Download Framework objectives - Coombeshead Academy Frog VLE

Coombeshead Academy
Mathematics Department
Homework : Year 8
Book 3
CHAPTER
1
Number and Algebra 1
2
Geometry and Measures 1
Statistics 1
3
1.1
Framework objectives – Multiplying and dividing
negative numbers
Multiply and divide integers.
1 Work out the answer to each of these:
a 5
b
c
e
f 5 × 11
g 9×2
I
j
k
l
m
o
q 16 ÷ 4
r
s
t
6
2 Find the missing number:
a
c
e
f
3 Work out:
a ( 6)2 ÷ +4
c ( 4 + 2)2 + ( 5 + 2)2
Framework objectives – HCF and LCM
1.2
Use multiples, factors, common factors, highest common factors, lowest
common multiples and primes.
1 Find the LCM of:
a 6 and 10 b 6 and 21
c 4 and 10
f 12 and 27 g 15 and 25
h 9 and 11
2 Find the HCF of:
a 16 and 20 b 15 and 20
c 8 and 12
f 8 and 20
g 15 and 25
h 9 and 11
d 6 and 27
e 8 and 18
d 6 and 10
e 3 and 18
6
7
3 a Two numbers have an LCM of 30 and an HCF of 3. What are they?
b Two numbers have an LCM of 12 and an HCF of 2. What are they?
Framework objectives – Powers and roots
Homework
1.3
Use squares, positive and negative square roots, cubes and cube roots, and
index notation for small positive integer powers.
1 Without using a calculator, write down the following:
a √1
b √64
c 3√8
d 3√27 e 3√64
2 Use a calculator to find the value of:
a 172
b 173
c 253 d 64
e 37
f 85
2
3
4
3 Given that 0.1 = 0.01, 0.1 = 0.001, 0.1 = 0.0001, write down the answers to:
5
a 0.15 b 0.18
6
Framework objectives – Prime factors
1.4
Find the prime factor decomposition of a number, for example, 8000 = 2 6 × 53
1 These are the products of the prime factors of some numbers. What are the numbers?
a 2×3×5
b 2×2×3×5
c 23 × 52
2 Using a prime factor tree, work out the prime factors of:
a 44
b 120
c 250
3 Using the division method, work out the prime factors of:
a 84
b 125
c 240
4 The prime factors of 100 are 2, 2, 5, 5. The prime factors of 150 are 2, 3, 5, 5.
Use this information to work out the HCF and LCM of 100 and 150.
6
Framework objectives – Sequences 1
1.5
Generate terms of a linear sequence using term-to-term and position-to-term
rules.
1 Write down four sequences beginning 1, 2, …, and explain how each of them is generated.
2 Describe how each of the following sequences is generated and write down the next two terms:
a 50, 48, 46, 44, 42, 40, … b 9, 12, 18, 27, 39, 54, … c 1, 3, 6, 10, 15, 21, …
d 2, 6, 8, 14, 22, 36, …
3 You are given a starting number and a multiplier. Write down at least the first six terms.
a start 1, multiplier 4
b start 2, multiplier –1
c start 20, multiplier 10
5
d start 40, multiplier
Framework objectives – Sequences 2
1.6
Use linear expressions to describe the nth term of a simple arithmetic
sequence, justifying its form by referring to the activity or practical context from
which it was generated.
1 Given the first term a and the constant difference d, write down the first six terms of each of these
a a = 2, d = 6
b a = 0.5, d = 2
c a = –8, d = 3
2 The nth term of a sequence is given by each of the rules below. Use this to write down the first
six terms of each sequence:
a 6n – 1
b 10n + 3
c 4n + 1
d 8n – 3
3 Write down the nth terms of these sequences:
a 4, 9, 14, 19, 24, …
b 6, 13, 20, 27, 34, …
c 1, 5, 9, 13, 17, …
d –3, 2, 7, 12, 17, …
e 7, 15, 23, 31, 39, …
5
6
Homework
2.1
Framework objectives – Alternate and
corresponding angles
Identify alternate angles and corresponding angles.
6
Work out the size of the lettered angles in these diagrams:
a
b
c
d
e
f
2.2
Framework objectives – Interior and exterior angles
of polygons
Explain how to find, calculate and use: the sums of the interior and exterior angles
of quadrilaterals, pentagons and hexagons; the interior and exterior angles of
regular polygons (Year 9 Framework Objective).
6
1 Find the sum of the interior angles for each of the following polygons:
a a pentagon
b a hexagon
c an octagon
2 Calculate the size of the lettered angle in each of the following polygons:
a
b
c
d
3 Write down the size of each exterior angle for the following regular polygons:
a a regular pentagon
b a regular hexagon
c a regular octagon
Framework objectives – Geometric proof
2.3
Understand a proof that: the angle sum of a triangle is 180° and of a quadrilateral
is 360°; the exterior angle of a triangle is equal to the sum of the two interior
opposite angles.
6
1 Write out a proof to show that the sum of the angles of a triangle is 180°.
2 Write out a proof to show that the exterior angle of a triangle is equal to the sum of the two interior
opposite angles.
Framework objectives – The geometric properties of
quadrilaterals
2.4
Solve geometrical problems using side and angle properties of special
quadrilaterals, explaining reasoning with diagrams and text; classify quadrilaterals
by their geometrical properties.
1 Copy and complete the table:
Square
Rectangle
Number of lines of symmetry
4
Order of rotational symmetry
2
All sides equal
All angles equal
Opposite sides parallel
Yes
2 a Which quadrilaterals have diagonals that bisect each other?
Parallelogram
Rhombus
5
No
No
b Which quadrilaterals have diagonals that intersect at right angles?
3 The instructions below are to draw the rectangle shown.
[FORWARD 5
TURN RIGHT 90°
FORWARD 12
TURN RIGHT 90°]
6
Write down a set of similar instructions to draw a rectangle that has sides twice the length of those
on the diagram.
Framework objectives – Constructions
Homework
2.5
Use a straight edge and compasses to construct: the mid-point and
perpendicular bisector of a line segment; the bisector of an angle; the
perpendicular from a point to a line; the perpendicular from a point on a line.
Use ICT to explore these constructions.
Use straight edge and compasses to construct triangles, given right angle,
hypotenuse and side (RHS) (Year 9 Framework Objective).
1 Draw a line XY 8 cm in length. Using compasses, construct the perpendicular bisector of the line.
2 a Draw an acute angle of any size. Using compasses, construct the angle bisector.
b Draw an obtuse angle of any size. Using compasses, construct the angle bisector.
3 Explain how to draw an angle of 45° without having to use a protractor.
4 Construct the following right-angled triangles.
a
b
6
Framework objectives – Probability
3.1
Interpret the results of an experiment using the language of probability;
appreciate that random processes are unpredictable.
6
1 I roll a dice 100 times. How many times should I expect to see:
a the number 3 b a number less than 3 c a number greater than 3?
2 A pack of cards is shuffled and one drawn at random. This happens 80 times.
How many times would you expect to see:
a a spade b a King c a red Ace d either a ten or a Jack?
3 If I toss a fair coin 200 times, how many tails should I expect to see?
4 The probability of winning a local weekly lottery is 0.005.
Wendy had a go on this lottery every week for 20 years.
How many times could she expect to have won this lottery?
Framework objectives – Probability scales
Homework
3.2
Know that if the probability of an event occurring is p, then the probability of it
not occurring is 1 – p.
1 The probabilities of different events happening are given. Write down the probability of these
events not happening:
a 0.1
h 1
b 0.25
i
1
5
c 0.5
j
1
4
d 0.6
k
1
10
e 0.85
l
2
3
f 0.91
m
9
10
6
g 0.001
n
4
7
2 There are eight outcomes when throwing three coins. Make a list of all the outcomes. Write down
the probability of obtaining:
a three heads b at least one tail c three tails d at least one head
Framework objectives – Mutually exclusive events
3.3
Use diagrams and tables to record in a systematic way all possible mutually
exclusive outcomes for single events and for two successive events.
1 A coin is tossed and a dice is rolled.
a Make a list of all the possible outcomes.
b Make a table of all the possible outcomes.
2 There are three pets in a house: a cat, a dog and a hamster. There are also three children, Mark, David and
Paul, who each own one of the pets. Make a table of all the possibilities of who owns each pet.
3 Five girls, Bev, Val, Lynne, Sarah and June, go to the cinema. Bev wants to sit next to Val and June
wants to sit on the end. Make a list of the possible seating arrangements.
6
Framework objectives – Calculating probabilities
Homework
3.4
Use diagrams and tables to record in a systematic way all possible mutually
exclusive outcomes for single events and for two successive events.
6
1 Two four-sided dice are thrown and the scores added together. Copy and complete the table of
scores:
1
2
3
4
1
2
2
3
3
4
Write down the probability of:
a 3
b 4
e greater than 5
f an even number
i a multiple of 3
c 8
g a prime number
d less than 4
h a square number
2 A room is painted using two different colours. The colours can be chosen from red, green, blue and
yellow.
a Make a list of the six different combinations that could be chosen.
Write down the probability of choosing:
b red and green
c green with any other colour
d no red
Framework objectives – Experimental probability
Homework
3.5
1
2
3
4
Compare estimated experimental probabilities with theoretical probabilities,
recognising that: if an experiment is repeated the outcome may, and usually will, be
different; increasing the number of times an experiment is repeated generally
leads to better estimates of probability.
This homework is connected to the extension work, which some pupils may already have started.
Having decided on an experiment of your own, collect your data. Try to collect as much as possible.
Write a brief report about the data you have collected.
Work out an experimental probability for your data.
If appropriate for your experiment, calculate a theoretical probability and compare this with your
experimental probability.
6
Number and Algebra 1 – Learning Checklist
Level 4
I can write down the multiples of any whole number.

I can work out the factors of numbers under 100.

Level 5
I can add and subtract negative numbers.

I can write down and recognise the sequence of square numbers.

I know the squares of all numbers up to 152 and the corresponding square roots.

I can use a calculator to work out powers of numbers.

I can find any term in a sequence given the first term and the term-to-term rule.

I know that the square roots of positive numbers can have two values, one positive
and one negative.

Level 6
I can multiply and divide negative numbers.

I can find the lowest common multiple (LCM) for pairs of numbers.

I can find the highest common factor (HCF) for pairs of numbers.

I can write a number as the product of its prime factors.

I can find any term in a sequence given the algebraic rule for the nth term.

I can find the nth term of a sequence in the form an + b.

I can investigate a mathematical problem.

Level 7
I can work out the LCM and HCF of two numbers using prime factors.

I can devise flow diagrams to generate sequences.

Geometry and Measures 1 – Learning
Checklist
Level 5
I know the symmetrical properties of quadrilaterals.

I can construct triangles from given information.

Level 6
I can use alternate and corresponding angles in parallel lines.

I can use the interior and exterior angle properties of polygons.

I can use proof in geometry.

I can solve problems using the geometrical properties of quadrilaterals.

I can draw constructions using a ruler and compasses.

Statistics 1 – Learning Checklist
Level 6
I can identify all the outcomes from two events.

I can use tables to show the outcomes.

I understand what is meant by mutually exclusive.

I can use the fact that the total of mutually exclusive events in a situation is 1.

Level 7
I know how to identify bias in an experiment.

I know how to compare outcomes of experiments.

Chapter 1 – Number and Algebra 1 End of Chapter Test
5
1
Work out the following.
(a)
2
–6 + 8
(b)
–3 – 4
(c)
–5 + –3 – –1
Complete the lists of factors for the numbers below.
(a)
12: 1, 2, ...
28:
(b)
3
4
6
5
6
Calculate:
(a)
52
(b)
23
(c)
34
(e)
√9
(f)
√64
(g)
3
√27
(d)
105
(h)
3
√1000
For each sequence below:
(i)
describe the term-to-term rule
(ii)
write down the next three terms in the sequence.
(a)
1, 5, 9, 13, ......................
(b)
2, 11, 20, 29, ......................
(c)
21, 19, 17, 15, ......................
(d)
10, 5, 0, –5, ......................
Calculate:
(a)
+2 × –4
(b)
–5 × +3
(c)
–7 × –4
(d)
–9  +3
(e)
–10  –2
(f)
+4  –1
(g)
4 × –9 × –2
(h)
–3 × –6  2
(i)
(15 – –6)  –3
Find the lowest common multiple of:
(a)
7
Use your lists from part a to find the highest common factor of 12 and 28.
3 and 5
(b)
9 and 12
Write down the first five terms of the sequence whose nth term is:
(a)
5n + 1
(b)
3n – 3
6
8
(a)
Ian only uses his mobile phone in the evening, when it is cheaper. The costs of
some calls he makes to his friend Martin’s mobile are shown in the table below.
Length of call
1 minute
2 minutes
3 minutes
4 minutes
Cost on Ian’s bill
7p
9p
11p
13p
The cost in pence is calculated as 2n + 5, where n is length of the call in minutes.
Write down the cost of:
(b)
(i)
a 5-minute call
(ii)
a 15-minute call.
The costs of some of Martin’s calls to Ian are shown in the table below.
Cost on Martin’s bill
4p
7p
10p
13p
Length of call
1 minute
2 minutes
3 minutes
4 minutes
Write down an expression for the cost in pence of an n-minute call from Martin to
Ian.
9
The first five terms of some sequences are given below. For each sequence, write down
the next three terms and find an expression for the nth term.
(a)
2, 4, 6, 8, 10
(b)
3, 5, 7, 9, 11
(c)
2, 5, 8, 11, 14
(d)
1, 6, 11, 16, 21
72
2
7
10
Complete the prime-factor tree for the number 72.
Check your answer by multiplying together all the prime
factors. (The answer should be 72.)
11
The diagram shows the prime factors of 66.
(a)
Complete the diagram with the prime factors of 72
that you have found in question 10.
(b) Use
12
your completed diagram to find:
(i)
the highest common factor of 66 and 72
(ii)
the lowest common multiple of 66 and 72.
*Write down two positive numbers with an HCF of 2
and an LCM of 30.
36
Chapter 2 – Geometry and Measures End of Chapter Test
5
1
Work out the size of each lettered angle in the diagrams below.
(a)
2
(b)
*Four types of quadrilateral are described on the left below. The quadrilaterals are drawn
on the right in a different order.
Match each description to the correct drawing.
(a)
Four right angles and four lines of symmetry.
(b)
Opposite sides parallel, opposite sides equal and no lines of
symmetry.
(c)
Four equal sides, opposite angles equal and two lines of
symmetry.
1
2
3
>
(d)
6
4
One pair of parallel sides.
>
3
Use alternate and corresponding angles to work out the size of each lettered angle in the
diagrams below. (The diagrams are not drawn accurately.)
4
Work out the size of each lettered angle in the diagrams below.
(a)
(b)
6
5
Look at the diagram :
(a)
Measure the angles a and b and check that they add up to 180°.
(b)
Prove, without using your measurements, that the sum of the angles a and b must
be 180º. Make a sketch of the diagram to help explain your proof.
6
Use a ruler and protractor to draw an angle of 70º. Now use ruler and compasses to
construct the bisector of the angle. Leave all construction lines on your diagram.
Check with your protractor that the two angles formed are both 35º.
7
You are going to construct the kite shown below.
(a)
Draw a horizontal line 5 cm long and label the ends A and C.
(b)
Construct the perpendicular bisector of AC.
Leave all construction lines on your diagram.
(c)
Set your compasses to 3 cm and draw an arc
centred at A to intersect the vertical line above
the horizontal line. Label this intersection B.
(d)
Set your compasses to 4 cm and draw an arc
centred at A to intersect the vertical line below
the horizontal line. Label this intersection D.
(e)
8
9
Join A, B, C and D to draw the kite. Leave all
construction lines on the diagram.
*Work out the lettered angles in each of the
diagrams below. (The diagrams are not drawn accurately.)
*Using ruler and compasses, construct the right-angled triangle shown below.
Chapter 3 – Statistics 1 End of Chapter Test
5
1
.
Alice has two bags of counters.
In Bag A there are 3 red, 2 blue and 1 yellow counter.
In Bag B there are 8 red, 4 blue and 3 yellow counters.
Alice is going to pick a counter at random from one of the bags.
Which bag should she choose to have the better chance of picking:
(a)
6
2
a red marble?
(b)
a blue marble?
Look at the probability scale below.
(a)
Write the labels ‘impossible’, ‘even chance’ and ‘certain’ in the correct places on the
scale.
(b)
Write the letter of each of the following events in the correct place on the scale.
1
.
2
1
event B The probability of getting two tails when tossing a coin twice is
.
4
event C It is certain that the sun will rise tomorrow.
event A The probability of getting a tail when tossing a coin is
event D It is impossible to get more than 6 when a normal dice is rolled.
3
The spinner shown below is divided into equal sections, each coloured black, grey or
white.
What is the probability of the pointer landing on:
4
(a)
a black section?
(b)
not a black section?
(c)
not a white or a black section?
Yasmin rolls a regular octahedral dice. The eight faces are numbered from 1 to 8.
(a)
(b)
Calculate the probability that Yasmin rolls:
(i)
1
(ii)
more than 1
(iii)
an even number
(iv)
an odd number
(v)
less than 4
(vi)
4 or more.
Add together the following pairs of probabilities from part a:
(i)
(c)
(i) and (ii)
(ii)
(iii) and (iv)
(iii)
(v) and (vi)
Explain why the sums in part b all gave the same answer.
6
5
A bag contains a large number of green balls and one red ball. There are no other balls
1
in the bag. The probability of picking a red ball from the bag is
.
30
(a)
What is the probability of picking a green ball from the bag?
(b)
How many green balls are there in the bag?
6
Jason eats a packet of chocolate spangles every day. He keeps a record of the number
of chocolates in the packet every day for two months. His results are shown in the table
below.
Number of chocolates in
Number of
(a) Estimate the probability that the next
packet
packets
packet of spangles that Jason opens will
have 10 or more chocolates in it.
Fewer than 10
22
(b) How could Jason improve this estimate?
7
The table shows the possible outcomes of
rolling two dice.
10 or more
38
First dice
Second
dice
1
2
3
1
1, 1
1, 2
1, 3
2
2, 1
2, 2
4
5
3
4
5
6
7
8
(a)
Complete the table.
(b)
Calculate the probability of rolling:
(i)
two sixes
(ii)
a one and a six
(iii)
two even numbers
(iv)
one even number and one odd number
(v)
two numbers that are both greater than three.
*Alex rolls a dice 300 times and records the score. The table shows his results.
Number on dice
Frequency
(c)
1
49
2
64
3
48
4
52
5
36
6
51
(a)
What is the theoretical probability of rolling a 5?
(b)
How many 5s would you expect Alex to get in 300 rolls of the dice?
Do you think his dice is biased? Give a reason for your answer.
6