Download ORANGE HIGH SCHOOL MATHEMATICS

Name: …………………………………………………
Class Teacher: ……………………………………..
ORANGE HIGH SCHOOL
MATHEMATICS
YEAR 9 2015 - Assignment 2
TIME ALLOWED - 2 weeks
Due: Monday 3rd August 2015
INSTRUCTIONS
•
Read all questions carefully and answer each question in the space provided.
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All working out should be shown in every question.
•
Correct grammar, punctuation and spelling should be ensured for questions requiring full sentence
answers.
•
Calculators may be used.
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The mark for each question is indicated.
Trigonometry:
/38
Linear Functions: /27
Evaluation:
/15
TOTAL:
/80
YEAR 9
TRIGONOMETRY
Engage
What is Trigonometry? Investigate the relationship between
sides and angles in right-angled triangles.
Explore
Students are to research the origins of Trigonometry and
investigate the occupations in which this area of mathematics
is useful.
Task 1
Explain
Students explain how Trigonometry is used to perform
calculations and make judgements in different situations.
Tasks 2 and 3
Students explain what they understand and question about the
applications of Trigonometry.
Elaborate
Evaluate
SECTION 1
Connect-ExtendChallenge sheet
Students conduct further investigation of applications of
Trigonometry including Australian standards for access
ramps and calculating the heights of objects of an
inaccessible height.
Tasks 2 and 3
Students reflect on their learning in the Evaluation Section.
Evaluation Tasks
Reflect using the learning grid and self assessment.
• What have I learned?
• What did I do well?
• What did not go so well?
• What could I do to improve next time?
Worksheet Reflection
Assessment Rubric
YEAR 9
LINEAR FUNCTIONS
Engage
Applying linear functions to solve real world problems.
Explore
Students investigate which Fitness Centre best suits them in
terms of value. The graph allows them to explore multiple
scenarios in making their decision.
Task 1
Explain
Using the graphs to aid them, students explain which Fitness
Centre is the best value for their needs.
Task 1
Elaborate
Apply linear function processes to unfamiliar problems.
Task 2
How does this connect to what you have learnt in class?
Evaluation Tasks
Reflect using the learning grid and self-assessment.
• What have I learned?
• What did I do well?
• What did not go so well?
• What could I do to improve next time?
Worksheet Reflection
Evaluate
SECTION 2
Assessment rubric
SECTION 1: TRIGONOMETRY (38 marks)
Task 1: Trigonometry Research (9 marks)
Use the internet to research the following questions.
•
•
All websites used should be referenced at the end of each question.
Marks will be allocated for full sentences. Ensure that you use correct punctuation,
grammar and spelling. An extra 3 marks will be allocated for this.
Your mark: _____/3
1. In your own words, give a formal definition of ‘Trigonometry’.
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2. Explain how the word ‘trigonometry’ was derived.
E.g. the word perimeter comes from the Greek words peri (around) and metron
(measure).
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3. List 6 occupations that use trigonometry. (Sentences not required.)
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4.
Choose one of these occupations. Describe an example of how trigonometry is used to
perform a task in this occupation. Make reference to sides and angles in your description.
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Task 2: Access Ramps (16 marks)
Just like designing the gradient (steepness) of roads, engineers need to follow guidelines
when designing access ramps.
Australian standards regulate that ramps have an inclined access way with a gradient
steeper than 1 in 20, but not steeper than 1 in 14.
(http://www.jobaccess.gov.au/content/ramps)
1.
Explain what a gradient of 1 in 14 means, in terms of vertical and horizontal distances.
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2.
Draw a scale diagram of a ramp with a gradient of 1 in 14.
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3.
Mark the angle of elevation of the ramp on your diagram with the symbol 𝜃𝜃.
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Using a protractor, carefully measure this angle to the nearest degree.
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Angle of elevation = ………..
4.
Now use trigonometry to calculate the size of the angle of elevation.
(Give your answer correct to the nearest degree.)
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5.
What was your measurement error?
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Measurement error = calculated angle – measured angle
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6.
A ramp is to be constructed to enable access to an entry that is 90 centimetres above the
ground.
Use trigonometry and the angle you calculated in Question 4 to determine the minimum
horizontal (base) length (b) of the ramp.
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Give your answer in metres correct to 2 decimal places.
Firstly, label the triangle with the side length and the angle to represent the situation.
b
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7.
Handrails are to be constructed on both sides of the slope of the ramp from Question 6.
They are to extend along the entire slope.
Calculate the length of steel that is required to make the horizontal section of the
handrails. Hint: use Pythagoras’ Theorem.
Give your answer in metres correct to 2 decimal places.
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8.
Another access ramp has a horizontal base length of 8.5 metres and vertical height of 0.4
metres. Does this ramp meet Australian standards as described at the start of this section?
Explain your answer using both clearly worked calculations and sentences.
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Task 3: Clinometers (10 marks)
An inclinometer, or clinometer, is an instrument for measuring angles of slope (or tilt) and
elevation or depression of an object with respect to gravity. They measure both inclines and
declines. In ancient times, clinometers called Astrolabes were used for navigation and
locating astronomical objects.
Your task is to create and use a clinometer using the instructions given to calculate the
height of an everyday object. Examples are not limited to but may include:
- A tree
- Goal posts
- A Building
Things to consider when choosing an object:
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You need to be able to see the top of the object from your viewpoint.
Flat ground is required and you must be able to measure the distance from you to the
object’s base.
As a clinometer requires two people for best accuracy, you may team up with another student
or get a guardian’s help.
Steps:
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Remove the last page of this Assignment and follow the instructions to make a
clinometer (take care as your clinometer will be allocated a mark out of 2).
Glue the clinometer template onto cardboard for greater strength and accuracy in
measurement.
Visit the Orange High website and read the Steps to using a clinometer.
Follow the steps to construct the relevant diagrams and calculate the height of the
object.
Clinometer
Complete the table below
Eye Height
(metres)
Distance from object
(metres)
/2
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Angle Measured
Angle of Elevation
(90º - Angle Measured)
Draw and label a neat diagram of the object and measurements using the information
collected (similar to step 5 in the link).
2
Simplify your diagram to a right- angled triangle and calculate the vertical side length of the
triangle using the appropriate trigonometric ratio (steps 6 and 7).
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Add your solution for the triangle to your eye height to calculate the height of the object (step
8).
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SECTION 2: LINEAR FUNCTIONS (27 marks)
Task 1: Fitness Centre (12 marks)
Marks will be allocated for full sentences. Ensure that you use correct punctuation, grammar
and spelling. An extra 3 marks will be allocated for this.
Your mark: _____/3
Two of the top fitness centres in Orange have the current deals.
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1.
Fitness Centre A is allowing anyone the opportunity to visit by offering classes at a
cost of $12 per class.
Fitness Centre B is a loyalty club where members are charged a fee of $20 per month
(paid at the start of each month) and are charged $8 per class.
Complete a table of values for both fitness centres:
Fitness Centre A
0
Number
of Classes
1
1
2
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4
5
6
7
Cost ($)
Fitness Centre B
Number
of Classes
0
1
1
2
3
4
5
6
7
Cost ($)
2.
Graph and label both options on the grid below
Give the graph a title, label both axes and label the options as Fitness Centre A and B.
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3.
(a) How much does it cost to attend 2 classes at Fitness Centre B?
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(b) How many classes need to be attended for the cost of each Fitness Centre to be the
same?
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(c) Determine the cost per month to attend 8 classes at Fitness Centre B.
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(d) If you were choosing between the two fitness centres, which would you choose?
Why?
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(e) Fitness Centre A is losing customers. To compete with Fitness Centre B they
create a new deal. They claim that the cost to attend 10 classes at their centre is
half that of Fitness Centre B. They also have a monthly fee of $20. Under this deal
what would be the new cost of each class?
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Task 2: Triangles On the Cartesian Plane (12 marks)
1.
(a) Plot the points A(-2, 1) and B(4, 5) on the number plane below.
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Draw a right-angled triangle joining these two points.
(b) Write the coordinates of the midpoint of A and B.
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(c) Determine the gradient of the line joining points A and B.
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(d) Calculate the length of the interval joining points A and B, correct to 1 decimal place.
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(e) Calculate the area of the right-angled triangle.
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2.
For the linear function (straight line graph) below:
(a) Write the y-intercept. …………………..
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(b) Determine the gradient.
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(c) Write the equation of the graph in gradient-intercept form. (𝑦𝑦 = 𝑚𝑚𝑚𝑚 + 𝑏𝑏 )
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(d) Calculate the area of the triangle formed by the x and y axes and the straight line
graph.
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3.
Consider the following information about a linear function (straight line graph):
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The distance from the origin to the y-intercept of a linear (straight line) graph is twice
the distance from the origin to the x-intercept.
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The area of the triangle formed by the line and the axes is 2.25 units2.
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The line has a negative gradient.
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The line has a negative y-intercept.
Determine the gradient and y-intercept of the line.
You may use the number plane below to assist you. (Use pencil so that you can make
changes.)
Show all of your thinking and calculations to fully communicate all steps of your answer.
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SECTION 3: EVALUATION (15 marks)
Activity Recording Sheet
Name: ...........................................................................................................................................................
Learning obstacle tasks: TRIGONOMETRY and LINEAR FUNCTIONS
1.
What were the activities asking you to FIND OUT for both tasks?
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2.
What STRATEGIES did you use to help solve the problems?
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 look for a pattern
 guess and check
 work backwards
 draw a picture
 make a list
 make a table or diagram
 use objects to act out the problem
 brainstorm
 use logical reasoning by eliminating some answers
3.
Did you get anyone to help you solve the problems? Describe the assistance given.
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4.
How do you rate the problems?
 The problems were easy to solve.
 The problems needed some brain power.
 The problems were hard to solve and needed lots of brain power.
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Connect–Extend–Challenge
(6 Marks)
Connect:
Extend:
Challenge:
How are the ideas and
information connected to
what you already know?
What ideas did you get that
extend your thinking in
new directions?
What is challenging or
confusing for you? What
questions do you have?
What puzzles you?
Worksheet Reflection
Reflection
• What have I learned? What did I do well?__________________________________________ 2
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What did not go so well? What could I do to improve next time?________________________ 2
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Assessment Rubric
Name: ………………………………………………………………………………………………………………………………………………Date: ___/___/20__
A
Criteria
• Look beyond
• Look within
• Look at
B
1.
Above the standard: expert
2.
At the standard: practitioner
E
C/D
3.
Approaching the standard:
moving forward
4.
Ungraded below standard:
travelling required
An efficient strategy is chosen and
progress towards a solution is
evaluated.
Adjustments in strategy, if
necessary, are made along the way,
and/or alternative strategies are
considered.
Note: The expert must achieve a
correct answer.
Deductive arguments are used to
justify decisions and may result in
more formal proofs.
A correct strategy is chosen based
on the mathematical situation in
the task.
Evidence of solidifying prior
knowledge and applying it to the
problem-solving situation is
present.
Note: The practitioner must
achieve a correct answer.
A systematic approach and/or
justification of correct reasoning
are present.
A partially correct strategy is
chosen, or a correct strategy for
solving only part of the task is
chosen. Evidence of drawing on
some relevant previous knowledge
is present, showing some relevant
engagement in the task.
No strategy is evident, or a strategy
is chosen that will not lead to a
solution. Little or no engagement
in the task is present.
Some correct reasoning or
justification for reasoning is
present with trial and error or
unsystematic trying of several
cases.
No correct reasoning or
justification for reasoning is
present.
Communication
• Look beyond
• Look within
• Look at
Precise mathematical language and
symbolic notation are used to
consolidate mathematical thinking
and to communicate ideas.
Communication of an approach is
evident through a methodical,
organised, coherent, sequenced and
labelled response.
No awareness of audience or
purpose is communicated. Little or
no communication of an approach
is evident.
Connections
Mathematical connections or
observations are used to extend the
solution.
Abstract or symbolic mathematical
representations are constructed to
analyse relationships and to extend
thinking.
Mathematical connections or
observations are clearly
recognised.
Appropriate and accurate
mathematical representations are
constructed and refined to solve
problems or portray solutions.
Communication of an approach is
evident through verbal/written
accounts and explanations, use of
diagrams or objects, writing and
mathematical symbols.
Some attempt is made to relate the
task to other subjects or to own
interests and experiences.
An attempt is made to construct
mathematical representations to
record and communicate problem
solving.
Problem solving
• Look beyond
• Look within
• Look at
Reasoning and proof in
support of conclusions,
opinions and claims.
• Look within
• Look at
Representations
Numerical grade [
No connections are made.
No attempt is made to construct
mathematical representations.
/ 80]; Letter grade [___] (A = 90–100%; B = 70–89%; C = 40–69%; D = 10–39%; E = 0-9%)
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