Transcript Yr7-WrittenCalculationsx (Slides)

Year 7 Written Calculations
Dr J Frost ([email protected])
www.drfrostmaths.com
Objectives: Be able to add, subtract, multiply and divide numbers of any
size (including decimals).
Extension: Solve problems involving missing digits within a
multiplication/addition.
Last modified: 12th April 2016
STARTER :: Long Multiplication
Use any preferred method (but preferably long multiplication!)
to multiply the following numbers.
?
284 × 32 = 𝟗𝟎𝟖𝟖
?
375 × 65 = 𝟐𝟒𝟑𝟕𝟓
2
?
111 = 𝟏𝟐𝟑𝟐𝟏
?
739 × 384 = 𝟐𝟖𝟑𝟕𝟕𝟔
?
806 × 608 = 𝟒𝟗𝟎𝟎𝟒𝟖
If you finish…
Put the digits from each
calculation into the triangle.
What do you notice?
110
111
112
113
114
=
=
=
=
=
1
?1 ?1
1 ?
2 1
1 3? 3 1
1 4 ?
6 4 1
This is known as Pascal’s
Triangle. Each number is
the sum of the two
above it.
Sadly we will no longer
see Pascal’s Triangle for
115 onwards (why?)
Multiplying Decimals
3.2 × 0.04
First ignore decimal points and multiply
as whole numbers.
32 × 4 = 128
3.2 × 0.04 = 0.128
Correct by counting total number of
decimal place jumps.
Further Example
84.1 × 0.05
841 × 5 = 4205
84.1 × 0.05 = 4.205
Test Your Understanding
0.3 × 0.2
𝟑×𝟐=𝟔 ?
𝟎. 𝟑 × 𝟎. 𝟐 = 𝟎. 𝟎𝟔
2
3.21
𝟑𝟐𝟏 × 𝟑𝟐𝟏 = 𝟏𝟎𝟑𝟎𝟒𝟏
?
𝟑. 𝟐𝟏 × 𝟑. 𝟐𝟏 = 𝟏𝟎. 𝟑𝟎𝟒𝟏
Decimal Addition/Subtraction
11.3 − 8.74
Fill any ‘gaps’ with 0s.
11.30
−8.74
2.56
Ensure decimal points are lined up so
that the digits in each column have
the same place value.
Check Your Understanding
18.43 + 192.71
103.8 − 48.91
𝟏𝟖. 𝟒𝟑
? 𝟕𝟏
+𝟏𝟗𝟐.
𝟐𝟏𝟏. 𝟏𝟒
𝟏𝟎𝟑. 𝟖𝟎
−𝟒𝟖.? 𝟗𝟏
𝟓𝟒. 𝟖𝟗
Exercise 1
1
Calculate:
3.4 × 5 = 𝟏𝟕 ?
4.56 × 0.2 = 𝟎. 𝟗𝟏𝟐
?
0.04 × 0.003 = 𝟎. 𝟎𝟎𝟎𝟏𝟐
?
48 − 13.63 = 𝟑𝟒. ?
𝟑𝟕
5.48 − 1.584 = 𝟑. 𝟖𝟗𝟔
?
42.7 × 0.56 = 𝟐𝟑. 𝟗𝟏𝟐
?
11.3 − 7.444 = 𝟑. 𝟖𝟓𝟔
?
45.1 × 0.0043 = 𝟎. 𝟏𝟗𝟑𝟗𝟑
?
2
3
4
[JMC 2007 Q1] What is the value of
0.1 + 0.2 + 0.3 × 0.4? Solution:
? 0.42
[Kangaroo Pink 2012 Q1] What is the
value of 11.11 − 1.111? Solution:
? 9.999
[JMC 2001 Q8] What is the difference
between the largest and smallest of the
following numbers?
A 0.89 B 0.9
C 0.17
D 0.72 E 0.73 Solution:
?E
(On provided sheet)
5
[IMC 2015 Q1] What is the value of
1 − 0.2 + 0.03 − 0.004?
Solution:
? 0.826
6
[IMC 2002 Q2] Which of the following has
the greatest value?
A 0.3 × 7
B 0.5 × 5
C 0.2 × 11
D 0.09 × 30
E 0.026 × 100
Solution:
?D
7
8
[IMC 2003 Q5] What is the value of 20032 ?
Solution: ?
4 012 009
[JMC 2012 Q11] In the following expression,
each □ is to be replaced with either + or – in
such a way that the result of the calculation is
100.
123 □ 45 □ 67 □ 89
The number of + signs used is 𝑝 and the
number of – signs used is 𝑚. What is the
value of 𝑝 − 𝑚?
Solution:
? -1
Exercise 1
9
[JMC 1997 Q10] Each day throughout
July 1995 I picked 300g of raspberries
from my garden. What was the total
weight of the raspberries I picked that
month?
A 10g B 9kg C 9.3kg
D 10kg E 9300kg
Solution:?C
10 [JMC 1998 Q7] The Mystery Prize at the
Bank of England Christmas Party was a
pile of crisp new £5 notes, numbered
from 659384 up to 659500. What was
the value of the prize?
A £116 B £117 C £580
D £585 E £1420
Solution:?D
(On provided sheet)
11
[JMC 2000 Q10] Each Junior
Mathematical Challenge answer sheet
weighs 6 grams. If 140000 pupils enter the
challenge this year, what will be the total
weight of all their answer sheets?
A 84kg B 840kg C 8 400kg
D 84 000kg
E 840 000kg
Solution: B
?
12 [JMC 2000 Q14] The DISPUTOR is similar
to a calculator, but it behaves a little
oddly. When you type in a number, the
DISPUTOR doubles the number, then
reverses the digits of this result, then adds
2 and displays the final result. I type in a
whole number between 10 and 99
inclusive. Which of the following could be
the final result displayed?
A 39
B 41
C 42
D 43
E 45
Solution: E
?
Exercise 1
13 [Kangaroo Grey 2014 Q6] Which of the
following calculations gives the largest
result?
A 44 × 777 B 55 × 666
C 77 × 444 D 88 × 333
E 99 × 222
(On provided sheet)
15
[JMC 1997 Q22] The Grand Old Duke of
York, he had ten thousand men, he
marched them up to the top of the hill, …
By 2pm they were one third of the way
up. By 4pm they were three quarters of
the way up. When did they set out?
A 12 noon
B 12.24pm
C 1.12pm
D 1.36pm
E 1.48pm
Solution:?B
16
[TMC Final 2014 Q7] Place exactly three
common mathematical operations (which
need not all be different) between the
digits below so that the result equals 100.
You are not allowed to rearrange the
order of the digits.
1 2 3 4 5 6 7 8 9For example,
1234 × 5 − 67 × 89. We know this example
is wrong because the result is 207
Solution: 𝟏𝟐𝟑 −?𝟒𝟓 − 𝟔𝟕 + 𝟖𝟗
Solution: B. We can ignore factors of 11
?
and 111 and just do 𝟒 × 𝟕, etc.
14 [JMC 1999 Q21] Granny says “I am 84
years old – not counting my Sundays”.
How old is she really?
A 90
B 91
C 96
D 98
E 99
Solution: D
?
Exercise 1
(On provided sheet)
17 [JMO 1999 A7] Before the decimalisation of money in the UK, there
were 12 pence (d) in 1 shilling (s) and 20 shillings in 1 pound (£). Thus
1 pound 3 shillings and 4 pence was written £1 3s 4d. What would
have been the total cost of 7 items each costing £1 6s 8d? Write your
answer in simplest £ s d form.
Solution: £8 ?
[JMC 2003 Q22] Two builders, Bob and Geri, buy bricks at the same
18
price. Bob sells 10 for £6 and Geri sells 12 for £7. Supposing they sell
equal numbers of bricks, what number has each sold when Bob has
gained £4 more than Geri?
A 42
B 60
C 72
D 120 E 240
Solution: E
?
Dividing Decimals
8 ÷ 0.2
It is fine in practice to divide decimals by whole numbers.
e.g. Dividing £1.68 between
? 4 people.
However, we don’t in general like dividing by decimals.
e.g. Dividing £3 between 5.6 people. We can’t do that!
?
= 80 ÷ 2
= 40 ?
?
Consider that for example
8÷4=2
and
80 ÷ 40 = 2
What does this suggest we
can do to both numbers in the
division without affecting the
result?
More Examples
1.47 ÷ 0.3
= 14.7 ÷ 3
04.9
3 | 14.7
2.53 ÷ 0.011
= 2530 ÷ 11
= 230
Multiply each number by 10
until we’re dividing by a
whole number.
Ensure decimal point goes in
same place in result.
Test Your Understanding
?
174.9 ÷ 0.03 = 𝟓𝟖𝟑𝟎
?
6.055 ÷ 0.7 = 𝟖. 𝟔𝟓
Exercise 2
1
[JMC 2012 Q2] What is half of 1.01?
Solution:?0.505
2
[Kangaroo Pink 2010 Q1] What is the
result of dividing 20102010 by 2010?
Solution:?10,001
3
[IMC 1999 Q5] 30 ÷ 0.2 equals
Solution:?150
4
Calculate:
a) 22.8 ÷ 0.5 = 𝟒𝟓. 𝟔 ?
b) 5.376 ÷ 0.06 = 𝟖𝟗. 𝟔 ?
c) 2825.2 ÷ 0.007 = 𝟒𝟎𝟑𝟔𝟎𝟎?
d) 10.593 ÷ 1.1 = 𝟗. 𝟔𝟑 ?
e) 1.00001 ÷ 0.11 = 𝟗. 𝟎𝟗𝟏 ?
5 A death laser can fire every 0.004
seconds. How many times can it fire in 3
minutes? Solution:?45000
6 A rectangle has area 21.242cm2
and height 1.3cm.
a) What is its length?
16.34cm?
b) What is its perimeter?
35.28cm?
7 Calculate:
a) 0.09104 ÷ 0.16 = 𝟎. 𝟑𝟖𝟏𝟓
?
b) 149.85 ÷ 1.5 = 𝟗𝟗. 𝟗?
c) 610.8 ÷ 0.12 = 𝟓𝟎𝟗𝟎
?
Missing Digit Puzzles
This lesson you will be able to work in groups on some puzzles to do with missing digits
within column addition/multiplication.
[JMC 2007 Q18] The letters 𝐽, 𝑀, 𝐶 represent three different non-zero digits. What is
the value of J + 𝑀 + 𝐶 ?
A 19
B 18
C 17
D 16
E 15
Clue:
𝐽 could only be 1
or 2, as greatest
result is ?
99 + 99 + 99 = 297
Putting this together:
𝑱 = 𝟏, 𝑴 = 𝟗, 𝑪 = 𝟖
Answer is B?
𝐽
𝐽
𝑀 𝑀
+ 𝐶 𝐶
𝐽 𝑀 𝐶
Clue:
These two digits are the
same. Thus ?
J and M
must add to 10.
Clue:
We’re adding the same digits in each of the
two columns, but get a different digit as the
result. There therefore?must have been a
carry, and hence M is one more than C.
Missing Digit Puzzles
Question 1
[JMC 2014 Q12] In this subtraction, 𝑃, 𝑄, 𝑅, 𝑆
and 𝑇 represent single digits.
What is the value of 𝑃 + 𝑄 + 𝑅 + 𝑆 + 𝑇?
A 30
B 29
C 28
D 27
E 26
Solution: B
?
Get into groups of 3 or 4.
Merits to the team with the
most correct answers at the
end.
Effective Team Maths Challenge
strategy: Have two people
independently work on each
problem, and each ‘initial’ the
problem on a shared sheet if you
have a solution. When both people
have solved the problem, compare
your two answers.
Missing Digit Puzzles
Question 2
[JMC 2003 Q16] In this multiplication each letter stands for a
different digit. Which letters stands for 3?
A
B
C
D
E
Solution:? D
Missing Digit Puzzles
Question 3
[JMC 2005 Q18] In the subtraction sum on the right 𝑎, 𝑏 and 𝑐
are digits, and 𝑎 is less than 𝑏. What is the value of 𝑐?
A 3
B 4
C 5
D 6
E a number greater
than 6
Solution:? A
Missing Digit Puzzles
Question 4
[TMC Regional 2011 Q10] 𝐽6𝐾4 × 7 = 𝐿9𝑀98
Each of 𝐽, 𝐾, 𝐿 and 𝑀 is a different digit.
Find the values of 𝐽, 𝐾, 𝐿 and 𝑀.
Solution: 𝑱 = 𝟓, 𝑲 =
? 𝟏, 𝑳 = 𝟑, 𝑴 = 𝟐
Missing Digit Puzzles
Question 5
[TMC Final 2010 Q3] Each of the letters used below is standing
for a different single digit.
Example: 𝐴𝐵 + 2 = 𝐴𝐶 [the answer could be 𝐴 = 1, 𝐵 = 3, 𝐶 =
5 because 13 + 2 = 15]
If 𝐴𝐵𝐶𝐷 × 9 = 𝐷𝐶𝐵𝐴
find 𝐴, 𝐵, 𝐶 and 𝐷.
Solution: 𝑨 = 𝟏, 𝑩?= 𝟎, 𝑪 = 𝟖, 𝑫 = 𝟗
Missing Digit Puzzles
Question 6
[Junior Kangaroo 2015 Q12] In the sum shown, different shapes
represent different digits. What digit does the square represent?
A 2
B 4
C 6
D 8
E 9
Solution:
?C
Missing Digit Puzzles
Question 7
[JMC 2002 Q24] In the multiplication on the right, each letter
represents a different digit and only the digits 1, 2, 3, 4, 5 are
used. Which of the letters represents 2?
A
B
C
D
E
Solution:
?E
Missing Digit Puzzles
Question 8
[JMC 2010 Q25] What is the value of 𝑃 + 𝑄 + 𝑅 in the
multiplication on the right?
A 13
B 12
C 11
D 10
Solution:
?A
E 9
Missing Digit Puzzles
Question 9
[JMC 1999 Q25] The two-digit by two-digit multiplication on the
right has lots of gaps, but most of them can be filled in by logic
(not by guesswork). Which digit must go in position ∗ ?
A 1
B 3
C 5
D 7
E 9
Solution:
?D
Missing Digit Puzzles
Question 10
[JMO 2000 B4] How many different solutions are there to the
letter sum on the right? Different letters stand for different
digits, and no number begins with a zero.
JMC
+ JMO
SUMS
Solution:
?6