Download EM unit notes - Hamilton Trust

Reasoning, explaining and properties of numbers
Year 6 Summer 8
Make and justify estimates and approximations of large numbers, e.g. how many pennies there might be a in a line 1 km long
Previous learning
Core for Year 6
Extension
Understand, read and begin to write these words:
Understand, read and write these words:
Understand, read and write these words:
round, nearest, … estimate, approximate, ...
round, nearest, … estimate, approximate, ...
round, nearest, … estimate, approximate, ...
Understand what each digit represents in whole numbers and
numbers with up to two decimal places, e.g.
Understand what each digit represents in large whole
numbers.
Respond to oral and written questions such as:
• A city has a population of five million and forty-seven
thousand. Write this number in figures.
• The first decimal place is tenths, and one tenth is 0.1.
• The second decimal place is hundredths,
and one hundredth is 0.01.
• The population of the United Kingdom is about fifty-nine
million. Write this number in figures.
• What number is one hundred less than ten thousand?
• The third decimal place is thousandths,
and one thousandth is 0.001.
Revise rounding numbers to the nearest 10, 100 or 1000, e.g.
Understand and use the rules for rounding whole numbers.
Understand and use the rules for rounding decimals.
• To round 3627:
If the first unwanted digit is 5, 6, 7, 8 or 9, add 1 to the last
digit that you keep. Then replace all the unwanted digits by
zeros, e.g.
Understand and use the rules for rounding decimals, i.e. if
the first unwanted digit is 5, 6, 7, 8 or 9, add 1 to the last digit
that you keep. Then leave off any unwanted digits, e.g.
Sketch a line from 3620 to 3630 and mark on 3627.
Ring the nearest multiple of 10.
So 3627 rounds to 3630 to the nearest 10.
Sketch a line from 3600 to 3700 and mark on 3627.
Ring the nearest multiple of 100.
So 3627 rounds to 3600 to the nearest 100.
Sketch a line from 3000 to 4000 and mark on 3627.
Ring the nearest multiple of 1000.
So 3627 rounds to 4000 to the nearest 1000.
5621 rounded to the nearest 100 is 5600;
26 900 rounded to the nearest 1000 is 30 000
Respond to questions such as:
• Round 56 275 to the nearest 10 000.
37.295
correct to one decimal place is 38.3
12.346
correct to two decimal places is 12.35
54.34427 correct to three decimal places is 54.344
Respond to questions such as:
• Round 56.275 to two decimal places.
• What is two hundred and seventy-six centimetres to the
nearest metre?
• How long is the rope to the nearest 10 cm? The nearest
cm? The nearest 100 cm? The nearest mm?
© 1 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Use knowledge of multiplication facts to multiply a multiple of
10 and a multiple of 10 or 100, e.g.
Use knowledge of multiplication facts to multiply a multiple of
10, 100 or 1000 by a multiple of 10, 100 or 1000, e.g.
30 × 40 = 3 × 10 × 4 × 10
= 3 × 4 × 100
= 12 × 100
= 1200
40 × 600 = 4 × 10 × 6 × 100
= 4 × 6 × 10 × 100
= 24 × 1000
= 24 00
Complete questions such as:
50 × 30 = …
Complete questions such as:
60 × … = 1200
… × 50 = 2000
Use knowledge of division facts to divide a multiple of 100 by
a multiple of 10 or 100, e.g.
1200
30
3 × 4 × 100
=
3 × 10
= 4 × 10
= 40
500 × 30 = …
60 × 5000 = …
Use knowledge of division facts to divide a multiple of 1000
by a multiple of 10, 100 or 1000, e.g.
12000
600
6 × 2 × 1000
=
6 × 100
= 2 × 10
= 20
1200 ÷ 30 =
12000 ÷ 600 =
Complete questions such as:
3000 ÷ 50 = …
Extension
Complete questions such as:
2000 ÷ 100 = …
30 000 ÷ 60 = …
200000 ÷ 400 = …
Make and justify estimates and approximations, e.g.
Make and justify estimates and approximations of large
numbers, e.g.
Make and justify estimates and approximations of large
numbers, e.g.
• Work out an estimate for the total cost of 112 televisions at
£198 each.
• About how many pennies might there be a in a line 1 km
long?
• What volume of liquid does a typical person drink in a day?
In a year?
100 × 200 = 20 000
£20 000 is an estimate for the total cost.
• The weight of a can of beans is 424.5 grams.
Estimate the weight of 18 cans of beans.
• The cost of a cinema ticket is £4.55.
Find the approximate cost of 13 tickets.
• There are 58 classrooms in a school.
There are 32 chairs in each room.
Estimate total number of chairs in all of these rooms.
A 1p coin has a diameter of about 2 cm.
1 km is 1000 × 100 cm = 100 000 cm.
100 000 ÷ 20 = 5000
So there might be 5000 pennies in a line 1 km long.
• Could 1 million people stand on a football pitch?
• It takes 1 second to jump once.
How long do 1 million jumps take?
Respond to questions such as:
• Estimate the number of minutes in a day… in a year…
• There are about 29 million households in the UK.
Roughly how many eggs are eaten each week in the UK?
In a year?
• 212 children went on a school trip.
The total cost was £425.
Work out the approximate cost for each child.
© 2 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Explain methods and reasoning orally and in writing
Previous learning
Core for Year 6
Extension
Understand, read and begin to write these words:
Understand, read and write these words:
Understand, read and write these words:
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, …
sequence, pattern, relationship, rule, general statement, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, … general statement, …
sequence, pattern, relationship, rule, formula, substitute, …
problem, puzzle, solution, method, justify, explain how you
know, give your reasons, … general statement, …
sequence, pattern, relationship, rule, formula, substitute, …
Explain reasoning and methods to solve problems such as:
Explain reasoning and methods to solve problems such as:
Explain reasoning and methods to solve problems such as:
• Complete this multiplication table.
• Jennie has a pack of cards numbered from 1 to 20.
She picks four different number cards.
• Give examples to show these statements are not correct.
?
• Each ✸ represents one of the digits 1 to 6.
Replace each ✸ to make a correct product.
✸✸ ×✸ =✸✸✸
• Each ✸ represents a missing digit.
Use your calculator to solve this.
✸ 2 ✸ × ✸ ✸ = 11 316
• Investigate how many three-digit numbers there are with a
digit sum of 25.
• One whole number divided by another gives 1.1818181…
What are the two numbers? Use your calculator.
• I think of a number, multiply by 6 then subtract 5.
My answer is 61.
What number did I start with?
?
?
When you multiply a number by 2,
the answer is always greater than 2.
?
Exactly three of the four numbers are multiples of 5.
Exactly three of the four numbers are even numbers.
All four of the numbers add up to less than 40.
Write what the numbers could be.
• 30 children are going on a trip. It costs £5 including lunch.
Children who take their own packed lunch pay only £3.
The 30 children pay a total of £110.
How many children are taking their own packed lunch?
A
A
• Look at this information.
Two numbers multiply to make zero.
Which one of the statements below is true?
A Both numbers must be zero.
B At least one number must be zero.
C Exactly one number must be zero.
D Neither number can be zero.
• Here are five number cards.
A
When you subtract a number from 2,
the answer is always less than 2.
B
B
• k, m and n each stand for a whole number.
They add together to make 1500.
k + m + n = 1500
A and B stand for two different whole numbers.
The sum of all the numbers on all five cards is 30.
What could be the values of A and B?
m is three times as big as n. k is twice as big as n.
Calculate the numbers k, m and n.
• I think of a number. I add 4. I multiply the result by 3.
Then I take away 9. My final answer is 90.
What number did I start with?
• I think of a number, multiply it by 8, then subtract 66
The result is twice the number that I was thinking of.
What is the number I was thinking of?
© 3 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Recognise squares of whole numbers to at least 12 × 12 and square multiples of 10, e.g. 50 × 50
Previous learning
Core for Year 6
Extension
Use, read and begin to write these words:
Use, read and write these words:
Use, read and write these words:
multiple, divides exactly by, factor, divisor, divisible by,
test of divisibility, … sequence, term, rule, …
square number, square root, multiple, factor, divisor, divisible
by, test of divisibility, … sequence, term, rule, …
square number, square root, multiple, factor, divisor, divisible
by, test of divisibility, … sequence, term, rule, …
and the notation 52 (five squared) and √5 (the square root of 5)
and the notation 52 (five squared) and √5 (the square root of 5)
Recognise 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144
as square numbers.
Calculate the values of larger squares, e.g.
Relate to drawings of squares.
• Calculate 152, 212.
Identify two-digit numbers which are the sum of two squares:
for example, 34 = 32 + 52.
Understand that finding the square root is the inverse of
squaring a number. If the square of 9 is 81, then the square
root of 81 is 9 (92 = 81, √81 = 9).
Respond to questions such as:
Using a calculator where appropriate, respond to question
such as:
• What is 4 squared?
• Which number, when multiplied by itself, gives 2809?
• What is the square of 6?
• Find two consecutive numbers with a product of 9506.
2
• What is 8 ?
2
• The area of a square is 256 cm .
What is the length of its side?
• Which number multiplied by itself gives 36?
• What is the area of a square whose side is 6 cm in length?
• What is the square root of 49?
• I think of a number, square it and add 4..
The answer is 40.
What is my number?
• 17 multiplied by itself gives a three-digit number.
17 × 17 = 289
What is the smallest two-digit number that can be
multiplied by itself to give a four-digit number?
• Is the statement below correct for all numbers?
The square of a number is greater
than the number itself.
Write Yes or No. Explain how you know.
• This information describes three different squares, A, B
and C.
2
The area of square A is 64 cm .
The side length of square B is 64 cm.
The perimeter of square C is 64 cm.
Put squares A, B and C in order of size, starting with the
smallest. Explain how you work out your answer.
© 4 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Use knowledge of multiplication facts and place value to
derive products of multiples of 10, e.g.
Use knowledge of squares of numbers 1 to 12 and place
value to derive squares of multiples of 10, e.g.
• 30 × 60 = 3 × 10 × 6 × 10
= 3 × 6 × 10 × 10
= 18 × 100
= 1800
• 50 × 50 = 5 × 10 × 5 × 10
= 5 × 5 × 10 × 10
= 25 × 100
= 2500
Use the square root key on a calculator
Previous learning
Core for Year 6
Extension
Use a calculator to work out square numbers, e.g.
Use the square key of a calculator, e.g.
• Find the 26th square number.
On most calculators, enter the number first, followed by the
square key.
Press these keys:
2 6 × 2 6 =
The display will show:
676
Use the square root key of a calculator, e.g.
On most calculators enter the number first, followed by the
square root key.
• Find the square root of 144.
Press these keys:
1 4 4 a
The display will show:
12
On some calculators you press the square-root key first like
this:
a 1 4 4
• Find the 26th square number.
Press these keys:
2 6 i
The display will show:
676
• Copy and complete this pattern.
1
1 +
1 +
1 +
F+
2
=1
3
= 22
2
3 + 5
=F
3 + 5 + 7
= F2
2
F+F+ F+ F=F
What is the answer to:
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19?
• Any whole number can be written as the sum of four
square numbers. For example:
23 = 1 + 4 + 9 + 9
Investigate different ways of writing 150 as the sum of four
square numbers. Use your calculator to help you.
© 5 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Revise finding factors of two-digit numbers and use factors to multiply when appropriate
Previous learning
Core for Year 6
Extension
Know that factors occur in pairs, e.g.
Know that factors occur in pairs, e.g.
Find the prime factors of any two-digit number, e.g.
Know that:
Know that:
Know that, for example:
• A factor is a number that divides exactly into a bigger
number, e.g.
• Factors occur in pairs so that if 3, say, is a factor of the
number 24, then 24 ÷ 3 = 8 is also a factor of 24.
5 is a factor of 15
because 5 divides exactly 3 times into 15.
• If a number is a multiple of, say, 2, then 2 is a factor of the
number.
6 is not a factor of 15
because 6 does not divide exactly into 15.
15 is not prime, but can be written as the product 3 × 5.
3 and 5 are prime numbers which are factors of 15.
They are called the prime factors of 15.
12 is not prime, but can be written as the product of its
prime factors:
12 = 2 × 2 × 3
• Factors occur in pairs, e.g. since 12 = 3 × 4, both 3 and 4
are factors of 12. The simplest factor pair of any number is
the number itself and 1.
Find factors of two-digit numbers, e.g.
Revise finding factors of two-digit numbers, e.g.
Write a number as the product of its prime factors, e.g.
• Find the factor pairs of, say, 20 by arranging 20 counters in
the shape of different rectangles:
• Find the factor pairs of 36 by arranging 36 counters in the
shape of different rectangles.
• To write 75 as the product of its prime factors, look for the
smallest prime factor of 75, which is 3.
Divide 75 by 3
75 ÷ 3 = 25
20 × 1, 10 × 2, 5 × 4
There are no other pairs of numbers which have a product
of 20.
So the six factors of 20 are 1, 2, 4, 5, 10 and 20.
The factor pairs of 36 are:
1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6
The nine factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.
• Make a ‘factor finder’.
Repeat with the new number (25).
Look for the smallest prime factor of 25, which is 5.
Divide 25 by 5
25 ÷ 5 = 5
Repeat with the new number (5).
Divide 5 by 5
5÷5=1
So 75 = 3 × 5 × 5
3 75
5 25
5 5
1
By colouring factors on a grid, show for example that
factors of 6 are also factors of 12.
You can also use factor trees, e.g. to show that
48 = 2 × 2 × 2 × 2 × 3.
48
4
2
2
© 6 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
4
12
3
2
2
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999
Previous learning
Core for Year 6
Extension
Solve problems such as:
Investigate problems such as:
Investigate problems such as:
• Which number(s) between 20 and 40 have:
• Give an example of a two-digit number with:
• I am thinking of a number which is a multiple of 4.
Which one of the statements below is true?
Explain how you know.
the most factors
exactly two factors (e.g. 17)
the least factors
exactly three factors (e.g. 49)
an odd number of factors?
exactly four factors (e.g. 10)
• Which numbers less than 100 have exactly three factors?
(the squares of prime numbers, e.g. the factors of 49 are 1,
7 and 49)
• Write a number in each circle so that the number in each
square is the product of the two numbers on either side of it.
A My number must be even.
B My number must be odd.
exactly five factors (e.g. 16)
C My number could be odd or even.
exactly ten factors (e.g. 48)
more than ten factors (e.g. 84)
• Which numbers have exactly five factors?
(the squares of the squares of prime numbers, e.g. the
factors of 81 are 1, 3, 9, 27 and 81)
• I am thinking of a number which is a factor of 20.
Which one of the statements below is true?
Explain how you know.
A My number must be even.
B My number must be odd.
C My number could be odd or even.
Use factors to multiply when appropriate, e.g.
15 × 6 = 15 × 2 × 3
= 30 × 3
= 90
Use factors to multiply when appropriate, e.g.
35 × 12 = 35 × 2 × 6
= 70 × 6
= 420
© 7 | Year 6 | Summer TS8 | Reasoning, explaining and properties of numbers
Use factors to multiply when appropriate, e.g.
35 × 120 = 35 × 2 × 6 × 10
= 70 × 6 × 10
= 4200
A few examples are adapted from the Framework for teaching mathematics from Reception to Year 6, 1999