for Middle Years Mathematics

1994

S105.15

- Analytic Geometry

- Lesson 1: Plotting Points on a Cartesian Grid using the Geoboard

- Lesson 2: Graphing Lines on the Cartesian Grid

- Lesson 3: Determining the Slope and Y-Intercept of the Line

- Lesson 4: Determining the Slope Using Coordinate Points

- Lesson 5: Parallel and Intersecting Lines in Relation to Slope

- Open Book Quiz

TOPIC: INTRODUCTION TO SLOPE FOR MIDDLE YEARS MATHEMATICS

** Precis:
**This topic deals with the Cartesian grid and linear equations with
some analysis of the graph of the line and its relationship to the equation
of the line. This topic was completed in five - fifty minute periods.

Recognize, Draw, Name:

- point, line, line segment Grade 3/4

- parallel, intersecting and perpendicular lines Grade 5

Locate and plot ordered number pairs in the first quadrant Grade 5

Grades 7/8 Geometry

Coordinate grid-plotting points

Definitions of Euclidean concepts

- geoboard

- TI-81 Emulator Program (Macintosh Computer)

- coordinate grid (I used a coordinate grid which was a Clarisworks creation
by Jonathan Elder, a Grade Eleven student at Fillmore)

- Grid Systems - examples

- Grid

- Locating Coordinates

- examples
- geoboard

**Relations:**

Forms

- Mapping Notation

- Function Notation

- Equations of Lines y = 2x, y = 3x, y = -2x, y = -3x with x, y charts.
- geoboard

- computer (TI-81 Emulator program)

**Slopes of Lines**

Use equations

y = 2x

y = 3x

y = -2x

y = -3x

to explain m = (rise)/(run)

- use y-intercept as a starting point

- positive, negative Ü horizontal and vertical and vertical slopes will
be covered ater.

y = + mx

y = + mx + b (slope-y-intercept equation)

Ax + By = C

Calculating slopes using coordinates.

m = y2 - y1

x2 - x1Horizontal and Vertical Lines

Parallel Lines-equations of lines used to illustrate the relationship between slope and parallel lines.

Intersecting Lines-equations of lines were used to indicate the relationship between slope and intersecting lines.

The students were instructed on use of the Emulator program and were to, with supervision, key in a couple of equations to graph. This topic took 4.5 classes of instruction with 0.5 of a class for an OPEN BOOK QUIZ.

- Graph Paper
- Straight-Edge
- Notebook

- Communication
- Creative and Critical Thinking
- Numeracy
- Technological Literacy

- with assignments
- instructional methods
- classroom management techniques

- Grid Systems

- grid road maps - Saskatchewan

- township maps or the local Rural Municipality Map

- games - battleship

- grid road maps - Saskatchewan
- Grid
**Key:**1. x axis

2. y axis

3. positive x

4 negative x

5. positive y

6. negative y

7. quadrant 1, 2, 3, 4

- Locating Coordinates

- example: (3,4)------ (x,y)

- example: (3,4)------ (x,y)
- Geoboard

- coordinate search (5 minutes)

- given points A-K (oral correction with instructor writing correct answer
on blackboard)

- the instructor has previously placed the points on the geoboard using a nonpermanent marker

- coordinate search (5 minutes)

**CLASSROOM MANAGEMENT**

when students plot points L, M, and N.

- plot 3 points L, M, N Ü hand out nonpermanent markers and when students are finished, take them back in.

- a comparison of numbers: from newspapers, etc.

Equations of Lines (results will be as ordered pairs with x, y elements
of the set of integers)

x, y charts

Results will be ordered pairs as (x,y), with x, y elements of the set
of integers

- Plot points on graph paper (different grids)

- Label graph using equation.

**Graphing Lines on the Cartesian Grid**

- The students will be able to graph line on a Cartesian coordinate
grid.

- The students will be able to determine the slope and y-intercept of the
line given:
- the graph of the line

- the equation in slope - y-intercept form

- the graph of the line

**Indirect Instruction: **reflective discussion on x, y charts using
the set of integers.

Students graph on blackboard using different colored chalk (plot check).

- examine relationships between equation

- positive and negative directions

- origin

- positive and negative directions

**Direct Instruction: **compare and contrast; demonstration

- use same equations where x y R

- geoboards - four elastics the students appreciated the use of the geoboards
as it gave them an idea of the rise and the run.

- TI-81 Mac Emulator program

- brainstorm with students various examples of lines used in architecture, house design, natural landscapes (ski slopes), engineering designs Ü to develop an intuitive idea of the applications of lines.

- geoboards - four elastics the students appreciated the use of the geoboards
as it gave them an idea of the rise and the run.

- Use equations from assignment 1 where x and y are real numbers.

- Slope

- y-intercept equation: y = ± mx

- use assignment 1 equations as examples.

- the y-intercept is the origin.

- y-intercept equation: y = ± mx

- Slope

- y-intercept equation: y = ± mx + b

- the y-intercept is not the origin.

- TI-81 Emulator program: y = mx + b

- instructor will key in sets of equations to illustrate different
equations

- instructor will key in sets of equations to illustrate different
equations

- y-intercept equation: y = ± mx + b
- the use of the TI-81 Emulator program by the students was somewhat reserved and they did not really understand the connection between the equation y = mx + b and the graph of the equation. This was corrected by some direct instruction on the blackboard and a paper and pencil assignment similar to the following:

**SET 1:
**a) y = x + 1

b) y = x - 1

c) y = -x + 1

d) y = -x - 1

examine similarities and differences.

**SET 2:
**a) y = 2x - 1

b) y = 2x - 3

c) y = 2x + 1

d) y = 2x - 3

examine similarities and differences.

**Determining the Slope and Y-Intercept of the Line**

- the graph of the line (review)

- the equation in slope-y-intercept form (review)

- the equation in standard form

**Independent Learning:** computer assisted instruction

- Supervise students computer use - 0.5 hour

- Geoboard: use to illustrate the determination of slope using the
y-intercept
as a starting point.

- Slope: m = RISE/RUN

- use of elastics to determine the run from the y-intercept to the next x coordinate. Determine the rise.

- paper and pencil graphing to show how slope y-intercept graphing works, (as opposed to using an x, y chart).

**Direct Instruction:** structured overview and demonstration.

- Standard Form equations.

- Ax + By = C

- Solve for y: y = -(A/B)x + C/B

- From any given equation in standard form select the numerical coefficients for A, B and C and determine slope using m = rise/run = -A/B. For the y-intercept b use C/B.

- Solve for y: y = -(A/B)x + C/B

- Ax + By = C

- Determine the slope and the y-intercept of the following equations:

a) y = x b) y = 7x

c) y = -4x d) y = 2x + 7

e) y = 8x + 6 f) y = -4x 2

g) y = -6 - 3x h) 2x + y = 4

i) 7x + 6y = 4 j) 12y + 3x = 14

- Determine the equation of the line in:

- slope - intercept form

a) m = 3 and b = -2

b) m = -7 and b = 4

c) b = 1 and m = 3

- standard form

d) -A/B = 6 and C/B = 2

e) -A/B = -3 and C/B = 2

f) -A/B = -1 and C/B = 1

- slope - intercept form

The instructor and the students went through these questions as a group (except 1hij and 2def).

**Determining the Slope Using Coordinate Points**

**Direct Instruction: **structured overview

- Finish correcting Assignment 2

- if any problems occur, use the geoboards or if time is a factor, use
the blackboard or computer.

- Calculating slopes using coordinates

m = (y2 - y1)/(x2 - x1)

- plot coordinates for some samples and then calculate the slope by substitution into the slope formula.

- if any problems occur, use the geoboards or if time is a factor, use
the blackboard or computer.

Determine the slope of:

AB where A(2,1) and B(-4,3)

CD where C(-3,5) and D(5,1)

EF where E(-11,5) and F(-5,7)

GH where G(-6,-5) and H(3,-1)

IJ where I(7,-5) and J(10,1)

KL where K(-3,8) and L(2,-2)

MN where M(3,3) and N(7,3)

OP where O(2,2) and P(2,4)

Indicate to students that the last two sets may have surprising results. Class time was given to work on the assignment and was corrected in the same class.

**Horizontal and Vertical Lines**

- the slopes of MN and OP were used to illustrate horizontal and vertical
lines (graph of each was shown).

- several samples were given on the blackboard and equations were used
to indicate connection with coordinates and equations.

**Parallel and Intersection Lines in Relation to Slope**

- the graphs of the lines

- the equations of the two lines in slope - y-intercept form.

- Parallel Lines

- equations of lines were used to illustrate the elationship between slope and parallel lines.

- Intersecting Lines

- equations of lines were used to indicate the relationship between slope and intersecting lines.

- QUIZ : OPEN BOOK / 20 MINUTES

The rationale for the open book nature was that this was the first visual representation of the equation of the line as it relates to the cartesian grid. GRID.

1. What are the coordinates for the points on the grid (3 marks)

(Use first graph)

A _____, B _____,

C _____, D _____,

E _____, F _____ .

2. Plot the points on the grid: (3 marks)

(Use second graph)

A(2,3); B(0,4); C(5,-3); D(2,0); E(-4,-6), F(-1,1)

3. Complete the x, y chart for the following equations: (2 marks)

4. What is the slope for the following equations: (3 marks)

a) y = 7x - 3

b) y = -3x + 3

c) 3x + 2y = 5

5. What is the slope and y-intercept of the following equations: (4 marks)

a) y = -2x

b) y = 7x + 6

6. Determine the slope: (6 marks)

a) (-3,6) (-1,3)

b) (7,0) (2,1)

c) (0,3) (0,6)

7. For the equation sets given determine which are parallel and which
intersect: (3 marks)

a) y = 7x - 2 b) y = 3x + 2 c) y = x - 1

y = -7x - 2 y = 3x - 3 y = x + 1

This unit comes from the The Stewart Resources Centre which provides library resources and teacher-prepared materials for teachers in Saskatchewan. To borrow materials or obtain a free catalogue listing unit and lesson plans contact :

Sask. Teachers' Federation,

2317 Arlington Avenue,

Saskatoon, SK S7J 2H8;

phone 306-373-1660; fax 306-374-1122,

e-mail src@stf.sk.ca.

http://www.stf.sk.ca/

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