Download Properties of numbers 1.1 - Pearson Schools and FE Colleges

N
1.1
Propertiesofnumbers
Previous learning
Objectives based on NC levels 2 and (mainly level )
Before they start, pupils should
be able to:
In this unit, pupils learn to:
count on and back in ones,
tens and hundreds
recognise multiples of 10
recall addition and
subtraction facts for each
number to 10
recognise odd and even
numbers.
solve problems in mathematics and in practical contexts
identify and visualise patterns and relationships
calculate accurately
record, explain and compare methods
engage in mathematical discussion of results
begin to generalise
and to:
count on and back in ones, fives, tens and hundreds from any number
recognise and extend sequences of consecutive numbers
recognise and use the properties of odd and even numbers
recognise 2- and 3-digit multiples of 2, 5 or 10
use mental methods to add and subtract a multiple of 10 or of 100.
Lessons
1 Multiplesof10
2 Multiplesof2and5
3 Oddandevennumbers
About this unit
Assessment
Common errors and
misconception
2 | N1.1
Properties of numbers
Pupils’ confidence with numbers is strengthened by having a good ‘feel
for number’.
In this unit they learn to use patterns to help them to recognise number
properties and recall number facts.
This unit includes:
an optional mental test which could replace part of a lesson (p. 00);
a self-assessment section (N1.1 How well are you doing? class book p. 00);
a set of questions to replace or supplement questions in the exercises
or homework tasks, or to use as an informal test (N1.1 Check up,
CD-ROM).
Look out for pupils who:
when counting on or back from a given number, include the given
number as the first number in the count;
have difficulty in counting on or back in tens through the hundreds
boundary;
don’t recognise a three-digit number such as 500 or 150 as a multiple of 10;
have difficulty in remembering addition and subtraction facts to 10, then 20;
confuse odd and even numbers.
Key terms and notation
Practical resources
Exploring maths
Useful websites
problem, solution, method, pattern, relationship, order, solve, explain,
represent
calculate, calculation, calculator, number facts, add, subtract, plus, minus,
sum, total, difference, how many more?, how many less?
property, consecutive, odd, even, multiple, sequence
calculators for pupils
pack of digit cards 0 to 9 for each
pupil
individual whiteboards
dice (enough for two per pair)
Tier 1 teacher’s book
N1.1 Mental test, p. 00
Answers for N1.1, pp. 00–00
Tier 1 CD-ROM
PowerPoint files
N1.1 Slides for lessons 1 to 3
Excel file
Find the multiples
Tools and prepared toolsheets
N1.1 100-square
Mask tool
Tier 1 class book
N1.1, pp. 00–00
N1.1 How well are you doing?, p. 00
Tier 1 home book
N1.1, pp. 00–00
Tier 1 CD-ROM
N1.1 Check up
Odd and even
www.crickweb.co.uk/ks2numeracy.html
Behind the blob
www.mathsonline.co.uk/nonmembers/gamesroom/grids/gridindex.html
Multi sequencer
www.amblesideprimary.com/ambleweb/mentalmaths/supersequencer.html
N1.1 Properties of numbers | 1 Multiples of 10
Learning points
Consecutive numbers follow each other in order, e.g. 7, 8, 9, 10, 11, …
Multiples of 10 end in zero.
To add or subtract a multiple of 10, count on or back in tens.
When you add or subtract a multiple of 10, the units digit stays the same.
When you add or subtract a multiple of 100, the tens and units digits stay the same.
Starter
Say that the three lessons in this unit are all about number patterns.
Tell the class that consecutive numbers follow each other in order, like 4, 5, 6. Say
that you are going to say three consecutive numbers and that you want them to
respond as a class by saying the next three. Try sequences such as:
16, 17, 18, …
97, 98, 99, …
504, 505, 506, …
64, 63, 62, …
803, 802, 801, …
Main activity
XL
Explain that a multiple of 10 divides exactly by 10. Launch Find the multiples.
Set the multiple to a multiple of 10 and ask pupils to write on their whiteboards
all the multiples of 10 in the list. Repeat several times by clicking on ‘New set of
numbers’.
What do you notice about the last digit of multiples of ten?
Establish that they all end in zero.
Repeat the starter activity with multiples of 10, saying three consecutive
multiples of 10, and asking pupils to say the next three. Try counting forwards
and counting backwards.
Say that counting on or back in tens is a useful way to add or subtract a multiple
of 10.
What is 40 plus 80? 120 take away 50?
Demonstrate how to count on or back in tens to find answers, then ask similar
questions. Vary the vocabulary to include add, subtract, plus, minus, sum, total and
difference.
Display N1.1 100-square Toolsheet. Try calculations such as 56 1 30, 93  50.
Confirm answers by referring to the 100-square, counting on or back in tens.
TO
What do you notice about the units digits when you add or subtract a
multiple of 10?
Move on to three-digit numbers, e.g.
| N1.1
Properties of numbers
What is 568 minus 60? What is 568 minus 70?
How can we find the difference between 82 and 32?
Establish that to find a difference you count up from the smaller to the larger
number. Refer again to the 100-square to count up in tens from 32 to 82.
Write 321 and 371 on the board, side by side.
What is the difference between 321 and 371?
How many more than 146 is 206?
Ask pupils to do N1.1 Exercise 1 in the class book (p. 00). Pairs will need a dice
for the game in question 5.
Review
Extend the questioning to adding and subtracting multiples of 100.
Add 500 to 200. Subtract 400 from 900.
What is 386 1 400? What is 715  600?
What do you notice about the tens digits and the units digits when you
add or subtract a multiple of 100?
Sum up with the points on slides 1.1 and 1.2.
Homework
Ask pupils to do N1.1 Task 1 in the home book (p. 00).
N1.1 Properties of numbers | 2 Multiples of 2 and 5
Learning points
Multiples of 5 end in 5 or 0; multiples of 2 end in 0, 2, 4, 6 or 8.
The even numbers are 0, 2, 4, 6, 8, 10, …; even numbers end in 0, 2, 4, 6 or 8.
The odd numbers are 1, 3, 5, 7, 9, 11, …; odd numbers end in 1, 3, 5, 7 or 9.
Starter
Say that this lesson is about multiples of 5 and multiples of 2.
Show the problem on slide 2.1. Tell the class that they have a purse full of 2p and
5p coins. They need to decide how many 2p coins and how many 5p coins they
will use to pay for a snack costing 25p.
How many different ways of paying are there?
Ask pupils to discuss the problem with a partner. Take feedback and record on
the board the different ways to pay:
five 5p coins
three 5p coins and five 2p coins
one 5p coin and ten 2p coins
Could we use four 5p coins? Why not?
Main activity
TO
Display N1.1 100-square Toolsheet. Get the class to count in fives from 0 to 100.
Click on each number on the 100-square to colour it.
What do you notice? Is there a pattern? [each number ends in 0 or 5]
Will 160 be in the pattern? How do you know?
Explain that the coloured numbers are all multiples of 5, that they all end in 0 or 5,
and that every other multiple of 5 is a multiple of 10.
Next, clear the colour, start at 2 and count on in fives, colouring the numbers.
What do you notice? Is there a pattern? [each number ends in 2 or 7]
What number comes after 97? How do you know? [it ends in 2; it is 5 more
than 97]
Will 317 be in the pattern? How do you know? What about 220?
Will there always be a pattern when we count in fives whatever number
we start at? Do all the numbers end in one of two digits?
Discuss responses, and test some different starting numbers, including some
two-digit numbers. Draw out that when you count in fives from any number, the
number always ends in one of two digits and one digit is the same as the units
digit in the start number.
What is 46 1 25? How can we work it out? [count on in fives]
What about 39 1 15?
| N1.1
Properties of numbers
On the 100-square, clear the colour again. This time start with 0 and count on in
twos, colouring the numbers as you go. Explain that the multiples of 2 are also
called even numbers, and that every other number is an even number.
What do you notice about the even numbers?
[they all end in 0, 2, 4, 6 or 8]
Establish that the numbers between the even numbers are called odd numbers.
What do you notice about the odd numbers?
[they all end in 1, 3, 5, 7 or 9]
Launch Find the multiples. Set the multiple to a multiple of 5. Ask pupils to
write on their whiteboards all the multiples of 5 in the list. Repeat several times
by generating new sets of numbers.
XL
Now set the multiples to a multiple of 2. Ask pupils to write on the whiteboards
all the even numbers in the list. Generate a new set of numbers and ask for the
odd numbers in the list. Repeat a couple of times.
Select individual work from N1.1 Exercise 2 in the class book (p. 00).
Review
Write a two-digit number on the board with a units digit of 0, 2, 4, 5, 6 or 8, e.g.
32. Ask pupils to write on their whiteboards to show if the number is a multiple
of 2, 5 or 10.
How do you know that this number is a multiple of …?
Repeat with other numbers, such as 65, 56, 130, and so on. Point out that
because it ends in zero, any multiple of 10 is also a multiple of 5 and a multiple
of 2.
Sum up with the points on slide 2.2.
Homework
Ask pupils to do N1.1 Task 2 in the home book (p. 00).
N1.1 Properties of numbers | 3 Odd and even numbers
Learning points
The sum or difference of:
–
two even numbers is even;
–
two odd numbers is even;
–
one odd and one even number is odd.
Starter
Say that this lesson is about odd and even numbers.
Display N1.1 100-square Toolsheet. Point to one of the numbers. Ask pupils
to tell their partner one property or fact about the number (e.g. it is odd; it is a
multiple of ten; its tens digit is 3; it is 10 less than 45). Take feedback, then repeat
with other numbers.
TO
Use the Mask tool to cover up a group of numbers in a rectangular block. Ask
pupils to identify the hidden numbers.
TO
What patterns are there in this group of numbers? [e.g. each number is 10
more than the previous one; all the tens digits in the top row are 3]
Repeat, this time using the Mask tool twice to cover up two strips of numbers
crossing each other at right angles.
Main activity
Write the numbers 1 to 10 on the board and draw this diagram:
Say that you want to put a number in each circle but that you want the difference
between the numbers to be even. Take suggestions.
What do you notice about the numbers that I should use?
Establish that they must both be even or both be odd.
What if I wanted the difference to be odd?
Take suggestions. Agree that one number must be even and one must be odd.
Show slide 3.1. Explain that the numbers 2 to 10 must go in the nine circles, one
number in each circle, and that the difference between each pair of numbers
joined by a line must be odd.
Show that a random approach to this type of problem is unhelpful by randomly
putting numbers in circles. Some thinking is needed first.
What do we know about the numbers that are joined?
Agree that one must be odd and one must be even. Point to the top left-hand
circle.
If this is an even number, what can you tell me about the numbers joined
to it?
| N1.1
Properties of numbers
Continue in this way to fill all nine circles.
Could we have solved the problem in a different way?
Establish that there is more than one answer but that the four odd numbers
must go in the middle of the sides.
Illustrate how to jot down possible solutions without wasting time drawing the
circles grid, and the importance of keeping these jottings for reference.
Show slide 3.2 with the numbers changed to 1 to 9. Ask pupils to work in pairs
to find a solution.
Select individual work from N1.1 Exercise 3 in the class book (p. 00).
Review
Take feedback on some of the problems in Exercise 3. Invite pupils to explain
one or more of their solutions and to say what they have learned from the
problem solving.
Sum up the lesson with the points on slide 3.3.
Round off the unit by referring again to the objectives. Ask pupils to find time to
try the self-assessment problems in N1.1 How well are you doing? in the class
book (p. 00).
Homework
Ask pupils to do N1.1 Task 3 in the home book (p. 00).
N1.1 Properties of numbers | N1.1 Mental test
Read each question aloud twice. Allow a suitable pause for pupils to write answers.
1 Write down the next odd number after ninety-nine.
2 What is the next multiple of ten after one hundred and sixty?
3 Write in figures the number five hundred and eight.
2005 KS3
4 I am counting back in steps of ten.
One hundred and twenty-five, one hundred and fifteen, one hundred and five, … Write down the next number.
2005 KS3
5 Write down two numbers that have a difference of three.
2003 PT
6 Write an even number which comes between sixty-one and seventy-one.
2003 KS2
7 What is the total of one hundred and twenty and seventy?
2003 KS2
8 What number is ten less than three hundred and two?
2003 KS2
9 Add thirty-six and seventy.
2002 KS2
10 Subtract forty from one hundred and twenty.
1999 KS2
11 What is one hundred and twenty add eighty?
2006 KS2
12 What number is two less than nine hundred and one?
2003 KS2
Key:
PT Progress test KS3 Key Stage 3 test KS2 Key Stage 2 test
Questions 1 to 12 are at level 3.
Answers
  1 101  2 170
  3 508  4 95
  5 any two numbers with a difference of 3
  6 62, 64,66,68 or 70
10 | N1.1
  7 190
  8 292
  9 106
  10 80
  11 200
  12 899
Properties of numbers
N1.1 Check up and resource sheets
Check up
N1.1
Check up [continued]
Write your answers on this sheet.
Properties of numbers (no calculator)
1
Write the next three numbers in this sequence.
5
1998 level 3
546 536 526 516 ... ... ...
48 362 93 101 228 409
2
4
Put a ring round each odd number.
1999 level 3
Write in the numbers missing from the white squares.
Do not write in the shaded squares.
Write the numbers missing from these sequences.
2
11
13
14
15
8
4
6
10
16
33
43
6
3
1996 level 3
2005 level 3
Here are the first two rows on a 100-square.
1
2
3
4
5
6
7
8
9
500
10
37
11 12 13 14 15 16 17 18 19 20
Here is another part of the 100-square.
Write the two missing numbers.
77
even
numbers
numbers less
than 100
Fill in three missing numbers.
Pearson Education 2008
Tier 1 resource sheets | N1.1 Properties of numbers | N1.1
Pearson Education 2008
Tier 1 resource sheets | N1.1 Properties of numbers | N1.1
N1.1 Properties of numbers | 11
N1.1 Answers
Class book
3 a 1, 3, 5, 7, 9, 11, 13, 15
Exercise 1
1 a 90
b 160
b 2, 4, 6, 8, 10, 12, 14, 16
c 15, 20, 25, 30, 35, 40, 45, 50
d 60, 55, 50, 45, 40, 35, 30, 25
c 360
d 410
e 76
f 154
4 a 65
g 522
h 317
b 90
5 a 25p, 20p, 10p
2 a 40
b 88
c 87
d 105
c 220
d 370
b 25p, 20p, 15p
e 113
f 211
c 20p, 10p, 5p
g 383
h 835
6 Any appropriate combination of 5s and 2s, e.g.
5 1 5 1 5  15 or 5 1 2 1 2 1 2 1 2 1 2  15
3 a 0, 10, 20, 30, 40, 50, 60, 70
b 320, 310, 300, 290, 280, 270, 260, 250
c 470, 460, 450, 440, 430, 420, 410, 400
Extension problem
d 590, 600, 610, 620, 630, 640, 650, 660
9 a 15
b 25
4 a 921
b 976
c 50
c 635
d 252
d 75. Count 15 lots of 5.
5 This is a game
Exercise 3
6 For example:
1 428, 550, 354
10
2 431, 253, 425
60
30
40
20
3 a 494, 496, 498, 500, 502, 504
50
b 605, 603, 601, 599, 597, 595
c 229, 231, 233, 235, 237, 239, 241
4 This is a game
7 a 30
c 100
e 500
b 70
5 1, 3 and 7
d 150
6 4, 6 and 10 or
2, 8 and 10 or
2, 6 and 12
8 For example:
40
20
50
30
10
7
Exercise 2
1 435, 865, 220
2 870, 146, 118
12 | N1.1
Properties of numbers
8 60p
2
3
4
9
1
7
8
5
6
TASK 3
Extension problem
9 a 10
b 20
c 30
d 40
e 100. Count on fifty 2s.
1 52
2 685 and 705 should be ringed.
N1.1 How well are you doing?
1 20, 580 and 400
3 a 7403
b 7430
c 0347
d 0374
2 11
CD-ROM
3 a 40 1 30  70 and 60 1 10  70
CHECK UP
b 60 1 6 1 4  70
c 100  30  70
1 93, 101 and 409 are odd numbers.
2
4 940
3
5 26
11
6 49, 87 and 503
12
13
7 303, 313, 323
23
Home book
33
TASK 1
43
1 210, 600 and 370 should be ringed.
2 a 165
b 291
c 503
d 345
e 140
f 270
g 171
h 387
3
20
40
10
4 506, 496, 486
30
1 a 240, 25 and 90 should be ringed.
b252, 240, 876, 90 and 134 should be ringed.
1
2
4
8
5
3
2
3
7
7
1
6
2
6
3
9
15
3 The missing numbers are 66 and 88.
TASK 2
14
5 The missing numbers are 0 and 30.
6The three numbers can be:
any even number;
any even number less than 100;
any number less than 100.
N1.1 Properties of numbers | 13