learning goals write stories to match up distancetime...

day 8 DistanceTime graphs answers.notebook

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May 16, 2014

Title Page

Story Graphs

Learning Goals interpret distancetime graphs write stories to match up distancetime graphs

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Minds On Page 1

5.2.1: Graphical Stories Below the following graphs are three stories about walking from your locker to your class.Three of the stories correspond to the graphs. Match the graphs and the stories using a line. Write a story for the other graph. Draw a graph that matches the forth story.

Graph Story

I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I went quickly at a constant rate to class.

I was rushing to get to class when I realized I wasn’t really late, so I slowed down a bit.

I started walking at a steady, slow, constant rate to my class, and then, realizing I was late, I ran the rest of the way at a steady, faster rate.

I walked to my friend’s locker, and stopped to talk to her for a few minutes. After she had collected all of her books, we walked (a little faster this time) to class together.

BLM

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HomeworkGizmos

GizmosBLM

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http://www.explorelearning.com/index.cfm?method=controller.dsppasstimeout&resourceid=625


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Minds On Page 2

BLM

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Action Page 1

With a partner, write a story to match 5 out of the 10 graphs below. Be creative!

BLM

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Action Page 2

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Attachments

G9_U5_L3_TeacherNote.pdf

G9_U5_L2_TeacherNote.pdf

BLM 5.2.1.pdf

BLM 5.2.2.pdf

BLM 5.2.3.pdf

BLM 5.2.4.pdf

G9_U5_L2_5.2.4Page1.pdf

U5_L2 TR.pdf

Word Wall L2.pdf

Gr9and10MathWordWall.pdf

Gr9and10WordWallCurriculumGlossary.pdf

Unit 5: Day 3: Ramps, Roofs, and Roads Grade 9 Applied

75 min

Math Learning Goals • Examine rate of change in a variety of contexts. • Calculate rate of change using rise

run and connect to the unit rate of change.

• Convert fractions ↔ decimals ↔ percents.

Materials • computer/data

projector • BLM 5.3.1

Assessment Opportunities

Minds On ... Whole Class Demonstration Review converting between fractions, decimals, and percents. Show the Rate of Change electronic presentation, summarizing the main ideas. Students make notes. With the students, complete the first example, Ramps, and the first two table rows on Roads (BLM 5.3.1).

Action! Pairs Problem Solving Students complete each page of BLM 5.3.1 in pairs and share answers in groups of four.

Learning Skill (Work habits)/Observation/Anecdotal: Observe students’ work habits and make anecdotal comments.

Consolidate Debrief

Whole Class Sharing Select students to share their answers to BLM 5.3.1. Draw out the mathematics, and clear up any misconceptions.

Rate of Change.ppt If a projection unit is not available, the pages in the electronic presentation can be made into transparencies. Word Wall pitch grade ramp incline

riserunrate of change =

Concept Practice Journal

Home Activity or Further Classroom Consolidation • Complete rate of change practice questions. • In your journal, give an example of where rate of change occurs in your

home.

Provide students with practice questions.

SMART Notebook

Unit 5: Day 2: Story Graphs Grade 9 Applied

75 min

Math Learning Goals Write stories related to piecewise graphs; demonstrate the connection between

the position, direction, speed, and shape of the graph.

Investigate a variety of graphs in contexts with respect to rate of change, e.g.,

filling containers, raising a flag, temperature.

Materials BLM 5.2.1, 5.2.2,

5.2.3, 5.2.4

Assessment Opportunities Minds On ... Whole Class Discussion

Explain the activity on BLM 5.2.1. Answer any questions. Use BLM 5.2.2 to

discuss what their stories must include.

Stress the difference between constant rate of change and variable rate of change.

An alternative is to have students write their story on the SMART board, OR copy the graph onto chart paper and write their story next to the graph. Students may wish to act out their story as well as give their oral presentation. A common student interpretation of these graphs involves going up and down hills. Explain that a hill is not necessary to explain the graph. Word Wall increasing rapidly increasing slowly decreasing rapidly decreasing slowly constant rate of change varying rate of change See Think Literacy, Mathematics, pages 62–68 for more information on reading graphs.

Action! Pairs Note Making/Presentation Using the graphs from BLM 5.2.3, students work in pairs to write 5 stories

and orally present 1-2 of them to the class.

Encourage students to think beyond the distance-time graphs done on the

CBR™ and think about raising a flag, filling containers, etc. Show some

examples.

Note: Most students will find it easier to think of time as the independent

variable rather than some other measure.

Curriculum Expectation/Observation/Checklist: Use BLM 5.2.2 as a tool

to assess communication.

Consolidate Debrief

Whole Class Discussion Review the graphs with students and clarify any information that students

may have misinterpreted (BLM 5.2.3).

Curriculum Expectations/Observation/Checklist: Assess student ability to

use proper conventions for graphing.

Concept Practice

Application

Home Activity or Further Classroom Consolidation Complete worksheet 5.2.4, Interpretations of Graphs.

NCTM has many activities that relate to rates of change and graphs at www.nctm.org.

http://www.nctm.org/

SMART Notebook

5.2.1: Graphical Stories Below the following graphs are three stories about walking from your locker to your class. Three of the stories correspond to the graphs. Match the graphs and the stories using a line. Write a story for the other graph. Draw a graph that matches the forth story.

Graph Story

I started to walk to class, but I realized I had forgotten my notebook, so I went back to my locker and then I went quickly at a constant rate to class.

I was rushing to get to class when I realized I wasn’t really late, so I slowed down a bit.

I started walking at a steady, slow, constant rate to my class, and then, realizing I was late, I ran the rest of the way at a steady, faster rate.

I walked to my friend’s locker, and stopped to talk to her for a few minutes. After she had collected all of her books, we walked (a little faster this time) to class together.

Graphs which compare DISTANCE FROM A POINT and TIME are called Distance-Time graphs. These graphs can be used to indicate direction, speed and total length of trip (from starting point to ending point). A story can be made from a distance-time graph.

SMART Notebook

5.2.2: Writing Stories Related to a Graph Names: As you create your story: Focus on the rate of change of each section of the graph and determine whether the rate of change is constant, varying from fast to slower or slow to faster or zero.

Criteria Does your story include:

Yes

the description of an action? (e.g., distance travelled by bicycle, change of height of water in a container, the change of height of a flag on a pole)

the starting position of the action?

the ending position of the action?

the total time taken for the action?

the direction or change for each section of the action?

the time(s) of any changes in direction or changes in the action?

the amount of change and time taken for each section of the action?

an interesting story that ties all sections of the graph together?

Scale your graph, and label each axis!

SMART Notebook

5.2.3: Oral Presentation Story Graphs

With a partner, write a story to match 5 out of the 10 graphs below. Be creative!

SMART Notebook

5.2.4: Interpretations of Graphs

Sunflower Seed Graphs Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl with the same amount of seeds. The graphs below all show the amount of sunflower seeds remaining in the person’s bowl over a period of time. Write sentences that describe what may have happened for each person.

a) b) c) d)

Multiple Choice Indicate which graph matches the statement. Give reasons for your answer.

1. A bicycle valve’s distance from the ground as a boy rides at a constant speed.

a) b) c) d)

2. A child swings on a swing, as a parent watches from the front of the swing.

SMART Notebook

5.2.4: Interpretations of Graphs Sunflower Seed Graphs Ian and his friends were sitting on a deck and eating sunflower seeds. Each person had a bowl with the same amount of seeds. The graphs below all show the amount of sunflower seeds remaining in the person’s bowl over a period of time. Write sentences that describe what may have happened for each person.

a) b) c) d)

Multiple Choice Indicate which graph matches the statement. Give reasons for your answer.

1. A bicycle valve’s distance from the ground as a boy rides at a constant speed. a) b) c) d)

2. A child swings on a swing, as a parent watches from the front of the swing.

SMART Notebook

Unit 5: Day 2: Story Graphs Grade 9 Applied

75 min

Math Learning Goals Write stories related to piecewise graphs; demonstrate the connection between the

position, direction, speed, and shape of the graph.

Investigate a variety of graphs in contexts with respect to rate of change, e.g., filling

containers, raising a flag, temperature.

Materials overhead

projector

BLM 5.2.1,

5.2.2, 5.2.3, 5.2.4

Whole Class Discussion Explain the activity on BLM 5.2.1. Answer any questions. Use BLM 5.2.2 to discuss

what their stories must include.

Stress the difference between constant rate of change and variable rate of change

Minds On…

Pairs Note Making/Presentation Using the graphs from BLM 5.2.3, students work in pairs to write 5 stories and orally

present 1-2 of them to the class.

Encourage students to think beyond the distance-time graphs done on the CBR™ and

think about raising a flag, filling containers, etc. Show some examples.

Note: Most students will find it easier to think of time as the independent variable

rather than some other measure.

Curriculum Expectation/Observation/Checklist: Use BLM 5.2.2 as a tool to assess

communication.

An alternative is to have students write their story on the SMART board, OR copy the graph onto chart paper and write their story next to the graph. Students may wish to act out their story as well as give their oral presentation.

Action!

Whole Class Discussion Review the graphs with students and clarify any information that students may have

misinterpreted (BLM 5.2.3).

Curriculum Expectations/Observation/Checklist: Assess student ability to use

proper conventions for graphing.

Whole Class Exploration

Using the Gizmo for Distance Time Graphs, explore the effect of changing conditions

(start and end point, adding additional points, horizontal lines, etc.) on the speed and

direction of the runner.

While this exercise is a repeat from the previous lesson, it is a further opportunity to

consolidate the concepts in these two lessons. Encourage students to add additional

segments and come up with real-life situations for the runner’s movement.

A common student interpretation of these graphs involves going up and down hills. Explain that a hill is not necessary to explain the graph. Word Wall increasing rapidly increasing slowly decreasing rapidly decreasing slowly constant rate of change varying rate of change

See Think Literacy, Mathematics, pages 62–68 for more information on reading graphs.

Consolidate Debrief

Concept

Practice

Application

Home Activity or Further Classroom Consolidation Complete worksheet 5.2.4, Interpretations of Graphs.

NCTM has many activities that relate to rates of change and graphs at www.nctm.org.

http://www.nctm.org/

SMART Notebook

Constant Rate of Change

A relationship between two variables

illustrates a constant rate of change

when equal intervals of the first variable

are associated with equal intervals of

the second variable.

Ex. If a car travels at 100 km/h, in the first

hour it travels 100 km, in the second hour

it travels 100 km, and so on.

SMART Notebook

Grade 9 & 10 Math Glossary Taken from Ministry of Education

Grades 9 & 10 Revised Math Curriculum

Acute Triangle

A triangle in which each of the three interior angles measures less than 90º.

Ex.

Algebra Tiles

Manipulatives that can be used to model operations involving integers,

polynomials, and equations. Each tile represents a particular

monomial, such as +1 , -1 , +x , -x , +x2 or –x2.

Ex. y = 3x2 - 4x + 7

Algebraic Expression

A collection of symbols, including one or more variables and possibly

numbers and operation symbols.

Ex.: 3x + 6, x , 5t , and 21 – 2w are all algebraic expressions.

Algebraic Modelling

The process of representing a relationship by an equation or a

formula, or representing a pattern of numbers by an algebraic expression.

Ex. The stars keep changing and match the term number while the pentagons stay constant.

s + 2 is the pattern or algebraic expression.

Algorithm

A specific set of instructions for carrying out a procedure.

Ex. y = mx + b a2 + b2 = c2

Alternate Interior Angle

A pair of angles on the inner sides of two lines cut by a transversal, but on opposite sides of the transversal

Ex. ∠3 & ∠ 6 and∠ 4 & ∠5 are alternate interior angles.

Alternate Exterior Angle

A pair of angles on the inner sides of two lines cut by a transversal, but on opposite sides of the transversal

Ex. ∠1 & ∠ 8 and∠ 2 & ∠7 are alternate exterior angles.

Altitude

A line segment giving the height of a geometric figure.

Ex.

Analytic Geometry A geometry that uses the xy-plane to determine equations that represent

lines and curves.

Ex:

The xy-plane helps us determine the equation of the line which is y = 2x + 1.

Angle Bisector

A line that divides an angle into two equal parts.

Ex.      

Angle of Elevation

The angle formed by the horizontal and the line of sight (to an object

above the horizontal).

Ex.

Wheelchair ramp

Angle of Elevation

Application

The use of mathematical concepts and skills to solve problems drawn

from a variety of areas.

Area

The number of square units needed to cover a given surface.

Ex:

The area is 9 square units.

Associative Property of Addition

The property which states that for all real numbers a, b, and c, their sum is always the same, regardless of their grouping: (a + b) + c = a + (b + c).

Ex: (2 + 3) + 4 = 2 + (3 + 4)

Associative Property of Multiplication

The property which states that for all real numbers a, b, and c, their

product is always the same, regardless of their grouping:

(a x b) x c = a x (b x c).

Ex: (5 x 6) x 7 = 5 x (6 x 7)

Axes

Two perpendicular lines that intersect to form the coordinate plane.

Base

A number used as a repeated factor

Ex.: 83 = 8 x 8 x 8

The base is 8. It is used as a factor three times.

The exponent is 3.

Bimedian The line that joins the midpoints of 2

opposite sides of a quadrilateral Ex.

Binomial

An algebraic expression containing two terms

Ex: 3x + 5y 7t – 12z2

Bisect

To divide into two congruent parts.

•

Ex: segment AB is bisected at point P.

A

P

B

Chord

A line segment joining two points on a curve.

Ex. chord RQ

Coefficient

The factor by which a variable is multiplied.

Ex.: In the term 5x, the coefficient is 5.

In the term ax, the coefficient is a.

R

Q

Communicative Property of Addition

The property of addition that allows two or more addends to be added in any order without changing the sum;

a + b = b + a.

Ex.: c + 4 = 4 + c (2 + 5) + 4r = 4r + (2 + 5)

Communicative Property of Multiplication

The property of multiplication that allows two or more factors to be multiplied in any order without

changing the product. a x b = b x a

Ex: 3 x c = c x 3 4 x 5 x y7 = 5 x 4 x y 7

Computer Algebra System (CAS)

A software program that manipulates and displays mathematical expressions (and equations)

symbolically.

Ex. CAS is used on the TI-Nspire calculators.

Congruence

The property of being congruent. Two geometric figures are congruent if

they are equal in all respects.

Ex:

Conjecture

A guess or prediction based on limited evidence.


A relationship between two variables illustrates a constant rate of change

when equal intervals of the first variable are associated with equal

intervals of the second variable.

Ex. If a car travels at 100 km/h, in the first hour it travels 100 km, in the second hour it travels 100 km, and so on.

Continuous

Can be broken into smaller and smaller parts that still have meaning.

Ex. the length of a piece of string

Continuous Variable

Variables that can have any value; represent quantities that you can

measure in parts; use a solid line to join continuous data.

Ex. time or distance

Coordinates

An ordered pair, used to describe a location on a grid labeled with an

x-axis and a y-axis.

Ex: y-axis

4

3 vertical 3 units

2

1

0 1 2 3 4 x-axis

horizontal 2 units

The coordinates (2, 3) describe this location.

Coordinate plane

A plane formed by two perpendicular number lines called axes; every point on the plane can be named by an ordered pair of

numbers.

Corresponding Angles

Angles that are in the same position and are formed by a transversal

cutting two or more lines

Ex. ∠2 and ∠6 are corresponding angles.

Cosine Law

The relationship, for any triangle, involving the cosine of one of the

angles and the lengths of the three sides; used to determine

unknown sides and angles in triangles. If a triangle has sides a, b, and c, and if the angle A is opposite side a, then:

a2 = b2 + c2 – 2bc cosA.

Ex.

a2 = b2 + c2 – 2bc cos A a2 = (3.4cm)2 + (3.2cm)2 - 2(3.4cm)(3.2cm)cos130o

a2 = 11.56cm2+10.24cm2–21.76cm2(-0.3672913304546) a2 = 21.8cm2 + 7.9922593506cm2 a2 = 29.7922593506cm2 a = √29.7922593506cm2 a ≅ 5.46 cm (rounded to the hundredths)

a

b = 3.4 cm c = 3.2 cm A

B C

130o

Cosine Ratio

For either of the two acute angles in a right triangle, the ratio of the length of the adjacent side to the length of

the hypotenuse.

Ex. cosine =

Counter-Example

An example that proves that a hypothesis or conjecture is false.

adjacent side hypotenuse

93.8 cm

x cm 38o

cosine = adjacent side hypotenuse cos38o = x cm 93.8 cm cos38o x 93.8 cm = x cm x 93.8 cm 93.8 cm 93.8cm x cos38o = x cm 73.2214 cm = x cm 73.22 cm ≅ x

Curve of Best Fit

The curve that best describes the distribution of points in a scatter plot.

Ex. Similar to the Line of Best Fit with the only difference being that the line is now curved.

Deductive Reasoning

The process of reaching a conclusion by applying arguments that have already been proved and using

evidence that is known to be true.

Dependent Variable

The variable in a relation whose value depends on the value of the independent variable.

Ex. Doing a science lab where you are comparing how far a car goes over time, time stays constant (so it is the independent variable) while distance changes over time (so it is the dependent variable).

Diagonal

In a polygon, a line segment joining two vertices that are not next to each

other (i.e., not joined by one side).

Ex.

Difference of Squares

An expression of the form a2 – b2, which involves the subtraction of two

squares.

Ex. a2 – b2 = 82 – 112 = 64 – 121 = -57

Direct Variation

A relationship between two variables in which one variable is a constant

multiple of the other. An equation of the form

y = kx, where k ≠ 0. There is NO y-intercept (b)

Ex. y = 4x y = 19x y = 248x

Discrete

Data that can be described by whole numbers or fractional values. Cannot be broken into smaller parts.

Ex. the number of cars in a parking lot

Discrete Variable

Whole numbers only; represent quantities that you cannot measure

in parts; use a dashed line to join discrete data.

Ex. number of students in a classroom

Distributive Property of Multiplication over Addition

The property which states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then

adding the products. a (b + c) = a x b + a x c

Ex: 3 (4 + 5) = 3 x 4 + 3 x 5

3 (b + c) = 3b + 3c

Dynamic Geometry Software

Computer software that allows the user to plot points and create graphs on a coordinate system, measure line segments and angles, construct two-

dimensional shapes, create two-dimensional representations of three-dimensional objects, and transform constructed figures by moving parts

of them.

Ex. Geometer’s SketchPad would be an example of this type of software.

Evaluate

To determine a value for the unknown.

Exponent

A special use of a superscript in mathematics.

Ex.: In 32, the exponent is 2.

An exponent is used to denote repeated multiplication.

Ex.: 94 means 9 x 9 x 9 x 9. In this example. 9 is multiplied by itself 4 times.

Expression

A mathematical phrase that combines operations, numerals,

and/or variables to name a number.

Ex: 35t – 15.5 32 x a

** Notice that there are NO EQUAL signs.

Exterior Angles

In the figure, transversal t intersects lines m and n.

Ex:

∠1, ∠2, ∠7, and ∠8 are exterior angles.

t

n

m

Extrapolate

To estimate values lying outside the range of given data.

To extrapolate from a graph means to estimate coordinates of points beyond those that are plotted.

Ex: The trend is an increase with no declines. From 1970 to 1990, the population increased about 25 million every 10 years.

Since the population increased about 25 million every 10 years from 1970 to 1990, a

good prediction for the year 2000 would be about 250 million + 25 million, or 275 million.

Factor

•1• To express a number as the product of two or more numbers.

Ex. 2 x 4 = 8 2 and 4 are factors of 8.

•2• An algebraic expression as the

product of two or more other algebraic expressions.

Ex. 3x x (4x + 5) = 12x2 + 15x 3x and 4x + 5 are factors of 12x2 + 15x.

•3• The individual numbers or

algebraic expressions in such a product.

Ex. 7 x (3h – 12) = 21h – 84 7 and 3h – 12 are factors of 21h – 84.

Finite Differences

Given a table of values in which the x-coordinates are evenly spaced, the

first differences are calculated by subtracting consecutive y-coordinates. The second differences are calculated

by subtracting consecutive first differences, and so on.

In a linear relation, the first differences are constant.

Ex. Refer to first differences for an example.

In a quadratic relation of the form y = ax2 + bx + c, where a ≠ 0,

the second differences are constant. d = at2 + bt + c

Time (s) t

Distance (m)

d

1

7

2

14

3

27

4

46

7

13

19

6

6

Second Difference (Change in Distance)

First Difference (Change in Distance)

First-Degree Equation

An equation in which the variable has the exponent 1.

Ex. 5(3x – 1) + 6 = –20 + 7x + 5

First-Degree Polynomial

A polynomial in which the variable has the exponent 1.

Ex. 4x + 20 8g – 35 9m + 2

First Difference The difference between two

consecutive y-values in a table when the differences between the x-values

are constant. Ex.

Time (s) Distance (m) 0 0 1 10 2 20 3 30

Fraction

A number representing some part of a whole or part of a set. A quotient

in the form .

Ex.

a b

3 4

63 92

329 872

10

10

10


Generalize

To determine a general rule or make a conclusion from examples.

Specifically, to determine a general rule to represent a pattern or

relationship between variables.

Graphing Calculator

A hand-held device capable of a wide range of mathematical

operations, including graphing from an equation, constructing a scatter plot, determining the equation of a curve of best fit for a scatter plot,

making statistical calculations, performing symbolic manipulation.

Many graphing calculators will attach to scientific probes that can be used

to gather data involving physical measurements (e.g., position,

temperature, force).

Graphing Software

Computer software that provides features similar to those of a graphing

calculator.

Greatest Common Factor (GCF)

The largest common factor of two or more given numbers

Ex: 18 : 1, 2, 3, 6, 9, 18

30 : 1, 2, 3, 5, 6, 10, 15, 30

1, 2, 3, and 6 are all common factors of 18 and 30. Only 6 is known as the GCF as it is

the largest of the common factors.

Hypotenuse

In a right triangle, the side opposite the right angle; the longest side in a

right triangle.

Ex:

Hypothesis

A proposed explanation or position that has yet to be tested.

Identity Property of One

The property which states that multiplying a number by 1 does not change the number's value.

Ex: 6 x 1 = 6 1 x a = a

Inequality

A mathematical sentence that shows the relationship between quantities

that are not equal, using , 12

Imperial System

A system of weights and measures built on the basic units of measure of the yard (length), the pound (mass),

the gallon (capacity), and the second (time). Also called the British system. The imperial system is used in

the United States.

Ex. Length: yard, feet, inches Mass: pound, ounces, pinch Capacity: gallon, fluid ounces

Improper Fraction

A fraction that has a numerator that is greater than or equal to the

denominator.

Ex.

7 4

63 59

931 70

Independent Variable

The variable in a function whose value is subject to choice.


Inductive Reasoning

The process of reaching a conclusion or making a generalization on the

basis of specific cases or examples.

Indirect Measurement

A method of measuring distances by solving a proportion.

Ex:

Inference

A conclusion based on a relationship identified between variables in a set

of data.

Integers

The set of whole numbers and their opposites. All positive and negative

numbers including zero.

Ex:

Intercepts

The place (or point) where a graph crosses the axis.

Ex:

The x-intercept is 1 and the y-intercept is -2.

Interior Angles

Angles on the inner sides of two lines cut by a transversal.

Ex: Angles 3 &5 and 4 & 6 are interior angles.

t

n

m

Interpolate

To estimate values lying between elements of given data.

To interpolate from a graph means to estimate coordinates of points

between those that are plotted.

Ex: Suppose you work for 2.5 hours and are paid $4.50 per hour. Use the graph to predict your earnings.

On the horizontal axis of the graph, locate 2.5

Draw a vertical line segment from this point to the line of the graph.

From there, draw a horizontal line segment to the vertical axis. It intersects the axis halfway between $9.00 and $13.50.

So, if you work 2.5 hr, you will earn $11.25.

Inverse Operations

Pairs of operations that undo each other.

Ex. • Addition and subtraction are inverse operations. ∴ a + b = c means that c – a = b

• Multiplication and division are inverse operations. ∴ d x t = s means that s ÷ t = d

• Squaring and square rooting are inverse operations. ∴ 52 = 25 means that √25 = 5

Irrational Numbers

Numbers that cannot be expressed as fractions, terminating decimals, or

repeating decimals.

Ex.

Kite

A quadrilateral with exactly two distinct pairs of adjacent congruent

sides.

Ex.

Isolate When solving an for a variable,

redistribute any other terms as well as the coefficient so that the variable is all by itself on one side of the equal

sign. Ex. 2x + 7 = 15

2x + 7 – 7 = 15 – 7

2x = 8

2x = 8 2 2

x = 4 * Use opposite operations to redistribute the terms and/or coefficient to the other side of

the equal sign.

Linear Equation

An equation whose graph is a straight line.

Ex:

The linear equation for the graph below is y = 2x + 1.

Linear Relation

A relation between two variables that appears as a straight line when

graphed on a coordinate system. May also be referred to as a linear

function.

Ex.

Points (0,1) and (1,3) are both on the line y = 2x + 1.

Least Common Multiple (LCM)

The smallest number, other than zero, that is a multiple of two or more given

numbers

Ex: The LCM of 6 and 9 is 18.

x 1 x 2 x 3 x 4 6 6 12 18 24 9 9 18 27 36

Like Terms

Expressions that have the same variables and the same powers of the

variables.

Ex: 8y, -4y, and 9.1y are like terms because they all involve the variable y.

Line of Best Fit

A straight line drawn through as much data as possible on a scatterplot.

Ex:

Manipulate

To apply operations, such as addition, multiplication, or factoring, on

algebraic expressions.

Mathematical Model

A mathematical description (e.g., a diagram, a graph, a table of values, an equation, a formula, a physical

model, a computer model) of a situation.

Mathematical Modelling

The process of describing a situation in a mathematical form. See also

mathematical model.

Mean (or Equal Sharing)

The sum of a set of numbers divided by the number of addends (or the

number of entries in our list)

Ex. 2, 3, 4, 5, 5, 8

(2 + 3 + 4 + 5 + 5 + 8) ÷ 6 = 4.5 6 is the number of entries in our list

The mean is 4.5

** Addends are the numbers that we add together. **

Median

In Geometry: The line drawn from a vertex of a triangle to the midpoint of

the opposite side. Ex.

In Statistics: The middle number in a

set, such that half the numbers in the set are less and half are greater when

the numbers are arranged in order. Ex. 1, 3, 4, 6, 7

The median is 4. 1, 2, 4, 5, 8, 9

The median is 4.5. (4.5 is the middle value between 4 and 5)

A

B C

Method of Elimination

In solving systems of linear equations, a method in which the coefficients of

one variable are matched through multiplication and then the equations are added or subtracted to eliminate

that variable.

Midpoint

The point that divides a line segment into two congruent line segments.

Ex:

M is the midpoint of .

Method of Substitution

In solving systems of linear equations, a method in which one equation is rearranged and substituted into the

other.

Ex. Solve for x where y=4x + 3 and y=5x - 7.

: Substitute y=4x + 3 into y=5x - 7 as the value of y.

: Solve the new equations by isolating for the variable x .

3rd: Once you have a value for x, test your answer by putting it back into the original questions.

: 4x + 3 = 5x - 7 : 4x + 3 – 3 = 5x - 7 – 3 4x = 5x – 10 4x – 5x = 5x – 10 – 5x -x = -10 -x = -10 -1 -1 x = 10

CHECK: 4x + 3 = 5x – 7 4(10) + 3 = 5(10) - 7 40 + 3 = 50 – 7 43 = 43

Since our CHECK works out,

we know that we did our work correctly!

Midsegment

A segment with endpoints that are the midpoints of two sides of a

triangle.

Ex.

Motion Detector

A hand-held device that uses ultrasound to measure distance. The data from motion detectors can be transmitted to graphing calculators.

Ex. CBR would be a motion detector.

Mixed Numbers

The indicated sum of a whole number and a fraction.

Ex. 1 7

3 4

2 7

Mode

The number or numbers that occur most frequently in a set of data; there can be one

mode, more than one mode or no mode.

2, 3, 4, 5, 5, 6, 7, 8, 8, 8, 9, 11 The mode is 8.

2, 3, 4, 5, 5, 5, 7, 8, 8, 8, 9, 11 The modes are 5 and 8.

2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17 There is no mode. Each entry appears an

equal number of times in the data set.

Monomial

An expression that is a number, a variable, or the product of a

number and one or more variables.

Ex: 3x + y 7y 5x2y

Multiple Trials

A technique used in experimentation in which the same experiment is done

several times and the results are combined through a measure such

as averaging. The use of multiple trials “smooths out” some of the random

occurrences that can affect the outcome of an individual trial of an

experiment.

Negative Reciprocal

The multiplicative inverse of a number but in it’s negative form.

Ex. X - = -1

- is the negative reciprocal of .

is the negative reciprocal of - .

2 3

3 2

3 2

2 3

3 2

2 3

Net

A connected arrangement of polygons in a plane that can be folded up to form a polyhedron.

Ex:

Nonlinear Function

A function whose graph is not a straight line.

Ex:

Non-Real Root of an Equation

A solution to an equation that is not an element of the set of real

numbers.

Non-Linear Relation

A relationship between two variables that does not fit a straight line when

graphed.

Ex:

Numerical Expression

An expression that includes numbers and at least one operation (addition,

subtraction, multiplication, or division).

Ex: 6 + 8.1 57 – 48 21.6 x 18.6

Outlier

A point that does not follow the pattern described by the line of best fit.

Opposite Integers

0 1 2 3 4 5 6 7 8 Time (s)

D

ista

nc

e (

m)

2

4

6

8

10

12

1

4

16

Distance versus Time

Opposite Integers

Two numbers that are represented by points on the number line that are the same distance from zero but are on

opposite sides of zero.

Ex:

4 and -4 are opposites

Optimal Value

The maximum or minimum value of a variable.

Order of Operations

The order in which the operations are done within an expression 1. Operate inside parentheses. 2. Multiply as indicated by exponents.

3. Multiply and divide from left to right. 4. Add and subtract from left to right.

Ex.: 10 ÷ (2 + 8) x 23 - 4 Add inside parentheses 10 ÷ 10 x 23 - 4 Solve the exponent 10 ÷ 10 x 8 – 4 Divide 1 x 8 – 4 Multiply (remember left to right) 8 – 4 Subtract 4 Final answer!

B E D M A S

B E D M A S

Ordered pair

A pair of numbers used to locate a point on a coordinate plane.

Ex:

(3,2) represents

3 spaces to the right of zero and 2 spaces up.

Origin

The point on the coordinate plane where the x-axis and the y-axis

intersect, (0,0).

Ex:

Parabola

The graph of a quadratic relation of the form y = ax2 + bx + c (a ≠ 0).

The graph, which resembles the letter “U”, is symmetrical.

Ex.

In this example, the equation is y=x2. The values of a, b, and c are as follows:

a=1, b=0, and c=0.

Parallel Lines

Lines in a plane that do not intersect.

Ex:

Line AB is parallel to line CD.

Partial Variation

A relationship between two variables in which one variable is a multiple of

the other, plus some constant number.

y = mx + b Ex.: The cost of a taxi fare has two

components, a flat fee and a fee per kilometre driven.

Fare = $0.15/km + $3.50 The flate fee is $3.50 and the fee per kilometer of travel is $0.15.

Perfect Square

A number that has an integer as its square root

Ex.: 16 is a perfect square.

Perpendicular Lines

Lines that intersect to form 90° angles, or right angles.

Ex:

Line RS is perpendicular to line MN.

Pi (π)

The ratio of the circumference of a circle to the length of its diameter.

Polygon

A closed figure formed by three or more line segments.

Ex.

Polynomial

A monomial or the sum of two or more monomials.

Ex: 3a2 + 8 a4 - 4a3 + 13 92b7t

Polynomial Expression

An algebraic expression taking the form a + bx + cx2 + . . . , where a, b,

and c are numbers.

Ex. 3 + 5x – 2x2

Population Statistics

The total number of individuals or items under consideration in a surveying or sampling activity.

Power

The value of a number represented by a base and an exponent.

Ex: 43 = 4 x 4 x 4 = 64

Primary Trigonometric Ratios

The basic ratios of trigonometry (i.e., sine, cosine, and tangent).

sinθ = = cscθ = = cosθ = = secθ = =

tanθ = = cotθ = =

Ratio Abbreviations and their Names sinθ = sine of θ cosθ = cosine of θ tanθ = tangent of θ

cscθ = cosecant of θ secθ = secant of θ cotθ = cotangent of θ

side adjacent to angle θ

hypotenuse

side

ad

jac

en

t to

an

gle

θ

θ

α

r

x

y

side opposite θ y hypotenuse r

side adjacent θ x hypotenuse r

hypotenuse r side opposite θ y


hypotenuse y side opposite θ r

hypotenuse r side adjacent θ x

Prism

A three-dimensional figure with two parallel, congruent polygonal bases. A prism is named by the shape of its

bases.

Ex. rectangular prism Ex. Cylindrical prism

Product

The answer in a multiplication problem.

Ex:

The product is 12.

Proportional Reasoning

Reasoning or problem solving based on the examination of equal ratios.

Pythagorean Theorem

The conclusion that, in a right triangle, the square of the length of the

longest side is equal to the sum of the squares of the lengths of the other

two sides. a2 + b2 = c2

Ex:

a2 + b2 = c2 Replace the variables (3cm)2 + (4cm)2 = c 2 with the known lengths. 9cm2 + 16cm2 = c 2 25cm2 = c 2 √25cm2 = c 5cm = c

Quadrant

One of the four regions of the coordinate plane.

Quadratic Equation

An equation that contains at least one term whose exponent is 2, and no term with an exponent greater

than 2.

Ex. x2 + 7x - 10 = 0 x2 - 4x + 2 = 9

Quadratic Formula

A formula for determining the roots of a quadratic equation of the form ax2 + bx + c = 0. The formula is phrased in

terms of the coefficients of the quadratic equation: x = –b +-(vb2 – 4ac)

2a.

Ex. x = –b +-(vb2 – 4ac) 2a. = -5 + -(4 x 52 – 4 x 3 x 7) 2 x 3 = -5 + -(4 x 100 – 84) 6 = -5 + -(400 – 84) 6 = -5 + -316 6 = -321 6 x = -53.5

Quadratic Relation

A relation whose equation is in quadratic form.

Ex. y= x2 + 7x - 10 y= x2 - 4x + 2

Quadrilateral

A polygon with four sides.

Ex.

Rate of Change

The change in one variable relative to the change in another. The slope of a line represents rate of change.

Ex. Driving 60 km per hour in the city. (60 km/h)

Growth spurt of 5 cm per year (5 cm/yr).

Paying $5 per dozen for cookies. ($5/dz)

Burning 100 calories in 30 minutes on the treadmill.

(100 cal/30 min OR 200 cal/h)

Rational Number

A number that can be expressed as the quotient of two integers where

the divisor is not 0.

Ex. 1= -2.3 = 2

Real Numbers

The set of numbers that includes all rational and all irrational numbers.

1 1

2 9

3 10

Realistic Situation

A description of an event or events drawn from real life or from an

experiment that provides experience with such an event.

Real Root of an Equation

A solution to an equation that is an element of the set of real numbers. The set of real numbers includes all

numbers commonly used in daily life: all fractions, all decimals, all negative

and positive numbers.

Reciprocal

One of two numbers whose product is 1.

Ex:

Regression

A method for determining the equation of a curve (not necessarily a straight line) that fits the distribution

of points on a scatter plot.

Regular Polygon

A polygon in which all sides and all angles are congruent.

Ex:

Relation

An identified relationship between variables that may be expressed as a

table of values, a graph, or an equation.

A description of how 2 variables are connected.

Ex. y = 3x – 14 y = x

Representivity

A principle of data analysis that involves selecting a sample that is typical of the characteristics of the population from which it is drawn.

Right Bisector

The line that is perpendicular to a given line segment and that passes

through its midpoint.

Ex. A Q B

R

Right Triangle

A triangle containing one 90º angle.

Ex.

Rise

The vertical change of a line.

Ex: The rise of line PQ is 2.

Run

The horizontal change of a line.

Ex: The run of line PQ is 1.

Sample

A small group chosen from a population and examined in order to make

predictions about the population.

Ex. The people in the Alta Vista area are asked who they will vote for in the upcoming elections. They are a sample group from Ottawa’s population.

Sampling Technique

A process for collecting a sample of data.

Ex. Questionnaire, phone interview, etc.

Scatter Plot

A graph that attempts to show a relationship between two variables

by means of points plotted on a coordinate grid. Also called scatter

diagram.

Ex.

Scientific Probe

A device that may be attached to a graphing calculator or to a computer

in order to gather data involving measurement (e.g., position,


Second-Degree Polynomial

A polynomial in which the variable in at least one term has an exponent 2,

and no variable has an exponent greater than 2

Ex. 4x2 + 20 x2 + 7x + 10

Second Differences.

See finite differences.

Similar Figures

Figures that have the same shape but may not have the same size.

Ex:

7 cm 3 cm 4 cm

14 cm 6 cm 8 cm

Similar Triangles

Triangles in which corresponding sides are proportional.

Ex.      

Simplest Form

•1• A fraction is in simplest form when the numerator and denominator have

no common factors other than 1.

Ex:

•2• The form of an expression when all like terms are combined.

Ex: 4x + 3x + 2 = 7x + 2 (in its simplest form)

Simulation

A probability experiment to estimate the likelihood of an event.

Ex. Tossing a coin is a simulation of whether the next person you meet will be male or female.

Sine Law

The relationships, for any triangle, involving the sines of two of the angles and the lengths of the

opposite sides; used to determine unknown sides and angles in

triangles. Ex.

a = b = c sinA sinB sinC

a

b = 3 cm c = 3 cm

B C

130o

A

Sine Ratio

For either of the two acute angles in a right triangle, the ratio of the length of the opposite side to the length of

the hypotenuse.

Please refer to PRIMARY TRIGONOMETRIC RATIOS for the equation.

Slant Height The altitude or height of each lateral

face of a pyramid.

Ex.

Slope

A measure of the steepness of a line, calculated as the ratio of the rise (vertical change between two

points) to the run (horizontal change between the same two points). Slope

Ex:

= vertical change horizontal change = y2 – y1 x2 – x1

Solution

The value or values that make an equation an inequality, or system of

equations true.

Ex: x – 4 = 16 x + 3 > 9 x – 4 + 4 = 16 + 4 x + 3 – 3 > 9 – 3 x = 20 x > 6 20 is the solution. Any number

greater than 6 is a solution.

Solution of a System

The values for the variables that make each of the equations in a

system of equations true; the coordinates of the point where the

graphs of the equations in the system intersect.

Ex: The equations of these lines are

y = 2x – 1 and y = x + 1

The solution of the system is (2,3).

Solve

To find the correct value of the variable in an equation.

Ex.: 2x + 5 = 17 *Isolate for 2x (-5) 2x + 5 – 5 = 17 – 5 2x = 12 *Isolate for x (÷2) 2x = 12 2 2 x = 6

** Notice how we how we have to ‘balance’ each side of our equation in each step. **

Spreadsheet

Computer software that allows the entry of formulas for repeated

calculation.

Ex. Excel, Quattro Pro, TinkerPlot, Fathom, etc.

Stretch Factor

A coefficient in an equation of a relation that causes stretching of the

corresponding graph.

Ex. The graph of y = 3x 2 appears to be

narrower than the graph of y = x 2 because its y-coordinates are three times as great for the same

x-coordinate. (In this example, the coefficient 3 causes the graph to stretch vertically, and is referred to as a vertical stretch factor.)

y = 3x 2 y = x 2

Square Number

A number that can be represented with a square array.

Ex:

Square Root

One of the two equal factors of a number.

Ex.: 6 is the square root of 36 since 62 = 6 x 6 = 36

So, √36 = 6 because we are looking for the one value or factor (which is 6 for this

example) that gives us the number below the square root sign.

Substitution

To replace a variable with a value (or number).

Ex: Which of the values 12, 20, and 21 are solutions of x - 4 = 16?

Substitute each of the values for x in the equation.

Supplementary Angles

Two angles are supplementary if the sum of their measures in 180 o.

∠1 ∠2

∠1 + ∠2 are equal to 180o.

Use x = 12. x - 4 = 16 12 - 4 = 16 8 = 16

not a solution

Use x = 20. x - 4 = 16 20 - 4 = 16 16 = 16

solution

Use x = 21. x - 4 = 16 21 - 4 = 16 17 = 16

not a solution

Surface Area

The sum of the areas of all the faces, or surfaces, of a solid figure.

Ex:

AreaA = 11m X 5m AreaD = 21m X 11m = 55m2 = 231m2 AreaB = 21m X 11m AreaE = 21m X 5m = 231m2 = 105m2 AreaC = 21m X 5m AreaF = 11m X 5m = 105m2 = 55m2

S. A. = AreaA+AreaB + AreaC + AreaD + AreaE + AreaF = 55m2 + 231m2 + 105m2 + 231m2 + 105m2 + 55m2 = 782m2

So, the surface area is 782 m2.

System of Linear Equations

Two or more linear equations involving two or more variables. The

solution to a system of linear equations involving two variables is

the point of intersection of two straight lines.

Ex.

Table of Values

A table used to record the coordinates of points in a relation.

Ex.

Tangent Ratio

For either of the two acute angles in a right triangle, the ratio of the length of the opposite side to the length of

the adjacent side.

Please refer to PRIMARY TRIGONOMETRIC RATIOS

for the equation.

Term Number Term Value y = 3x + 5

1 8 2 11 3 14 4 17 5 20

Term

•1• A real number, a variable, or a product of real numbers and

variables.

Ex.: In the expression 5x + 4, the terms are 5x and 4.

•2• A number in a sequence.

Ex.: 3, 6, 12, 24, . . . 6 is a term in the sequence

•3• One of the numbers in a ratio.

Ex.:

Transformation

A change in a figure that results in a different position, orientation, or size.

The transformations include translation, reflection, rotation, and

dilatation (compression and stretch).

Ex.

Trinomial

The sum of three monomials.

Ex: 3x + 5y + 7 6x2 – 14x + 8

Translation Reflection

Dilatations

Compression

Stretch

Rotation (90o clockwise)

Center of rotation

Trapezoid

A quadrilateral with one pair of parallel sides.

Ex.

Variable

A symbol used to represent an unspecified number.

Ex.: x and y are variables in the expression x + 2y.

Vertex

The point at which two sides of a polygon meet.

Ex.

Volume

The number of cubic units needed to occupy a given space.

Ex:

The volume of the cube is 8 cubic units.

vertex vertex

x-axis

The horizontal axis on the coordinate plane.

Ex:

x-coordinate

The first number in an ordered pair; tells whether to move right or left

along the x-axis of the coordinate plane.

Example: (3, 2)

3 is the x-coordinate.

x-intercept

The x-coordinate of a point at which a line or curve intersects the x-axis.

Ex:

The x -intercept is 1.

xy-plane

A coordinate system based on the intersection of two straight lines called axes, which are usually

perpendicular. The horizontal axis is the x-axis, and the vertical axis is the y-axis. The point of intersection of the

axes is called the origin.

Ex:

y-axis

The vertical axis on the coordinate plane.

Ex:

y-coordinate

The second number in an ordered pair; tells whether to move up or

down along the y-axis of the coordinate plane.

Ex: (3, 2) 2 is the y-coordinate.

y-intercept

The y-coordinate of the point at which a line or curve intersects the y-

axis.

Ex:

The y-intercept of 2x + 3y = 6 is 2.

Zeros of a Relation

The values of x for which a relation has a value of zero. The zeros of a

relation correspond to the x -intercepts of its graph.

See also x-intercept.

SMART Notebook

Grade 9 & 10 Math Glossary Most definitions are taken from Ministry of Education

Grades 7, 8, 9, & 10 Revised Math Curriculums All diagrams were created by Linda LoFaro (OCSB).

Acute Triangle

A triangle in which each of the three interior angles measures less than 90º.

Ex.

Algebra Tiles

Manipulatives that can be used to model operations involving integers,

polynomials, and equations. Each tile represents a particular

monomial, such as +1 , -1 , +x , -x , +x2 or –x2.

Ex. y = 3x2 - 4x + 7

Algebraic Expression

A collection of symbols, including one or more variables and possibly

numbers and operation symbols.

Ex.: 3x + 6, x , 5t , and 21 – 2w are all algebraic expressions.

Algebraic Modelling

The process of representing a relationship by an equation or a

formula, or representing a pattern of numbers by an algebraic expression.

Ex. The stars keep changing and match the term number while the pentagons stay constant.

s + 2 is the pattern or algebraic expression.

Algorithm

A specific set of instructions for carrying out a procedure.

Ex. y = mx + b a2 + b2 = c2

Alternate Interior Angle

The pair of equivalent angles located at the inside corners of two parallel lines that are cut by a transversal line. This is also known as a Z-pattern.

Ex.

∠3 & ∠ 6 and∠ 4 & ∠5 are alternate interior angles.

Alternate Exterior Angle

The pair of equivalent angles located at the outside corners of two parallel lines that are cut by a transversal line.

Ex. ∠1 & ∠ 8 and∠ 2 & ∠7 are alternate exterior angles.

Altitude

A line segment giving the height of a geometric figure.

Ex.

Analytic Geometry A geometry that uses the xy-plane to determine equations that represent

lines and curves.

Ex:

The xy-plane helps us determine the equation of the line which is y = 2x - 1.

Angle Bisector

A line that divides an angle into two equal parts.

Ex.      

Angle of Elevation

The angle formed by the horizontal and the line of sight (to an object

above the horizontal).

Ex.

Wheelchair ramp

Angle of Elevation

BG bisect ∠ABC

m∠ABG = m∠GBC

Application

The use of mathematical concepts and skills to solve problems drawn

from a variety of areas.

Area

The number of square units that cover a surface.

Ex:

A = 9 units2

Associative Property of Addition

The property whereby the sum of any real numbers a, b, and c, is always

the same. The order does not matter: (a + b) + c = a + (b + c)

Ex: (2 + 3) + 4 = 2 + (3 + 4)

Associative Property of Multiplication

The property whereby the product of any real numbers a, b, and c, is

always the same. The order does not matter:

(a x b) x c = a x (b x c).

Ex: (5 x 6) x 7 = 5 x (6 x 7)

Axes

Two perpendicular lines intersect to form a coordinate plane with four

quadrants.

Base

A number that is repeatedly multiplied (a repeated factor)

Ex.: 94 = 9 x 9 x 9 x 9

The base is 9. It is used as a factor three times.

The exponent is 4.

Bimedian A line segment that joins the

midpoints of opposite sides in any quadrilateral

Ex.

Binomial

An algebraic expression containing two terms

Ex: 3x + 5y 7t – 12z2

Bisect

To divide an item (line, angle, etc.) into two congruent (or equal) parts.

Ex: segment AB is bisected at point M.

Chord

A line segment joining two points on a curve.

Ex. chord RQ

Coefficient

The factor by which a variable is multiplied.

Ex.: In the term 5x, the coefficient is 5.

In the term ax, the coefficient is a.

R

Q

Commutative Property of Addition

The property whereby the sum of two or more addends is always the same.

The order does not matter: a + b = b + a.

Ex.: c + 4 = 4 + c (2 + 5) + 4r = 4r + (2 + 5)

Commutative Property of Multiplication

The property whereby the product of two or more factors is always the

same. The order does not matter: a x b = b x a

Ex: 3 x c = c x 3 4 x 5 x y7 = 5 x 4 x y 7

Computer Algebra System (CAS)

A software program that manipulates and displays mathematical expressions (and equations)

symbolically.

Ex. CAS is used on the TI-Nspire calculators.

Congruence

The property of being congruent. Two geometric figures are congruent if

they are equal in all respects.

Ex: RST is congruent to ABC

Conjecture

A guess or prediction based on limited evidence.


A relationship between two variables illustrates a constant rate of change

when equal intervals of the first variable are associated with equal

intervals of the second variable.

Ex. If a car travels at 100 km/h, in the first hour it travels 100 km, in the second hour it travels 100 km, and so on.

Continuous

Data that can be broken down into smaller and smaller segments. The

data still makes sense in these smaller parts.

Ex. Time – can be measured in days, minutes, milliseconds, etc.

Continuous Variable

A variable that represents quantities that can be measured in different

segments. A solid line is used to join continuous data.

Ex. Distance – can be measured in kilometers, meters, centimeters, etc.

Coordinates

Two values, known as an ordered pair, used to describe a specific

location on a Cartesian Plane with an x-axis and a y-axis.

Ex: y-axis

4

3 vertical 3 units

2

1

0 1 2 3 4 x-axis

horizontal 2 units


Coordinate plane

Two perpendicular lines intersect to form a coordinate plane with four

quadrants. Each point is known as an ordered pair.

Corresponding Angles

The pair of equivalent angles found below the parallel lines that are

created at the intersections of the parallel lines and the transversal line.

It is also known as the F-pattern. Ex. ∠4 and ∠8 are corresponding angles.

Cosine Law

The relationship, for any triangle, involving the cosine of one of the

angles and the lengths of the three sides; used to determine

unknown sides and angles in triangles. If a triangle has sides a, b, and c, and if the angle A is opposite side a, then:

a2 = b2 + c2 – 2bc cosA.

Ex.

a2 = b2 + c2 – 2bc cos A a2 = (3.4cm)2 + (3.2cm)2 - 2(3.4cm)(3.2cm)cos130o

a2 = 11.56cm2+10.24cm2–21.76cm2(-0.3672913304546) a2 = 21.8cm2 + 7.9922593506cm2 a2 = 29.7922593506cm2 a = √29.7922593506cm2 a ≅ 5.46 cm (rounded to the hundredths)

a

b = 3.4 cm c = 3.2 cm A

B C

130o

Cosine Ratio

For either of the two acute angles in a right triangle, the ratio of the length of the adjacent side to the length of

the hypotenuse.

Ex. cosine =

Counter-Example

An example that proves that a hypothesis or conjecture is false.

adjacent side hypotenuse

93.8 cm

x cm 38o

cosine = adjacent side hypotenuse cos38o = x cm 93.8 cm cos38o x 93.8 cm = x cm x 93.8 cm 93.8 cm 93.8cm x cos38o = x cm 73.2214 cm = x cm 73.22 cm ≅ x

Curve of Best Fit

The curve that best describes the distribution of points in a scatter plot.

Ex. Similar to the Line of Best Fit with the only difference being that the line is now curved.

Deductive Reasoning

The process of reaching a conclusion by applying arguments that have already been proved and using

evidence that is known to be true.

Dependent Variable

The variable in a relation whose value depends on the value of the

independent variable.


Diagonal

In a polygon, a line segment joining two vertices that are not next to each

other (i.e., not joined by one side).

Ex.

Difference of Squares

An expression of the form a2 – b2, which involves the subtraction of two

squares.

Ex. a2 – b2 = 82 – 112 = 64 – 121 = -57

Direct Variation

A relationship between two variables in which one variable is a constant

multiple of the other. An equation of the form

y = kx, where k ≠ 0. There is NO y-intercept (b)

Ex. y = 4x y = 19x y = 248x

Discrete

Whole number or fractional value data. The data pieces cannot be cut

into smaller parts.

Ex. the number of cars in a parking lot

Discrete Variable

Data that can be represented by whole numbers only. The data pieces

cannot be cut into smaller parts. A dashed line is used to join discrete

data on a graph.

Ex. number of students in a classroom

Distributive Property of Multiplication over Addition

The property whereby the product of a sum is equivalent to the result of

multiplying each added separately and then adding the products.

a (b + c) = a x b + a x c

Ex: 3 (4 + 5) = 3 x 4 + 3 x 5

3 (b + c) = 3b + 3c

Dynamic Geometry Software

Computer software that allows the user to plot points and create graphs on a coordinate system, measure line segments and angles, construct two-

dimensional shapes, create two-dimensional representations of three-dimensional objects, and transform constructed figures by moving parts

of them.

Ex. Geometer’s SketchPad would be an example of this type of software.

Evaluate

To determine a value for the unknown.

Exponent

A special use of a superscript in mathematics.

Ex.: In 32, the exponent is 2.

An exponent is used to denote repeated multiplication.

Ex.: 94 means 9 x 9 x 9 x 9. In this example. 9 is multiplied by itself 4 times.

Expression

A mathematical statement where operations, numbers, and/or

variables are combined. There is no set resultant.

Ex: 35t – 15.5 32 x a

** Notice that there are NO EQUAL signs.

Exterior Angles

Line segment t creates an external ∠K to the interior angle ∠D

Ex: ∠K is an exterior angle to ∠D.

t

Extrapolate

To estimate values lying outside the range of a given data. To

extrapolate from a graph means to estimate coordinates of points beyond those that are plotted.

The population has continued to increase each year.

From 1961 to 2001, the population increased about 3 million every 10 years.

Since there has been continued growth within Canada at a fairly steady rate, we

can make a prediction that Canada’s population in 2011 will be approximately 34

million.

Ex.

Factor

•1• To express a number as the product of two or more numbers.

Ex. 2 x 4 = 8 2 and 4 are factors of 8.

•2• An algebraic expression as the

product of two or more other algebraic expressions.

Ex. 3x x (4x + 5) = 12x2 + 15x 3x and 4x + 5 are factors of 12x2 + 15x.

•3• The individual numbers or

algebraic expressions in such a product.

Ex. 7 x (3h – 12) = 21h – 84 7 and 3h – 12 are factors of 21h – 84.

Finite Differences

Given a table of values in which the x-coordinates are evenly spaced, the

first differences are calculated by subtracting consecutive y-coordinates. The second differences are calculated

by subtracting consecutive first differences, and so on.

In a linear relation, the first differences are constant.

Ex. Refer to first differences for an example.

In a quadratic relation of the form y = ax2 + bx + c, where a ≠ 0,

the second differences are constant. d = at2 + bt + c

Time (s) t

Distance (m)

d

1

7

2

14

3

27

4

46

7

13

19

6

6

Second Difference (Change in Distance)


First-Degree Equation

An equation in which the variable has the exponent 1.

Ex. 5(3x – 1) + 6 = –20 + 7x + 5

First-Degree Polynomial

A polynomial in which the variable has the exponent 1.

Ex. 4x + 20 8g – 35 9m + 2

First Difference The difference between two

consecutive y-values in a table of values.

Ex. Time (s) Distance (m)

0 0 1 10 2 20 3 30

Fraction

A number representing part of a whole or set, or a measure. It is

written in the form .

Ex.

a b

3 4

63 92

329 872

10

10

10


Generalize

To determine a general rule or make a conclusion from examples.

Specifically, to determine a general rule to represent a pattern or

relationship between variables.

Graphing Calculator

A hand-held device capable of a wide range of mathematical

operations, including graphing from an equation, constructing a scatter plot, determining the equation of a curve of best fit for a scatter plot,

making statistical calculations, performing symbolic manipulation.

Many graphing calculators will attach to scientific probes that can be used

to gather data involving physical measurements (e.g., position,


Graphing Software

Computer software that provides features similar to those of a graphing

calculator.

Greatest Common Factor (GCF)

The biggest common factor of two or more numbers.

Ex: 18 : 1, 2, 3, 6, 9, 18

30 : 1, 2, 3, 5, 6, 10, 15, 30

1, 2, 3, and 6 are all common factors of 18 and 30. Only 6 is known as the GCF as it is

the largest of the common factors.

Hypotenuse

The side opposite to the right angle in a right-angled triangle. It is also the

longest side.

Ex:

Hypothesis

A proposed explanation or position that has yet to be tested.

Identity Property of One

When a value is multiplied by the number 1, it does not change the

value.

Ex: 6 x 1 = 6 1 x a = a

Inequality

A mathematical statement showing the relationship between two or more

values (numbers or variables). The symbols used include: , 12

Imperial System

A system of weights and measures built on the basic units of measure of the yard (length), the pound (mass),

the gallon (capacity), and the second (time). Also called the British system. The imperial system is used in

the United States.

Ex. Length: yard, feet, inches Mass: pound, ounces, pinch Capacity: gallon, fluid ounces

Improper Fraction

A fraction with a numerator that is greater than the denominator.

Ex.

7 4

63 59

931 70

Independent Variable

The variable in a relation whose value you choose.

Ex. A CBR activity where you are comparing how far someone walks over time. In this case, time stays contant (independent variable) and the distance changes over time (dependent variable).

Inductive Reasoning

The process of reaching a conclusion or making a generalization on the

basis of specific cases or examples.

TM

Indirect Measurement

Using ratios to find a missing distance.

Ex.

=

5 x h = 2 x 15

5h = 30

=

h = 6 So the height of the tree is 6 m.

Inference

A conclusion based on a relationship identified between variables in a set

of data.

2 h

5 15

5h 30 5

Integers

A set of positive and negative whole numbers and the number zero.

Ex:

Intercepts

The place where a line crosses an axis.

Ex:

The x-intercept is 2 and the y-intercept is 1.

Interior Angles

The pair of adjacent angles between the parallel lines that share part of the transversal. Together, the two angles

have a sum of 180o.

Ex: Angles 4 & 6 are interior angles.

Interpolate To estimate values lying between

elements of given data. To interpolate from a graph means to

estimate coordinates of points between those that are plotted.

Ex:

You now know that you have earned $12.00 for working 3.5 hours.

Suppose you work for 3.5 hours at $4.00 per hour. Using the graph, you can predict your earnings. On the x-axis, create a vertical line to meet with the line of best fit. Then, extend a horizontal line, from this point, to the y-axis.

Inverse Operations

Pairs of operations that undo each other.

Ex. • Addition and subtraction are inverse operations. ∴ a + b = c means that c – a = b

• Multiplication and division are inverse operations. ∴ d x t = s means that s ÷ t = d

• Squaring and square rooting are inverse operations. ∴ 52 = 25 means that √25 = 5

Irrational Numbers

Numbers that are not whole numbers, integers, or rational numbers.

Ex.

Kite

A quadrilateral with two pairs of congruent sides. Each pair of

congruent sides are adjacent to each other.

Ex.

Isolate When solving for a variable, any other

terms and/or coefficients found on the same side of the equal sign are moved to the other side allowing for

a solution to be found. Ex. 2x + 7 = 15

2x + 7 – 7 = 15 – 7

2x = 8

2x = 8 2 2

x = 4 * Use opposite operations when moving

terms and/or coefficients to the other side.

Linear Equation

An equation with a graph that is a straight line.

Ex:

The linear equation for the graph below is y = 2x + 1.

Linear Relation

A relation between two variables that appears as a straight line when

graphed on a coordinate system. May also be referred to as a linear

function.

Ex.

Points (0,1) and (1,3) are both on the line y = 2x + 1.

Least Common Multiple (LCM)

A smallest multiple of two or more numbers, other than zero.

Ex: The LCM of 4 and 6 is 12.

x 1 x 2 x 3 x 4 4 4 8 12 16 6 6 12 18 24

Like Terms

Terms that share a common variable and/or the same powers of the

variables.

Ex: 7y, -3y, and 5.9y are like terms because they all involve the variable y.

Line of Best Fit

A straight line drawn through as much data as possible on a scatterplot.

Ex:

Manipulate

To apply operations, such as addition, multiplication, or factoring, on

algebraic expressions.

Mathematical Model

A mathematical description (e.g., a diagram, a graph, a table of values, an equation, a formula, a physical

model, a computer model) of a situation.

Mathematical Modelling

The process of describing a situation in a mathematical form. See also

mathematical model.

Mean (or Equal Sharing)

An equal sharing of values in a set of numbers. Take the sum of the set of

numbers and then divide by the number of entries (or addends) in the

list.

Ex. 2, 3, 5, 5, 7, 8

(2 + 3 + 5 + 5 + 7 + 8) ÷ 6 = 5 6 is the number of entries in our list

The mean is 5

** Addends are the numbers that we add together. **

Median

In Geometry: The line drawn from a vertex of a triangle to the midpoint of

the opposite side. Ex.

In Statistics: The middle number in a

set, such that half the numbers in the set are less and half are greater when

the numbers are arranged in order. Ex. 1, 3, 4, 6, 7

The median is 4. 1, 2, 4, 5, 8, 9

The median is 4.5. (4.5 is the middle value between 4 and 5)

A

B C

Method of Elimination

In solving systems of linear equations, a method in which the coefficients of

one variable are matched through multiplication and then the equations are added or subtracted to eliminate

that variable.

Midpoint

A point the splits a line segment into two equivalent or congruent portions.

Ex:

G is the midpoint of line segment FH.

Method of Substitution

In solving systems of linear equations, a method in which one equation is rearranged and substituted into the

other.

Ex. Solve for x where y=4x + 3 and y=5x - 7.

: Substitute y=4x + 3 into y=5x - 7 as the value of y.

: Solve the new equations by isolating for the variable x .

: Once you have a value for x, test your answer by putting it back into the original questions.

: 4x + 3 = 5x - 7 : 4x + 3 – 3 = 5x - 7 – 3 4x = 5x – 10 4x – 5x = 5x – 10 – 5x -x = -10 -x = -10 -1 -1 x = 10

CHECK: 4x + 3 = 5x – 7 4(10) + 3 = 5(10) - 7 40 + 3 = 50 – 7 43 = 43

Since our CHECK works out, we know that we did our work correctly!

Midsegment

A line segment that is created by joining two midpoints on different

sides of a triangle.

Ex.

Motion Detector

A hand-held device that uses ultrasound to measure distance. The data from motion detectors can be transmitted to graphing calculators.

Ex. CBR would be a motion detector.

Mixed Numbers

The sum of a whole number and a fraction.

Ex. 1 7

3 4

2 7

Mode

The value that appears most often in a set of data.

2, 3, 5, 5, 6, 7, 9, 9, 9, 10, 11 The mode is 9.

2, 3, 4, 4, 4, 5, 7, 7, 7, 10, 11 The modes are 4 and 7.

2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 17 In this case, there is no mode because each

value appears the same number of times.

Monomial

An expression that is a number, a variable, or the product of a

number and one or more variables.

Ex: 3x + y 7y 5x2y

Multiple Trials

A technique used in experimentation in which the same experiment is done

several times and the results are combined through a measure such

as averaging. The use of multiple trials “smooths out” some of the random

occurrences that can affect the outcome of an individual trial of an

experiment.

Negative Reciprocal

The negative inverse of a number displayed as a fraction. When both fractions are multiplied together, the

product is -1.

Ex. X - = -1

- is the negative reciprocal of .

is the negative reciprocal of - .

2 3

3 2

3 2

2 3 3 2

2 3

Net

A pattern design that, once cut out and folded up, creates a three-

dimensional object or polyhedron.

Ex:

Nonlinear Function

An equation that does not have a straight line on the graph.

Ex:

Non-Real Root of an Equation

A solution to an equation that is not an element of the set of real

numbers.

f(x) = x2

Non-Linear Relation

A relationship between two variables that does not fit a straight line when

graphed.

Ex:

Numerical Expression

An expression with numbers and at least one operation

Ex: 4 + 1.9 39.2 – 12.7 27.2 x 17.9

f(x) = x2

Outlier

A point that does not follow the pattern shown on a graph. It does not follow

the line of best fit.

Ex. In this graph, the green point is an outlier.

It does not follow the line of best fit.

Opposite Integers

Values that are equal distance from zero – one is a positive value and the

other is a negative value.

Ex:

3 and -3 are opposites

Optimal Value

The maximum or minimum value of a variable.

Order of Operations

A strategic sequence to be followed to simplify an expression.

1. Operate inside parentheses. 2. Multiply as indicated by exponents.

3. Multiply and divide from left to right. 4. Add and subtract from left to right.

Ex.: 10 ÷ (2 + 8) x 23 - 4 Add inside parentheses 10 ÷ 10 x 23 - 4 Solve the exponent 10 ÷ 10 x 8 – 4 Divide 1 x 8 – 4 Multiply (remember left to right) 8 – 4 Subtract 4 Final answer!

B E D M A S

B E D M A S

Ordered pair

Two values, known as a coordinate pair, used to describe a specific

location on a Cartesian Plane with an x-axis and a y-axis.

Ex:


y-axis

Origin

The intersection point of the x-axis and y-axis on the Cartesian plane.

It’s coordinate is (0,0).

Ex:

Parabola

The graph of a quadratic relation of the form y = ax2 + bx + c (a ≠ 0).

The graph, which resembles the letter “U”, is symmetrical.

Ex.

In this example, the equation is y=x2. The values of a, b, and c are as follows:

a=1, b=0, and c=0.

Parallel Lines

Lines that will not intersect as they share a common rate of change or

slope.

Ex:

Line segment MD is parallel to line segment TQ.

Partial Variation

A relationship between two variables in which one variable is a multiple of

the other, plus some constant number.

y = mx + b Ex.: The cost of a taxi fare has two

components, a flat fee and a fee per kilometre driven.

Fare = $0.15/km + $3.50 The flate fee is $3.50 and the fee per kilometer of travel is $0.15.

Perfect Square

A whole number whose square root is an integer.

Ex.: 16 is a perfect square.

Perpendicular Lines

Lines that intersect at a right angle or at 90o.

Ex: Line segment RZ is perpendicular to line

segment JW.

Pi (π)

The ratio of the circumference of a circle and its diameter.

Polygon

A closed figure formed by three or more line segments.

Ex.

Polynomial

An expression involving one or more terms involving various operations.

Ex: 3a2 + 8 a4 - 4a3 + 13 92b7t

Polynomial Expression

An algebraic expression taking the form a + bx + cx2 + . . . , where a, b,

and c are numbers.

Ex. 3 + 5x – 2x2

Population Statistics

The total number of individuals or items under consideration in a surveying or sampling activity.

Power

The resulting value of a base number raised to an exponent.

Ex: 43 = 4 x 4 x 4 = 64

Primary Trigonometric Ratios

The basic ratios of trigonometry (i.e., sine, cosine, and tangent).

sinθ = = cscθ = = cosθ = = secθ = =

tanθ = = cotθ = =

Ratio Abbreviations and their Names sinθ = sine of θ cosθ = cosine of θ tanθ = tangent of θ

cscθ = cosecant of θ secθ = secant of θ cotθ = cotangent of θ

side adjacent to angle θ

hypotenuse

side

op

po

site

to

an

gle

θ

θ

α

r

x

y


side adjacent θ x hypotenuse r

hypotenuse r side opposite θ y


hypotenuse y side opposite θ r

hypotenuse r side adjacent θ x

θ

Prism

A three-dimensional figure with two parallel, congruent polygonal bases. A prism is named by the shape of its

bases.

Ex. rectangular prism Ex. Cylindrical prism

Product

The answer to a multiplication problem.

Ex: 3 x 5 = 15

The product is 15.

factor

factor

product

Proportional Reasoning

Reasoning or problem solving based on the examination of equal ratios.

Pythagorean Theorem

The conclusion that, in a right triangle, the square of the length of the

longest side is equal to the sum of the squares of the lengths of the other

two sides. a2 + b2 = c2

Ex:

a2 + b2 = c2 Replace the variables (3cm)2 + (4cm)2 = c 2 with the known lengths. 9cm2 + 16cm2 = c 2 25cm2 = c 2 √25cm2 = c 5cm = c

Quadrant

One of the four regions or sections of the Cartesian Plane. They rotate in a counter-clockwise direction around

the origin.

Quadratic Equation

An equation that contains at least one term whose exponent is 2, and no term with an exponent greater

than 2.

Ex. x2 + 7x - 10 = 0 x2 - 4x + 2 = 9

Quadratic Formula

A formula for determining the roots of a quadratic equation of the form ax2 + bx + c = 0. The formula is phrased in

terms of the coefficients of the quadratic equation: x = –b +-(vb2 – 4ac)

2a.

Ex. x = –b +-(vb2 – 4ac) 2a. = -5 + -(4 x 52 – 4 x 3 x 7) 2 x 3 = -5 + -(4 x 100 – 84) 6 = -5 + -(400 – 84) 6 = -5 + -316 6 = -321 6 x = -53.5

Quadratic Relation

A relation whose equation is in quadratic form.

Ex.

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