Teaching historically higher tier content to foundation

Teaching historically higher tier content to foundation tier

students

Christian Seager, Trinity High School

List of resources

1. Averages from grouped data – whodunit?

Data sheet and worksheet – use the grouped data to work out who committed the crime.

2. Vectors – whodunit?

Data sheet and worksheet – use the vectors to work out who committed the crime.

3. Expanding brackets

Worksheet – examples, practice questions, exam questions and checklist

4. Scatter graphs


5. Ratio


6. Stem and leaf diagrams


7. Reverse percentages number grid

Worksheet – find the answers to the questions in the number grid

8. Standard form connect 4

Worksheet – tackle the standard form questions to connect four answers on the grid

9. Standard form follow me

Worksheet – use standard and ordinary form to complete the sentence

10. Trigonometry dominos

Worksheet – solve the trig problems to play dominos

11. Functional skills strategies and revision table mat

Problem-solving strategies for multi-step functional questions

12. Revision table mat or poster

Tips for effective revision

Christian has taught at his current school, Trinity High School for nine years. He is Head of

Mathematics after starting at Trinity as an NQT, and has helped Trinity rise from National

Challenge in 2007 to the most improved school in England in the January 2013 league tables.

Christian is supported by a fantastic team, including Melanie. Along with Melanie, Christian

has set up JustMaths, the fantastic support for fellow maths teachers, with loads of resources

available on www.justmaths.co.uk.

http://www.justmaths.co.uk/

www.justmaths.co.uk ©JustMaths 2012

Now you need to work out where and when the crime was committed....

Who, where and when?

Who?

One of the following four people has committed a crime.

The criminal made 4 errors, the victim has made 0

errors and two suspects have made 1, 2 or 3 errors.

The ICT teacher said

- The modal class

of B is

160 < h ≤ 170

- The median of E is

18000 < S ≤ 20000

- The median of D is

4 < h ≤ 6

- The median of B is

140 < h ≤ 150

The music teacher

said:

- The modal class

and median of A

is 20 < t ≤ 30

- The modal class of F is

150 < h ≤ 160


14000 < S ≤ 16000

- The modal class of B is

150 < h ≤ 160

The English teacher said:

- The modal class

and median of D

are the same

- The median of F is

150 < h ≤ 160

- The median of C is 37 to 39


18000 < S ≤ 20000

The Maths teacher

said:

- The modal class

and median of B

are the same

- The median of F is

140 < h ≤ 150

- The modal class and median of

C are NOT the same

- The modal class of E is

50000 < S ≤ 100000


Where? The murder was committed at one of the locations below, but which one?

It happened where ALL the answers are true.

The maths classroom

The total number of girls is 30.

The total number of people in Table A is 40.

The total number of trainers is 4.

The dining hall

The total number of boys is 100.

The total hours using a PS3 is 720.

The total boys’ height is 14,900cm.

The gym

The total time to get to work is 1130 hours.

The total number of boys is 100.

The total salaries are £1,565,000.

The playing fields

The total girls’ height is 4430cm.

The total in table C is 20.

The total number of salaries is 7.

When? Find the day where all the estimates of the means are correct:

Monday If table A is 28.25 and table C is 38 (to 1 significant figure)

and table E is £17,388.89 (to 2 decimal places)

Tuesday If table B is 147.7 (to 1 decimal place) and table C is 40

(to 1significant figure) and table D is 6.1 (to 1 significant figure)

Wednesday If table A is 28.3 (to 1 decimal place) and table F is 200 (to 1 significant figure) and table D is 6.1 (to 1 decimal place)

Thursday If table C is 38.15, and table F is 100 (to 1 significant

figure) and table B is 147.7 (to 1 decimal place)

Friday If table F is 149 and table D is 6.1 (to 1 decimal place) and

table B is 150 (to 1 significant figure)

The Accusation

Who

Where

When

Time to get to work (t minutes) Time

Frequency 0 < t ≤ 10 3 10 < t ≤ 20

8 20 < t ≤ 30 11 30 < t ≤ 40 9 40 < t ≤ 50

9

TABLE A

Height of girls

Height

(h, cm(

Frequency

120 < h ≤ 130 3

130 < h ≤ 140 6

140 < h ≤ 150 7

150 < h ≤ 160 8

160 < h ≤ 170 6

Trainer shoe sizes

Shoe size Frequency 34 to 36 4 37 to 39

12 40 to 42 3 43 to 45 1

TABLE B TABLE C

Hours using a PS3

No of hours

(h, hours)

Frequency

0 < h ≤ 2 10

2 < h ≤ 4 15

4 < h ≤ 6 30

6 < h ≤ 8 35

8 < h ≤ 10 25

10 < h ≤ 12 5

Annual Salary

Salary (S, £) No of people 10000 < S ≤ 14000 32 14000 < S ≤ 16000 24 16000 < S ≤ 18000 16 18000 < S ≤ 20000 6 20000 < S ≤ 40000 9 40000 < S ≤ 50000 2

50000 < S ≤ 100000 1

Boys height

Height

(h, cm)

Frequency

120 < h ≤ 130 8

130 < h ≤ 140 16

140 < h ≤ 150 25

150 < h ≤ 160 30

160 < h ≤ 170 21

TABLE D

TABLE E TABLE F

©JustMaths 2012 www.justmaths.co.uk


Now you need to work out where and when the crime was committed....

Who, where and when?One of the following four people has committed a crime.

The criminal made 2 errors, the victim has made 0errors and the other two suspects have made 1 error.

a = b = c = d = e =

Victor said:

a + b =

c + d =

e = 3a

a + 2b =

The girls said:

a + c =

a + d =

5d =

2c + a =

The minions said:

d + e =

a + b =

a + 2b =

2d + c =

Gru said:

a + e =

d + b =

2e =

½ c =


Where and when?Use the questions on the accompanying sheet

The murder was committed at one of the locations below, but which one?It happened where the most mistakes have been made.

Gru’s Lab on Monday

Q1. AB = b – aQ2. AP = b - a

Q3. AF = (c – a)

Q4. FA = - b

Vector’s house onWednesday

Q1. BA = a – bQ2. PB = (b – a)

Q3. OF= a + c

Q4. EB = 2b

Miss Hattie’s Home forGirls on Saturday

Q1. CD = 2b – 2aQ2. BA = b - aQ3. OE = c + 2aQ4. AC = b – 2a

Bank of Evil onThursday

Q1. OC = 2aQ2. AB = a + bQ3. AC = c - aQ4. BD = a + b

The AccusationWho

Where &When

A and B are midpoints ofOD and OC respectively.

OB = bOA = a

©JustMaths 2014www.justmaths.co.uk

Q. 1.

A

C D

B

O Q. 2.

P is one third of the wayalong AB

OA = aOB = b

O

A

B

P

OA = AD = CB = BE = a

OC = AB = DE = c

F is one third of the way along AC

Q. 3.

ABCDEF is a regular hexagon

OA = a

OB = b

Q. 4.

O

A

BE

CD

F

O C

A B

D E

F

Now have a go yourself .. . .

How to . . .

Expanding Brackets

(a) Expand 3(2 + t)

(b) Expand 3x(2x + 5)

(c) Expand (m + 3)(m + 10)

(5)

Sorted it

a) 2(p + 3) b) 3(p - 3) c) 4(p + q)

d) 3(5 - p) e) 2(2x + y - 3) f) 5(3c + 1)

g) 4(x2 + 1) h) 3(x2 - 2) i) 3(n2 - 2n + 1)

NAILED IT

a) p(p + 2) b) q(q - 3) c) 2p(p + 5)

d) x(4 - x) e) x(y + z) f) d(3d - 4)

g) -2(x + 3) h) -3(2p + 2) i) -2d(d - 4)

MASTERED IT

a) (x + 2)(x + 3) b) (x + 3)(x + 4)

c) (x + 1) (x + 2) d) (y + 2)(y - 5)

e) (x - 2)(x + 3) f) (y + 1) (y - 2)

g) (x - 2)(x - 3) h) (x - 4)(x - 5)

i) (x + 2)2 j) (y + 1)(2y + 1)

k) (x - 1)(3x + 1) l) (2y + 3) (y + 4)

m) (3p + 2)(2p + 5) n) (x - y)(x - 2y)


Checklist

Ready to be marked ? Expand and Simplify (i) 2(x - 4) + 3(x + 2) (ii) x(x + 3) (iii) y(2y - 3) (iv) (x + 3)(x + 4) (v) (x - 3)(x + 9) (vi) (x - 3)(x - 7)

Answer checked

Working out shown

Keywords

Things to remember ...

What went well ...

Teacher comment ..

Exam Questions


How to . . .

Scatter Graphs The scatter graph shows information about 10 apartments in a city.

The graph shows the distance from the city and the monthly rent of each apartment


The table shows the distance from the city centre and the rent of two other apartments

a) On the graph plot these two points.

(2)

Distance

km

Rent

£

2 250

3.2 180

b) Describe the relationship between the distance from the centre and the monthly rent.

(1)

An apartment is 2.8km from the city centre.

c) Find an estimate for the monthly rent for this apartment.

(2)

Every Saturday for 5 weeks in the Autumn the number of centime-tres of rainfall and the percentage of cloud cover were recorded by a group of students. The results are shown in the table:

a) On the graph, draw a scatter diagram of the results . (1)

b) Draw, by eye a line of best fit. (2)

c) Describe the relationship between the percentage of cloud cover and the amount of rain. (1)

d) Find an estimate for the percentage of cloud cover on a day with 0.6

cm of rainfall. (2)

Cloud cover %

Rainfall cm

55 0.48 10 0.24 60 0.52 85 0.84 5 0.10

Checklist

Ready to be marked ?

Answer checked

Points plotted

Line of best fit

Keywords


What went well ...

Teacher comment ..

Exam Question The scatter diagram shows the height, in cm, and the weight, in kg, for each of 20 members of a sports club.

a)Write down the height and weight of the heaviest of the 20 members of the sports club.

Weight ....................kg

Height ....................cm

(2)

b) Write down the type of correlation shown in the scatter diagram. (1)

c) Draw, by eye, a line of best fit on the scatter diagram. (1)

d) Estimate the weight of a person of height 155 cm. (2)

e) Is it possible to estimate the weight of a person with a height of 210 cm from the scatter diagram. You must give a reason (1)


How to . . .

Ratio

5 schools sent some students to a conference. One of the schools sent boys and girls. This school sent 16 boys. The ratio of the number of boys it sent to the number of girls it sent was 1:2 The other 4 schools sent only girls. Each of the 5 schools sent the same number of students. Work out the total number of students sent to the conference by these 5 schools.

(4)

MUST - share the following in the ratio shown

a) £40 in the ratio 3:2 b) £35 in the ratio 4:1

c) £54 in the ratio 5:1 d) £35 in the ratio 4:3

e) £42 in the ratio 2:5 f) £65 in the ratio 2:3

SHOULD - share the following in the ratio shown

a) £30 in the ratio 2:2:1 b) £84 in the ratio 3:3:1

c) £990 in the ratio 7:2:2 d) £64 in the ratio 5:2:1

e) £240 in the ratio 5:3:2 f) £140 in the ratio 6:3:1

COULD

a) Mr A, Mr B and Mr C own 2, 3, and 6 parts of a busi-ness. They share the profit according to how many parts of the business they own. If Mr C gets £132 how much profit did the business make?

b) To make suet you need fat to flour in the ratio 1: 3. Jane has 180 g of flour. How much fat does she need to make the suet?

c) The sides of a triangle are in the ratio 2 : 4 : 5. The middle sized side is 28 cm.

a). Find the length of the other two sides.

b). Find the perimeter of the triangle.


Checklist

Ready to be marked ? Q1. Last year Kerry’s take home pay was £15 000 She spent 40% of her take home pay on rent. She used the rest of her take home pay for living expenses, clothes and entertainment in the ratio 3 : 1 : 2 How much did Kerry spend on entertainment last year?

Q2. Talil is going to make some concrete mix. He needs to mix cement, sand and gravel in the ratio 1 : 3 : 5 by weight. Talil wants to make 180 kg of concrete mix. He has:

15 kg of cement 85 kg of sand

100 kg of gravel Does Talil have enough cement, sand and gravel to make the concrete mix?

Answer checked

Working out shown

Keywords


What went well ...

Teacher comment ..

Exam Questions


How to . . .

Stem and Leaf

Sixteen babies are born in a hospital.

Here are the weights of the babies in kilograms.

2.4 2.7 3.5 4.4 4.5 4.1 4.4 2.8

4.1 3.8 3.8 4.2 3.3 3.0 3.7 3.3

Show this information in an ordered stem and leaf diagram.

(3)

Q1. Here are the times, in minutes, taken to solve a puzzle.

5 10 15 12 8 7 20 35 24 15

20 33 15 24 10 8 10 20 16 10

In the space below, draw a stem and leaf diagram to show these times.

Find the median time to solve this puzzle.

Q2. Jim did a survey on the lengths of caterpillars.

Information about the lengths is given in the stem and leaf dia-gram.

a) Work out the median. ............ cm b) Work out the range. ............ cm c) Work out the mode. ............ cm d) Work out the inter-quartile range ............ cm


Key:

1 3 5 7 7

2 0 6 8 8 8 9

3 1 5 5 5 5 6 8 9

4 1 5

5 2

Key: 5│2 means 5.2cm

Checklist

Ready to be marked ? .The numbers below list the ages of the members of a tennis club.

71 39 40 16 57 12 63 34 41 45 65

27 16 59 40 60 14 22 48 43 38 52

35 23 25 52 36 38 26 31 27 17 16

a) Construct a stem and leaf diagram with these ages.

b) Use it to find the following:

How many members the club has.

The modal age of the members.

Their median age.

The range of their ages.

The fraction of members who are over 40.

(8)

Order correct

Key included

Keywords


What went well ...

Teacher comment ..

Exam Questions


esreveR segatnecreP (!)

Work out the answers to the questions then search for the answers in the grid

(Ignore any decimal points e.g. 42.7 becomes 427)

1. In a sale, all prices are reduced by 30%. The sale price of a jacket is £33.60

Work out the original price of the jacket.

2. The price of a new washing machine is £376 which includes Value Added Tax (VAT) at 17.5% Work out the cost of the washing machine before VAT was

added.

3. Employees at a firm get a pay increase of 4% After the pay increase, Mel earns £24,960 How much did Mel earn before the pay increase?

4. Top Shop is having a 20% off sale and I have treated myself to a new handbag

– it cost £40, how much was it before the sale?

0 9 3 1 7 2

2 2 1 8 5 0

5 7 4 7 0 9

5 3 0 0 0 7

1 3 2 6 0 2

2 1 9 3 2 0

3 5 8 7 4 9


5. A limited edition version of perfume contains 10% more than the normal bottle. The special bottle contains 88 ml. How much does the normal bottle contain?

6. The recent advert for Jaffa cakes makes the claim that the boxes are 24%

bigger. The new boxes contain 31 biscuits, how many did it have before?

7. I want to sell vegetable boxes at a farmers market - the vegetables will cost me £7.60. How much would I need to sell them for to get 15% profit?

8. Christian invests some money in a bank account. Interest is paid at a rate of 8% per annum. After 1 year, there is £291.60 in the bank account. How much

did Christian invest?

9. A new car drops in value by 30% in its first year. After a year, it is worth £8,400 what was the cost of the car?

10. Christian has bought some shares. The value of the shares fell by 4% since he

purchased them and they are now worth £5200. What was their original value?

11. Fize has just reduced his personal best for the 100 m by 25% to 12.9 seconds. What was his previous personal best?

THINK YOU’VE NAILED IT? .....

How can you check your answers?

1. In a class there are 9 people out on a school trip to the Museum of Maths. This

is 20% of the class.

o How many are there in the class when no one is off ill?

o How many are in the class today?

2. If 35% of an amount is £70 what is 100%?

3. On the 1st May the Museum of Maths increased it prices by 25% to £9 and visitors in May dropped by 8% to 55,476

o What was the price in April?

o How many visitors were there in April?

o Their costs haven’t changed but the museum made 30% profit in May.

How much profit did they make in May?

o How much profit did they make in April?

o Was it a good idea to raise the prices?

Write 47,500,000 in standard form.

Write 4.56 x 104 as

an ordinary number

Write 8.43 x 10-4 as

an ordinary number0.000803 4.56 x 10

41.6 x 10

-5 0.0000456 4.7 x 10-3

Write 5.6 x 10-4 as an

ordinary number

Write 16 x 106 in standard form

Write 5 x 107 as an ordinary number

Write 45,600 in standard form.

0.002047 45,600 1.6 x 108

4.56 x 105

Write 0.0047 in standard form

Write 50,000,000 in standard form.

Write 160 x 106 in standard form

Write 2.047 x 10-3 as

an ordinary number1.6 x 10

-49.87 x 10

-1 47,500,000 0.000843

Write 4.56 x 105 as

an ordinary number

Write 4.56 x 10-5 as

an ordinary number


Write

0.0016x10-1 in standard form

50,000,000 4.75 x 107

8.03 x 10-4

1.6 x 107

Write

0.016x10-3 in standard form


Write 456,000 in standard form.

Write 4.75 x 107 as

an ordinary number2.3 x 10

-3 0.00056 5 x 107 456,000

www.justmaths.co.uk

STANDARD FORM (1) - Connect 4

© JustMaths 2013

Answer GridQuestion Grid

Working in pairs – each person takes it in turns to choose a question from the question grid to answer. The correct solution will be found in the answer grid (if your solution is not in the grid, you need to reconsider your answer), and you can colour that box on the answer grid. To win, you need to connect four answers in a line (horizontally, vertically or

diagonally) on the answer grid.

... ever wondered why?

START 1.44 x 104 9.9 x 10-1 6.4 x 10-2 20.4 64000 990 17.6 320 203.6

T I R S N H Y I U H6.4 X 104 1.24 x 10 3.2 X 100 9.9 x102 3.56 x 101 3.2 X 103 3.2 X 10-2 6.4 x 103 324 0.036

3.6 x 10-2 144 1.44 x 10-2 1760 3.6 3.6 x 101 176 14.4 0.0124 2.036 x10-2

T D K S A U S R N A6.4 x 102 3.60 x 100 0.0324 0.0036 1.76 x 10-1 0.99 1.76 X 10-2 0.144 0.064 0.0124

0.02036 64 1.79 x 103 3.24 x 102 1.76 x 10 1.24 x10-2 0.176 3.6 x 10-3 3.24 x 10-2 3200

R B N T O I R Y E E6.4 x 101 3.2 X 102 1.76 X 103 1.44 x 102 6.4 x 10-2 2.036 x10-2 0.0144 17.6 1.24 x10-2 1.76 X 102

0.0176 640 0.032 1.44 x 10-1 6400 3.2 0.064 35.6 3.6 x 102 12.4

U E O H G S U L K N2.04 x 101 1790 36 0.02036 2.036 x102 360 1.44 x 101 1.76 x 101 (1.2 x 102)2 FINISH


© JustM

aths 2013

START

17 cm

31o

x

8.76 3.72 40.24

22.65 47.46 y

19 m

33o

31.01 18 cm

FIN

ISH

49.06

In triangle ABC

BAC = 90o

AB = 11 cm

AC = 13 cm

Find angle ACB

7 m ladder leans

against a house mak-

ing an angle of 63o

with the floor. How

far up the house does

the ladder reach?

17o

40.37 6.24

Find angle BCA

y

13

.8 c

m

28o

Find the area

x

15m

70.41

7

x

x

41o

29.39

A

C

B

21 cm

16

cm

6

11

22o

19 cm

14

m

x

28o

NOTE: the maths involved is

not always difficult. The problem is work-

ing out what is being asked, deciding what

maths you need to use and remembering to

explain your answer. Pro

ble

m-s

olvin

g

strate

gie

s f

or

MU

LTI—

STE

P

fu

nc

tio

nal

qu

es

tio

ns

Read the question

Extract the key information

Always look at diagrams

Decide what the question is

Underline the key bits

Sounds obvious I know!!

Pick the maths to use

Link your working

Allow for different maths skills

Now do the work...

Work must be organised

One sentence at the end

Re-read the question

Kalculations must be shown

Think of a rough estimate

Is your answer sensible?

Check your answer

Korrect units used?

The question will

probably include

aspects of several

different maths

topics Try to get the spelling

right! ... Oops!

/

REVISING

MATHS

Practising

Know what

topics to

focus on

To revise maths you need to DO maths!

Make sure you

have and USE a

recommended

revision guide

BUT do g

et

them

marke

d

using a

mark-

schem

e

Do a little bit

of “practice”

every day

Study with a friend - teach them a topic & vice versa

Make a

timetable

Ask for

help if

unsure

Wo

rk t

hro

ugh

past

pap

ers

Find a

quiet

worksp

ace

Its no good just ownin

g

a guide ... USE IT!

Take regular

breaks

Follow

@ReviseJustMaths Make a list and update it regularly

Know the formula you need to remember & what is in the paper

Use or make revision cards, a popplet or a prezi

Watch the tutorials or revision clips your teacher suggests

Google it! Work through past papers

/

REVISING

MATHS

Practising

Know what

topics to

focus on

To revise maths you need to DO maths!

Make sure you

have and USE a

recommended

revision guide

BUT do g

et

them

marke

d

using a

mark-

schem

e

Do a little bit

of “practice”

every day

Study with a friend - teach them a topic & vice versa

Make a

timetable

Ask for

help if

unsure

Wo

rk t

hro

ugh

past

pap

ers

Find a

quiet

worksp

ace

Its no good just ownin

g

a guide ... USE IT!

Take regular

breaks

Follow

@ReviseJustMaths Make a list and update it regularly

Know the formula you need to remember & what is in the paper

Use or make revision cards, a popplet or a prezi

Watch the tutorials or revision clips your teacher suggests

Google it! Work through past papers

Documents

Teaching historically higher tier content to foundation