MATHEMATICS 10C

UNDERSTANDING SLOPE

High School collaborative venture withJasper Place, Ross Sheppard and Victoria Schools

Jasper Place: Martin Fechner, Elisha Pinter, Nic Ryan, Suzan SaadRoss Sheppard: Tim Gartke, Jeremy Klassen, Don SymesVictoria: Kevin BissoonFacilitators: Greg McInulty (Consulting Services), Gail Drouin (Alberta Education)Writer: Rosalie Mazurok (Ross Sheppard High School)

Spring, 2009

Mathematics 10C Understanding Slope Page 2 of 69

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea

Enduring Understandings

Essential Questions

Knowledge

Skills

4

4

5

6

7

Stage 2 ASSESSMENT EVIDENCE.

Teacher Notes For Transfer Tasks 8

Transfer Tasks The Miracle At McInulty Summit Teacher Notes for The Miracle At McInulty Summit and Rubric Transfer Task Rubric Possible Solution

Logo Assignment Teacher Notes for Logo Assignment and Rubric Transfer Task Rubric Possible Solution

910 - 1112 - 1314 - 15

1617 - 1819 - 2021 - 23

Stage 3 LEARNING PLANS

Lesson #1 Slope, But Not The Equation 24 - 28

Lesson #2 Slope With A Value 29 - 36

Lesson #3 The Slope Formula 37 - 41

Lesson #4 Applications Of Slope 42 - 44

Lesson #5 Linear Vs. Nonlinear

Lesson #6 Rate of Change

Lesson #7 Parallel Lines

Lesson #8 Perpendicular Lines

45 - 47

48 - 52

53 - 55

56 - 59

APPENDIX – Handouts

Grid Paper 61

Mathematics 10C Understanding Slope Page 3 of 69

Mathematics 10C Understanding Slope

STAGE 1 Desired Results

Big Idea:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

Enduring Understandings:

The student will understand that…

The slope represents the orientation of a line segment.

The value of a slope can be determined in a variety of ways.

The slope represents a rate of change.

All line segments on a given line will have the same slope.

Lines that do not intersect have the same slope, and lines that intersect have different slopes.

Mathematics 10C Understanding Slope Page 4 of 69

Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

Essential Questions:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

o Why is the product of the slopes of oblique perpendicular lines equal to -1?

o How can you show that horizontal and vertical lines are perpendicular?

o What relationships are there between parallel lines?

o What relationships are there between perpendicular lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

Mathematics 10C Understanding Slope Page 5 of 69

Implementation note: Ask students to consider one of the essential questions every lesson or two.Has their thinking changed or evolved?

Knowledge:

Enduring Understanding Specific Outcomes

Knowledge that applies to this Enduring Understanding

The student will understand that…

the slope represents the orientation of a line segment.

*RF3Students will know…

the definition of slope as .

The student will understand that…

the value of a slope can be determined in a variety of ways.

*RF3

Students will know…

the definition of slope as ,

percentages, grade, gradient, direct variation, etc…

The student will understand that…

the slope represents a rate of change.

*RF3Students will know…

the definition of linear and non-linear relations.

To connect rate of change to slope.

The student will understand that…

all line segments on a given line will have the same slope.

*RF3 Students will know…

the definitions of slope, line segment, line and collinear.

The student will understand that…

lines which do not intersect have the same slope; lines which intersect have different slopes.

*RF3

Students will know…

the definitions of parallel and perpendicular lines.

that parallel lines have equal slopes.

that perpendicular oblique lines have slopes which are negative reciprocals and the product of their slopes is -1.

the intersection of horizontal and vertical lines forms a special case.

*RF = Relations and Functions

Mathematics 10C Understanding Slope Page 6 of 69

Skills:

Enduring Understanding

Specific Outcomes

Skills that apply to this Enduring Understanding

The student will understand that…

the slope represents the orientation of a line segment.

*RF3

Students will be able to…

create a line given a slope. determine the slope of a given line. compare and interpret slope values. plot points on a coordinate plane.

The student will understand that…

the value of a slope can be determined in a variety of ways.

*RF3Students will be able to…

calculate slopes using rise and run, slope formula and by visual inspection of a graphed line.

The student will understand that…

the slope represents a rate of change.

*RF3

Students will be able to…

compare and interpret slopes – greater values result in a steeper rise.

compare units when finding rate of change.

The student will understand that…

all line segments on a given line will have the same slope.

*RF3Students will be able to…

determine the slope of a line. find another point on a line given one

point on the line and the slope.

The student will understand that…

lines which do not intersect have the same slope; lines which intersect have different slopes.

*RF3Students will be able to…

compare and interpret slope values. predict whether two lines intersect

based on their slopes.

*RF = Relations and Functions

STAGE 2 Assessment Evidence

Mathematics 10C Understanding Slope Page 7 of 69

Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.

1 Desired ResultsDesiredd Results

2 The Miracle at McInulty Summit or Logo Assignment

Teacher Notes

There are two transfer tasks to evaluate student understanding of the concepts relating to slope. The teacher (or the student) will select one for completion. Photocopy-ready versions of the two transfer tasks and rubric are included in this section.

Each student will:

Be able to calculate slopes of line segments. Interpret the results of calculations. Identify parallel segments and perpendicular line segments.

Mathematics 10C Understanding Slope Page 8 of 69

Implementation note:Students must be given the transfer task & rubric atthe beginning of the unit. They need to know howthey will be assessed and what they are workingtoward.

Teacher Notes for The Miracle at McInulty Summit Transfer Task

Part A

Students need to use the slope formula to determine the slopes of each segment of the journey, AB, BC, CD, etc. They need to compare slopes of different magnitudes and signs and identify those that are parallel or perpendicular. Part of this analysis requires students to interpret this information as a rate of change (where the sign is irrelevant).

Part B

Students are encouraged to go online and find their own pictures to illustrate the trip or in the absence of internet connectivity, students can use the pictures provided below. (Links to the pictures on page 11 are given on pages 62 – 64.) They should be able to defend why they chose certain pictures and the order in which they placed them. A completed project should have a slope calculation, a picture and an explanation for each segment.

Students are encouraged to use their creativity in the plausible presentation of the material.

Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.

Mathematics 10C Understanding Slope Page 9 of 69

Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

The Miracle at McInulty Summit - Student Assessment Task

Your friend has just returned from a hiking adventure. He has kept a record of his locations along his path using a Chronological GPS locator. He has presented you with a list of coordinates that indicate his elevation in terms of meters from his starting point and the time elapsed. He fell and now has amnesia and developed amnesia and wants you to help him remember what happened on his trip. He gives you his pictures and now you need to put them in order based on the information that he has given you. Can you recreate his trip for him so that his pictures make sense?

Here is his list of points.

Point Time (hours) Elevation (m)A 0 0B 2 100C 3 300D 6 400E 8 300F 11 200G 11.2 200H 16 400I 17 100J 20 100K 22 300L 23 200M 26 400N 28 300O 30 500P 30 0

…and this is where things got foggy. (Nobody expected him to live. This was the Miracle at McInulty Summit!)

Part A

You need to use the slope formula to determine the slopes of each segment of the journey, AB, BC, CD, etc. Compare slopes of different magnitudes and signs and identify those that are parallel or perpendicular. Part of your analysis requires interpretation of this information as a rate of change (where the sign is irrelevant).

Part B

You are encouraged to go online and find pictures to illustrate the trip or in the absence of internet connectivity, you can use the pictures provided below. You should be able to defend why certain pictures were chosen and the order in which they are placed. A completed project should have a slope calculation, a picture and an explanation for each segment.

Use your creativity in the plausible presentation of the material.

Optional Pictures

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22. 23. 24. 25.

Assessment

MATHEMATICS 10C

Understanding Slope

Rubric

Level

CriteriaExcellent

4Proficient

3Adequate

2Limited*

1Insufficient /

Blank*

Performs Calculations

Performs precise and explicitcalculations.

Performs focused and accuratecalculations.

Performs appropriate and generally accuratecalculations.

Performs superficial and irrelevant calculations.

No score is awarded because there is no evidence of student performance.

Presents Data Presentation of data is insightful and astute.

Presentation of data is logical and credible.

Presentation of data is simplistic and plausible.

Presentation of data is vague and inaccurate.

No data is presented.

Explains Choice

Shows a solution for the problem; provides an insightful explanation.

Shows a solution for the problem; provides a logical explanation.

Shows a solution for the problem; provides explanations that are complete but vague.

Shows a solution for the problem; provides explanations that are incomplete or confusing.

No explanation is provided.

Communicates findings

Develops a compelling and precise presentation that fully considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a convincing and logical presentation that mostly considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a predictable presentation that partially considers purpose and audience; uses some appropriate mathematical vocabulary, notation and symbolism.

Develops an unclear presentation with little consideration of purpose and audience; uses inappropriate mathematical vocabulary, notation and symbolism.

No findings are communicated.

Glossary

accurate – free from errors

astute – shrewd and discerning

appropriate – suitable for the circumstances

compelling – convincing and persuasive

complete – including every necessary part

convincing – impressively clear or definite

credible – believable

explicit – expressing all details in a clear and obvious way

focused – concentrated on a particular thing

incomplete – partial

inaccurate – not correct

inappropriate – not suitable

insightful – a clear perception of something

irrelevant – not relevant or important

logical - based on facts, clear rational thought, and sensible reasoning

precise - detailed and specific

plausible – believable

predictable - happening or turning out in the way that might have been expected

simplistic – lacking detail

superficial - having little significance or substance

unclear – ambiguous or imprecise

vague - not clear in meaning or intention

Partial Possible Solution to the Miracle at McInulty Summit

1. Calculation of Slopes.Segment Substitution SlopeAB

= 50

BC= 200

CD

DE= 50

EF

FG= 0

GH

HI= –300

IJ= 0

JK= 100

KL= –100

LM

MN= –50

NO= 100

OPUndefined slope

Mathematics 10C Understanding Slope Page 14 of 69

2. Slope Analysis

The following segments are truly parallel:FG and IJ (both flat: slope = 0)JK and NO (m = 100)

The following segments are truly perpendicular:FG and OP, and IJ and OP (vertical and horizontal lines)

The following segments have the same steepness (but different direction):AB and DE (50,-50)AB and MN (50,-50)CD and EF (100/3,-100/3)JK and KL (100,-100KL and NO (100,-100)

3. JPEG Journal of Journey (incomplete at this point)

AB BC CD

Moderate hike up Steepest Climb: met the goats on the way up.

Easy walk up

DE EF FG

Moderate hike downhill Easy walk downhill; didn’t realize we’d gone so far!

Perfectly flat

Mathematics 10C Understanding Slope Page 15 of 69

1. 3.

4. 5. 6.

2.

Mathematics 10C Understanding Slope Page 16 of 69

Teacher Notes for Logo Assignment Transfer Task

The student is asked to design or recreate a logo on graph paper. The completed logo would presumably be used to cut out a screen print template from a piece of steel. It would be more economical to use straight line segments.

Differentiated Instruction

Calculate the y-intercept of at least 4 lines.

Find the “equation of the line” ( ) for your lines (one that is vertical and one that is horizontal)

Calculate the length of 4 line segments in your drawing. Express the length both as an exact radical and as a decimal approximation.

Communicate which lines belong to which slope, y-intercept, equation, and length. Consider creating a legend that associates a set of calculations with a given line or segment.

Calculate manufacturing costs given a cost per mm for cutting the steel logo template. Different costs could be used for parallel lines, perpendicular lines etc. A given budget could be set.

Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions about appropriate intervention to help the student improve.

Mathematics 10C Understanding Slope Page 17 of 69

Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

Background:Logos are used everyday as an icon to promote and market products, companies and ideas. Logos such as the Nike “swoosh” and the "golden arches" are known universally. Logo design has even become an art form in its own right, with artists reproducing impressions of landscapes, portraits and other established works of art.

Your task:Design or recreate a logo on graph paper to create your own piece of art. Your completed logo will be used to cut out a screen print template from a piece of steel. For this reason, it will be more economical to use straight lines.

Note: You will need to modify your design to suit the requirement of the stencil cutter The stencil cutting machine requires the slope of each line to be cut. Parallel and Perpendicular lines are more economical to cut. You should include at

least 2 pairs of each.

Procedure:1. Draw a rough sketch of your logo.2. Transfer your initial design to a blueprint on the graph paper.3. Label each point where a line segment changes direction.4. Calculate the slope of each line segment.5. Label each line segment with its slope.6. Identify line segments which are parallel or perpendicular.

Logo Assignment - Student Assessment Task

Alternate source for graph paperhttp://www.printfreegraphpaper.com/

Assessment

MATHEMATICS 10C

Understanding Slope

Rubric

Level

CriteriaExcellent

4Proficient

3Adequate

2Limited*

1Insufficient /

Blank*

Performs Calculations

Performs precise and explicitcalculations.

Performs focused and accuratecalculations.

Performs appropriate and generally accuratecalculations.

Performs superficial and irrelevant calculations.

No score is awarded because there is no evidence of student performance.

Presents Data Presentation of data is insightful and astute.

Presentation of data is logical and credible.

Presentation of data is simplistic and plausible.

Presentation of data is vague and inaccurate.

No data is presented.

Explains Choice

Shows a solution for the problem; provides an insightful explanation.

Shows a solution for the problem; provides a logical explanation.

Shows a solution for the problem; provides explanations that are complete but vague.

Shows a solution for the problem; provides explanations that are incomplete or confusing.

No explanation is provided.

Communicates findings

Develops a compelling and precise presentation that fully considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a convincing and logical presentation that mostly considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism.

Develops a predictable presentation that partially considers purpose and audience; uses some appropriate mathematical vocabulary, notation and symbolism.

Develops an unclear presentation with little consideration of purpose and audience; uses inappropriate mathematical vocabulary, notation and symbolism.

No findings are communicated.

Glossary

accurate – free from errors

astute – shrewd and discerning

appropriate – suitable for the circumstances

compelling – convincing and persuasive

complete – including every necessary part

convincing – impressively clear or definite

credible – believable

explicit – expressing all details in a clear and obvious way

focused – concentrated on a particular thing

incomplete – partial

inaccurate – not correct

inappropriate – not suitable

insightful – a clear perception of something

irrelevant – not relevant or important

logical - based on facts, clear rational thought, and sensible reasoning

precise - detailed and specific

plausible – believable

predictable - happening or turning out in the way that might have been expected

simplistic – lacking detail

superficial - having little significance or substance

unclear – ambiguous or imprecise

vague - not clear in meaning or intention

Mathematics 10C Understanding Slope Page 21 of 69

Possible Solution to Logo Design Project

Logo Design Blueprint

LOGO DESIGN CALCULATIONS

Horizontal Line Segments

Mathematics 10C Understanding Slope Page 22 of 69

A (0, 5)

b (-9 , 3)

c (-10, 5)

d(-8.6) e (-4, 6)

R (-3, 0)

S (-4,-3)

T (-4,-6)

U(-5,-8)V (-7,-8)

W (-8,-7)

X (-7,-5)

Y (-6,-2)

Z (-4, 4)

a(-7,3)P (2, 3)Q (-1, 3)

J (4,-7)

K (2,-9)

L (0,-8)

M (0,-6)

N (1,-3)

O (2, 0)

B (3, 5)

C (6, 6)

G (4, 3)

D (8, 7)

E (8, 5)

F (7, 4)

I (3,-3)

H (4, 1)

Vertical Line Segments

Remaining Line Segments

Mathematics 10C Understanding Slope Page 23 of 69

Parallel Line Segments

All horizontal line segments are parallel.All vertical line segments are parallel.Line segments with the same slope are parallel:

Perpendicular Line Segments

All vertical - horizontal pairs are perpendicular. Line Segments with negative reciprocal slopes are perpendicular:

Mathematics 10C Understanding Slope Page 24 of 69

STAGE 3 Learning Plans

Lesson 1

Slope, But Not the Equation

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

.

KNOWLEDGE:

Students will know…

the definition of slope as .

to connect rate of change to slope.

SKILLS:

Students will be able to…

compare and interpret slope values. compare and interpret slopes – greater

values result in a steeper rise.

Mathematics 10C Understanding Slope Page 25 of 69

Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

Lesson Summary

Slope describes the steepness of a line. Students will understand that the larger the slope the steeper the line.

Lesson Plan

Lesson Goal

Students will recognize and make a connection between the magnitude of the slope and the steepness of a line. The larger the magnitude, the steeper the line is.

Activating prior knowledge

In grade 9 students plotted points and graphed linear relations from a table of values. Have students look at different tables of values and describe the patterns they see. Give several sets of data. Give a table of values where the x-values go from 1 to 5 and then create the y values to match the pattern you want. One set can go up by 1, the other goes up by 4, the other goes down by 1 and another that goes down 3. You might want to include a table which creates a slope of ½.

Lesson

Challenge:

How would you describe the difference between the following three lines?

Mathematics 10C Understanding Slope Page 26 of 69

Have students look at the tables and describe the pattern using words underneath the tables. Plot the points on grids with the same scale and ask how the rule which was described is visible in the graph. (We should be getting to the conclusion that the larger the absolute value of the number, the steeper the line.) Then label the slope on each.

Examples

Assessment

Show a variety of photos with different slopes of lines like pitches of roofs, ramps, mountain sides etc. Discuss the difference between the lines irrespective of their Mathematics 10C Understanding Slope Page 27 of 69

x y1 22 33 44 55 6 .

The pattern: y-values go up by one.

x y1 22 63 104 145 18

The pattern: y-values go up by four. (Students should notice that, when graphed, this line is steeper)

The pattern: y-values go down by two.(Students should notice that, when graphed, this line is steeper that the first, not as steep as the second, but it is falling – not rising.)

x y1 62 43 24 05 -2

Teachers should use as many tables as required.

directions and lengths. Do not expect exact answers, only comparisons here. Then, the students can share several results and put them on the board. Discuss the reasonableness of the values and the comparisons. Students should know that the word we use to discuss steepness is slope and the greater the slope the steeper the line. This is not referring to a specific numerical value at this point. You might choose to include a line that has a negative slope to see how students deal with it.

Or, split the class into two groups and give half the students a set of 5 pictures and the other half a different set of 5 pictures. Everyone assigns approximate values to the slopes of the lines (include at least one with negative slope for enrichment). The group should be given time to discuss their values and come to a consensus. Then put the results of each group on the board and the other group gets to agree or disagree with what decided.

Going Beyond

Have students go out with their cell phones and take their own pictures depicting a variety of slopes. Have them share their pictures with a partner or you could project

Mathematics 10C Understanding Slope Page 28 of 69

1...

2... 3.

..

4...

5...

6...

7...

8...

them for the class to compare. You could project the images and superimpose a grid over the picture. You could also use a document camera to project images with various lines.

You could introduce the idea of horizontal and vertical lines in their pictures.

Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)Math 10 (McGraw Hill: sec 6.5)

Slope Applethttp://www.ronblond.com/M10/slopes.APPLET/index.html

cell phones (if allowed)camerasdigital projector or an interactive whiteboard to display photosa set of photos that can be handed outgraph paper

Supporting

Use an applet to illustrate how the value of a slope changes when you change the steepness of a line and direction.

You could show pictures of ski slopes and discuss the steepness and the speed that a skier would be going down. Make a direct correlation between the two.

Glossary

slope – incline or grade or

Mathematics 10C Understanding Slope Page 29 of 69

Mathematics 10C Understanding Slope Page 30 of 69

Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary (http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more than one division. Some terms have animations to illustrate meanings.

Lesson 2

Slope With a Value

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

KNOWLEDGE:

Students will know…

the definition of slope as .

to connect rate of change to slope.

SKILLS:

Students will be able to…

compare and interpret slope values.

compare and interpret slopes – greater values result in a steeper rise.

Lesson Summary

Students know that the numerical value of a slope can be determined by using the ratio, rise over run. They should also be able to produce a line with a specific slope.

Mathematics 10C Understanding Slope Page 31 of 69

Lesson Plan

Lesson Goal

Get the students to make connections between slope and the numeric value of slope

using the ratio . Students should produce a line with a given slope.

Activating prior knowledge

Previously students associated values with the slopes of lines, but the use of these numbers was comparative. There was no standard approach to slope. Show another picture and ask the students to suggest a value for the slope. There should be a bit of disagreement because at this point they have no standard method for determining slope.

Lesson

Now we superimpose a grid on a picture.

Mathematics 10C Understanding Slope Page 32 of 69

Ask the students how they might assign a numeric value for the slope of this line.

Students should be guided towards slope as the ratio .

(Notice the slope of the above is less than 1.)

Now determine the slope of the frame from the pedal to the handle bar.

Here the slope of the bar is exactly one.

Mathematics 10C Understanding Slope Page 33 of 69

Find the slope of the steeple.

Notice that the slope here is greater than one.

Students should now be able to find the slope of line segments on the grid without reference to a specific object. Examples of horizontal and vertical lines should be included. (The slope of a horizontal line is zero, and the slope of a vertical line is undefined).

Students should draw a line with a given slope. This can be used to assess a student’s understanding of the slope. (Have students keep the sheet for future reference – we will be making a point about the location of line segments).

Mathematics 10C Understanding Slope Page 34 of 69

To see if students can go beyond what has been discussed, some students may be given the challenge of drawing a line with a negative slope. Students are not expected to know what is meant by this yet.

Applets that may help:

http://www.members.shaw.ca/ron.blond/slopes.APPLET/index.htmlhttp://www.analyzemath.com/Slope/Slope.html

Challenge:

Given pictures, a student should be able to reasonably estimate a slope value, regardless of direction, including vertical and horizontal lines.

Look at two lines on a grid and give students the opportunity to discuss how to calculate the slope. Identify that we have established that one line has a greater slope than the other, and ask students to derive a method for illustrating this property numerically (segments should be the same length so as not to introduce another

variable). Students should be guided toward the ratio .

Have students generalize that a slope of one generates a line with an angle of elevation of 45°.

Use this generalization to estimate, and then use the previous relationship to determine the slopes of various lines, including vertical and horizontal lines, and those of varying lengths.

Going Beyond

Students can relate the slope of line to the tangent of the angle of elevation.

Mathematics 10C Understanding Slope Page 35 of 69

Resources

Math 10 (McGraw Hill: sec 6.5)Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)

slope comparison applet

grid paperdigital projector or an interactive whiteboardgraph paper with photos providedgrids on transparencies to be placed over photos

Supporting

Math 10 (McGraw Hill: “Key Ideas” pg. 324)

slope comparison applet

Assessment

This is a formative assessment. Give students a photo and ask them to discuss the characteristics of the slope of the line and approximate numerical value for the slope.

Mathematics 10C Understanding Slope Page 36 of 69

For example,

Give another photo superimposed on a grid and have students provide an exact numerical value of the slope of a line segment (i.e. the banister).

Mathematics 10C Understanding Slope Page 37 of 69

Have the students compare and contrast the slopes (students should be using words like “steepness” and “greater than” or “less than”, they should also be referring to the slope value).

Glossary

horizontal – a line or segment parallel to the horizon

rise – change in the vertical

run – change in the horizontal

vertical – a line or segment perpendicular to a horizontal line or segment

Mathematics 10C Understanding Slope Page 38 of 69

Lesson 3

The Slope Formula

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

.

KNOWLEDGE:

Students will know…

the definitions of slope, line segment, line and collinear.

SKILLS:

Students will be able to…

determine the slope of a given line.

find another point on a line given one point on the line and the slope.

Lesson Summary

The formula, can be used to determine slope. The letter denotes

the slope.

Lines that rise from left to right have a positive slope, and lines that fall from left to right have a negative slope.

Horizontal lines have a slope of zero, and vertical lines have an undefined slope.

Mathematics 10C Understanding Slope Page 39 of 69

Lesson Plan

Learning Goal

Students will develop the slope formula and use it to develop slope. They will also understand the difference between positive and negative slopes.

Activating Previous Knowledge

Earlier, students determined the slope of a line segment in terms of . Give

students a grid and have them determine the slope of a given line segment.

Mathematics 10C Understanding Slope Page 40 of 69

x

y x

O

Lesson

Determine the coordinates of the endpoints of the given line segment. Identify the coordinates of the right angle in the corresponding triangle.

Use an example with two points A & B in quadrant I.

Have student extend a vertical line segment from A until it intersects a horizontal line segment through B at point C (the right angle).

Have students determine the coordinates for C.

Have students determine the slope of AB, and then connect to the formula for slope. Next, the students should be exposed to an example using all quadrants, as well as examples involving various slopes.

Lead the class in a discussion about how to use the coordinates to determine the rise and the run of the line segment. Develop a formula to represent the slope.

Note that the rise is the difference in the y-coordinates of the vertical line segment and that the run is the difference in the x-coordinates of the horizontal line segment.

Mathematics 10C Understanding Slope Page 41 of 69

A (2, 4)

B (10, 8)

C (10, 4)

4 18 2

slope

y

xO

Consider the line segment on the grid below.

Apply the formula you developed previously to the above graph. Students should notice that the value of the slope is the same, but the sign is different. Students should discuss what the change in signs means and the significance of the same

value applied to the slopes. The first should have a slope of and the second has a

slope of . They have the same steepness, but the first rises and the second falls as

we move to the right.

Connect the idea of positive and negative slope to the graph and then direct students to the importance of order when using the slope equation.

Slope =

At this point students should be able to extend their application of the slope formula to line segments in other quadrants, and examples with horizontal and vertical lines.

Present students with four line segments, including lines with positive, negative, zero and undefined slope.

Have students determine the slope of each line segment.

Mathematics 10C Understanding Slope Page 42 of 69

(2, 9)

(10, 5)

x

y

O

Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)Math 10 (McGraw Hill: sec 6.5)

calculating slopes applethttp://www.ronblond.com/M10/sl.APPLET/index.html

grid paper

Mathematics 10C Understanding Slope Page 43 of 69

Lesson 4

Applications of Slope

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

.

KNOWLEDGE:

Students will know…

the definitions of slope, line segment, line and collinear.

SKILLS:

Students will be able to…

determine the slope of a line.

find another point on a line given one point on the line and the slope.

Lesson Summary

A line can be produced given any point on the line and the slope of the line.

Mathematics 10C Understanding Slope Page 44 of 69

Lesson Plan

Lesson Goal

A line can be produced given any point on a line and the slope of the line.

Activating prior knowledge

Students should have the following knowledge: the slope formula, what positive and negative slopes imply, and what zero and undefined slopes imply. Students should be

able to draw a line given a slope. Have students draw a line with a slope of

Students should identify two points on the line, then demonstrate that the slope formula produces the proper slope.

Lesson

Ask students to draw a line with a given slope through a particular point. This can be modeled using the computer and an applet or on an interactive whiteboard. Students should then be asked to identify another point on the line.

Give students the coordinates of a point and a slope value. Have students provide the coordinates at any other point that would be on the line, and have them connect the two points to create a line.

Next, give students a point and a slope, and just one coordinate of another point. Students will find the missing coordinate.

Draw a line through A (-4,-1) with a slope of 2/3.

Mathematics 10C Understanding Slope Page 45 of 69

•

The second point has the coordinates (x, 5). Find the value of x.

We can find the missing coordinate using the formula.

x

y

Resources

Math 10 (McGraw Hill: sec 6.5)Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)

graph papera digital projector or an interactive whiteboard to view applets

Assessment

A roof has the coordinates A (0, 0) and B (4, 3.5) and C (C is not known).

1. Determine the slope of AB2. Given that BC has a slope at , find the coordinates of point C.

Mathematics 10C Understanding Slope Page 46 of 69

A (0, 0)

B (4, 3.5)

C (x, 0)

Lesson 5

Linear vs. Nonlinear

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

What is the impact of constant vs. variable rates of change concerning different graphical representations?

.

KNOWLEDGE:

Students will know…

the definition of linear and non-linear relations.

to connect rate of change to slope.

SKILLS:

Students will be able to…

determine the slope of a given line.

compare and interpret slope values.

Lesson Summary

If a series of line segments have a constant slope value, then they form a line (linear).

If a series of line segments does not have a constant slope value, then they do not form a line (nonlinear).

Any two points on a line can be used to determine slope.

Mathematics 10C Understanding Slope Page 47 of 69

Lesson Plan

Activate prior knowledge

Using diagram #1, determine the slope of the line segments AB, BC, CD and DE. Determine the slope of the line segments AF, FG, GH and HI.

Lesson

What do you notice about the first set of slopes? The 2nd set?

What is the difference between the two sets of line segments?

Discuss with students what they expect the slope from A to D to be, then calculate it. The slopes of every line segment along a line will have the same slope.

Mathematics 10C Understanding Slope Page 48 of 69

.... •••• ....

B

CDE

F

G

H

I

AA B

CD

E

F

G

H

I

x

y

Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)Math 10 (McGraw Hill: sec 6.5)

slope applet

Supporting

Use an applet to explore the properties of linear and nonlinear relations. Connect this to the activity from the Explore section.

OR

Recreate the Explore activity on a giant Cartesian coordinate system on the floor of the classroom or gym. Have students play the role of the points A through I. Have students move into the coordinate system in pairs while other students determine the slope between those two points. Ensure students make the connection that constant slope linear, non-constant slope nonlinear, and that for lines, any two points can be used to determine the slope.

Glossary

constant – a term that does not change

line segment – part of a line between two endpoints on the line

linear – capable of being represented by a straight line

non-linear – not capable of being represented by a straight line

Mathematics 10C Understanding Slope Page 49 of 69

Lesson 6

Rate of Change

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

lines which do not intersect have the same slope, and lines which intersect have different slopes.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

o Why is the product of the slopes of oblique perpendicular lines equal to -1?

o How can you show that horizontal and vertical lines are perpendicular?

o What relationships are there between parallel lines?

o What relationships are there between perpendicular lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

What is the impact of constant vs. variable rates of change concerning different graphical representations?

.

KNOWLEDGE:

Students will know…

to connect rate of change to slope.

the definitions of parallel and perpendicular lines.

that parallel lines have equal slopes.

that perpendicular oblique lines have slopes which are negative reciprocals and the product of their slopes is -1.

that the intersection of horizontal and vertical lines forms a special case.

SKILLS:

Students will be able to…

compare and interpret slope values.

predict whether two lines intersect based on slopes.

Mathematics 10C Understanding Slope Page 50 of 69

Lesson Summary

Students are interpreting slopes in the context of the given units to describe slopes as rates of change.

Lesson Plan

Activating Prior Knowledge

Look at the challenge question below. We activate prior knowledge using the first question given. Students should be able to calculate the slopes of the line segments.

Challenge

Students should be able to work through a problem similar to the following:

This graph illustrates the distance that someone is away from their home at a given time.

a) Calculate the slope of each segment.b) Interpret the meaning of the slope of each segment, providing a plausible

explanation or description of the motion.

Mathematics 10C Understanding Slope Page 51 of 69

(1, 10)

(4, 8)

(8, 8)

(9, 0)

km

(0, 10)

t

O hours d

This is a good opportunity to do an experiment with rates of change.

One example would be a student’s pay cheque at their part-time job. Discuss how much the student gets paid over a work day. Build a chart and graph the line. Then offer the student a raise and build the chart and graph the line again. Discuss the difference between the lines and relate it to the scenario.

Number of Hours Worked

Pay Earned ($)

0 012 10520 17528 24536 31540 350

Plot this data on the grid below.

Determine the slope of the line, and using the units provided; determine the significance of the slope. Students should realize that the slope ($/h) represents the worker’s hourly wage.

Mathematics 10C Understanding Slope Page 52 of 69

Pay

Ear

ned

($)

Number of Hours Worked (h)20 40

60

120

180

240

300

360

Provide students with various straight line graphs and include the units. Have them interpret the graphs keeping the units in mind (the inclusion of the units in their discussion should be emphasized). Their explanations do not necessarily need to include specific values at this point. They should understand that the slope of the line is a relationship between two values, such as speed and time.

Another example would be a graph of earnings vs. sales. The units represent a percent commission for the slope.

For Example

The slope is a rate of commission. This also allows for an opportunity to discuss with the class how someone can make money without any sales.

Mathematics 10C Understanding Slope Page 53 of 69

Students could be encouraged to do research online and look for other examples of a rate of change. They should specifically be encouraged to find examples that are not related to time.

Sales ($)

Earn

ings

($)

Resources

Math 10 (McGraw Hill: sec 6.5)Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.1)

a worksheet with several good questions near its end http://www.teacherweb.com/NY/Arlington/AlgebraProject/2L3SlopeasRateofChange.pdf

grid paperdigital projector or an interactive whiteboard

Glossary

parallel – lines in the same plane that do not intersect

perpendicular – lines which intersect at right angles (90°) to each other

Mathematics 10C Understanding Slope Page 54 of 69

Lesson 7

Parallel Lines

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

lines which do not intersect have the same slope, and lines which intersect have different slopes.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

o Why the product of the slopes of oblique perpendicular is lines equal to -1?

o How can you show that horizontal and vertical lines are perpendicular?

o What relationships are there between parallel lines?

o What relationships are there between perpendicular lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

What is the impact of constant vs. variable rates of change concerning different graphical representations?

KNOWLEDGE:

Students will know…

the definition of parallel lines.

that parallel lines have equal slopes.

SKILLS:

Students will be able to…

to compare and interpret slope values.

predict whether two lines intersect based on slopes.

Mathematics 10C Understanding Slope Page 55 of 69

Lesson Summary

Students understand that the slopes of parallel lines are equal. They are able to draw lines parallel to a given line through given points, and they are also able to indicate coordinates of some points on the lines.

Lesson Plan

Activating prior knowledge

Have students graph two horizontal lines. They can either identify or calculate the slopes of the lines.

Challenge

Given two slopes, determine whether the lines segments are parallel.Given a slope and a point determine another point on the line with the given slope. Students should also be able to draw this situation.

Give the students grid paper and have them draw any two line segments. Discuss whether or not they are parallel and link this to their slopes. Students can then take one endpoint of both line segments (like the right endpoint) and vertically translate them the same number of units. The lines seem to remain parallel and the students can then calculate the slopes. Do this for a few different vertical translations to verify the pattern.

Mathematics 10C Understanding Slope Page 56 of 69

x x

y y

Conclusions: Parallel lines have equal slopes, and if two lines have equal slopes, then they are parallel.

After students understand the relationship between the slopes of parallel lines give them a line and a point not on the line. They should be able to draw a line parallel to the given line through the given point. Then students could be challenged to algebraically determine another point on that parallel line.

Be sure to include a discussion of vertical and horizontal lines.

Resources

Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.2)Math 10 (McGraw Hill: sec 7.4)

http://www.ronblond.com/M10/slopes.APPLET/index.html

Grid paperDigital projector or an interactive whiteboardGeoGebra

Supporting

Students could access some applets online that explore the relationships between parallel lines. They can be created in GeoGebra and their slopes can be measured. The lines can then be adjusted and their slopes will change appropriately. This can also be done in Geometer’s Sketchpad and on the TI-84 using Cabri Jr.

Glossary

direction – the way in which a line or segment goes or points

magnitude – numerical value that describes the size of something

Mathematics 10C Understanding Slope Page 57 of 69

Lesson 8

Perpendicular Lines

STAGE 1

BIG IDEA:

An understanding of slope will lead to the ability to interpret rate of change in real world applications.

ENDURING UNDERSTANDINGS:

The student will understand that…

the slope represents the orientation of a line segment.

the value of a slope can be determined in a variety of ways.

the slope represents a rate of change.

all line segments on a given line will have the same slope.

lines which do not intersect have the same slope, and lines which intersect have different slopes.

ESSENTIAL QUESTIONS:

What is the meaning of slope?

How do we interpret the differences between the magnitudes and signs of the slopes, and the orientation of the lines?

o Why is the product of the slopes of oblique perpendicular lines equal to -1?

o How can you show that horizontal and vertical lines are perpendicular?

o What relationships are there between parallel lines?

o What relationships are there between perpendicular lines?

What are the connections between trigonometry and slope?

o What is the relationship between the tangent ratio and the slope?

What is the impact of constant vs. variable rates of change concerning different graphical representations?

.

KNOWLEDGE:

Students will know…

the definition of perpendicular lines.

that perpendicular oblique lines have slopes which are negative reciprocals and the product of their slopes is -1.

that the intersection of horizontal and vertical lines forms a special case.

SKILLS:

Students will be able to…

calculate slopes using rise and run, slope equation and by visual inspection of a graphed line.

compare and interpret slope values.

predict whether two lines intersect based on slopes.

Mathematics 10C Understanding Slope Page 58 of 69

Lesson Summary

Students understand that slopes of perpendicular lines are the negative reciprocals of each other and the product of their slopes is -1. They are able to determine points on lines perpendicular to a given line through a given point.

Lesson Plan

Activating prior knowledge

Have students draw two lines that are perpendicular. The most straightforward example would be to draw a vertical and a horizontal line together on the grid. They should identify the slopes of these two lines. Then let the next part of the example show the endpoints of the lines equidistant from the point of intersection.

Challenge

Given two slopes, determine whether the lines segments are perpendicular.Given a slope and a point, determine algebraically another point on the line with a slope perpendicular to the one given. Students should also be able to draw this situation.

Students can rotate the line segments by moving the endpoints. Take the top endpoint of the vertical line and translate it horizontally to the right a specified number of spaces. Then take the right endpoint of the horizontal line and translate it vertically down the same number of units. As long as they are moved in the same relative direction this should keep them perpendicular.

Mathematics 10C Understanding Slope Page 59 of 69

xx

y y

This can also be done using a set perpendicular rays and then rotate them as shown.

Another option would be to have students create an L shape on a grid and then cut it out. They can then place it on another grid and calculate the slopes of the line segments. Then they can rotate the L and calculate the new slopes.

After calculating the slopes for several rotations of the shape which the students know to be perpendicular, the students should be encouraged to multiply the slopes together.

They should notice that the slopes are the negative reciprocals of each other and that the product of the two slope values is -1.

This can be modeled using GeoGebra or Geometer’s Sketchpad or an applet.

Students can also be told to draw the most obvious perpendicular lines – horizontal and vertical lines – specifically the axes. Then rotate the lines around the origin. Choose a point on the y-axis and a point on the x-axis and move them appropriately. Now we can explore the relationship between perpendicular lines.

After they understand the relationship between the slopes of perpendicular lines give them a line and a point not on the line. They should be able to draw a perpendicular line through the given point. Then determine another point on the perpendicular line.

Note: Be sure to include a discussion of vertical and horizontal lines. This is an exception to the rule. The slopes are not the negative reciprocals of each other

and their product is not -1.

Mathematics 10C Understanding Slope Page 60 of 69

x x

y y

Resources

Math 10 (McGraw Hill: sec 7.4)Foundations and Pre-calculus Mathematics 10 (Pearson: sec 6.2)

negative reciprocal slopes (perpendicular lines) applethttp://www.ronblond.com/M10/perp.APPLET/index.html

grid paperdigital projector or a an interactive whiteboardGeoGebra

Supporting

Students could access some applets online that explore the relationships between perpendicular lines. They can be created in GeoGebra and their slopes can be measured. The lines can then be adjusted and their slopes will change appropriately. This can also be done in Geometer’s Sketchpad and on the TI-84 using Cabri Jr.

Mathematics 10C Understanding Slope Page 61 of 69

Appendix

Handouts

Mathematics 10C Understanding Slope Page 62 of 69

Alternate source for graph paperhttp://www.printfreegraphpaper.com/

ACKNOWLEDGEMENTS

Pictures or Digital Images

P age 11

1. http://images.google.ca/imgres?imgurl=http://image62.webshots.com/62/3/35/0/413933500aUnjhO_fs.jpg&imgrefurl=http://travel.webshots.com/photo/1413933500046516165aUnjhO&usg=___0jtb-7WZ6AVkxNXwGXBZ8y3yU0=&h=1536&w=2048&sz=661&hl=en&start=10&tbnid=9aq8Oh78WqIM3M:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26hl%3Den%26safe%3Dactive%26sa%3DG

2. http://images.google.ca/imgres?imgurl=http://www.canadianrockieshiking.com/images/mount_assiniboine_hike_6.jpg&imgrefurl=http://www.canadianrockieshiking.com/backpacking-trips/mount-assiniboine/&usg=__alBBxwGwY8Vz05gYQ0cWe1QRKTA=&h=290&w=480&sz=27&hl=en&start=8&tbnid=OWD_CEbXPGi91M:&tbnh=78&tbnw=129&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26hl%3Den%26safe%3Dactive%26sa%3DG

3. http://images.google.ca/imgres?imgurl=http://www.canadianrockieshiking.com/images/mount_assiniboine_hike_6.jpg&imgrefurl=http://www.canadianrockieshiking.com/backpacking-trips/mount-assiniboine/&usg=__alBBxwGwY8Vz05gYQ0cWe1QRKTA=&h=290&w=480&sz=27&hl=en&start=8&tbnid=OWD_CEbXPGi91M:&tbnh=78&tbnw=129&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26hl%3Den%26safe%3Dactive%26sa%3DG

4. http://images.google.ca/imgres?imgurl=http://besthike.com/blog/wp-content/uploads/2007/12/mountain-goats.jpg&imgrefurl=http://besthike.com/blog/2007/12/23/3-best-hiking-region-in-the-world-is-the-%25E2%2580%25A6/&usg=__G8C5fFs-34mQsSKMifh_6YbhjBU=&h=354&w=500&sz=97&hl=en&start=19&tbnid=h-WKbfu53RoqKM:&tbnh=92&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

5. http://images.google.ca/imgres?imgurl=http://www.i-needtoknow.com/assiniboine/images/2003_mccharles/images/027nub_et.jpg&imgrefurl=http://www.i-needtoknow.com/assiniboine/details/index.html&usg=__oAxrzxOLO6DCz4NsqbxcGS3mr44=&h=300&w=400&sz=17&hl=en&start=21&tbnid=kdmFIYV7MSqhqM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

6.http://images.google.ca/imgres?imgurl=http://www.i-needtoknow.com/assiniboine/images/2005_long/bus2.jpg&imgrefurl=http://www.i-needtoknow.com/assiniboine/details/03.html&usg=__OIWhEt8KfSNBfA38QiZlQsDPR9g=&h=300&w=400&sz=29&hl=en&start=23&tbnid=wD2kOK1P0W0ZJM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

7.http://images.google.ca/imgres?imgurl=http://www.i-needtoknow.com/assiniboine/images/2005_long/bus2.jpg&imgrefurl=http://www.i-needtoknow.com/assiniboine/details/03.html&usg=__OIWhEt8KfSNBfA38QiZlQsDPR9g=&h=300&w=400&sz=29&hl=en&start=23&tbnid=wD2kOK1P0W0ZJM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D188.http://images.google.ca/imgres?imgurl=http://www.mchalepacks.com/gallery/customer%2520gallery/Cindy%2520with%2520flowers.jpg&imgrefurl=http://www.mchalepacks.com/gallery/detail/Red%2520Helmut%2520Whitney.htm&usg=__v1P-

Mathematics 10C Understanding Slope Page 64 of 69

78VwQD_rhbUxj2vjYPvnRyo=&h=698&w=465&sz=132&hl=en&start=32&tbnid=TQIn4r5psARljM:&tbnh=139&tbnw=93&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

9.http://images.google.ca/imgres?imgurl=http://image44.webshots.com/45/4/25/98/2565425980049167929KTjuPs_ph.jpg&imgrefurl=http://outdoors.webshots.com/photo/2565425980049167929KTjuPs&usg=__Oi374vjQEuAJPGQGpZA7_HA6U1g=&h=640&w=480&sz=110&hl=en&start=109&tbnid=Gctt4rSizOgNcM:&tbnh=137&tbnw=103&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108

10.http://images.google.ca/imgres?imgurl=http://image18.webshots.com/19/4/69/64/197346964bOWnSi_fs.jpg&imgrefurl=http://travel.webshots.com/photo/1197346964030289480bOWnSi&usg=__uxz60MycWlxg-97C-APRd3YIIbw=&h=960&w=1280&sz=194&hl=en&start=113&tbnid=t1uQ-g48Fow-AM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108

11.http://images.google.ca/imgres?imgurl=http://image18.webshots.com/19/4/69/64/197346964bOWnSi_fs.jpg&imgrefurl=http://travel.webshots.com/photo/1197346964030289480bOWnSi&usg=__uxz60MycWlxg-97C-APRd3YIIbw=&h=960&w=1280&sz=194&hl=en&start=113&tbnid=t1uQ-g48Fow-AM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108

12.http://images.google.ca/imgres?imgurl=http://farm1.static.flickr.com/21/30147636_aa17d27340.jpg%3Fv%3D0&imgrefurl=http://flickr.com/photos/refrozenseabass/30147636/&usg=__zzvekwF5VR0mCrywMIbPCqgjC1I=&h=333&w=500&sz=57&hl=en&start=122&tbnid=dpZuNvcF_KQSEM:&tbnh=87&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108

13.http://images.google.ca/imgres?imgurl=http://www.diadav.com/travel/canadian-rockies/images/d12-marveltrail.jpg&imgrefurl=http://www.diadav.com/travel/canadian-rockies/chapter05.htm&usg=__j7sYPfdVL1ev9Z1L1LJxMHWxnDU=&h=322&w=500&sz=33&hl=en&start=2&tbnid=WKHhg-_fIGZ3mM:&tbnh=84&tbnw=130&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26hl%3Den%26safe%3Dactive

14.http://images.google.ca/imgres?imgurl=http://lh4.ggpht.com/_aQYEj8tugVU/SLSxlbvuaPI/AAAAAAAACMw/Pe-NGVqfvQg/P1010806.JPG&imgrefurl=http://picasaweb.google.com/lh/photo/O2X5VKDiAzCdrkph6-8Bvw&usg=__tcpMFI_E-BBDkPHa03DwxKF49BY=&h=1200&w=1600&sz=18&hl=en&start=29&tbnid=wwxQQdZq5Z-hqM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

15.http://images.google.ca/imgres?imgurl=http://lh4.ggpht.com/_aQYEj8tugVU/SLSxlbvuaPI/AAAAAAAACMw/Pe-NGVqfvQg/P1010806.JPG&imgrefurl=http://picasaweb.google.com/lh/photo/O2X5VKDiAzCdrkph6-8Bvw&usg=__tcpMFI_E-BBDkPHa03DwxKF49BY=&h=1200&w=1600&sz=18&hl=en&start=29&tbnid=wwxQQdZq5Z-hqM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dmarvel%2Blake%2Bpass%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D1816.http://images.google.ca/imgres?imgurl=http://www.ontariowildflower.com/images/andy_fyon_photographer2.jpg&imgrefurl=http://www.ontariowildflower.com/andyfyon.htm&usg=__RmqD70eghPNsvfxFH_N8s0gaXo4=&h=525&w=700&sz=242&hl=en&start=2&tb

Mathematics 10C Understanding Slope Page 65 of 69

nid=8w2a8oRIcDKchM:&tbnh=105&tbnw=140&prev=/images%3Fq%3Dtable%2Bmoutain%2Balberat%26gbv%3D2%26hl%3Den%26safe%3Dactive

17.http://images.google.ca/imgres?imgurl=http://www.rpmgroundworks.co.uk/images/steps_1.jpg&imgrefurl=http://www.rpmgroundworks.co.uk/groundworks.htm&usg=__EKv6Vef5O_UpvoMsvqmJZx0RFXU=&h=300&w=400&sz=86&hl=en&start=26&tbnid=dTH1_kPDo4BhjM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dsteep%2Bincline%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

18.http://images.google.ca/imgres?imgurl=http://www.rpmgroundworks.co.uk/images/steps_1.jpg&imgrefurl=http://www.rpmgroundworks.co.uk/groundworks.htm&usg=__EKv6Vef5O_UpvoMsvqmJZx0RFXU=&h=300&w=400&sz=86&hl=en&start=26&tbnid=dTH1_kPDo4BhjM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dsteep%2Bincline%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

19.http://images.google.ca/imgres?imgurl=http://www.pilotguides.com/images/content/destination_guide/north_america/american_rockies/americanrockies_teton_incline.jpg&imgrefurl=http://www.pilotguides.com/destination_guide/north-america/american-rockies/climbing_the_grand_tetons.php&usg=__3fKWzUndOlARaOOVRnGW6818pW8=&h=169&w=260&sz=8&hl=en&start=24&tbnid=q36y6BcImf2WYM:&tbnh=73&tbnw=112&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

20.http://images.google.ca/imgres?imgurl=http://www.perryzip.com/hawk_rock_2006/here_we_go.jpg&imgrefurl=http://www.perryzip.com/hawk_rock_2006/hawk_rock_2006.htm&usg=__OLaXbgrVJyvy0rgTcCja3OzwJcc=&h=1704&w=2272&sz=477&hl=en&start=67&tbnid=px9BPHNdS73fwM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54

21.http://images.google.ca/imgres?imgurl=http://wanderus3.files.wordpress.com/2008/04/half-dome.jpg&imgrefurl=http://wanderus.com/2008/04/16/3-things-to-do-to-climb-half-dome/&usg=__Gkqax0WHXTfrjoiy51DwSDaf8eE=&h=500&w=333&sz=158&hl=en&start=70&tbnid=DZC29Xu8HEHdmM:&tbnh=130&tbnw=87&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54

22.http://images.google.ca/imgres?imgurl=http://1.bp.blogspot.com/_uXDMPKN0Bjo/STrW1Zqf-FI/AAAAAAAABq0/j8pBhJ_m060/s400/Scree%2Bslope.jpg&imgrefurl=http://cookjmex.blogspot.com/2008/12/climbing-mexicos-nevado-de-colima.html&usg=__xWYs1At6ix6Yj9EQdby7PAw5CaI=&h=300&w=400&sz=39&hl=en&start=71&tbnid=HHadrqjPtxOfBM:&tbnh=93&tbnw=124&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54

23.http://images.google.ca/imgres?imgurl=http://www.ultrarob.com/blog/uploaded_images/ManitouInclineP4221457-764791.JPG&imgrefurl=http://www.ultrarob.com/blog/labels/Colorado%2520Springs.php&usg=__6so3sXOV7ZUbjo0fbc6gOJ9TMfo=&h=1600&w=1200&sz=505&hl=en&start=79&tbnid=bBkxJLsZW7S8kM:&tbnh=150&tbnw=113&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D72

24.

Mathematics 10C Understanding Slope Page 66 of 69

http://images.google.ca/imgres?imgurl=http://www.hallscountryhouse.com/images/ThukelaFalls.jpg&imgrefurl=http://www.hallscountryhouse.com/drakensberg.html&usg=__i1tlr1K0ohcdCGTfhDJv0BeoiQA=&h=452&w=350&sz=61&hl=en&start=98&tbnid=7Q0Q9bKtJVJrBM:&tbnh=127&tbnw=98&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D90

25.http://www.destination360.com/asia/china/great-wall-of-china

P age 15

1. http://images.google.ca/imgres?imgurl=http://www.perryzip.com/hawk_rock_2006/here_we_go.jpg&imgrefurl=http://www.perryzip.com/hawk_rock_2006/hawk_rock_2006.htm&usg=__OLaXbgrVJyvy0rgTcCja3OzwJcc=&h=1704&w=2272&sz=477&hl=en&start=67&tbnid=px9BPHNdS73fwM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D54

2.http://images.google.ca/imgres?imgurl=http://besthike.com/blog/wp-content/uploads/2007/12/mountain-goats.jpg&imgrefurl=http://besthike.com/blog/2007/12/23/3-best-hiking-region-in-the-world-is-the-%25E2%2580%25A6/&usg=__G8C5fFs-34mQsSKMifh_6YbhjBU=&h=354&w=500&sz=97&hl=en&start=19&tbnid=h-WKbfu53RoqKM:&tbnh=92&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D18

3.http://images.google.ca/imgres?imgurl=http://farm1.static.flickr.com/21/30147636_aa17d27340.jpg%3Fv%3D0&imgrefurl=http://flickr.com/photos/refrozenseabass/30147636/&usg=__zzvekwF5VR0mCrywMIbPCqgjC1I=&h=333&w=500&sz=57&hl=en&start=122&tbnid=dpZuNvcF_KQSEM:&tbnh=87&tbnw=130&prev=/images%3Fq%3Dhiking%2Bassinaboine%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D108

4.http://images.google.ca/imgres?imgurl=http://www.ultrarob.com/blog/uploaded_images/ManitouInclineP4221457-764791.JPG&imgrefurl=http://www.ultrarob.com/blog/labels/Colorado%2520Springs.php&usg=__6so3sXOV7ZUbjo0fbc6gOJ9TMfo=&h=1600&w=1200&sz=505&hl=en&start=79&tbnid=bBkxJLsZW7S8kM:&tbnh=150&tbnw=113&prev=/images%3Fq%3Dsteep%2Bincline%2Bhiking%2Bpicture%26gbv%3D2%26ndsp%3D18%26hl%3Den%26safe%3Dactive%26sa%3DN%26start%3D72

5.http://www.destination360.com/asia/china/great-wall-of-china

6. http://www.terragalleria.com/pictures-subjects/pastures/picture.pastures.usco44179.html

P age 27

1.http://images.google.com/imgres?imgurl=http://www.propools.com/images/40-271a.jpg&imgrefurl=http://www.propools.com/family.php/40-271&usg=__U0bf8zswxhxhUyUUkxjDG7C02Gg=&h=300&w=386&sz=53&hl=en&start=1&um=1&itbs=1&tbnid=Sh3tayUR_1F8NM:&tbnh=96&tbnw=123&prev=/images?q=slope&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&um=1

Mathematics 10C Understanding Slope Page 67 of 69

2.http://images.google.com/imgres?imgurl=http://www.kidsgeo.com/images/landform-slope.jpg&imgrefurl=http://www.kidsgeo.com/geology-for-kids/0035-landform-slope.php&usg=__6nZW6CsWOW84dSbUauMEUgvScVw=&h=300&w=400&sz=31&hl=en&start=5&um=1&itbs=1&tbnid=ir-KgTctHDMqIM:&tbnh=93&tbnw=124&prev=/images?q=slope&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&um=1

3. http://images.google.com/imgres?imgurl=http://www.dudh.gov.bt/Thimphustructural/images/slope_view.jpg&imgrefurl=http://www.dudh.gov.bt/Thimphustructural/partthree/3.4.html&usg=__vP-NjpFei_k9cxn4NLESrRORFrk=&h=308&w=591&sz=27&hl=en&start=11&um=1&itbs=1&tbnid=hOuGl5YTFZGA_M:&tbnh=70&tbnw=135&prev=/images?q=slope&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&um=1

4. http://images.google.com/imgres?imgurl=http://upload.wikimedia.org/wikipedia/commons/9/93/Toyota_Land_Cruiser_at_a_80Percent_slope_at_the_IAA_2005.jpg&imgrefurl=http://commons.wikimedia.org/wiki/File:Toyota_Land_Cruiser_at_a_80Percent_slope_at_the_IAA_2005.jpg&usg=__5nfukZAlzHsVvFqjms_AYg9R0U0=&h=656&w=849&sz=143&hl=en&start=38&um=1&itbs=1&tbnid=wNF-vcG-zsqwGM:&tbnh=112&tbnw=145&prev=/images?q=slope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=21&um=1

5. http://images.google.com/imgres?imgurl=http://www.ridemorzine.org/wp-content/uploads/2009/07/chatel-slope-style.jpg&imgrefurl=http://www.ridemorzine.org/tag/slopestyle/&usg=__GtQpRWV8yJxiMu_mzHP1CiKKWw8=&h=484&w=645&sz=79&hl=en&start=114&um=1&itbs=1&tbnid=qggxXsknMBljaM:&tbnh=103&tbnw=137&prev=/images?q=slope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=105&um=1

6. http://images.google.com/imgres?imgurl=http://tinyfarmblog.com/wp-content/uploads/2009/03/win08_south-slope-start-of-march.jpg&imgrefurl=http://tinyfarmblog.com/tag/long-shots/&usg=__ctge6tOcERRKAUrrjvR9LkWnNwc=&h=375&w=500&sz=69&hl=en&start=131&um=1&itbs=1&tbnid=oD5baSzp6PY-SM:&tbnh=98&tbnw=130&prev=/images?q=slope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=126&um=1

7. http://images.google.com/imgres?imgurl=http://media-cdn.tripadvisor.com/media/photo-s/01/04/6c/c4/laax-ski-slope.jpg&imgrefurl=http://www.tripadvisor.com/Hotel_Review-g580353-d290148-Reviews-Hotel_Signina-Laax_Grisons_Swiss_Alps.html&usg=__zqQN3T47HhozQXv2VA1Jn_HTPgQ=&h=412&w=550&sz=33&hl=en&start=157&um=1&itbs=1&tbnid=D08HChtZdFFmTM:&tbnh=100&tbnw=133&prev=/images?q=slope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=147&um=1

8. http://images.google.com/imgres?imgurl=http://www.absolutesteelhorse.com/images/cont-slope640.jpg&imgrefurl=http://www.absolutesteelhorse.com/continuous_slope.html&usg=__EJT0B8Pmp0mrfYzbm1h2iW5Wsko=&h=375&w=500&sz=21&hl=en&start=172&um=1&itbs=1&tbnid=8gc2UdZl7mL9UM:&tbnh=98&tbnw=130&prev=/images?q=slope&ndsp=21&hl=en&safe=active&rls=com.microsoft:en-US&sa=N&start=168&um=1

Page 30

http://marciabonta.wordpress.com/2007/12/01/golden-eagle-days-part-2/

Page 31

http://www.bendelectricvehicles.com/yahoo_site_admin/assets/images/Sprint_slope.327223715.gif

Page 32

Mathematics 10C Understanding Slope Page 68 of 69

http://traylork.home.comcast.net/~traylork/slopepix/pages/YellowChurch2.html

Page 35

Snow scene

http://images.google.ca/imgres?imgurl=http://www.background-wallpapers.com/d/88256/-/snow-slope_1152_x_864.jpg&imgrefurl=http://www.background-wallpapers.com/nature-wallpapers/snow/snow-slope.html&usg=__8fmxheQO72d6lrVGDBSUBXTc8z0=&h=864&w=1152&sz=247&hl=en&start=422&um=1&itbs=1&tbnid=fn5LhSPzjA-bBM:&tbnh=113&tbnw=150&prev=/images%3Fq%3Dslope%26start%3D420%26um%3D1%26hl%3Den%26safe%3Dactive%26sa%3DN%26rls%3Dcom.microsoft:en-US%26ndsp%3D21%26tbs%3Disch:1

Stair

http://imagesme.net/homedosh/systema-modula-staircase.jpg

Mathematics 10C Understanding Slope Page 69 of 69