CAMPUS PLACEMENTS PREDICTOR about .pptxx

CAMPUS PLACEMENTS PREDICTOR about .pptx

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YEAR 9/10/10A
ALGEBRA INQUIRY
TASKS

Goal Setting & Personalized learning
Compiled by Mr Soo

2. - To apply algebraic concepts,

| N T 3 N a - = ; manipulate variables, linear
y ner m.

algebraic relationships and
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nonlinear algebraic relationships.
E TECLA
- To consider the trajectory of

learning of algebra through goal
setting.

Sample Foater Tent 2

3. BIG IDEAS — CONCEPT MAPPING

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7. Option 1:

Task: (Guided Inquiry)

+ Select your goals from the
list of options:

+ Year 9, Year 10 and Year
10A

SEITE er
TRAJECTORY

AND GOAL

SETTING

Sample Foater Text

8. OPEN INQUIRY REN

Open Inquiry: Algebraic functions
and applications in real-life.
Please see your teacher for the
structure of your research

Sample Foater Text

9. CURIOSITIES

- Factual
+ Conceptual
+ Debatable

12. Analysis

1 Janessa is driving at an average speed of 80 km/h from
her workplace to her home, a distance of 25 km.

a Write an equation in terms of the distance Janessa
has travelled, d kilometres, and the time elapsed
since leaving work, ı minutes.

b State the dependent and independent variables.

€ Determine the expected time it will take for
Janessa to drive home in minutes and seconds.

d After 10 minutes, Janessa has to drive at 40 km/h
for 2 km of roadwork before returning to her
average speed of 80 km/h. How much longer will it
take for her journey home?

e Draw a distance-time graph of Janessa's journey
home.

13. OPEN INQUIRY

+ Taxi Task:

+ Can we investigate how this concept can be used to answer our curiosities of taxi fares
around the world?

14. 2 Ben has a simple interest bank account carning 6.2% per annum calculated and paid monthly on the minimum
monthly balance. Ben's bank statement for October is shown below,

11316,50
13 196.50

31/10 | Interest

Calculate the interest that Ben’s account earned in October.
b If Ben worked four 40-hour weeks, what is his hourly net wage?
When grocery shopping on the 5/10, Ben had to decide between two options:
* 12 rolls of toilet paper for $12.96
* 24 rolls of toilet paper for $25.68
© Which option is the best buy?
d What percentage of the grocery shopping was the best buy option, correct to one decimal place?
The EFTPOS purchase on the 19/10 was after a 30% discount. Determine the price before the sale.

15. 3 Lisa has modelled the profit she will ely carn im her candle business, $2", whem selling m canales (units) with
the equation "= ~n* + 410 ~ 11000.

ae et

= Calculate the profit earned! ce lows mache when she sells these units:

1 100 M 200 M 400 lv so vo
Dd 1. Determine the number of units that would make the company the ment profit. Hint: Find the migpetet

between $0 ant 360.

AL State the amount of profit Lina woud mak.
€ Use your amer for part b to write the equation for the peofi in the form P= a(n + Dt e
Ifthe selling price per unis is $500 and the number of units in part is sold, determine:

A the sonal cout

A the percentage the profit i of the total revenue eue, correct to one decimal place
© Ifthe setting price per unit iv $500, write an equation for:

1 the total revenue, SA

Mabe total cost, SC

16. YEAR 10 GOALS

17. Analysis

1 "The cross-section of a building is drawn on a Cartesian plane with the scale on the axes showing length in

metres. The x-axis represents ground level,

a On the same Cartesian plane, sketch the graph of:

1 &-x=12 5 y=5-Le

b To represent the cross-section of the building, shade the area between the graph of dy — x = 12 and the
axis from x = 010 =, as well as the arca between the graph of y = $— x and the xaxis between
x=dandx=8.
How tall is the building at its highest point?
What is the distance from the top of the roof to the lower edge of the roof, correct to one decimal place?
What is the positive gradient of the roof?

Ifa chimney is to be placed halfway along the slope of the roof on the side with the positive gradient,
describe its position on the Cartesian plane.

mane

18. 2 A rectangle DEFG has vertices at D(-2, -1), E(0, 1), FG, -2) and GC, 4).

Draw the rectungle on the Cartesian plane.

Calculate the lengths of all the sides of the rectangle, correct to one decimal place.
Using your sketch, identify the intercepts of the line segments:
1 DE a OF i DG le FO

4 Find the gradients of the lines through:

i um m Be te FO

If point Pis the midpoint of DE, point Q is the midpoint of EF, R is the midpoint of FG and $ is the
midpoina of DO, Find the coandinates of:

ip uo mR ws
Describe the shape of the figure PORS, Justify the statements you make,

If the original figure DEFT had been à square instead of a rectangle, explain how this would affect the
shape of PORS. Support your answer with mathematical evidence.

as

19. 510A GOALS |

Urea:

20. Analysis

3 The parabolas you have looked at so far are vertical parabolas, because they cach have a vertical axis of
symmetry. There are also horizontal parabolas, for which the axis of symmetry is horizontal. Similar to the way
we have used y = x" as the basic rule for a vertical parabola, we can use x = y' as the basic rule for a horizontal
parabola.

A horizontal parabola can be represented by the general equation x h = aly —A)?, where (h, À) is the turning
point of the parabola, and a is a constant.
a Consider the graph of x + 4 = (y+ 1)’.
1 Provide values for a, hand k
4 What are the coordinates of the turning point?
fii There is one x-intercept. Calculate its coordinates,
iv There are two y-intercepts. Calculate the coordinates of both points.
w Use the coordinates of the turning point and the intercepts to sketch the graph on a Cartesian plane.
vi Describe the transformations that need to be performed on the graph of x = y” to produce the same
graph you sketched in part v.
b Follow the procedure in part a to sketch the graph of x 4=-(y+ 1)* on the same Cartesian plane as the
previous graph.
€ Compare the two graphs from parts a and b. What is the effect of the negative value of a?

21. 2 By now, you are familiar with the quadratic formula expressed in the form x = EST

£ : ae 2e
An alternative form of the quadratic formula is x = Be per
a Show that both quadratic formulas give the same solution for the quadratic equation € - x — 3 = 0. Give
your solutions correct to one decimal place.
Ge b Start with the first formula given above. Show how algebraic methods can be used to transform the

standard quadratic formula into the second formula. (Hint: Take the positive value of the square root, then
rationalise the numerator.)

22. BI 2 ‘The graph of hyperbola consists of two branches, which are rotationally symtpetrical around a centre point.
“That is, there is a point in the centre of the hyperbola and when the graph is rotated 180° about this point, the
result is the same graph.

“The shortest distance between the two branches of a hyperbola can be found by drawing a straight line angled
at 45” (that is, with a gradient of 1 or —1) that passes through the centre of the hyperbola.

A hyperbola with equation y = = + à has its centre at (Ay),

‘Consider the hyperbola described by the equation y = y + 2.

a What is the centre of this hyperbola?

Sketch the graph of y = + 2.

Find, using tial and error, the equation for the circle with the same centre as the hyperbola, that just
twxiches each branch of the hyperbola,

Une the equation of the circle you found in part © 10 find the shortest distance between the two branches of

+2
€ Consider the hyperbola y = § where a > 0, and the circle x? + ym e,
Find the relationship between a and r, such that the circle just touches each branch of the hyperbola. I is
possible to find this relationship algebrascalty, but it might be easier to find it with tial and error using
technology, by first sketching the circle for specific values of rand then finding corresponding values of a,
£ Find the shortest distance between the two branches of the general hyperbola: y = E + A, where a> 0,

oe

23. Analysis
AX: You should be familiar with the infinity symbol. It looks like the number 8 on its side. Its shape is similar to the
graph of a cubic function with three x-intercepts, combined with its reflection in the x-axis,
a Itis possible to model the infinity sign using two cubic relationships graphed for the same set of x values.
4 Copy and complete the table below for y= xx.

05
06
1

Ü Repeat part for y

iii On a Cartesian plane, plot the points from the tables in parts 4 and i. Join the points with a smooth curve,

iv Describe the domain of the two graphs.

v Do the turning points for the two relationships occur at x =-0.5 and x = 0.5? Explain.

vi Use digital technology to draw the graphs of these two relationships for the set af x values described in
part iv. Find the coordinates of the turning points,

+x,

24. b Ir is also possible to model the infiedty syenbol by tracing the path a point tales an moves from an angle of 0°

‘with the z-axis to an angle of 360" with the a-axis,

"The coordinates of al points Gr, 9 om the infinity symbol can be represented by the values (cos 19, 221201)

where is the angle made with the x-axis by the line joining a point with the oeigin.

|. Conmiructa table like the cese below. for cand using angles of D increasing by 15%, been (7 and 360%,
(This number of points is necewary for an accurate representation of the graph.)

Complete the table, giving your answers correct tn two decimal places if nocesary. You might find some of
the trigonometric values 10 be negative as the angle increases beyond 90". You will understand the reason
for this when you study the trigonometric ratios for angles greuser than 90" in Chapter 8.

HA Plot the poines (x,y) on a Cartesian plane. Join theen with a smooth curve. (Alternatively, you could use a
spreadsheet to plot the graph.)

fv Describe the shape of the graph.

Y. Describe the domain and range ofthe graph,

25. Analysis

1 The cross-section of a building is drawn on a Cartesian plane with the scale on the axes showing length in

metres. The x-uxis represents ground level,

a On the same Cartesian plane, sketch the graph of:

i dy-x=12 u y=5-}x

b To represent the cross-section of the building, shade the area between the graph of 4y — x= 12 and the
acaxis from x = 0 to x = 4, as well as the area between the graph of y = 5 - Lx and the x-axis between
x=4andx=8.
How tall is the building at its highest point?
What is the distance from the top of the roof to the lower edge of the roof, correct to one decimal place?
What is the positive gradient of the roof?
Ifa chimney is to be placed halfway along the slope of the roof on the side with the positive gradient,
describe its position on the Cartesian plane,

mene

26. AOPEN INQUIRY

27. USTRA

EXTENSION

2 Borders

The design shown I formed by removing one square from each corner of

a rectangular grid and shading a border of thickness 1 unit, In this 6 x 8

example, there are 24 shaded squares in the border and 20 unshaded squares

in the interior.

a Ignoring rotations, we want to find the number of such designs that have
the same numbers of border and interior squares. Follow these steps:

4 Let the dimensions of the original rectangular grid be x x y. Find
expressions, in terms of x and y, for the number of border squares and
the number of interior squares.

14 Scuting the expressions equal, rearrange the equation to get 0 on the right-hand side.

MA The left-hand side can almost be fully factorised as a binomial product of the form (x ~__}(y - —
What number do you need to add to both sides of the equation to make this work?

lv Remember that x and y are positive integers, so the factorised expression on the left-hand side of your
equation should match up with a factorisation of the number on the right-hand! side. Deduce that there are
six possible pairs of positive integer solutions for x and y.

v Why does this mean that there are only three possible designs?

b Adapt the above methods to find, ignoring rotations, the number of such designs in which there are:

À twice as many interior squares as border squares

AL twice as many border squares as interior squares.

28. EE
OTHER POSSIBLE
INQUIRIES

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29. REFLECTION

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( Concrete Models ) ( Pictures \

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