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MA5.3-8NA Linear Relationships
Summary of Sub Strands Duration
S4 Linear Relationships (plotting points on the Cartesian plane)S5.1 Linear Relationships
Duration: 4 weeks
Start Date:
Completion Date:
Teacher and Class:
Unit overview Outcomes Big Ideas/Guiding Questions
Students review Stage 4 Linear Relationships – plotting points on the Cartesian plane.For Stage 5.1, students find the distance, midpoint and gradient of a line, using a variety of techniques, including graphical software.They sketch linear graphs using the coordinates of two points and solve problems using parallel lines.In Stage 5.2, students use the gradient-intercept form of a straight line to interpret and graph linear relationships. Reference is also made to perpendicular lines.
› MA5.3-1WM uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures
› MA5.3-2WM generalizes mathematical ideas and techniques to analyse and solve problems efficiently
› MA5.3-3WM uses deductive reasoning in presenting arguments and formal proofs
› MA5.1-6NA determines the midpoint, gradient and length of an interval, and graphs linear relationships
› MA5.2-9NA uses the gradient-intercept form to interpret and graph linear relationships
› MA5.3-8NA uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard
This may be an area where we could link to Asia and compare and contrast various data types
Key Words
Cartesian plane, coordinates, vertical, horizontal, linear, relationships, plot, right-angled triangle, parallel, perpendicular, distance, midpoint, gradient, slope, line segment (interval), line, scale, mean, average, positive, negative, Pythagoras’ Theorem, x-intercept, y-intercept, rise, run, substitution, x-axis, y-axis, general form, gradient-intercept form, prove, derive, collinear.Students also need to be aware of symbols such as ∥ for parallel to and ⊥ for perpendicular. The letter m represents gradient and b or c represents the y-intercept.
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In Stage 5.3, students use formulae to find the distance, midpoint and gradient of a line and present proofs. This unit is targeted at students who will be doing the Mathematics Course in Senior Years.
forms of the equation of a straight line
Catholic Perspectives School Free Design
The Mathematics teachers undertake to uphold the ethos and teachings of the Catholic church and to support the liturgical life of the College. They promote in the classroom a sense of compassion, respect between students and staff and a positive and supportive learning environment.
Assessment Overview
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Content Teaching, learning and assessment Resources
Stage 5.1 - Linear RelationshipsStudents
Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)
determine the midpoint of an interval using a diagram
use the process for calculating the 'mean' to find the midpoint, M, of the interval joining two points on the Cartesian plane
explain how the concept of mean ('average') is used to calculate the midpoint of an interval (Communicating)
plot and join two points to form an interval on the Cartesian plane and form a right-angled triangle by drawing a vertical side from the higher point and a horizontal side from the lower point
use the interval between two points on the Cartesian plane as the hypotenuse of a right-angled triangle
and use the relationship to find the gradient of the interval joining the two points
describe the meaning of the
Students are shown various lines joining two points and students determine the midpoint of each line by working out the halfway point for both the x-coordinate and y-coordinate.
Discuss the concept of finding the mean (average) of two numbers.
Ask students why we discussed finding the mean of two numbers? How does this relate to finding the midpoint of an interval?Give students various points to plot on the Cartesian plane. Afterwards form a right-angled triangle.Students calculate the vertical displacement (rise) and horizontal displacement (run) for the given diagrams.Make sure that they are not only in one direction. Describe the concept of gradient and where they may have seen it, for example, on the sign on Mt Victoria Pass.Discussion of such objects as a ramp at a skate park. Why is the gradient important?Could look at Scenic Railway Katoomba to discuss the gradient and steepness of a curve.Risks involved if the gradient is too high, etc – for example, accidents on skateboard.When is it a positive gradient and when is it negative? Use ideas such as uphill and downhill on a bike.Use graphing software such as Autograph or Geogebra
Grid paper.Graphing software such as Autograph or Geogebra.If using iPads, grid lines can be drawn on to assist students in finding the midpoint visually.Students could view the following website to discuss changes in gradient and also to understand the concept of gradient:http://www.scenicworld.com.au/experience/scenic-railway/
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Content Teaching, learning and assessment Resources
gradient of an interval joining two points and explain how it can be found (Communicating)
distinguish between positive and negative gradients from a diagram (Reasoning)
use graphing software to find the midpoint and gradient of an interval
Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)
use the interval between two points on the Cartesian plane as the hypotenuse of a right-angled triangle and apply Pythagoras' theorem to determine the length of the interval joining the two points (ie 'the distance between the two points')
describe how the distance between (or the length of the interval joining) two points can be calculated using Pythagoras' theorem (Communicating)
use graphing software to find the
to find the midpoint and gradient of an interval.AdjustmentGrid lines would be helpful for some in seeing the midpoint of an interval.AssessmentInformally assess students understanding through the discussion of concepts.
Recap Pythagoras’ Theorem. Ask students to explain why we are doing Pythagoras’ Theorem in this topic? What is the link to finding the distance between two points?Using the right-angled triangles drawn to calculate the gradient, now use them to calculate the distance between them.Also use graphing software to find the distance between two pointsAdjustmentCare should be taken with finding the square root of numbers that are not perfect squares. For Stage 5.1, only use numbers that are perfect squares.Assessment
Assess students through the graphics software used to find the distance between two points on the Cartesian plane
http://www.scootle.edu.au/ec/ 4
Content Teaching, learning and assessment Resources
distance between two points on the Cartesian plane
Sketch linear graphs using the coordinates of two points (ACMNA215)
construct tables of values and use coordinates to graph vertical and horizontal lines, such as , ,
, identify the x- and y-intercepts of
lines identify the x-axis as the line y = 0
and the y-axis as the line x = 0 explain why the x- and y-axes
have these equations (Communicating, Reasoning)
graph a variety of linear relationships on the Cartesian plane, with and without the use of digital technologies, eg
, , , ,
compare and contrast
equations of lines that have a negative gradient and equations of lines that have a positive gradient (Communicating, Reasoning)
Using a table of values, students substitute different values of x into a given linear relationship. They then plot points from the table of values to graph the line.
Define the x-intercept as the value of x when y = 0 and the y-intercept as the value of y when x = 0. Determine the x-intercept and y-intercept for your given lines. Why is this so?
Graph various linear relationships using graph paper as well as digital technologies such as graphics calculators and graphics software.Links could be established to other KLAs such as the Graphics elective in Year 9.
What determines the way that the line slopes? Students investigate various lines to answer this question.
Does a point lie on a line. Yes if the equation works, no if it does not.AdjustmentUse of graphics calculators or similar to first see how to graph a linear relationshipAssessmentCollect student work samples of various graphs to determine whether they have understood the concept.
viewing/L6552/index.html
Grapher inhttp://nlvm.usu.edu/en/nav/category_g_4_t_2.htmlGraphics calculatorExcel spreadsheet
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Content Teaching, learning and assessment Resources
determine whether a point lies on a line by substitution
Solve problems involving parallel lines (ACMNA238)
determine that parallel lines have equal gradients use digital technologies to
compare the graphs of a variety of straight lines with their respective gradients and establish the condition for lines to be parallel (Communicating, Reasoning)
use digital technologies to graph a variety of straight lines, including parallel lines, and identify similarities and differences in their equations (Communicating, Reasoning)
Stage 5.2 - Linear RelationshipsStudents:Interpret and graph linear relationships using the gradient-intercept form of the equation of a straight line
Show various linear relationships.What is in common if they are parallel?Students investigate and discover that if the lines are parallel, their gradients are the same.
Graph equations given in the form, for example, y = 2x + 1.
Using the equations just drawn, calculate the gradient of each line and the y-intercept of each line.Through investigation, ask students what does the m stand for in and what does the b stand for in
?
Function Transformations inhttp://nlvm.usu.edu/en/nav/frames_asid
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Content Teaching, learning and assessment Resources
graph straight lines with equations in the form ('gradient-intercept form')
recognise equations of the form as representing straight
lines and interpret the x-coefficient as the gradient, and the constant as the y-intercept, of a straight line
rearrange an equation of a straight line in the form ('general form') to gradient-intercept form to determine the gradient and the y-intercept of the line
find the equation of a straight line in the form , given the gradient and the y-intercept of the line
graph equations of the form by using the gradient and the y-intercept, and with the use of digital technologies use graphing software to graph a
variety of equations of straight lines, and describe the similarities and differences between them, eg
(Communicating)
explain the effect of changing the gradient or the y-intercept on the
Recap equations first using examples such as x + 4 = 6., 3 + 2y = 13, etcAfter this, give students equations such as x + y = 6, x + 2y = 13 and then rearrange the equation into the form . After this, find the gradient and y-intercept of the line.Do the corollary: give students the gradient and y-intercept and ask them what the equation of the line is.
Using graphics calculators or graphics software, graph a variety of lines and see similarities and differences, for example, y = x and y = x + 1 have the same gradient but they have a different y-intercept.Also, look at changing the gradient and explain what happens to the line as the gradient increases and decreases. Also discuss the effect of changing the y-intercept of the line.
Calculate the gradient and y-intercept of a line from its graph and use these to write the equation of the line.
Match up equations with their graphs using only the necessary information – y-intercept and 2 points.AdjustmentIn the matching up of equations to lines, give equations that don’t fit any graph such as equations of non-linear relationships so that there are more
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Content Teaching, learning and assessment Resources
graph of a straight line (Communicating, Reasoning)
find the gradient and the y-intercept of a straight line from its graph and use these to determine the equation of the line match equations of straight lines
to graphs of straight lines and justify choices (Communicating, Reasoning)
Solve problems involving parallel and perpendicular lines (ACMNA238)
determine that straight lines are perpendicular if the product of their gradients is –1
graph a variety of straight lines, including perpendicular lines, using digital technologies and compare their gradients to establish the condition for lines to be perpendicular (Communicating, Reasoning)
recognise that when two
equations than graphs to choose from. This is to extend students.AssessmentGiven various equations and their graphs, students match them up. Teacher collects answers and corrects them.
Revise conditions necessary for straight lines to be parallel, that is, their gradients are the same.
Calculate the gradients of straight lines and then through investigation, students discover the conditions necessary for lines to be perpendicular, that is, that the product of their gradients is -1.
Using this, students are asked to write down equations of lines parallel or perpendicular to another line given in the form .
AdjustmentSome prompting may be necessary for some students to see the link, such as asking them to calculate the gradient of each line and then multiply them.Once discovered the link, students could be asked to prove whether two lines are perpendicular or not.
The following websites could be used at this point. It asks distance, midpoint, gradient, parallel lines, etc.
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Content Teaching, learning and assessment Resources
straight lines are perpendicular, the gradient of one line is the negative reciprocal of the gradient of the other line (Reasoning)
find the equation of a straight line parallel or perpendicular to another given line using
Stage 5.3 - Linear Relationships §Students:
Find the midpoint and gradient of a line segment (interval) on the Cartesian plane (ACMNA294)
use the concept of an average to establish the formula for the midpoint, M, of the interval joining two points and on the Cartesian plane:
explain the meaning of each of the pronumerals in the formula for midpoint (Communicating)
use the formula to find the midpoint of the interval joining two points on the Cartesian plane
Students should be introduced to the concept of midpoint as the average of the x-coordinate and the average of the y coordinate first using grid paper or digital technologies.After this, they can be shown the formula for the midpoint. Begin with positive whole numbers first before moving onto a mixture of positive and negative numbers.Ask students to explain the meaning of each of the pronumerals in the formula for the midpoint.Calculate the midpoint given the formula. Students should be able to recall this formula when being assessed.
Students should have been introduced to the concept of gradient and then using two given points be able to discover the formula for the gradient of a straight line.Show that the order that the formula is written in
or will give you the same answer.
Videos are available for those that do not know how to do these distance, midpoint, etc.https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/e
http://www.amsi.org.au/teacher_modules/Introduction_to_coordinate_geometry.htmlLine plotter and grapher inhttp://nlvm.usu.edu/en/nav/category_g_4_t_2.html
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Content Teaching, learning and assessment Resources
use the relationship to establish the formula for the gradient, m, of the interval joining two points and on the
Cartesian plane: use the formula to find the gradient
of the interval joining two points on the Cartesian plane explain why the formula
gives the same value
for the gradient as (Communicating, Reasoning)
Find the distance between two points located on the Cartesian plane (ACMNA214)
use Pythagoras' theorem to establish the formula for the distance, d, between two points and on the Cartesian plane:
explain the meaning of each of the pronumerals in the formula for distance (Communicating)
use the formula to find the distance between two points on the Cartesian plane explain why the formula
Adjustment
Assessment
The concept of finding the distance between two points using Pythagoras’ Theorem should be established first.Then students need to be able to show that it can be found using the formula and be able to explain what each pronumeral means.
Show that
and will give you the same answer.Adjustment
Assessment
Substitute x = 0 into a given equation to find the y-intercept and substitute y = 0 to find the x-intercept. Using these two points, graph the line.
For showing the link between finding the equation of a line passing through a point with a given gradienthttp://skwirk.com.au/esa/
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Content Teaching, learning and assessment Resources
gives the same value for the distance as
(Communicating, Reasoning)
Sketch linear graphs using the coordinates of two points (ACMNA215)
sketch the graph of a line by using its equation to find the x- and y-intercepts
Solve problems using various standard forms of the equation of a straight line
describe the equation of a line as the relationship between the x- and y-coordinates of any point on the line recognise from a list of equations
those that can be represented as straight-line graphs (Communicating, Reasoning)
rearrange linear equations in gradient-intercept form ( ) into general form
find the equation of a line passing
Adjustment
Assessment
Students are asked to recognise various forms of the equation of a line(ax +by +c = 0 and y = mx +b) .Students identify from a list of equations which ones are linear and which ones are not by whether they are written in the standard form.Students need to be able to rearrange linear equations from one form into another. Begin by solving equations as shown in Stage 5.2.
Find the equation of a line that passes through a point with a given gradient using the point-gradient form of a line or the gradient-intercept from of a line, .This can be done using the given link or visually using graph paper
By determining the gradient of the line passing through two points and then using one of these points and the point-gradient formula, find the equation of the line.Rearrange the formula to write it in the general form.
Parallel_Lines.html
For links between perpendicular lines and passing
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Content Teaching, learning and assessment Resources
through a point , with a given gradient m, using:
point-gradient form:
gradient-intercept form:
find the equation of a line passing through two points
recognise and find the equation of a line in general form
Solve problems involving parallel and perpendicular lines (ACMNA238)
find the equation of a line that is parallel or perpendicular to a given line
determine whether two given lines are perpendicular use gradients to show that two
Adjustment
Assessment
Students are asked to recognise various forms of the equation of a line(ax +by +c = 0 and y = mx +b) .Students identify from a list of equations which ones are linear and which ones are not by whether they are written in the standard form.Students need to be able to rearrange linear equations from one form into another. Begin by solving equations as shown in Stage 5.2.
Find the equation of a line that passes through a point with a given gradient using the point-gradient form of a line or the gradient-intercept from of a line, .This can be done using the given link or visually using graph paper
By determining the gradient of the line passing through two points and then using one of these points
through a point:http://skwirk.com.au/esa/Perpendicular_Lines_Linear_Equations.html
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Content Teaching, learning and assessment Resources
given lines are perpendicular (Communicating, Problem Solving)
solve a variety of problems by applying coordinate geometry formulas derive the formula for the
distance between two points (Reasoning)
show that three given points are collinear (Communicating, Reasoning)
use coordinate geometry to investigate and describe the properties of triangles and quadrilaterals (Communicating, Problem Solving, Reasoning)
use coordinate geometry to investigate the intersection of the perpendicular bisectors of the sides of acute-angled triangles (Problem Solving, Reasoning)
show that four specified points form the vertices of particular quadrilaterals (Communicating, Problem Solving, Reasoning)
prove that a particular triangle drawn on the Cartesian plane is right-angled (Communicating, Reasoning)
and the point-gradient formula, find the equation of the line.Rearrange the formula to write it in the general form.
Adjustment
Assessment
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Content Teaching, learning and assessment Resources
Registration Evaluation
Class: __________________________Start Date: _______________________Finish Date: ______________________Teacher’s Signature: _______________________
Teachers evaluate the extent to which the planning of the unit has remained focused on the syllabus outcomes. After the unit has been implemented, there should be opportunity for both teachers and students to reflect on and evaluate the degree to which students have progressed as a result of their experiences, and what should be done next to assist them in their learning.Evaluation reflects:
The effectiveness of the program in meeting the diverse needs of students and identifies curriculum adjustments
Level to which syllabus outcomes have been demonstrated by students The effectiveness of pedagogical practices employed Suggested program adjustments Elements of the school’s Contemporary Learning Framework
Sample questions
Highlight the response that best describes your view to the following statements and provide comments in the spaces provided.
1. The set text/s (if relevant) were suitable for the student needs and interests:
STRONGLY AGREE AGREE UNSURE STRONGLY DISAGREE
2. There were sufficient and suitable resources to teach the unit:
STRONGLY AGREE AGREE UNSURE STRONGLY DISAGREE
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3. There was sufficient time to teach the set content:
STRONGLY AGREE AGREE UNSURE STRONGLY DISAGREE
4. Assess the degree to which syllabus outcomes have been demonstrated by students in this unit:
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5. Evaluate the degree to which the diverse needs of learners have been addressed in this unit:
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6. Comment on the effectiveness of pedagogical practices employed in this unit:
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7. Assessment was meaningful and appropriate to reflect student learning and achievement:
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8. Suggested program adjustments / other comments:
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