FRACTIONAL INDICES · 3. Brian is using an iterative equation to find the roots of the equation = 2 + – 5. He rearranges the equation to give the following interative formula. Is

10M1 © Kangaroo Maths 2016

FRACTIONAL INDICES

Name:

BAM Indicator: Manipulate fractional indices

1. a) Evaluate:

i. 161/2

ii. 81/3

iii. 3-3

iv. 2563/4

v. 81-3/4

b) Find the value of x in each of the following equations:

i. 3x = 81

ii. 625x = 5

iii. 256x = 0.25

2. Henry says that 6-2 = -12.

Henry is incorrect. Explain what Henry has done wrong and correct his answer.

3. Put these numbers in ascending order:

80 −4 1612 64

23 2−2 8−

13

F M P


4. Find the area of this triangle:

5. State whether the following statements are always true, sometimes true or never true.You should justify your answer in each case.

a) Raising a number to a fractional power results in a smaller answer.

b) Any number can be raised to the power of 0.5.

c) You cannot achieve a negative answer by raising a positive number to an index.

A

R

Overall, I think my success level is: Low High

F = Fluency R = Reasoning P = Problem-solving A = Application M = Misconception

Q FRACTIONAL INDICES

I can evaluate a number raised to a fractional power.

I know that a negative power means you find the reciprocal.

I can solve problems involving fractional and negative indices.

Improvements I could make:

Mathematical presentation Method Accuracy Units


DIRECT AND INVERSE PROPORTION

Name:

BAM Indicator: Solve problems involving direct and inverse proportion

1. The gravitational force, F, between two masses is inversely proportional to the square

of the distance d between them.

When d = 8, F = 10.

a) Find a formula for F in terms of d

b) Calculate the value of F when d = 10

2. Steve is solving the following problem:

In a factory, chemical reactions are carried out in spherical containers.

The time, T minutes, the chemical reaction takes is directly proportional to the square

of the radius, R cm, of the spherical container.

When R = 120, T = 32

Find the value of T when R = 150

Steve writes the following: T α R T = kR 32 = 120k k = 120 ÷ 32 k = 3.75 T = 3.75R Hence if R = 150, then T = 3.75 x 150 = 562.5 minutes.

Explain why Steve is wrong.

3. Find and correct any errors in the statements below:

a) y is inversely proportional to the cube of x so y α 1/x3 so y = k/ x3

b) x is inversely proportional to y so x α 1/y so x = 1/ky

c) y is directly proportional to x so x α y so y = kx

F

R

M


4. Write a function to explain the proportional relationship between y and x shown on

each of the graphs below:

5. The time required to build a house is inversely proportional to the number of builders,all working at the same rate.

If there are 6 builders, it takes 80 days to complete the house.

How many builders must be employed to build the house in just 16 days?

P

A



Q DIRECT AND INVERSE PROPORTION

I can write an equation to represent a situation of direct proportion or inverse proportion.

I can use proportional relationships to find other values.

I can recognise a graph of a function showing direct proportion, inverse proportion and quadratic proportional relationships.




ITERATIVE METHODS

Name:

BAM Indicator: Solve equations using iterative methods

1. Rearrange the following equations to form an iterative formula that can be used to find aroot of the equation.

a) 𝑥2 + 10𝑥 + 24 = 0

b) 𝑥2 + 10𝑥 + 14 = −10

c) 2𝑥2 − 2𝑥 − 12 = 0

d) 6𝑥2 + 3𝑥 = 18

2. Maya says that it is more accurate to use surds in iterative formulae. Explain why Maya iswrong.

3. Brian is using an iterative equation to find the roots of the equation 𝑦 = 𝑥2 + 𝑥 – 5.

He rearranges the equation to give the following interative formula.

Is Brian’s formula correct?

𝑥𝑛+1 = √𝑥𝑛 − 5

4. Anthony says that you can use an iterative formula to find the root of a cubic.Rearrange the following equation and use it to find a root of the equation.

𝑥3 − 3𝑥2 + 𝑥 − 5 = 0

F

R

M

P


5. Holly has designed a box in Design Technology.The box is 1cm longer than it is tall and is 2cm wider than it is tall.Holly wants the box to hold 100 cm3.

Construct and then use an iterative formula to find the height of the box to the nearestmillimetre.

A



Q ITERATIVE METHODS

I can rearrange an equation to form an iterative formula.

I know how to use the ANS button on a calculator to use the iterative formula efficiently.

I can find a root of a cubic using an iterative formula.

I know that the choice of the first approximation may affect the root found by the iterative process.




FACTORISING QUADRATIC EXPRESSIONS

Name:

BAM Indicator: Manipulate algebraic expressions by factorising a quadratic expression of the form ax2 + bx + c

1. Factorise the following expressions completely

a) 2x2 + 7x + 3

_____________________________

b) 3x2 + 5x + 2

_____________________________

c) 6n2 – 7n – 10

_____________________________

d) 4a2 + 8a - 12

_____________________________

2. Explain why it is possible to factorise 2x2 – 5x – 3 but not 3x2 – 5x – 3.

3. John says the answer to the question ‘Factorise fully 3x2 – 12’ ” is 3(x2 – 4).

Explain why John is wrong and correct his answer.

F

R

M


4. Fill in the missing numbers to complete a correct factorisation:

6y2 – 17y – = ( y – 10)(2y + )

5. Simplify the following algebraic fractions:

a) 𝑥2−4 𝑥 +3

2𝑥−6

b) 𝑥2−2𝑥−15

2𝑥2−9𝑥−5

P

A



Q FACTORISING QUADRATIC EXPRESSIONS

I can factorise quadratic expressions in the form ax2 + bx + c

I recognise when it is not possible to factorise an expression.

I can use my understanding of the difference of 2 squares to factorise fully.

I can apply my skills of factorising to simplifying algebraic fractions.




SOLVING QUADRATIC EQUATIONS

Name:

BAM Indicator: Solve quadratic equations by factorising

1. Solve the following equations by factorising:

a) 𝑥2 + 3𝑥 = 0

b) 𝑥2 − 3𝑥 + 2 = 0

c) 𝑥2 − 49 = 0

d) 𝑥2 − 6 = 5𝑥

e) 3𝑥2 − 18𝑥 − 21 = 0

2. Melanie says that the solutions to 𝑥2 − 5𝑥 + 4 = 0 are 𝑥 = -4 and 𝑥 = -1.Explain why she is wrong.

3. Jane and Elliot each choose a whole number.

Jane multiplies her number by three and subtracts 2.Elliot multiplies his number by 2 and adds 3.When they multiply their results together, the product is 110.

They both chose the same number to start with. What was their number?

F

M

P


4. This football pitch has an area of 7140 m2.

a) Show that 12𝑥2 − 13𝑥 − 7175 = 0

b) Find the values of p and q given that 12𝑥2 − 13𝑥 − 7175 = (𝑥 − 𝑝)(𝑞𝑥 + 289).

c) Find the dimensions of the pitch.

5. How many solutions are there to the equation 𝑥2 + 12𝑥 = −36 ?

A

R



Q SOLVING QUADRATIC EQUATIONS

I can solve a quadratic where the coefficient of x2 is 1

I can solve a quadratic where the coefficient of x2 is greater than 1

I know I may have to rearrange a quadratic to be able to solve it

I recognise and can use the difference of two squares

I can identify quadratics that cannot be solved by factorising.



4x + 5

3x – 7


QUADRATIC GRAPHS

Name:

BAM Indicator: Link graphs of quadratic functions to related equations

1. Draw the graph y = x2 + 3x – 6, for -5 ≤ x ≤ 5 on graph paper.

Use it to solve the equation x2 + 3x – 6 = 2.

2. Look at these pairs of functions and graphs.

Do each of these graphs match their equations?

Explain how you know for each one.

y = (x – 2)(x+3) y = x2 + 7x + 10

y = x2 – 2x – 15 y = x2 – 3x – 4

F

R


3. To solve the equation x2 + 3x – 4 = 0, George was given the graph of y = x2 + x + 1.

He was told to add an equation of a line to this graph to help solve the original equation.

George chose to add the line y = 2x + 5.

Explain why George has made an error and suggest what he should do instead.

4. What equation, of the form ax2 + bx + c = 0, does the point of intersection of this graphsolve?

5. A ball is thrown in the air and falls back down again.

Its height is modelled by the equations h = 12t - 3t2, where h is the height above the

ground and t is the time in seconds.

By using an appropriate graph, calculate how long the ball is at least 8m above the

ground.

M

P

A



Q QUADRATIC GRAPHS

I can plot the graph of a quadratic function

I can relate the factorised form of the expression to the roots of the graph

I can solve quadratic equations graphically by considering intersections



RATES OF CHANGE

Name:

BAM Indicator: Interpret a gradient as a rate of change

1. Here is a velocity-time graph for a bus travelling between two sets of traffic lights.

a) Calculate the average acceleration over the first 20 seconds.

____________

b) Calculate an estimate of the acceleration of the bus at when the time is equal to 20

seconds.

____________

2. A ball is thrown in the air. Its height, ℎ, at time 𝑡 can be modelled as ℎ = 8𝑡 − 5𝑡2.

a) Plot a graph of ℎ against 𝑡 for 0 ≤ 𝑡 ≤ 1.6. Attach your graph paper to this sheet.

b) Use the graph to find the speed of the ball when 𝑡 = 0.6____________

c) Use the graph to find the greatest speed of the ball____________

3. Natalie is sketching graphs. Natalie says:

“This graph shows that the rate of change of velocity is constant throughout”

Is Natalie correct? Explain how you know.

F

A

R


4. Here is a distance-time graph of Jenni’sjourney.

Decide whether each statement is true orfalse? Explain how you know.

a) Jenni travelled 80m altogether

b) Jenni travelled backwards after 20 seconds

c) Jenni finished in the same place that she started

d) Jenni was travelling at approximately 4m/s after 30 seconds.

5. Is the following statement true or false?

“The average rate of change between two points A and B is calculated. There is always apoint between A and B where the instantaneous rate of change is equal to this average

rate of change.”

Explain your answer fully. You can use diagrams to help you.

M

P



Q RATES OF CHANGE

I can calculate an average rate of change from a graph

I can estimate an instantaneous rate of change by drawing a tangent to a curve

I can interpret the meaning of the gradient at a point in the context of the problem




EQUATION OF A CIRCLE

Name:

BAM Indicator: Recognise and use the equation of a circle with centre at the origin

1. State the radius of each circle:

a) x2 + y2 = 49 b) x2 + y2 = 1002 c) y2 = 81 – x2 d) x2 + y2 + 17 = 138

2. x2 + y2 = 25 is a circle centred on the origin.

Tick the statements that are correct:

The circle has a radius of 25

The points (3, 4) and (-3, 4) lie on the circumference of the circle

The circle has a diameter of 10

The coordinate (-3, 2) lies within the circle

3. Draw arrows to match the equations to their graph

F

M

A


4. x2 + y2 = 25 and x2 + y2 = 49 are both circles centred on the origin.

Does the point (4, 3.5) lie within both circles, only one circle, or neither circle?

Give reasons for your answer.

5. Point A lies at (3,7), on the circumference of the circle x2+ y2 = 58.

Stacey says:

‘The gradient of the line segment joining A to the origin is 21

3’

Do you agree with Stacey? Give reasons for your answer.

P

R



Q EQUATION OF A CIRCLE

I can recognise the equation of a circle

I can find the radius of a circle when given its equation

I can solve problems involving the equation of a circle




TRIGONOMETRY

Name:

BAM Indicator: Apply trigonometry in two dimensions

1. a) Find the missing lengths marked 𝑥 in these diagrams:

b) Find the missing angles marked 𝜃 in these diagrams:

2. Explain the two mistakes that were made when calculating the solution to the problembelow.Then find the correct solution.

F

M

a = 13 ÷ sin50 = 16.97029… a = 16.97 cm (2 d.p.)


3. A ladder, 6 metres long, rests against a wall. The foot of the ladder is 2.5 metres fromthe base of the wall. What angle does the ladder make against the ground?

4. A building, shown in orange, has a ledge half way up.

Amy is standing at point A looking at the building.

She measures the length AB as 100m, the angle CAB as 31o and the angle EAB as 42o.

Use this information to calculate the width of the ledge CD.

5. Jason says:

I only need to know two pieces of information about a triangle and I can find any missing side or angle.

Do you agree or disagree? Explain why.

A

P

R



Q 2-D TRIGONOMETRY

I can find missing sides of right-angled triangles using trigonometry

I can find missing angles of right-angled triangles using trigonometry

I can correctly label a triangle so as to select the appropriate trigonometric ratio

I can apply trigonometry to problems involving multiple triangles.



10M11 (v2) © Kangaroo Maths 2016

VOLUME

Name:

BAM Indicator: Calculate volumes of spheres, cones and pyramids

1. Calculate the volume of the solids, leaving your answers in terms of π where appropriate.

a) Sphere with radius 4 cm b) Cone

c) Square-based pyramid

2. Explain how to find the volume of a regular-octagonal-based pyramid.

3. A sphere and cone have the same volume and the same radius.Calculate the height, h, of the cone as a proportion of the radius, r.

F

R

P

10 cm

12 cm 13 cm

12 cm

8 cm

10M11 (v2) © Kangaroo Maths 2016

4. A metal cone of radius 5 cm and perpendicular height 10 cm is melted down to form asphere. What is the radius of the sphere?

5. a) Bob is finding the volume of a sphere with radius 6 cm. Here is his working:

𝑉 = 4

3𝜋 𝑟3

∴ 𝑉 = 4

3× 𝜋 × 6 = 8𝜋 𝑉 = (8𝜋)3 = 15 875𝑐𝑚3 (to nearest integer)

What has Bob done wrong?

b) Tracey is finding the volume of a triangular based pyramid with base area 50 cm2 andperpendicular height 10 cm. Here is her working:

𝑉 = 1

2𝐴ℎ

∴ 𝑉 = 1

2× 50 × 10 = 250𝑐𝑚3

What has Tracey done wrong?

A

M



Q VOLUME

I can calculate the volume of a sphere, cone and pyramid.

I calculate a missing dimension of a sphere, cone or pyramid given its volume.

I can calculate in terms of π




VECTORS

Name:

BAM Indicator: Understand and use vectors

1. If a = (31) and b = (

−1 2

), find:

a) a + b = b) 2a + b = c) 1

2 (a + b) = d) a − 2b =

2. a = ( 4

−2), b = (

−1 2

) and c = 3a + 2b

Explain why c can also be written as (10 −2

).

3.

𝐴𝐵⃗⃗⃗⃗ ⃗ = a

M divides AB in the ratio 2:1.

Brian says that 𝐴𝑀⃗⃗⃗⃗ ⃗⃗ = 1

2 a

Explain why Brian is wrong.

F

R

M

A

B

M

a


4. A boat is travelling across a river at a speed of 3 ms-1.

The current is flowing at 4 ms-1.

What is the actual speed of the boat?

5.

The line DE is 4 times longer than BC.

𝐷𝐸 ⃗⃗⃗⃗⃗⃗ ⃗is in the opposite direction of 𝐵𝐶 ⃗⃗ ⃗⃗ ⃗⃗

Express 𝐷𝐸 ⃗⃗⃗⃗⃗⃗ ⃗in terms of a and b.

A

P



Q VECTORS

I can add and subtract column vectors, as well as multiply by a scalar.

I can express a required vector in terms of known vectors.

I can find a resultant from given vectors.



A

B

C4a + 2b

2a − b


MEASURES OF CENTRAL TENDENCY

Name:

BAM Indicator: Analyse data through measures of central tendency and spread, including quartiles

1. The boxplots show the lengths of reigns of the last 19 popes and (English) monarchs.

Compare fully the lengths of reigns of monarchs and popes.

2. Two cricket players both have a median score of 38, based on their last 20 matches.

Player A’s data shows an interquartile range of 41.Player B’s data shows an interquartile range of 20.

Which player would you prefer to select for the cricket team and why?

3. The cumulative frequency graph shows thetime taken to respond to online help requestson a website for 120 customers.

a) Find the median time taken to respond toan online help request

b) The website has a target to ensure that 80% of online help requests receive responseswithin 30 seconds.

Did the website meet its target in this sample? Explain your answer fully.

F

R

A


4. Match a statement to each boxplot and complete the missing boxplot.

5. Will says, “The interquartile range is less accurate than the range because not all data isincluded”.

Do you agree or disagree with Will? Explain why

P

M



Q MEASURES OF CENTRAL TENDENCY

I can find statistics from a cumulative frequency curve or boxplot

I can compare the central tendency and spread of two data sets

I can use quartiles and other measures to make estimates about the population



The middle 50% of both subjects' results were

equally spread.

Results in English were more consistent than in

Maths

25% of the students achieved a higher

percentage in English than any of them

achieved in Maths.

The maximum mark in English was half that in

Maths.

On average, students achieved a higher

percentage in English than in Maths.

Documents

FRACTIONAL INDICES · 3. Brian is using an iterative equation to find the roots of the equation = 2 + – 5. He rearranges the equation to give the following interative formula. Is