91028 SINCOS10M – 4 credits 91028 – Investigate relationship between tables, equations and
graphs
You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE (a ) George is making a post and rail design, as shown in the picture below. Design 1 Design 2 Design 3 He begins a table for the number of wood pieces in the design patterns Design (n) Number of pieces of wood used in the design (W) 1 5
2 9
3 4 5 6 25
(i) Give the rule for calculating the number of wood pieces, W, that George will use in making the nth design. (ii) On a grid, sketch a graph showing the number of wood pieces used for up to the 10th design. (Axes should be ‐3 to 11 for n (x) and ‐10 to 50 for W (y)) (iii) Give the rule for the total number of wood pieces that George would need if he was to continue following the pattern and complete n designs. (iv) Use this rule to find the total number of wood pieces needed to complete the first 15 designs using George’s pattern. (b) Alison decides to make a different post and rail pattern. She begins with this basic design using 4 pieces of wood: Each new design has one section added to the previous design as shown: (i) Give the rule for the number of pieces of wood required to make the nth design in Alison’s pattern. (ii) Use the rule to find the number of pieces of wood required for the 8th pattern of this design. (iii) Describe how the graph for the number of pieces of wood Alison uses for n designs relates to George’s graph. 91028 SINCOS10M – 4 credits QUESTION TWO (a) Graham and Kirsty are making hot potato chips as a fund raiser for their school basketball team. They are going to buy 20 kg of peeled potatoes to sell in bags of hot potato chips. [Assume that there is no wastage.] Graham draws a graph of the profit that they hope to make against the number of kilograms of potato chips sold. Profit ($)
(i) From the graph, give the equation for the profit 200
made in terms of the number of kilograms of potato chips sold. 150
(ii) Find the cost of the potatoes for a 500g bag of hot potato chips. (iii) How much profit do they hope to make on each of 100
the 500 g bag of hot potato chips, and how is this shown on the graph? (iv) Suppose Kirsty is able to get the potatoes at a 20% 50
the discount from a grower. If they sell the hot potato chips at same price, explain, in detail, how this will affect the graph. 2
4
6
8 10 12 14 16 18 20 kg sold
– 50
Profit ($)
(b) Graham graphs the actual profit they make from the 200
sale of their chips. Identify the changes between this graph and the estimated 150
profit graph on the previous page, AND explain in detail why these changes may have occurred. 100
50
2
4
6
8 10 12 14 16 18 20 kg sold
– 50
y
QUESTION THREE 8
(a) For this graph give: (i) the intercepts 6
(ii) the equation (iii) The graph of the parabola above is moved 2 4
units to the left and 3 units up. Give the equation of the parabola in the new position, in 2
simplified form, AND also give the y‐intercept. (b) An archway in the Botanic Gardens is – 6
– 4
– 2
2
4
6
8 x
modelled by the equation y = x (7 − x) where y is – 2
the height of the archway above ground, in metres, and x is the distance, in metres, from the left hand side of the archway. – 4
(i) Sketch the graph of the archway on a grid (ii) What is the maximum height of the archway? – 6
(iii) A horizontal metal support is put on the archway at a height of 6 metres above the – 8
ground. How long is this support across the archway? You must support your answer with algebraic reasoning/equation solving. 91028 SINCOS10M – 4 credits ASSESSMENT SCHEDULE
91028 (1.3) Investigate relationships between tables, equations and graphs
Achievement
Achievement with Merit
Achievement with Excellence
Investigate relationships between tables,
equations and graphs:
Investigate relationships between tables,
equations and graphs, using relational
thinking.
Investigate relationships between tables,
equations and graphs, using extended abstract
thinking.
Investigate relationships involves
• making links between tables, equations and
graphs
• demonstrating knowledge of concepts and
terms
• communicating using appropriate numeric,
symbolic or graphical representations .
Relational thinking involves one or more of:
• selecting and carrying out a logical
sequence of steps
• connecting different concepts and
representations
• demonstrating understanding of concepts
• forming and using a model;
Extended abstract thinking involves one or more
of:
• devising a strategy to investigate a situation
• identifying relevant concepts in context
• developing a chain of logical reasoning, or
proof
• forming a generalisation;
and also relating findings to a context, or
communicating thinking using appropriate
mathematical statements.
and also using correct mathematical statements,
or communicating mathematical insight
Q1
1a(i)
Expected overage
W = 4n + 1
1a(ii)
Graph drawn
1a(iii)
Rule is
T = 2n 2 + 3n
Achievement
Merit
Excellence
Investigate
relationships
between tables,
equations and
graphs by:
Investigate
relationships between
tables, equations and
graphs, using
relational thinking by:
Investigate relationships
between tables,
equations and graphs,
using extended abstract
thinking by:
THREE of :
TWO of :
TWO of :
•
Correct discrete graph
from
n = 1 to n = 10 only
•
formed correct model
with correct answer
using correct
mathematical
statements
•
Demonstrated an
understanding of both
important concepts
with clear and
complete description
•
Writing a correct
equation
•
Continuous graph
with correct
gradient and y
intercept
•
correct graph (but
continuous) but has
only from n =1 to n
= 10
•
Correct answer of
495
•
formed a quadratic
model with correct
answer but minor
errors
T (15) = 495 pieces of wood
1b(i)
P = 3n + 1
•
Correct symbolic
relationship
1b(ii)
P (8) = 25
•
Correct answer of
25
•
The rule given and
correct answer given
1b(iii)
The gradient is less steep, decreasing
from 4 to 3.
The graph starts at a point 1 lower at
(1,4) instead of (1,5)
•
Demonstrated an
understanding of
one concept
correctly
•
Demonstrated an
understanding of
both concepts but
with incomplete
description(s)
91028 SINCOS10M – 4 credits Q2
2a(i)
Expected overage
P = 12 x − 40
Achievement
Merit
Excellence
Investigate
relationships
between tables,
equations and
graphs by:
Investigate relationships
between tables,
equations and graphs,
using relational thinking
by:
Investigate relationships
between tables,
equations and graphs,
using extended abstract
thinking by:
TWO of :
TWO of :
TWO of :
•
Writing a correct
equation
or
2a(ii)
$1 per 500g bag
•
$1 answer given
•
Linking graph to the
equation correctly and
solving one problem
correctly
or
2a(iii)
$5 per 500g bag
2b
Graph has a new y- intercept at
-$32 instead of -$40
The gradient will be the same.
After 15 kg of hot potato chips are
sold the gradient changes to 6
instead of 12.
This means the selling price halves
from $12 per kg to $6 per kg with
the total profit now $170.
The changes have occurred because
after selling 15 kg they changed the
selling price. [This could have
arisen to ensure that they did sell all
the potatoes and were not left with
some unsold etc]
Linked the graph to
the equation correctly
and used it to find
solutions to both
problems
•
Correct answer of
$5
(accept answer
given per kg)
•
Demonstrated an
understanding of
one important
concept (perhaps
vaguely
expressed)
•
Demonstrated an
understanding of both
important concepts but
with incomplete
description(s)
•
Demonstrated an
understanding of both
important concepts
with clear and
complete descriptions
•
Demonstrated
some
understanding of
gradient change
correctly (perhaps
vaguely
expressed)
•
Demonstrated a correct
understanding of
gradient change with
specific values quoted
but not in context
•
Demonstrated a correct
understanding of
gradient change with
specific values quoted
clearly in context
Shown from the total profit is $200
for 20 kg
2a(iv)
•
91028 SINCOS10M – 4 credits Q3
3a(i)
3a(ii)
Expected overage
x- intercepts at -1 and 2
y-intercepts at 2
(accept coordinates)
Achievement
Merit
Excellence
Investigate
relationships
between tables,
equations and
graphs by:
Investigate
relationships between
tables, equations and
graphs, using
relational thinking by:
Investigate relationships
between tables,
equations and graphs,
using extended abstract
thinking by:
THREE of :
TWO of :
ONE of :
•
Correct intercepts
or
y = −( x − 2)( x + 1)
= −x2 + x + 2
•
correct equation
•
correct equation and
intercepts
•
sketching new
graph and giving
correct yintercept
•
found new correct
equation but
incorrect y-intercept
•
Graph drawn
with correct
shape and correct
x-intercepts
•
Graph drawn with
correct shape and
intercepts showing
vertex between 12
and 12.5
•
Consistent with
graph drawn
•
CAO
•
Solving correct
equation but not
finding the length of
the support
(or equivalent)
3a(iii)
y = − x 2 − 3x + 3 or
1
1
= −( x + 1 ) 2 + 5
2
4
•
found new equation
correctly and gave
correct y-intercept
•
Solving correct
equation and using it
to correctly find the
length of the support
as 5 m
(or equivalent)
y-intercept at 3
3b(i)
y
12
9
6
3
2
4
3b(ii)
12.25 m
3b(iii)
6 = x (7 − x )
( x − 1)( x − 6) = 0
x =1 x = 6
6
8 x
Length of support is 5 m
Judgement Statement:
Achievement
Achievement with Merit
Achievement with Excellence
Minimum of
2A
Minimum of
2M
Minimum of
2E