoretical probability of covering a
space when there are 36 uncovered spaces (36 out of 36 or 1), 12 uncovered
spaces (12 out of 36 or 31 ), 1 uncovered space (1 out of 36 or 361 ).
Learning Skills (class participation)/Question & Answer/Checklist:
Discuss the results from BLM 19.1 using students’ answers to questions 3 to 5.
Discuss the results of the tally charts on BLM 19.2. Relate the results to
theoretical and experimental probability. Which columns represent these
probabilities?
Probability = number of favourable outcomes
number of possible outcomes
Theoretical Probability = the predicted probability of an event
Experimental Probability = the probability of an event based on actual trials
from experiments.
Application
Concept Practice
Reflection
Encourage the use
of likely, unlikely,
probable, and
possible.
Students’
vocabulary should
be moving from
‘luck’ towards
theoretical
probability terms.
Probability of a 7 is
6 out of 36 or 1 .
6
Probability of each
of 2 and 12 is 1 out
of 36 or 1 .
36
Remind students
that experimental
probabilities would
be closer to the
theoretical
probabilities if the
sample space
were larger.
Home Activity or Further Classroom Consolidation
Solve this problem as an entry in your math journal:
In a game, players are asked to choose 5 numbers from 1 - 25. The numbers
are drawn at random. You choose 1, 16, 18, 24, and 25. Your friend chooses
the numbers 1, 2, 3, 4, and 5. Who do you think has a better chance of
winning the game? Explain.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 62
19.1: Number Cube Game
Names:
Date:
Mark 12 spaces on the game board and predict the number of rolls it will take to fill the 12
spaces. One partner rolls the number cubes; the other places markers until the 12 spaces are
full. Compare your prediction with your results. More than one marker may be on a space.
1. Predict how many rolls it will take you to cover each space on the board with at least one
marker.
Our prediction is _________________________ .
2. Working in pairs, one player rolls the cubes and the other player places a marker on the
corresponding board space for that roll. If a combination is rolled that has already been
recorded on the board, place another marker on top of the marker(s) that are already on that
space.
Colour: ______
1
2
3
4
5
6
1
2
Colour:
______
3
4
5
6
3. When every space is filled with at least one marker, count the markers to find your total
number of rolls.
Our total number of rolls ________________________.
4. Compare this total to your prediction. Why are they different?
5. Are some numbers “luckier” than others?
TIPS: Section 3 – Grade 8
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Page 63
19.2: Recording Chart
Name:
Date:
Game 1
Possible
total of
two
coloured
number
cubes
2
Number of
combinations
that could
yield that
total
Theoretical
probability
of that total
(out of 36)
Number
of rolls
that did
yield this
total
Total
number of
rolls in the
experiment
Experimental
Fraction of
total number
of rolls
Experimental
Percent of
total number
of rolls
3
4
5
6
7
8
9
10
11
12
Play the game again.
Game 2
Possible total of
two coloured
number cubes
2
Number of rolls
that did yield this
total
Total number of
rolls in the
experiment
Experimental
Fraction of total
number of rolls
Experimental
Percent of total
number of rolls
3
4
5
6
7
8
9
10
11
12
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 64
Day 20: Theoretical and Experimental Probability of Events: Part 3
Grade 8
Materials
• disks/tiles/cubes
(two colours)
• paper bags
• BLM 20.1
Assessment
Opportunities
Description
• Analyse a game of chance to demonstrate understanding of theoretical and
experimental probability.
Minds On…
Whole Class Æ Reflecting on Prior Learning and Orientating
students to Activity
Recall the concepts of theoretical and experimental probability discussed on
Day 19. Provide each pair with three green and three red tiles/cubes/disks (or
any two colours) and a paper bag. Introduce the game Green is a Go by having
students read BLM 20.1. Ensure that students understand how the game is
played.
Action!
Pairs Æ Investigation
Students play the game and each student completes all the questions. Working
with a partner, students consider changing the probability of the outcome. How
can the rules be changed in order to make the theoretical probability of winning
a 1 in 4 chance? (BLM 20.1, Answers).
Curriculum Expectations/Question and Answer/Mental Note: Listen to
pairs’ discussions, making mental notes of all of the ideas that need to be
discussed during whole class consolidation and debriefing.
Consolidate
Debrief
Whole Class Æ Demonstrate Understanding and Extend Thinking
Discuss the students’ answers to BLM 20.1 Green is a Go. As a class, decide
on the rules for a new game that will change the theoretical probability of
winning to 1 in 4.
Pairs Æ Game
In pairs, students conduct the new investigation. How does the experimental
probability compare with the theoretical probability the class discussed? (BLM
20.1, Answers.)
Encourage students to review the past few days’ work, in preparation for an
assessment (Day 22).
Concept Practice
Exploration
Reflection
Students may refer
to the results of
their coin toss
simulation.
Home Activity or Further Classroom Consolidation
Suppose you play the game with 3 green, 3 red, and 3 yellow tiles. Write a
summary in your math journal explaining how to find the theoretical
probability of drawing 2 green tiles from the bag. If you were to play the game
40 times, what result would you expect? Suggest possible reasons to support
your prediction.
TIPS: Section 3 – Grade 8
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Page 65
20.1: Green is a Go
Names:
Date:
With a partner, play a simple game involving six tiles in a bag, e.g., three red and three green
tiles. Take two tiles from the bag during your “turn.”
Rules
You may not look in the bag. Draw one tile from the bag and place it on the table. Draw a
second tile from the bag and place it on the table. Return the tiles to the bag.
You win if the two tiles drawn during your turn are both green.
Predict the number of wins if you play the game 40 times. Record and explain your prediction.
Play the Game
1. Take turns drawing two tiles from the bag, following the rules above. Record your wins and
losses on the tally chart. Continue this until you have played a total of 40 times.
Green, Green (win)
Red, Red (loss)
Red, Green (loss)
Totals
2. After you have played 40 times, use your results to find the experimental probability of
winning. (Remember that probability is the number of wins divided by the total number of
times the game was played.)
3. How does this compare with your predictions? Explain.
4. Find the theoretical probability of winning. (Hint: Use a tree diagram to show all possible
draws).
5. Write a paragraph to compare the theoretical probability you just calculated to the
experimental probability you found earlier. Are these results different or the same? Why do
you think they are the same/different?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 66
20.1: Green is a Go
(Answers)
Many students may predict 20 wins, thinking there are 2 possible outcomes, i.e., 2 green or not
2 green. Those with a bit more knowledge will predict 10 wins, basing their prediction on the
possibility of 2 heads resulting from the tossing of 2 coins. This would lead to thinking that the
possible outcomes are gg rr gr rg.
The table below shows all possible outcomes with tiles labelled g1, g2, g3, r1, r2, and r3.
1st pick / 2nd pick
g1
g2
g3
r1
r2
r3
n/a
win
win
loss
loss
loss
g1
win
n/a
win
loss
loss
loss
g2
win
win
n/a
loss
loss
loss
g3
loss
loss
loss
n/a
loss
loss
r1
loss
loss
loss
loss
n/a
loss
r2
loss
loss
loss
loss
loss
n/a
r3
Students may draw a tree diagram or list all possibilities. Students may show more or less
organization in their analysis of the outcomes, depending on their level of understanding.
There are 6 wins and 24 losses. Wins + losses = 30 (all possible outcomes).
6
1
Number of favourable outcomes
=
=
Probability =
Number of possible outcomes
30
5
Post-activity discussion:
Students may suggest different ideas to change the game to get a 1 in 4 chance of winning.
They may suggest rule changes or equipment changes. Each suggestion can lead to a rich
discussion or a new experiment to test whether it will produce the desired results and why it
does or does not. Some students may suggest placing only two of each colour in the bag.
1
However, a table of possibilities will show that this change leads to a probability of
for
6
winning. Based on this result and the emerging pattern of 2 of each colour yielding a probability
1
1
of winning of , and 3 of each colour yielding a probability of , students may suggest 4 of
6
5
3
.
each colour. However, this change of equipment yields a probability of
14
A suggestion that does not change the equipment for the game but strictly the rules (method of
play) is to pick the first tile from the bag, record its colour, return it to the bag and pick a second
tile. The analysis of this method of play is shown in the chart below. There are 9 wins out of the
36 total possible picks, producing the desired 1 in 4 chance of winning. From the discussion the
teacher can introduce the terms ‘with replacement’ (after the first tile is drawn out of the bag and
its colour noted, the tile is returned to the bag before the second tile is drawn from the bag) and
‘without replacement’ (one tile is drawn out of the bag and its colour noted; without returning the
drawn tile to the bag, a second tile is drawn from the bag and its colour noted).
1st pick / 2nd pick
g1
g2
g3
r1
r2
r3
TIPS: Section 3 – Grade 8
g1
win
win
win
loss
loss
loss
g2
win
win
win
loss
loss
loss
g3
win
win
win
loss
loss
loss
© Queen’s Printer for Ontario, 2003
r1
loss
loss
loss
loss
loss
loss
r2
loss
loss
loss
loss
loss
loss
r3
loss
loss
loss
loss
loss
loss
Page 67
20.2: A Probability Game
Name:
Date:
You want to develop a game, using red and green tiles, so that you have a 1 in 3 chance of
winning the game. Using up to 10 red and 10 green tiles, decide how many of each colour to put
in the bag and calculate the theoretical probability of drawing 2 green tiles.
Repeat this process by adjusting the number of red and green tiles until you arrive at a suitable
number of each colour in order to get the desired results of drawing two green tiles. Using the
number of each colour you decided on, play the game at least 30 times.
Use your data to compare the theoretical probability to the experimental probability. Explain why
there may be a difference between the two.
Green tiles
TIPS: Section 3 – Grade 8
Red tiles
Probability of drawing
two green tiles
© Queen’s Printer for Ontario, 2003
Decimal equivalents
Page 68
20.2: A Probability Game
(Answer)
As students work on their new game design, some may be content with probabilities that are
close to
1
; others may be exact. There are 121 possible combinations with 10 or fewer green
3
and 10 or fewer red tiles.
Access to a calculator or computer would be useful. If this is not possible you may need to guide
students to the conclusion to test only games with more green than red tiles.
Students may use charts, tree diagrams, or actual listing of combinations (as done in BLM
20.1). Be sure to allow adequate time for students to complete their work.
The combinations of red and green tiles that have probabilities close to or equal to
Green tiles
Red tiles
10
7
9
6
8
5
7
5
6
4
5
3
4
3
3
2
2
1
Probability of drawing
two green tiles
45
⎛ 10 9 ⎞
⎜ × ⎟ =
⎝ 17 16 ⎠ 136
8 ⎞ 12
⎛ 9
⎜ × ⎟ =
⎝ 15 14 ⎠ 35
14
39
7
22
1
3
5
14
2
7
3
10
1
3
1
:
3
Decimal equivalents
0.331
0.343
0.359
0.318
0.333
0.357
0.286
0.300
0.333
Students would likely only have time to find one or two combinations of red and green tiles to
lead to the desired results.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 69
Day 21: Checkpoint
Grade 8
Materials
• BLM 20.2, 21.1
Description
• Consolidate concepts of theoretical and experimental probability.
Assessment
Opportunities
Minds On…
Whole Class Æ Connecting to previous group of lessons
Invite students to ask any questions about the work from the previous class and
then to prepare for their assessment.
Action!
Individual Æ Demonstrating Skills and Understanding
Curriculum Expectations/Test/Marking Scheme: Students complete the test
individually (BLM 21.1).
Consolidate
Debrief
Pairs Æ Summarizing
As students finish the test, they can play games developed on Day 20.
Concept Practice
See BLM 21.1
Checkpoint for
Understanding
Probability
Answers.
Peer tutoring
would be
appropriate if a
student has
difficulty.
Home Activity or Further Classroom Consolidation
Finish designing your game.
Complete textbook questions: (teacher identifies exercise).
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 70
21.1: Checkpoint for Understanding Probability
Name:
Date:
Show full solutions in spaces provided. Read the questions carefully!
1. Tria had one each of five different-shaped number solids having 4, 6, 8, 12, and 20 sides.
She rolled two at a time and found probabilities of the sum of the numbers that came up.
She recorded the probabilities in the first column of the table. When it came time to fill in the
second column, she had forgotten which number solids she had used. Figure out which
number solids she must have used and explain your thinking. The first one has been done
for you.
She found that Using these number solids
Probability of a The total number of possible combinations was 48.
Both the 4 and 12, and the 6 and 8 combinations would have given 48 possible
5
combinations.
6 was
48
If a 4-sided and a 12-sided number solid were rolled and the sum was 6, the possible
combinations were 1 and 5, 2 and 4, 3 and 3, and 4 and 2 on the respective number
solids. That gives 4 rolls totalling 6 and a probability of rolling a 6 as
4
48
or
2
24
, not
5
48
If a 6-sided and an 8-sided number solid were rolled and the sum was 6, the possible
combinations were 1 and 5, 2 and 4, 3 and 3, 4 and 2, and 5 and 1 on the respective
number solids. That gives 5 rolls totalling 6 and a probability of rolling a 6 as
5
48
.
Therefore, Tria must have used the 6- and 8-sided number solids.
Probability of a
1
2
or
3 is
80
40
Probability of a
1
3 is
80
Probability of
a 4 is less than
probability of
a5
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 71
.
21.1: Checkpoint for Understanding Probability (continued)
2. Henry, Toshi, Lizette, Anna, and Vance were all scheduled to give oral reports in their
history class on Tuesday. However, when the class met, the teacher announced that only
two people would give their presentations that day. To determine which two, all of their
names were placed in a hat and two names were drawn out. What is the probability that
Henry and Anna were the names picked to give presentations? Show how you arrived at
your conclusion.
3. Claire has two bags of coloured cubes, one marked A and the other marked B. In bag A
there are 3 yellow and 4 green cubes. In bag B there are 2 blue and 5 red cubes. Without
looking, Claire picks one cube from bag A and then one cube from bag B. Answer the
questions below based on this information. Assume that after each part all cubes are
replaced in their appropriate bag.
8
a) What is the question, if the answer is
?
49
b) What is the question, if the answer is 0?
c) What is the question, if the answer is
3
?
7
Sources: Transforming Traditional Tasks, 2000; Explain It, 2001; Roads to Reasoning, 2002
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 72
21.1: Checkpoint for Understanding Probability
(Answers)
Probability of a
1
2
or
3 is
80
40
Probability of a
1
3 is
80
Probability of
a 4 is less than
probability of
a5
The 4 and 20 combination gives 80 possible outcomes. A sum of 3
results from rolling 1 and 2 or 2 and 1. That gives 2 rolls totalling 3 and a
1
2
or
Therefore, Tria must have used the
probability of rolling a 3 as
80
40
4- and 20-sided number solids.
The sum of 3 results from rolling 1 and 2 or 2 and 1. For these 2 rolls to
2
1
, we must have had
probability. Only the
yield a probability of
80
160
combination of 8 and 20 gives 160 possibilities. Therefore, Tria must
have used the 8- and 20-sided number solids.
A sum of 4 results from rolling 1 and 3, 2 and 2, or 3 and 1. A sum of 5
results from rolling 1 and 4, 2 and 3, 3 and 2, or 4 and 1. All of these
rolls are possible using any of the number solids. Since the sum of 4 can
occur in fewer ways then a sum of 5 for any pair of number solids,
probability of a 4 is less than probability of a 5 for any pair of these
number solids. Therefore, Tria cannot tell from this information which
number solids she used.
2. Students may list all possible outcomes using a tree diagram.
1st name
drawn
2nd name
drawn
There are 20 outcomes in all and the 2 circled outcomes represent Henry and Anna being
picked. Therefore the probability of Henry and Anna being picked is
2
1
or
20 10
OR
Students may reason that the probability that the first name drawn from the 5 names in the
2
, and the probability that the second name drawn will be the
5
1
2 1 1
other of Henry/Anna is
Therefore, the probability of both draws happening is × =
4
5 4 10
hat will be Henry or Anna is
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 73
21.1: Checkpoint for Understanding Probability
(Answers)
3. a) Each of the 7 cubes from A could be picked along with each of the 7 cubes from B. This
gives 7 x 7 = 49 possibilities in all. If 8 of these 49 outcomes are favourable, then we want
the 4 green cubes from A with the 2 blue cubes from B. Therefore, the question is, “What is
the probability of picking 1 green and 1 blue cube?”
b) If the probability is 0, the outcome is impossible. There are many possible answers to this
question, e.g., What is the probability of picking a purple cube? What is the probability of
picking a yellow and a green cube?
3
21
3
and there are 49 possible outcomes, I’ll think of
as
.
7
7
49
To get 21 favourable outcomes, I could pick yellow from A and any colour from B in
3 × 7 = 21 ways. Therefore, the question could be, “What is the probability of picking 1
yellow cube?”
c) Since the answer is
Sources: Transforming Traditional Tasks, 2000; Explain It, 2001; Roads to Reasoning, 2002
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 74
Day 22: Revisiting Rolling Number Cubes for Pythagoras
Grade 8
Description
• Investigate order of outcomes on theoretical and experimental probability.
Materials
• completed
BLM 17.1
• BLM 22.1
Assessment
Opportunities
Minds On…
Action!
Whole Class Æ Connecting to a Previous Lesson and to Prior
Knowledge
Recall the concepts from Rolling Number Cubes for Pythagoras (Day 17). How
must three side lengths be related to form a triangle?
Discuss with the class the types of triangles they know - equilateral, isosceles,
scalene, and right-angled. Could you consider 1, 1, and 2, or 3, 4, and 7 to be
the sides of a triangle? Why or why not?
Individual Æ Applying Concepts
Refer students back to BLM 17.1 and have them add a column using Type of
Triangle as the heading. Students complete this new column and prepare to
discuss their criteria for deciding on the type of triangle.
Whole Class Æ Inquiry
Students consider the following question and record their answers. How many
outcomes are possible when rolling three number cubes? Explain. Circulate to
identify students who have argued A (order matters): 6 × 6 × 6 = 216 and
students who have argued B (order does not matter): (6 ways to get all 3
numbers the same) + (30 ways to get 2 numbers the same) + (20 ways to get all
3 numbers different) = 56. Identify students to explain both viewpoints.
Pose the question: Now that you have heard both arguments, which do you
prefer?
Pairs Æ Exploring
Pair students who prefer argument A together and students who prefer
argument B together. Students complete BLM 22.1, being consistent with their
preferred argument – either the order of the numbers does matter (argument A)
or the order does not matter (argument B). Both groups should arrive at the
same answer.
P (equilateral) =
P (scalene) =
1
36
P (isosceles) =
7
36
P (right-angled) =
P (impossible triangle) =
Consolidate
Debrief
Concept Practice
Exploration
7
24
1
36
105
216
Whole Class Æ Communicate Understanding
Arrange student presentations of BLM 22.1 by a pair that followed argument A
and by a pair that followed argument B.
Curriculum Expectations/Presentation/Checkbric: Assess students on their
presentations.
Home Activity or Further Classroom Consolidation
Roll three number cubes 50 times and record the results of each roll. Create a
tally chart of the outcomes according to no triangle possible or the type of
triangle that could be formed.
Calculate the experimental probability of each type of triangle.
Compare the theoretical probabilities to the experimental probabilities and
explain differences.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Is the outcome 2,
3, 4 the same as
outcome 4, 2, 3?
OR
Does the order of
the numbers
matter in this
context?
For two numbers
the same, there
are 6 choices for
the repeated
number and 5
choices for the
different number,
making 6 × 5 = 30
ways. Of these 30
ways, 21
combinations yield
isosceles triangles;
9 yield impossible
triangles. Each of
the 21 isosceles
combinations can
be rolled in 3 ways
(with the different
st
number rolling 1
nd
rd
2 or 3 ).
If there are too few
students preferring
one of the
arguments, assign
some of the
strongest students
to work with the
other argument.
Scalene triangles
include rightangled triangles.
For all three
numbers to be
different, it does
not matter what
number is chosen
first. After that,
there are 5 ways
for the second
number to be
different, then 4
ways for the third
number to be
different from the
first and second.
This makes 5 × 4 =
20 ways for all
three numbers to
be different.
Page 75
22.1: Analysing the Number Cube Data
Name:
Date:
1. What is the total number of possible outcomes when rolling three number cubes? Explain.
2. Fill in the following chart using the data from the Home Activity on Day 17.
Type of Triangle
Frequency
Number of rolls that
resulted in this type of
triangle
Equilateral
Isosceles
Right-Angled
Scalene
3. a) When rolling three number cubes to determine the three possible side lengths for a
triangle, what is the theoretical probability of forming:
i)
an equilateral triangle?
ii) an isosceles triangle?
iii) a right-angled triangle?
iv) a scalene triangle?
b) When rolling three number cubes, what is the theoretical probability of being able to form
a triangle of any type?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 76
Day 23: Investigating Probability Using Integers
Grade 8
Materials
• integer tiles
• number cubes
• BLM 23.1
Description
• Review addition and subtraction of integers.
• Link probability to the study of integers.
Assessment
Opportunities
Minds On…
Whole Class Æ Connecting to Other Strands
Gather all student data from the Day 22 follow-up activity to calculate
experimental probabilities for more trials. Generally, the larger the number of
trials, the closer experimental probability should approach theoretical
probability.
Review the representation of integers, using integer tiles. Identify opposites,
several models of zero, and several models of +2.
Ask students to model adding and subtracting integers, using integer tiles.
For (+3) + (−2) show
Using the zero principle, the result is +1.
For (−2) – (−5) show
and ask if it is possible to take away −5. Ask
for a different model of -2 that would make it possible to take away −5.
[
]
Once -5 is removed, the result of +3 is obvious.
What is the result of adding an integer and its opposite? Does the order matter
when we add integers? When we subtract?
Model a series of questions like: (+2) – (+5) and (+2) + (−5); (−1) – (+4) and
(−1) + (−4) to show that subtracting an integer is like adding the opposite
integer. Since addition is easier to envision mentally, practise changing
subtraction questions to addition questions.
Ask: What addition question and answer are modelled by…?
e.g.,
giving (+ 1) + (−2) = −1
giving (−1) – (−2) = +1
What subtraction question and answer are modelled by…?
Action!
Pairs Æ Exploring to Develop Concepts
Using BLM 23.1, students conduct a simple probability experiment with
integers and two different-coloured number cubes and record their results on a
tally chart.
Learning Skills/Observation/Checklist and Curriculum
Expectations/Observation/Mental Note: Circulate while students analyse the
experimental and theoretical probabilities up to and including question 6 on
BLM 23.1. Make a note of students whose integer skills need review.
Consolidate
Debrief
Whole Class Æ Making Connections
Several students describe and compare their experimental results. Lead a
discussion to compare the experimental probabilities with each of the
theoretical probabilities. What have the students found? Are the results close?
Pose the questions: What do you think would happen to your experimental
probabilities if you did more than 25 trials or if you combined the trials from
every group in the class? How might expertise with integers have affected
findings?
Home Activity or Further Classroom Consolidation
Write a math journal entry using questions in the journal entry part of
worksheet 23.1.
Reflection
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Use Integer tiles
cut out of coloured
transparencies to
visually reinforce
concepts.
Many students will
easily use the zero
principle where
simple matching
and removing
“zeros” is required.
Situations
requiring the
addition of one or
more zeros to
facilitate an
operation may
require extra
practice.
Students can use
the integer tiles to
assist in finding
sums.
Some students
may focus on the
subtraction
operation itself and
not inspect the role
subtraction plays
in the experiment.
Since subtraction
is not
commutative, rules
for order are
needed. For
example: if using a
red and a white
number cube, the
student subtracts
the roll of the red
die from the roll of
the white number
cube.
Page 77
23.1: Integer Number Cubes
Name:
Date:
Use two different-coloured number cubes. Choose one cube to be negative numbers and the
other to be positive numbers. Record all of your results in the table.
Experimental
Sum
Tally
Frequency
4
TIPS: Section 3 – Grade 8
Probability
Possible
Combinations
6, −2; 5, −1
© Queen’s Printer for Ontario, 2003
Theoretical
Number of
Possible
Combinations
Probability
2
Page 78
23.1: Integer Number Cubes (continued)
1. Roll the number cubes and add the two numbers together, remembering which cube
represents positives and which cube represents negatives. Note the sum.
2. Repeat rolling the number cubes and finding the sum until you have a variety of sums.
3. What possible sums can you get? Fill out the first column of the table with the possible
sums.
4. Roll the number cubes 25 times and record each outcome of the sum in the tally column of
the table.
5. Total the tallies to find the frequency of the various sums.
6. What sum did you get the most? Why do you think this is so?
7. What sum did you get the least? Why do you think that is?
8. The experimental probability of an event happening is given by the fraction
number of times the event happened
.
total number of trials
For example, if you rolled the number cubes 25 times, and you got a sum of “3” 5 times,
5
1
then the experimental probability of getting a sum of 3 is
= . Find the experimental
25
5
probability of each of the sums and enter these experimental probabilities in the table.
9. Fill in the 5th column of the table with all of the combinations of numbers that you could roll to
yield each sum. For example, a sum of 4 would have possible combinations of: 6 and −2, 5
and −1.
10. Fill in the 6th column with the number of combinations in the 5th column.
11. The theoretical probability of an event is given by the ratio
number of possible ways of the event happening
.
total possible outcomes
For example, there are two possible ways of getting a sum of 4 (see the first chart you
completed). There are 36 possible combinations with the number cube, so the theoretical
2
1
probability of getting a sum of 4 is
=
. Find the theoretical probability of rolling each
36
18
of the possible sums. Enter your results in the last column of the table.
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 79
23.1: Integer Number Cubes (continued)
Math Journal Entry
1. What is the theoretical probability of:
a) rolling a negative sum?
b) rolling an even sum (positive or negative?)
c) not rolling a sum of 3?
d) rolling anything other than a sum of 0?
2. How might errors in integer calculations be prevented in this experiment?
3. What extra considerations would be needed if the roll of the number cubes were subtracted
rather than added?
4. What other words do we use to indicate positive and negative? Suggest a variety of
situations where these words might be used.
5. How might addition and subtraction of integers be shown using a number line?
TIPS: Section 3 – Grade 8
© Queen’s Printer for Ontario, 2003
Page 80
Multi-dart
Summative Task
Grade
Total time
Materials
8
240 minutes
• The Geometer’s Sketchpad® (dynamic geometry software) file (cones.gsp),
• Calculators, dart board or diagram, BLMs
Using manipulatives and technology students collect data to investigate the relationship
between circumference and the area of various circles within a boundary. Students
discover a pattern and use it to solve a problem. They submit a report that justifies and
explains their conclusions.
Description
Expectations Number Sense and Numeration
8m31 – *explain the process used and any conclusions reached in problem solving and
investigations;
Assessed*
8m32 – *reflect on learning experiences and interpret and evaluate mathematical issues
and
using appropriate mathematical language (e.g., in a math journal).
addressed
Measurement
8m37 – *solve problems related to the calculation of the radius, diameter and
circumference of a circle;
8m42 – *make increasingly more informed and accurate measurement estimations
based on an understanding of formulas and the results of investigations;
8m44 – *measure the radius, diameter, and circumference of a circle using concrete
materials.
8m47 – estimate and calculate the radius, diameter, circumference, and area of a circle,
using a formula in a problem solving context;
Patterning and Algebra
8m75 – identify, create, and discuss patterns in algebraic terms;
8m77 – identify, create, and solve simple algebraic equations;
8m78 – *apply and defend patterning strategies in problem-solving situations.
Ontario Catholic School Graduate Expectations
CGE3c – thinks reflectively and creatively to evaluate situations and solve problems
CGE5a – works effectively as an interdependent team member
Prior
• Understanding of circle relationships
Knowledge
• Skills with The Geometer’s Sketchpad®
Students should be able to:
• Draw parallel and perpendicular lines
• Create a table
• Calculate area and perimeter/circumference
• Construct points, circles, and line segments
• Create a scale drawing
• Estimate
• Develop and use Ratio and Proportion
• Apply circle formulas
• Use tables to organize data and thinking
Assessment
Tools
Extensions
Rubric
What would be the impact on total costs of changing the cost of curved plastic trim to
$1.50/m regardless of its length?
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 81
Pre-task Instructions
Read the following script to students on the day before the investigation begins.
Teacher Script
1. We will be working on an investigation over the next three mathematics classes.
2. An investigation is an extended problem designed to allow you to show your ability to undertake an
inquiry: to make a hypothesis, formulate a plan, collect data, model and interpret the data, draw
conclusions, and communicate and reflect on what you have found.
3. For this investigation you will be using The Geometer’s Sketchpad®. As well you will need pencils,
pens, an eraser, a ruler, notepaper and a graphing calculator to complete the work.
4. As you do the investigation you will work as part of a group and also individually. (Distribute an
envelope or folder to each student.) I am giving each of you an envelope in which you can store your
notes for the duration of the investigation. Write your name on the front of the envelope.
5. On the third day you will write a report giving your conclusions and summarizing the processes you
have followed to arrive at them.
6. Be sure to show your work and include as much explanation as needed.
7. Each section of the investigation has a recommended time limit that I will tell you so you can manage
your time.
8. You will be assigned to the following groups for the three days of the investigation. (You may wish to
assign students to their groups at this time – recommended group size is four students.)
9. Are there any questions you have regarding the format or the administration of the investigation?
Teacher Notes
• This summative task could be used for gathering summative assessment data or for providing formative
feedback to students before they complete another task for assessment purposes.
• If a Home Activity or Further Classroom Consolidation task is to be used for gathering assessment data,
it may be most appropriate for students to work on it independently under teacher supervision.
• Some suggested Home Activity or Further Classroom Consolidation tasks help prepare students for
later assessments.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 82
Day 1: Introducing the Problem
Grade 8
Description
the complexity of the problem.
• Discuss various strategies for solving the problem.
• Understand
Minds On ...
Whole Class Æ Guided
Summarize the experiences of the Canadians who have invented a new game.
(BLM S1.1)
Discuss the game of darts.
The purpose of the game is to score the greatest number of points. Players earn
points by throwing a dart at a circular board. Points are earned when a dart lands
within one of the concentric circles. The maximum number of points are in the
centre of the circle. Each concentric circle radiating from the centre contains
fewer points. If possible, simulate a game of darts with magnets.
Explain the investigation task: Over the next four days it is your role to help a
manufacturer with the design of a new and different dart game called Multi-dart.
Small Groups Æ Discussion
Students read the problem together and highlight the key information
(BLM S1.2). They paraphrase the problem in their own words.
Whole Class Æ Discussion
Read and discuss BLM S1.3. Students ask questions for clarification of the task
and their final submission.
Action!
Small Groups Æ Discussion
Students explore and discuss strategies and plans for solving the problem and
develop hypotheses about their potential preferred design.
Consolidate
Debrief
Whole Class Æ Share
The groups share their ideas with the class.
Individual Æ Assessment
Students record their hypotheses and plans for this investigation.
Reflection
Materials
• BLM S1.1, S1.2,
S1.3
• dart board
An optional table
has been provided
for teachers to make
available to students
if required
(BLM S1.3)
Home Activity or Further Classroom Consolidation
Answer the following questions in your journal.
• How did the small group discussion help you better understand the problem?
• What further ideas came out in the whole class sharing?
• What further considerations have you thought of since the class sharing?
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 83
S1.1
Article
Canadian game and toy inventors have an amazing history. In 1891, James Naismith was
credited with the invention of the game of basketball. In 1909, T.E. Ryan of Toronto developed
five pin bowling. In 1980, University of Toronto students Chris Hardy and Scott Abbott created
the hit board game Trivial Pursuit. More recently, a trio of young inventors, Anton Rabie, Ben
Varadi, and Ronnen Harary created Air Hogs, and later, finger bikes and finger skateboards.
Now, just coming on to the market is a new set of games called “Toss’ems.” In these games
pog-like magnetic pieces are tossed at a metal board to simulate the games of soccer, hockey,
football, basketball, and baseball. The creator, John MacEachern was inspired as he watched
his nephews toss magnets at a fridge to amuse themselves during a period of bad weather at
the cottage. “I thought if I could package this activity it would catch on,” he explained.
What is interesting, particularly about these last inventions, is that they are not entirely unique,
but rather variations on existing games/activities. So for this mathematical investigation, you will
be examining a new variation of an old game – we’ll call it “Multi-darts.”
http://www.tossems.com
Toronto Star, Monday Dec. 6, 1999. Section E pp. 1 and 3 “Air Hogs Creators Betting Little Bikes Will Fly.”
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 84
Introducing the Problem
The Problem
Our company is marketing a new product called Multi-dart. It's a game featuring a number of
congruent small dartboards inside a 60 x 60 cm square frame, a new twist to the original game.
We are considering several plans, starting with the 2 x 2 and 3 x 3 designs shown below.
Different designs will be considered as long as the small dartboards fill the entire frame with an
equal number of rows and columns.
original
2×2
3×3
Market research has decided that the small dartboards will be more attractive with a coloured
plastic trim form-fitted around the outside curves of the exterior circles. The cork circles will be
mounted on square plywood backing outlined with a wooden frame.
Important Information
The cost of materials is:
Cork for each circular dartboard
Plastic curved trim
Wooden frame
Plywood backing
$ 0.25 per 100 cm2
$ 2.00 for less than 2 metres
$ 1.50 for 2 to 3 metres
$ 1.20 for more than 3 metres
$ 3.00 per metre
$ 5.00 per 60 × 60 cm board
Your Task
The company needs your help in deciding on the final design based on this criteria:
• Cost for each design
• Customer appeal (style)
• Ease of scoring
Your final selection should be compared to the original design of one large circular dartboard,
60 × 60 cm frame.
1.
2.
3.
4.
Calculate the cost of three different designs.
Create labelled diagrams to explain your designs.
Collect and organize your data.
Use the information you collect to justify the design you prefer based on cost, style, and
ease of scoring.
Present the results of your investigation using The Geometer's Sketchpad®.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 85
S1.2
Questions to Guide Your Final Submission
Your final submission should be a portfolio comprised of four sections:
• A reflection
• Diagrams used in investigating the dart board designs
• Organized data
• Conclusion and justifications as to which design is best based on cost, style, and ease of
scoring
Guiding Questions for the Reflection
Describe the processes and problem-solving strategies you employed in investigating MultiDart.
What patterns and relationships did you notice?
Did anything surprise you? Explain.
If you were going to conduct this same investigation again, would you use the same processes
and strategies, or would you try different ones? Explain.
Diagrams
The Geometer’s Sketchpad® diagrams should be printed and included in your portfolio. Include
a text box with your name and date on each diagram. When printing a file, first select PRINT
PREVIEW from the FILE menu, and then SCALE to make it fit the page.
Data Organization
Include a summary of the data use these headings:
• Design type
• Square frame perimeter
• Frame cost
• Plywood backing cost
• Cork area
• Cork cost
• Curved perimeter of outside circles
• Plastic trim cost
• Total cost
Recommendation
Make a specific recommendation for one of the three designs, explaining your choice by
referring to, cost, style, and ease of scoring.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 86
S1.3
Optional Data Organization Table
A Comparison of Multi-dart Designs
Design
Square
Frame
Perimeter
(cm)
Frame
Cost
TIPS: Section 3 – Grade 8 Summative Task
Plywood
Backing
Cost
Cork
Area
(cm2)
Cork
Cost
Circular
Trim
Perimeter
(cm)
© Queen’s Printer for Ontario, 2003
Curved
Plastic
Trim
Cost
Total
Cost
Page 87
Days 2 and 3: Exploring Patterns
Grade 8
Description
a GSP representation of various dart boards.
• Use GSP tools to gather data and compute costs.
• Summarize findings.
• Develop
Minds On ...
Action!
Consolidate
Debrief
Reflection
Evaluation
Whole Class Æ Guided Discussion
Students reiterate what the investigation is about and pose questions.
Individual Æ Investigation
Students work with GSP to solve the problem. Invite them to ask any technical
questions throughout the investigation.
As students work through the investigation, the following questions may be used
to probe their thinking:
• What patterns and relationships have you noticed?
• How does the amount of cork you need compare to the original design? …to
your other designs?
• How does the amount of plastic trim compare to the original design? …to your
other designs?
• Is it easier to score on your board than on the original dartboard? Why or why
not?
• How much more or less will your preferred design cost than the original
dartboard design? …than the other designs?
Students record the data for area, perimeter and circumference. Students may
want to use the tabulate feature on GSP.
Students also need to calculate the cost for the designs they have examined, as
well as for the original dartboard version.
Students begin to analyse the data looking for patterns and relationships.
(See solutions on BLM S1.6)
Materials
• The Geometer’s
Sketchpad®
• BLM S1.4, S1.5,
S1.6
Three different entry
points for GSP are
provided for the
students depending
on their expertise
with the technology.
• Partially pre-made
sketches provided
on GSP
(BLM S1.4)
• Explicit written
instruction to
construct their own
sketches
(BLM S1.5)
• Students
designing their
own methods for
construction of
sketches
Keep anecdotal
notes as you
observe and
conference with
students for
assessment
opportunities.
Whole Class Æ Guided Discussion
Revisit how to organize the summary.
Individual Æ Assessment
Students write their summary.
Home Activity or Further Classroom Consolidation
Answer the following question in your journal.
How is use of The Geometer’s Sketchpad® saving time in solving this problem?
What new GSP skills have you learned doing this investigation?
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 88
S1.4 Partial Pre-made Sketch Instructions for GSP 4.03
*Note that these instructions represent only one way to complete the sketches and solve the
problem using The Geometer's Sketchpad®. As there are many other possible approaches,
you may explore and deviate from the steps provided below.
Standard Single Circle Dartboard Design
• Begin with the pre-made sketch showing a square with a dot in the centre.
• Select one of the sides of the square. Measure its length. What is the relationship between the
length of the side and the dimensions of the dartboard described in the investigation? Why do
you think the sketchpad measure and the actual dartboard measure are different? How will
this impact your results?
• Select one of the sides of the square again. Go to the CONSTRUCT menu and select
MIDPOINT.
• Left click in a blank area to deselect the midpoint you have just constructed.
• Select the centre of the circle and then the midpoint you just constructed. Go to the
CONSTRUCT menu and select CIRCLE BY CENTRE + POINT.
• Go to the MEASURE menu and select CIRCUMFERENCE. Drag this measurement to a
convenient location.
• Select the circumference, and go to the CONSTRUCT menu and select CIRCLE INTERIOR.
Then go to the MEASURE menu and select AREA. Once again, drag this to a convenient
location.
Completing the 2 × 2 Multi-dart Design
• Begin with the pre-made sketch showing a 2 × 2 square with a circle in the top left square.
• Using the SELECT TOOL, double click on the point at the centre of the entire 2 × 2 square to
establish that as the centre of rotation.
• Still using the select tool, double click on the heavy red lined portion of the circumference of
the circle, followed by a single click on the remaining thin-lined section of the circumference.
• Go to the TRANSFORM menu and select ROTATE. It should say 90 degrees. Click ROTATE.
• Go directly back to the TRANSFORM menu and again select ROTATE and click ROTATE.
Repeat this procedure one more time to complete the Multi-dart 2 × 2 sketch.
• From this diagram, you may construct circle interiors, measure area, arc lengths, etc., to
complete this portion of the investigation.
Completing the 3 × 3 Multi-dart Design
• Begin with the pre-made sketch showing a 3 × 3 square with a circle in the top left square.
• To make a point at the exact centre of the entire diagram, use the SELECT TOOL to highlight
two opposite vertices of the small middle square. From the CONSTRUCT menu choose
SEGMENT. Click in a blank area to "deselect" these items, then repeat the process to
construct a segment joining the other two opposite vertices of the small centre square.
• Highlight only the two crossing segments you have just constructed. From the CONSTRUCT
menu choose Point at Intersection. Click in a blank section to "deselect" these items. Then
double click on the central point you have just constructed to set it as the centre of rotation.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 89
S1.4 Partial Pre-made Sketch Instructions for GSP 4.03
(continued)
• Still using the select tool, double click on the heavy red lined portion of the circumference of
the circle, followed by a single click on the remaining thin-lined section of the circumference.
• Go to the TRANSFORM menu and choose ROTATE. Repeatedly rotate this section 90
degrees until all four corners are filled.
• Now, single click on the thin lined-portion of the circumference of the original circle, followed
by a single click on each of the four points on that circumference. Double click on the vertical
line connected to the right hand side of the circle. This will set the line of reflection.
• Go to the TRANSFORM menu and choose REFLECT.
• Double click on the same centre of rotation we used originally. From the TRANSFORM menu
choose ROTATE, and repeatedly rotate this circle until all outside squares contain circles.
• To fill the centre circle, highlight the circumference of the top centre circle, double click on the
horizontal line attached to its bottom point, and from the TRANSFORM menu choose
REFLECT.
• To complete the sketch, click on the west, north, and east points of the top centre circle. Go to
the CONSTRUCT menu and choose ARC THROUGH 3 POINTS. Go to the DISPLAY menu
and choose LINE WIDTH. Choose THICK.
• Continue to define this outer arc on the remaining three circles, either by rotating the arc you
have just constructed, or by constructing arcs on each of the individual circles.
• Measure, calculate, and tabulate as needed to complete the investigation.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 90
S1.5
GSP 4.03 Instructions
Follow the directions below to create a 2 × 2 Multi-dart Board
using The Geometer's Sketchpad® 4.03:
*Note that these instructions represent only one way to complete the sketches and solve the
problem using The Geometer's Sketchpad®. As there are many other possible approaches,
you may explore and deviate from the steps provided below.
Constructing a circle with a radius of 2 cm.
2x2
1. Go to the EDIT menu and select PREFERENCES.
2. Change the units: angles to degrees; distance to cm. Change all three precision measures
to units. Place a check in the box beside apply to this sketch. (The use of units rather than
using decimals maintains the length measurements as whole numbers.)
3. Go to the GRAPH menu and select SHOW GRID. A coordinate grid will appear with a point
constructed at 1,0 and 0,0.
4. Go to the GRAPH menu and select PLOT POINTS.
5. Highlight the space on the left and enter –1.0 for a 3 × 3 Multi-dart. This will create a 2 cm
segment. For a 2 × 2 Multi-dart, enter – 2. This will create a three cm segment. In the space
on the right, highlight the number and type 0. Click on PLOT and DONE. A third point will
appear.
6. Click to select the two axes and go to the DISPLAY menu. Select HIDE AXES. The x and y
axes will disappear.
7. Go to the GRAPH menu and select HIDE GRID. The grid should disappear leaving the three
points.
8. Click on the two points, (-1, 0) and (+1, 0) and go to the CONSTRUCT menu. Choose
SEGMENT. A pink line segment should appear with endpoints and a midpoint.
9. Drag the segment to a corner. Select the POINT TOOL. Click once on the page to create a
point around the middle of the page.
10. The point will be bright pink. This shows that the point is selected. Click on the segment to
select it as well.
11. Go to the CONSTRUCT menu and select CIRCLE BY CENTRE + RADIUS. A circle will
appear. Its radius will be the same length as the line segment you placed in the corner.
12. Click somewhere on the blank sketch to deselect the circle you have selected. Measure the
line segment by selecting it. (It will be pink when selected.) Then go to the MEASURE menu
and select LENGTH. A small pink box will appear with the measurement inside it. m AB = 2 cm
13. Click on the circumference of the circle and go to the CONSTRUCT menu. Click on POINT
ON OBJECT. A point will appear on the circumference. Drag the point to the top of the
circle.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 91
S1.5
GSP 4.03 Instructions (continued)
Creating the 2 × 2 dartboard from a single circle.
14. With both the point and the CENTRE OF THE CIRCLE selected, go to the CONSTRUCT
menu and select LINE. This allows you to create points on your circle. (Note: a line has no
beginning and no end.)
15. Select the circumference and the line and go to the CONSTRUCT menu. Choose
INTERSECTIONS.
16. Go to the DISPLAY menu and select HIDE LINE. Highlight the two points and go to
CONSTRUCT. Select SEGMENT. This will create a diameter for the circle and will help you
to duplicate the circle.
17. In order to create a second circle for the Multidart Board, click on the point at the top of the
circle then the point at the bottom of the circle. Go to the TRANSFORM menu and select
MARK VECTOR. The segment will briefly be highlighted black. This line indicates the
distance and direction of the translation.
18. Using the point tool click just above and to the left of the circle and drag to create a
rectangle. This selects the circle and all points and segments.
19. Then go to the TRANSFORM menu and select TRANSLATE. A dialogue box will appear. If
needed, move the dialogue box by clicking on the blue band at the top and dragging it to the
side. This will allow you to see the faint outline of a circle.
20. The marked box will be checked. Click on TRANSLATE to create a second circle directly
below the first. (If you are drawing a Multidart Board with three circles on each side, repeat
this translation by marking the diameter of the second circle as a vector and repeating the
process above.)
21. Select the line segment or diameter in the top circle and the centre point, then go to the
CONSTRUCT menu. Choose PERPENDICULAR LINE. Do the same in the bottom circle so
that two parallel lines are created perpendicular to the existing diameters. These lines allow
you to create points where they intersect with the circle.
22. Click on a white part of the page to deselect all points and segments. Then, click on the
circumference of the top circle and the perpendicular line through the centre and choose
INTERSECTIONS. Do the same thing for the bottom circle.
23. Click on one of the circle diameters and one of the intersection points on the circumference.
Go to the CONSTRUCT menu and choose PARALLEL LINE. Click on the white space to
deselect the highlighted objects. This will create a line of reflection.
24. To reflect the two circles across the parallel line, double click on the parallel line to mark it as
the mirror line. Then click and drag to draw a box around the two circles and select
everything on or inside them.
25. Go to the TRANSFORM menu and click on REFLECT. (If you are drawing the 3 x 3
Mulitdart Board, create another parallel line and reflect two of the circles again.)
26. Select the two points on the top of the circles. Go to the CONSTRUCT menu and select
LINE. Do the same on the bottom and each side of the circles. These lines will help create
the frame for the Multidart Board.
27. Click on two intersection lines, at the corners of the board. Go to the CONSTRUCT menu
and select INTERSECTION to create a corner point for the Multidart frame. Repeat to create
a point for each corner of the Multidart frame.
28. Click on each of the lines and segments and go to DISPLAY menu. Select HIDE LINES.
29. Select the four points in order around the corners of the Multidart and go to the
CONSTRUCT menu to select SEGMENTS.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 92
S1.5
GSP 4.03 Instructions (continued)
30. Outline the outside of the circles by creating arcs on the circles. This will outline the area to
be form-fitted with plastic trim. Click on the point between two adjacent circles (A), then one
of the points on the circumference (B), and a third point where adjacent circles join (C). See
the illustration below. This will select the points needed to create an arc around part of the
circle.
A
B
C
31. With the three points selected choose CONSTRUCT and ARC THROUGH 3 POINTS. This
will select the part of the first circle which is trimmed with plastic. Repeat this process to
construct arcs on the other circles. See illustration below.
B
C
A
32. Select the appropriate parts of the Multidart Board and use the MEASURE menu to find the
measurements.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 93
S1.6
Solutions (Days 2 and 3)
The calculations are based on circles inscribed on a 60 x 60 cm square
Design
Circumference
Perimeter
Area
C = πd
= 3.14 (60)
= 188.4 cm
P = πd
= 3.14 (60)
= 188.4 cm
A = πr2
= 3.14 (302)
= 2826 cm2
C = 4 πd
= 4 (3.14 ) (30)
= 376.8 cm
P = 3πd
= 3 (3.14)(30)
= 2.826 cm
A = 4πr2
= 4 (3.14) (152)
= 2826 cm2
double the single circle
circumference
1½ times the single
circle “perimeter”
C = 9 πd
= 9 ( 3.14) (20)
= 565.2 cm
P = 5πd
= 5 (3.14)(20)
= 314 cm
triple the single circle
circumference
1 32 times the single
C = 16 πd
= 16 ( 3.14) (15)
= 753.6 cm
P = 7πd
= 7 (3.14)(15)
= 329.7 cm
quadruple the single
circle circumference
1¾ times the single
circle “perimeter”
C = 188.4 n
where “n” represents
the design number
The general formula
would be
P = (2n-1) πd
circle “perimeter”
(See Calculating
Perimeter on next
page.)
TIPS: Section 3 – Grade 8 Summative Task
A = 9 πr2
=9 (3.14) (102)
= 2826 cm2
A = 16πr2
= 16 (3.14) (7.52)
= 2826 cm2
Area is constant,
regardless of the
number of congruent
circles enclosed in the
space. This means
chances of “scoring”
are equal regardless of
design.
© Queen’s Printer for Ontario, 2003
Page 94
S1.6
Solutions (Days 2 and 3) (continued)
If youCalculating
construct a square
joining the centres of
the by
Perimeter
each circle, it becomes easier to see that the
“perimeter” of this figure is composed of four ¾
circumferences. That is,
3
(circumference of single circle)
4
3
= 4 × ( πd)
4
P=4 ×
= 3πd
This time the four ¾ circumferences still exist but
there are an additional four ½ circumferences
along the perimeter, or
3
1
⎡
⎤ ⎡
⎤
P = ⎢4 × ( πd)⎥ + ⎢4 × ( πd)⎥
4
2
⎣
⎦ ⎣
⎦
= 3πd + 2πd
= 5πd
Similarly, with a 4 x 4 array, the four ¾
circumferences still exist on the corners
and this time there are eight ½
circumferences along the perimeter, or
3
1
⎡
⎤ ⎡
⎤
P = ⎢4 × ( πd)⎥ + ⎢8 × ( πd)⎥
4
2
⎣
⎦ ⎣
⎦
= 3πd + 4πd
= 7πd
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 95
Day 4: Extending the Pattern
Grade 8
Description
GSP skills to a new problem.
• Investigate a change in the parameters to the multi-dart problem.
• Apply
Minds On ...
Whole Class Æ Guided
Read the problem together (BLM S1.7) and give students an opportunity to ask
questions for clarification.
Small Group Æ Discussion
Students highlight the key information and paraphrase the problem in their own
words.
Discuss how students think the change in size of the dartboard will
proportionally affect the area, perimeter, and circumference, as well as how it
might affect the cost of the larger board.
Whole Class Æ Sharing Æ Discussion
Each group shares its predictions.
Action!
Independent Æ Investigation
Students work on GSP to adjust their preferred design to fit the new dimensions
for the frame. They look at the values for area, perimeter, and circumference and
compare them to the similar smaller multi-dart board and to their predictions.
Students analyse how the results proportionally changed for area, perimeter, and
circumference.
They calculate the new cost for the larger game and analyse how the cost
changes compared to the smaller version and their predictions.
Consolidate
Debrief
Whole Class Æ Guided Discussion
Review the expectations for the written report.
Independent Æ Assessment
Students write a report on their findings.
Application
Concept Practice
Differentiated
Reflection
Materials
• The Geometer’s
Sketchpad®
• BLM S1.7, S1.8
Any values for
perimeter and
circumference
double whereas any
values for area
quadruple; as a
result, the cost of
the larger board will
be more than double
the smaller one.
Home Activity or Further Classroom Consolidation
Answer one of the following questions in your journal based on whether you are
satisfied with your solution to their multi-dart problem. If so, do A, or you think
you could have done a much better job, do B.
A: If the price of plastic curved trim was $1.50/m regardless of the length
needed, what would be the impact on your answer to today’s problem –
doubling the frame size?
OR
B: What could you have done differently to improve your solution? What about
this experience will you remember when you encounter a similar situation?
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 96
S1.7
Extending the Pattern
What if the frame size is doubled?
Our marketing department has decided it needs a larger model of your preferred design to sell
to public places such as restaurants and clubs. We need to double the size of the frame to 120
cm × 120 cm. It is important we know the following information to help with the ordering of
materials and calculating our costs:
• How will doubling the dimensions of the frame size affect:
• the area of the plywood backing?
• the area of the cork?
• the perimeter of the wooden frame?
• the perimeter of the curved plastic trim?
• What do you think the new cost of plywood backing will be? Justify your answer.
• How will the cost increase for the larger game?
• Record your results.
• Explain the relationships you noticed.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 97
S1.8
Solutions (Day 4)
A Comparison of Multi-dart Design Costs
Costs for a 60 cm × 60 cm board
Design
1 circle
4 circle
9 circle
12 circle
Square
Frame
Perimeter
(cm)
240
240
240
240
Frame
Cost
($)
Plywood
Backing
cost ($)
Cork
Area
(cm2)
Cork
Cost
($)
7.20
7.20
7.20
7.20
5.00
5.00
5.00
5.00
2826
2826
2826
2826
$7.07
$7.07
$7.07
$7.07
Frame
Cost
($)
Plywood
Backing
cost ($)
Cork
Area
(cm2)
Cork
Cost
($)
14.40
14.40
14.40
14.40
20.00
20.00
20.00
20.00
11304
11304
11304
11304
28.26
28.26
28.26
28.26
Circle –
based
Perimeter
(cm)
188.4
282.6
314
329.7
Curved
plastic
trim cost
($)
3.77
4.24
3.77
3.96
Circle –
based
Perimeter
(cm)
376.8
565.2
628
659.4
Curved
plastic
trim cost
($)
4.52
6.78
7.54
7.91
Total
Cost
($)
23.03
23.50
23.03
23.22
Costs for 120 cm × 120 cm board
Design
1 circle
4 circle
9 circle
12 circle
Square
Frame
Perimeter
(cm)
480
480
480
480
Total
Cost
($)
67.18
69.44
70.20
70.57
Note: When the board dimensions changed from 60 × 60 to 120 × 120, the perimeter measures
doubled but the area measures quadrupled. Thus the new cost for plywood backing would,
assuming the rate stays constant, be four times the original price, or $20.00. In this extended
situation, all designs benefit from the “bulk discount” for the curved trim material, so the design
with the fewest circles is the least expensive.
If the cost of the curved plastic trim was $1.50/metre regardless of the length:
Design
1 circle
4 circle
9 circle
12 circle
Curved plastic trim
cost for a
60 cm × 60 cm board
2.83
4.24
4.71
4.95
Curved plastic trim
cost for a
120 cm × 120 cm board
5.65
8.48
9.42
9.89
Total cost for a
60 cm × 60 cm
board ($)
22.10
23.50
23.98
24.22
Total cost for a
120 cm × 120 cm
board ($)
68.33
71.16
72.10
72.57
The cost of the 1-circle game on a 60 cm × 60 cm board goes down.
The cost of the 4-circle game on a 60 cm × 60 cm board goes down.
All others cost more.
TIPS: Section 3 – Grade 8 Summative Task
© Queen’s Printer for Ontario, 2003
Page 98