Download Course: Foundations and Pre-Calculus 10 Stage 1 ~ Desired

Course: Foundations and Pre-Calculus 10
Stage 1 ~ Desired Results
Outcome:
FM 10.9 Demonstrate understanding of the writing and application of equations of linear
relations, given:
● a graph of a relation
● a point that satisfies a relation and the slope of the relation
● a point that satisfies the relation and the equation of a line parallel or perpendicular to
the relation
Key Indicators:
. Develop, generalize, explain, and apply strategies for writing an equation for a linear
relation
using data obtained from a graph.
2. Develop, generalize, explain, and apply strategies for writing an equation for a linear
relation
when given:
a point that satisfies the relation and the slope of the relation
two points that satisfy the relation
the coordinates of a point that satisfy the relation and the equation of a line parallel or
perpendicular to the line
3. Compare and critique the structure and purposes of different forms of linear relations,
including y = mx + b, Ax + By = C, and y – y
1
= m (x – x
1
) (e.g., there is not way to write a
vertical linear relation in the form y = mx + b
4. Graph and write equations for linear data generated within an experiment or collected
from a
situation.
5. Apply knowledge and skills of linear relations and their equations to solve situational
questions.
Enduring Understandings:
Students will understand that…
Essential Questions:
What information do you need to write the
equation of a line?
The equation of a line can be determined
from many different pieces of information.
Students will know:
● Slope intercept form is y = mx + b
● Standard form is Ax + By = C
● When to use y = mx + b or y – y
1
= m(x – x
1
) to write the equation of a line
● Equations of an oblique, vertical and
horizontal line
Students Will Do:
Students will do:
● Analyze
the graph of a linear function for
slope and y-intercept
● Find the slope from a graph, from 2 given
points, equations, from parallel and
perpendicular
equations
● Write the equation of a line, given the
graph and slope
● Find y-intercept
2
Key Vocabulary:
Common Misunderstandings:
Prior Skills:
Stage 2 ~ Assessment Evidence
Performance Tasks:
Matching Activity:
Students are put into small groups and are given 3 different coloured decks of pre-made
cards (one
deck containing 10 equations in general form, one deck containing 10 in slope-intercept
form and one
containing 10 graphs).
Their task is to match the two types of equations with the corresponding graph.
HINT – they should have one of each colour per row.
Evaluation - Rubric
Summative:
Unit Exam
Performance Task
Formative (Pre-Assessment, Self-Assessment, etc.)
Textbook assignments
Exit cards
Introductory activities
3
Stage 3 ~ Learning Plan
Learning Activities:
Introduction:
Do Math Lab 6.3 (p. 354) ~ discuss as a class
Do Construct Your Understanding (p. 355) ~ with a partner
Do Assess Your Understanding (p. 356) ~ individually
Lesson 1: Writing an Equation Using Data Obtained from a Graph
Give the students an equation (in slope-intercept form) and discuss the parts of the
equation
(y-intercept and slope). Then, as a class, graph the equation.
ex) y = 2/3x - 2
Ask the students what they could do to come up with an equation, if they were only given
the graph.
Explain that by using the equation format y = mx + b, we can look at the graph and
identify the slope
and y-intercept and fill in those values to create the equation.
Ask the students how we could check to see if it is correct.
Explain that we choose a point on the line to use and substitute it into the equation to
determine if it
satisfies the equation.
4
Work through the following example:
ex) Write an equation to describe the following function. Verify the equation.
slope (m) = rise = 3 y-intercept = -2
run 2
*Since the line is falling from left to right, the slope is negative.
**Now that we have acquired all of the necessary information, we can write the equation:
y = mx + b
y = -3 x - 2
2
Verify:
Choose a point from the line (-2, 1)
LHS RHS
y_3x–2
2
1
_ 3 (-2) – 2
2
3–2
1
5
Discuss Example #3 on p. 360
Assign: p. 362 – 364 #12, 13, 14, 16, 17, 18
Lesson 2: Writing an Equation Given a Point that Satisfies and the Slope of
a Relation
Review the concepts from the previous day regarding equation formation and verifying.
Ask the students how they could use any point on the line in the equation format y = mx
+b
ie) where would you insert those values? (x and y)
Ask the students: “If you are only given a point that satisfies and the slope, what DON'T
you know?”
(y-intercept or b)
Ask the students: “What variable in y = mx + b represents the y-intercept? (b)
Work through an example together to illustrate a strategy to find the equation.
ex) Find the equation of the line that passes through (-10, 4) and has a slope of -8.
First, list what you do know:
x = -10
y=4
m = -8
b = ???
Next, solve for what you don't know:
y = mx + b
4 = (-8)(-10) + b
4 = 80 + b
-76 = b
Finally, insert values m and b into the slope-intercept format:
y = mx + b
y = -8x – 76
Discuss Example #2 on p. 368
Assign: p. 372 – 373 #5, 8, 9, 10, 18
6
Lesson 3: Writing an Equation Given Two Distinct Points that Satisfy a
Relation
Discuss the meaning of slope ~ the rise between two given points on a line
run
Introduce the equation format: y
1
–y
2
= m(x
1
–x
2
)
Discuss what we know vs. we don't know.
● We
know 2 points that satisfy
● We know the slope (using the above formula)
● We DO NOT know the y-intercept (b)
Ask the students: “How can we find b, once we know m and 2 points? (Encourage them
to think back
to when we were give only one point and the slope.)
Explain that by knowing 2 points that satisfy, we can find slope. Then, once we know
slope (and 2
points) we can find our y-intercept by isolating b.
Illustrate steps through the following example:
ex) Find the equation of the line that passes through the points (8, -4) and (-4, -13).
Find slope:
m=y
1
–y
2
= - 4 - (-13) = - 4 + 13 = 9 = 3
x
1
–x
2
8 - (- 4) 8 + 4 12 4
Using one of the given points and the slope, find y-intercept (b):
m = ¾ y = mx + b
x = 8 - 4 = ¾ (8) + b
y=-4-4=6+b
b = ?? -10 = b
Now, insert values of slope and y-intercept into y = mx + b to create your equation:
y = mx + b
y = ¾ x – 10
Discuss Example #3 on p. 369
Assign p. 372 – 74 #11, 18, 19
7
Lesson 4: Writing an Equation Given a Point that Satisfies the Relation and
the Equation of a
Line Parallel or Perpendicular to the Relation
Review the concept of slope of parallel lines and slope of parallel lines.
● If the lines are parallel, then their slopes are equal.
● If the lines are perpendicular, then their slopes are negative reciprocals of one another
Explain that because we can identify (by sight) the slope of the parallel or perpendicular
line, we can
then determine the slope of the relation, given the above information.
Ask the students: “How can we find the equation of a linear relation if we know the slope
and a point
on the line that satisfies it?”
Illustrate with an example:
ex) Find the equation of the line that passes through the point (2, -5) and is
a) perpendicular to the line y = 2x – 12
slope = 2 ~ perpendicular slope = negative reciprocal
So, slope is _ 1
2
List what we know and solve for what we don't know:
m = _ 1 y = mx + b
2 -5 = (-1/2) (2) + b
x = 2 -5 = -1 + b
y = - 5 -4 = b
b = ??
Now, use what we know to build the equation: y = mx + b
y=_1x–4
2
b) parallel to the line y = 2x – 12
slope = 2 ~ parallel slope is equal
So, slope is 2
List what we know and solve for what we don't know:
m = 2 y = mx + b
x = 2 -5 = (2)(2) + b
y = -5 -5 = 4 + b
b = ?? -9 = b
8
Now, use what we know to build the equation: y = mx + b
y = 2x – 9
Pose the following question: “What would you do differently if you were given your
equation in the
following format, instead of the y = mx + b format?
-2x + y – 9 = 0
**Discuss how you could change it into a format that meets the needs of the question.
Discuss the commonalities between all of the equation formats ~ what are we always
needing to find
out before we can build an equation?
(we are always finding m and b)
Assign: p. 374 # 20 – 25
9
Lesson 5: Comparing and Critiquing the Structure and Purposes of
Different Forms of Linear
Relations
Have the students recall the format we have been using and discuss why that format is
useful for
graphing and for creating an equation from a graph or from the other information we
have been given
throughout this unit.
Introduce the other structures:
6x – 2y = 64 Standard
5x + 4y – 20 = 0 General
y = 7x + 3 Slope-Intercept
y – 3 = 5(x + 7) Slope-Point
Discuss how one structure can be rearranged into another through the use of our equation
solving
methods.
Explain how there is no way to write a vertical linear relation in the form y = mx + b.
(Recall slope of
a vertical line is infinity.)
Discuss Examples #1 – 3 on p. 379 – 380
Assign: # 4 – 6, 9 – 13; 8, 24
Lesson 6: Graph and write equations for Linear Data Generated Within an
Experiment or
Collected from a Situation
Discuss Example #4 on p. 382
Have the students complete “Check Your Understanding” and then discuss as a class.
Assign: p. 384 # 10, 11, 16