UNIT FOUR
COORDINATE GEOMETRY
MATH 421A
23 HOURS
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UNIT 4: Coordinate Geometry
Previous Knowledge
With the implementation of APEF Mathematics at the Intermediate level, students should be able
to:
- Grade 7 - analyze graphs to determine a rate of change (slope) and base amount (y intercept)
- Grade 8 and 9 - construct and analyze tables and graphs to describe how change in one quantity
affects a related quantity
- Grade 8 - link visual characteristics of slope with its numerical value
- Grade 9 - represent patterns using a table of values, a graph and writing an equation describing
the relationship
- Grade 9 - given slope and y intercept, determine the equation of a line
- Grade 9 - determine the equation of a line by obtaining its slope and y intercept from a graph
- Grade 9 - sketch graphs using slopes and y intercepts
Overview:
- simplify radicals
- operations with radicals
- lengths of line segments
- midpoints of line segments
- slope
- point-slope form and standard form of a linear equation
- slope-y intercept form of a linear equation
- parallel and perpendicular lines
- graph linear equations using
- any 2 points
- the intercepts
- the slope and y intercept
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SCO: By the end of grade
10 students will:
A7 demonstrate and apply
an
understanding of discrete
and continuous number
systems
Elaboration - Instructional Strategies/Suggestions
Real Number System
Review number systems which students are familiar with (natural,
whole, integer, rational, irrational and real). Design activities that
have students discover properties of rational and irrational numbers.
Explain how these sets are nested within each other.
D7 apply the Pythagorean
Theorem
Pythagorean Theorem (p.2)
Investigate the Pythagorean Theorem as a means of demonstrating the
need for irrational numbers. Students can actually see lengths
representing irrational numbers.
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Worthwhile Tasks for Instruction and/or Assessment
Pythagorean Theorem (p.2)
Activity/Manipulatives
Mathpower has a good activity which can include the use of
geoboards to demonstrate the need for irrational numbers.
See Math 10 for problems relating to the Pythagorean
Theorem.
Find the length of a roof where the ratio of height to length is
4:12. (4 in 12 pitch)
Pencil/Paper
Draw a line that is
cm in length using centimetre grid
paper. Once it is drawn as accurately as you can, measure it
with a ruler. Then use a calculator to find the length to two
places of decimals.
Geoboard
Assuming the distances between the pins on a geoboard is 1
unit, use an elastic to represent a length of
units.
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Suggested Resources
Pythagorean Theorem
Mathpower 10 p.2 # 1, 2
Math 10 p.96-100
See an interactive applet at:
http://www.ies.co.jp/math/java/
samples/pytha2.html
SCO: By the end of grade
10 students will
Elaboration - Instructional Strategies/Suggestions
Estimation of Square Roots (1.3)
Estimation skills with radicals is a good indicator as to a student’s
understanding of square root values on a real number line.
B2 develop algorithms and
perform operations on
irrational numbers
Simplifying Square Roots (1.4)
a) Entire to Mixed Radicals
For square roots only convert radicals which are not perfect squares
into simplest radical form. Determine the highest perfect square
factor of the radicand.
Note: Students should completely simplify answer
C12 solve linear, simple
radical, exponential,
or absolute value
equations or linear
inequalities
Simplifying square roots can also be looked at pictorially. Examining
the diagrams below, one notices that each small square has an area of
2. The lengths of the sides of the three diagrams are:
Look at this formula and the below diagrams, each small square has
an area of 2:
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Worthwhile Tasks for Instruction and/or Assessment
Journal
Ask students to explain, using Pythagorean Theorem, how
they could draw a line segment which has the exact value of
the square root of 13.
Research
Have students prepare a brief presentation, paper or poster
about Pythagoras
Radicals
Pencil/Paper/Technology
Evaluate with and without using a calculator.
Suggested Resources
Estimation of Square Roots (1.3)
Estimation of Square Roots
Mathpower 10 p.14 # 2-13
http:/aleph0.clarku.edu/~djoyce
/mathhist/chronology.html
Pencil/Paper
Estimate the following:
a)
b)
Simplifying Square Roots
Mathpower 10 p. 19#1-15
Simplifying Square Roots (1.4)
Pencil/Paper
Write each of the following in simplest form:
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SCO: By the end of grade
10 students will be
expected to:
E20 develop and apply
formulas for distance
and midpoints
Elaborations - Instructional Strategies/Suggestions
Distance between two points (6.1)
Allow time for student groups to explore methods of finding the
length of a horizontal, vertical or diagonal line segment on a Cartesian
Coordinate Plane.
Enrichment - collinearity
Invite students to construct a triangle using a compass and straight-edge
with sides of
7cm, 9cm and 12cm
5cm, 8cm and 12cm
5cm, 7cm and 12cm
Challenge students to see the trend here. The 3 points become collinear
when AB + BC = AC.
The Triangle Inequality Theorem states that the sum of any 2 sides of
a triangle is greater than the third side. The limiting case of this theorem
is when the points become collinear and no longer form a triangle.
Midpoint of a line segment (6.2)
Invite students to work on problems on p.258 where midpoints of
horizontal and vertical line segments are to be calculated. By studying
p.259, students can understand the midpoint formula.
Challenge students to attempt mathpower 10 p.261 #29. Students must
research the term geometric median discuss it in their groups and apply
that understanding to the problem.
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Worthwhile Tasks for Instruction and/or Assessment
Distance between two points (6.1)
Pencil/Paper/Activity
Find the distance from one corner of your classroom to the
corner diagonally opposite.
Pencil/Paper
How long must a cable be to go from the top of a 15m power
pole to the ground at a point 5m from the base of the pole?
(Assume the pole and ground meet at right angles).
Suggested Resources
Distance between two points
Mathpower 10 p.252 # 1-3
p.256 #1,3,5,11,13,14
16,20,24,25
Pencil/Paper/Presentation
Find the distance from one corner of your classroom to the
diagonally opposite corner of the room at the ceiling.
Journal
Write a few sentences explaining how three points can be
shown to be collinear or not.
Midpoint of a line segment (6.2)
Midpoint of a line segment
Mathpower 10 p.261#1,5,6,11,13,14
21, 27,29
a) Find the coordinates of the midpoints of AB and AC above.
Label these points as D and E respectively.
b) Find the lengths of DE and BC. How do they compare?
Note to teachers:
DE is known as a mid-segment. You may want to research
this term and find out two properties of a mid-segment.
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SCO: By the end of grade
10 students will be
expected to:
B24 analyze rates of
change from graphs
and calculate slope at
various points
B25 develop and apply a
procedure to calculate
the rate of change
B27 interpret positive,
negative and zero
slopes
C58 demonstrate an understanding of the
concept “rate of
change” in a variety
of
situations
Elaborations - Instructional Strategies/Suggestions
Slope (6.3)
Student groups can brainstorm their ideas on the concept of slope. The
main ideas to be brought out are:
- the number represents steepness
- the sign represents direction
Any diagonal motion is broken down into 2 standard components( just
as with vectors) - vertical or rise
- horizontal or run
the slope formula is:
other ways of expressing slope are:
Slope as a rate of change (6.4)
In the real world the graphical axes are not x and y but represent real
life quantities such as velocity, time, flow of electricity, productivity of
farmlands, efficiencies of assembly lines, EEG’s and many other
quantities from numerous fields. Most fields of study do graphical
analysis of data. Even your car is computerized and mechanics must be
able to hook your engine up to a computer and analyze the graphs
generated by the engine. One of the most important ways to analyze a
graph is to investigate its slope or rate of change.
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Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Slope
Slope (6.3)
Pencil/Paper
Determine the slope of each side of the triangle below.
Mathpower 10 p.267 #1-4,9,11,13,18,
24,30,31,41,42
Slope as rate of change
Mathpower 10 p.272 #5,6,8,9
Slope as rate of change (6.4)
Communication/Presentation
Give three examples of rates of change that you have
experienced in your everyday life.
Research Project/Pencil/Paper
Use your research skills to determine the number of family
farms on PEI in 1981, 1986, 1991, and 1996. Plot the data and
draw a line of best fit. What is the rate of change(number of
farms lost per year) If this rate of change continues, how
many family farms will there be in 20 years time? Do you
have any evidence in your community or family to support
this data?
Note to teachers:
As a wrap-up this data could be graphed on the TI-83, then
determine a regression line and get the slope using the TI-83.
A discussion could ensue on the agreement between the
students’ results and those on the TI-83.
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SCO: By the end of grade
10 students will be
expected to:
C17 determine if a graph is
linear by plotting in a
given situation
F9 determine and apply a
line of best fit using
linear regression with
technology
Elaborations - Instructional Strategies/Suggestions
Point - Slope Linear Equation (6.5)
Allow student groups time to read the section and discuss the point slope formula and its applications. Hopefully, through collaboration,
students will see that the objective is to get the equation of a straight
line using the point - slope formula y ! y1 = m(x ! x1) when given the
slope m and the coordinates of any point on the line (x1,y1) or given two
ordered pairs on the line.
The two most common ways of writing the answer are:
1) slope - y-intercept formula
2) Standard form
y = mx + b
Ax + By + C = 0
Where
A, B, C , I
A>0
A and B are not both zero
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Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Point - Slope Linear Equation
Mathpower 10 p.28#1,4,5,10,11,18,
21,25,40,41,47,
49
Point - Slope Linear Equation (6.5)
Pencil/Paper/Technology
Write an equation of the line that passes through the points
(1,1) and (3, !3). Express the answer in standard form.
Pencil/Paper/Technology
Surface temperatures have warmed over the past century. For
the following graph draw the line of best fit, determine two
ordered pairs on the line, calculate the slope and get the
equation of the line. Use that equation to predict what the
surface air temperature will be in the year 2050.
Transparency at end of unit
A web-site for stock market graphs
is below and a sample graph is at the
end of the unit.
http://www.imoney.com/bm_apps/
graphs/cgi-bin/stocks.egi
Families of lines
Mathpower 10 p.285 #3,4
F
amilies of lines
Journal
Explain the two characteristics that determine families of lines.
Group Activity
Use the following graphs to get the equation for each of the
lines shown.
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SCO: By the end of grade
10 students will be
expected to:
C14 determine the slope
and y-intercept of a
line from a table of
values or a graph
Elaborations - Instructional Strategies/Suggestions
Slope - y intercept Linear Equation (6.6)
Challenge student groups to develop definitions for x and y intercepts.
If the general coordinates of the y - intercept are represented as (0,b),
invite student groups to modify the point - slope formula to obtain the
slope - y intercept linear equation.
y ! y1 = m(x ! x1)
y ! b = m(x ! 0)
y ! b = mx
C15 determine the equation
of a line using the
slope and y-intercept
Allow students to investigate the problems in Math Power 10 p.288 to
discover the applications of the formula.
They are:
C54 sketch graphs from
words and tables, and
from real data
collected in
experiments
-given an equation - re-arrange in y = mx + b form to determine
m and b. Then use these to draw the graph
of the equation.
- given the m and b values - write the equation of the line and
re-arrange in standard form
- draw the graph
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Worthwhile Tasks for Instruction and/or Assessment
Slope - y intercept Linear Equation (6.6)
Pencil/Paper/Technology
If you climb a mountain the atmosphere gets thinner and the
air temperature drops. Assuming the air temperature at sea
level is 200 C, re-arrange the formula below in slope - y
intercept form, determine the m and b values and use these
values to draw the graph. Using the graph complete the
following table.
a + 150t = 3000
a = altitude in metres a is the independent variable
t = temperature in 0 C t is the dependent variable
Suggested Resources
Slope - y intercept Linear Equation
Mathpower 10 p.288 #9-14,19,21,23,
24,28,30,31,35
40,42-44
Written assignment/Communication
Jane accepts a position where she gets a base salary of $300
and commission at the rate of 5% on all sales. Write an
equation describing her pay, p, in relation to sales, s. Sketch
the graph and describe what the $300 and the 5% actually
represent on the graph. Communicate this to the rest of the
members in your group or to the class as a whole.
Group activity
Write an equation for the graph below and describe a realworld scenario it could be picturing.
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Possible scenario could be:
The x axis represents the price of a
suit and the y axis the number of suits
sold in a month at a certain store.
SCO: By the end of grade
10 students will be
expected to:
Elaborations - Instructional Strategies/Suggestions
Parallel and Perpendicular lines (6.7)
The activity in Math Power 10 p.291 is a simple way to draw parallel or
perpendicular lines.
C14 determine the slope
and y-intercept of a
line from a table of
values or a graph
A second method could be as follows:
For the given equations, draw the graph, which lines appear to be
parallel or perpendicular. Calculate the slope of each line and make a
conjecture about the slopes of parallel and perpendicular lines.
1) 5x ! 2y + 4 = 0
2) 2x ! y + 1 = 0
3) x + 2y ! 4 = 0
4) y = 2x + 3
5) 2x + 5y ! 5 = 0
6) x + 2y ! 2 = 0
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Worthwhile Tasks for Instruction and/or Assessment
Parallel and Perpendicular lines (6.7)
The activity in Math Power 10 p.291 is a simple way to draw
parallel or perpendicular lines.
In Junior High students would have learned about slides or
translations. These are a type of transformation or mapping.
Perhaps students could graph a line then slide it up or down
by changing the y coordinates of the ordered pairs.
Example: (1,1) and (4,3) yield a line with slope m = 2/3. If we
map
(x,y) ± (x,y !2) we get the ordered pairs (1,!1) and (4,1)
yielding again a slope of m = 2/3.
A transformation called rotation rotates a figure. Using the
Corel Presentations Cartesian Coordinate template students
could draw a line segment. Use Ctrl C and Crtl V to copy the
line and then select the second line segment and use edit
rotate; to rotate the line 900 right click one of the corner
handles, enter 900 and OK. Students can then get the slopes
of the two lines and see the negative reciprocal relationship.
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Suggested Resources
Parallel and Perpendicular lines
Mathpower 10 p.294 #1,3,5,7,9,17,19
21, 27(a),(f),41
43-48,60,63
Note to teachers: An alternate
activity could be to use the Corel
Presentations Cartesian Coordinate
template to duplicate ( click the line
to select it, then Ctrl C and Ctrl V to
copy it. Then click and drag the line,
where ever you choose to place it, the
line will be parallel to the first line
drawn.
Worthwhile Tasks for Instruction and/or Assessment
Parallel and Perpendicular lines (cont’d) (6.7)
Pencil/Paper
Find the slope of line segments AB and BC. How do they
compare?
Note to teachers: An angle inscribed in a semicircle is a right
angle(ie. Sides are perpendicular)
Pencil/Paper
Find the slope of line segments AD and BC; how do they
compare? Then find the slopes of line segments AB and CD;
compare them. Can you guess the name of this quadrilateral?
Note: A parallelogram is a quadrilateral with both pairs of
opposite sides parallel.
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Suggested Resources
Parallel and Perpendicular lines
Problem Solving Strategies
Math Power 10 p.303 #5
SCO: By the end of grade
10 students will be
expected to:
C15 determine the equation
of a line using the
slope and y-intercept
C16 graph by constructing
a table of values, by
using graphing
technology and, when
appropriate, by the
slope y-intercept
method
C36 describe real world
relationships depicted
by graphs, tables of
values and written
descriptions
Elaborations - Instructional Strategies/Suggestions
Graphing linear equations (6.8)
This section is a recap of work done in this and the previous unit.
Various methods are reviewed here:
Given the formula for a linear relationship ( equation of a line), graph;
< using the x and y intercepts
< using any two points
< using the slope and y intercept
Given linear data or a linear graph; determine the linear formula.
Perhaps the problems could be real world examples of linear situations
and not just problems with x and y.
An experiment is conducted to examine rates of water loss in plant
leaves. A student applied petroleum jelly to the underside of a plant and
used a second plant as a control. Use the graph below to answer these
questions:
< which plant may have petroleum jelly on its underside
< determine the equation of that line
< what would the water loss be after 6 hours
C54 sketch graphs from
words and tables, and
from real data
collected in
experiments
Source: Biology Living Systems p.447
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Worthwhile Tasks for Instruction and/or Assessment
Graphing linear equations (6.8)
Pencil/Paper/Technology
Graph the data below, get the line of best fit, determine the
equation of the line and extrapolate to get the height of a
seedling after 30 days.
Pencil/Paper/Technology
Below is data on the effects of nicotine. The effects seem to
indicate a linear relationship. Determine a formula for this
relationship. Interpolate to find the heart rate for 1.5 drops
and extrapolate for the heart rate for 6 drops of nicotine.
Source: Biology Living Systems p.727
Internet Research/Journal
Try to find data on the amount of nicotine in various products
such as a pack of cigarettes.
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Suggested Resources
Graphing linear equations
Mathpower 10 p. #35,36,40,41
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Art with a Graphing Calculator
The design below consists of seven lines. Determine the equations of these lines using the screens below.
Then check your work by graphing the lines on a TI-83 using the same window dimensions.
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