Maths K–6
Stage 3B
Stage 3B – Unit 14
Number
Multiplication and Division
Entry 2: Division Strategies
This booklet includes:
• Teacher notes
(to be detached before sending to the student and supervisor)
• Supervisor notes
• Student and supervisor guide
P/M 3B 43859
Centre for Learning Innovation
Number: 43859
Title: Using Maths Tracks Stage 3B Unit 14
This publication is copyright New South Wales Department of Education and Training (DET), however it may contain
material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Extracts from Mathematics Syllabus Years K-6 © Board of Studies, NSW 2002
Maths Tracks Teacher’s Resource Book, Harcourt Education, 1st ed., 2004, by Trish Leigh and
Jennifer Vincent.
Maths Tracks Student Book, Harcourt Education, 1st ed., 2004, by Trish Leigh and Jennifer
Vincent.
The copyright in the Maths Tracks material is vested in the publisher, Reed International Books
Australia Pty Ltd, trading as Harcourt Education Australia. Maths Tracks for NSW has been
published under the Rigby imprint and the series covers seven stages from Early Stage 1 to
Stage 3B. Each stage has a Teacher’s Resource Book, Student Book and Homework Book.
For professional development and support, view online at www.rigby.com.au/pd/event.asp
Photograph of a school building © Barbara Gurney
Teacher notes p 5,
Supervisor notes p 5
Supervisor notes
p 7, Student sheet 1
p 17, sheet 2 p 19,
sheet 5 p 25
Student and
supervisor guide
p 15
COMMONWEALTH OF AUSTRALIA
Copyright Regulations 1969
WARNING
This material has been reproduced and communicated to you on behalf of the
New South Wales Department of Education and Training
(Centre for Learning Innovation)
pursuant to Part VB of the Copyright Act 1968 (the Act).
The material in this communication may be subject to copyright under the Act.
Any further reproduction or communication of this material by you may be the
subject of copyright protection under the Act.
CLI Project Team acknowledgement:
Writer:
Editors:
Illustrators:
Desktop publishing:
Averil Griffith
Rae Lister and Alan Barnes
Barbara Gurney and David Stanley
Esta Tserpes
All reasonable efforts have been made to obtain copyright permissions. All claims will be settled in good faith.
Published by
Centre for Learning Innovation (CLI)
51 Wentworth Rd
Strathfield NSW 2135
________________________________________________________________________________________________
Copyright of this material is reserved to the Crown in the right of the State of New South Wales. Reproduction or
transmittal in whole, or in part, other than in accordance with provisions of the Copyright Act, is prohibited without
the written authority of the Centre for Learning Innovation (CLI).
© State of New South Wales, Department of Education and Training 2005.
Stage 3B – Unit 14
These Teacher notes support ‘Using Maths Tracks’. The teacher should detach them
before sending the Supervisor notes and the Student and supervisor guide to the
supervisor and student. They contain:
•
•
•
•
•
•
•
•
•
•
•
Student outcomes
Prior knowledge
Language
What is needed
Preparation
Interactivity
Resources (including websites)
Assessment
Returns
Checking up answers
Assessment record
Student outcomes
Outcomes from the Mathematics K–6 Syllabus, © Board of Studies NSW 2002
Number
NS3.3
Multiplication and Division
Selects and applies appropriate strategies for multiplication and division
Working Mathematically
WMS3.4
Reasoning
Gives a valid reason for supporting one possible solution over another
Students will learn about:
•
applying appropriate mental, written or calculator strategies to solve multiplication
and division problems
•
dividing a number with three or more digits by a single-digit divisor using mental or
written strategies
Prior knowledge
•
•
Applying appropriate mental, written or calculator strategies to solve multiplication
and division problems
Dividing a number with three or more digits by a single-digit divisor using mental or
written strategies
Language
division, divide, divisor, divisible, divisibility rule, rounding, estimate, counting pattern
Using Maths Tracks, Stage 3B, Unit 14
1
Teacher notes
What is needed
Introduction
•
calculator
Activity 1
•
coloured pencils
Activity 2
•
Microsoft Word and colour printer
Activity 3
•
Maths Tracks Student Book Stage 3B, page 35
Activity 4
•
Maths Tracks Student Book Stage 3B, page 36
Maths Tracks Homework Book Stage 3B, page 13 (if you are using it)
Preparation
Select the activities you think suitable for the student by ticking the boxes beside the
activity numbers in the Student and supervisor guide.
Introduction (explicit teaching) – for all students
Activity 1 (beginning) – can provide extra support
Activity 2 (additional assistance) – can provide extra support
Activity 3 (consolidating) – for all students
Activity 4 (establishing) – for all students
Activity 5 (problem solving) – can provide extra challenge
Activity 6 (extension) – can provide extra challenge
Reflection – for all students
Checking up – for all students
Interactivity
Activity 6: Students are required to play the game Divisibility rules. Students could
play the game during a satellite lesson. Use the ‘How to play’ information in Activity 6.
The teacher can use the camera to turn over the two cards. First student to call out,
or type, ‘divisibility’ then match and state why the divisibility rule can be applied, gets
a point. The teacher can also keep a track of points. The student with the most points
when all the cards have been used, wins the game.
Resources
Add any you find suitable.
Websites
Check all websites before recommending them to students.
Add any you find suitable.
Using Maths Tracks, Stage 3B, Unit 14
2
Teacher notes
Returns
Student sheet 3 – Divisibility Rules – Activity 2
Student sheet 4 – Problem-solving – Activity 5
Checking up sheet
personal tape or recording – Reflection and Checking up
Student and Supervisor Feedback sheets
the guide (if you ask for it)
Checking up answers
Recording answers:
•
Think of a 4-digit number and then select a divisibility rule that can be applied to
that number. Discuss the number and divisibility rule with your teacher. Teacher
to mark, as answers will vary.
•
How does knowing about the divisibility rules help you solve problems?
Knowledge of divisibility rules helps you to work out division problems
faster and without a calculator.
Checking up sheet:
1
a
b
c
d
2,
3,
2,
2,
3, 4, 6, 9
5
3, 4, 5, 6, 9, 10
3, 4, 6, 7, 9
2
a
b
c
d
192, 564
75, 450, 2565
96, 132, 672, 6348
576, 6372
3
a
b
c
d
90
100
800
21 000
4
Teacher to check as answers will vary.
Estimated answer should be in the vicinity of 800.
Using Maths Tracks, Stage 3B, Unit 14
3
Teacher notes
Using Maths Tracks, Stage 3B, Unit 14
4
Teacher notes
Student's name:
Assessment record
Using Maths Tracks, Stage 3B – Unit 14
Numbers: Multiplication and Division
Entry 1: Division Strategies
Circle the numbers of the activities the student was asked to complete.
1
2
3
4
5
6
The student:
Indicator
selects and applies the
appropriate divisibility rules
to divide 3-digit or 4-digit
numbers by 1-digit or 2-digit
numbers (NS3.3)
•
estimates answers to problems
and checks to justify solutions
(WMS3.4)
Comment
All
Introduction,
1, 3, 4
Adapted from: Mathematics K–6 Syllabus, © Board of Studies NSW 2002.
•
Activity
Using Maths Tracks, Stage 3B, Unit 14
5
Teacher notes
Using Maths Tracks, Stage 3B, Unit 14
6
Teacher notes
Maths K–6
Stage 3B – Unit 14
Number
Multiplication and Division
Entry 2: Division Strategies
Supervisor notes
and
Student and supervisor guide
P/M 3B 43859
Centre for Learning Innovation
Number: 43859
Title: Using Maths Tracks Stage 3B Unit 14
This publication is copyright New South Wales Department of Education and Training (DET), however it may contain
material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Extracts from Mathematics Syllabus Years K-6 © Board of Studies, NSW 2002
Maths Tracks Teacher’s Resource Book, Harcourt Education, 1st ed., 2004, by Trish Leigh and
Jennifer Vincent.
Maths Tracks Student Book, Harcourt Education, 1st ed., 2004, by Trish Leigh and Jennifer
Vincent.
The copyright in the Maths Tracks material is vested in the publisher, Reed International Books
Australia Pty Ltd, trading as Harcourt Education Australia. Maths Tracks for NSW has been
published under the Rigby imprint and the series covers seven stages from Early Stage 1 to
Stage 3B. Each stage has a Teacher’s Resource Book, Student Book and Homework Book.
For professional development and support, view online at www.rigby.com.au/pd/event.asp
Photograph of a school building © Barbara Gurney
Teacher notes p 5,
Supervisor notes p 5
Supervisor notes
p 7, Student sheet 1
p 17, sheet 2 p 19,
sheet 5 p 25
Student and
supervisor guide
p 15
COMMONWEALTH OF AUSTRALIA
Copyright Regulations 1969
WARNING
This material has been reproduced and communicated to you on behalf of the
New South Wales Department of Education and Training
(Centre for Learning Innovation)
pursuant to Part VB of the Copyright Act 1968 (the Act).
The material in this communication may be subject to copyright under the Act.
Any further reproduction or communication of this material by you may be the
subject of copyright protection under the Act.
CLI Project Team acknowledgement:
Writer:
Editors:
Illustrators:
Desktop publishing:
Averil Griffith
Rae Lister and Alan Barnes
Barbara Gurney and David Stanley
Esta Tserpes
All reasonable efforts have been made to obtain copyright permissions. All claims will be settled in good faith.
Published by
Centre for Learning Innovation (CLI)
51 Wentworth Rd
Strathfield NSW 2135
________________________________________________________________________________________________
Copyright of this material is reserved to the Crown in the right of the State of New South Wales. Reproduction or
transmittal in whole, or in part, other than in accordance with provisions of the Copyright Act, is prohibited without
the written authority of the Centre for Learning Innovation (CLI).
© State of New South Wales, Department of Education and Training 2005.
Stage 3B – Unit 14
These Supervisor notes support the Student and supervisor guide for ‘Using Maths
Tracks’. The supervisor should detach them before giving the guide to the student.
They contain information on:
•
•
•
•
•
How to use this unit
Support and extension
Answer guide
Feedback
Checking up.
How to use this unit
Read
•
•
•
with your student:
What you’ll do
What you need
Words you need to know.
Your student’s teacher may have selected the appropriate activities from 1 to 6 by ticking
them in the list of What you’ll do. See also Support and extension.
The boxes on the right-hand side of the pages in the Student and supervisor guide
contain information and suggestions to help you support your student.
There is also space for you to make notes about how your student managed.
You can use your notes to help you fill in the Feedback sheet at the end of the unit.
An icon
shows when to refer to the Maths Tracks Student Book pages.
page x
After completing the unit, ask your student to complete the Checking up sheet
independently and return it to the teacher. Complete the supervisor side of
the Feedback sheet. Discuss the student side of the Feedback sheet and help
your student complete it.
Support and extension
The activities following the Introduction are at different levels. Your student’s teacher
may have selected the activities for your student. If activities have not been selected in
the guide, choose activities as below:
Introduction – for all students
Activities 1 and 2 – can provide extra support
Activities 3 and 4 – for all students
Activities 5 and 6 – can provide extra challenge
Reflection and Checking up – for all students.
Using Maths Tracks, Stage 3B, Unit 14
1
Supervisor notes
Answer guide
This guide helps you give your student feedback on questions and tasks in the unit or the
Maths Tracks Student Book, especially where answers will vary.
Introduction
538 ÷ 9 = 540 ÷ 9 = 60
782 ÷ 3 = 780 ÷ 3 = 260
1223 ÷ 6 = 1200 ÷ 6 = 200
6485 ÷ 5 = 6500 ÷ 5 = 1300
Activity 1
Number pattern
Divisibility rule
10, 20, 30, 40, 50, 60, 70, 80, 90, 100
Numbers end in 0
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55,
60, 65, 70, 75, 80, 85, 90, 95, 100
Numbers end in 5 or 0
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,
26, 28, 30, etc
All even numbers
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33,
36, 39, etc
Digits, when added together, are divisible
by 3
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44,
48, etc
Numbers divisible by 4
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66,
72, 78, 84, 90, 96
Numbers are even;
digits, when added together, are divisible
by 3
8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88,
96
Digits are divisible by 8
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99
Digits, when added together, are divisible
by 9
386 ÷ 5; 390 ÷ 5 = 78
6032 ÷ 2; 6030 ÷ 2 = 3015
Activity 2
Numbers divisible by 3, 6 and 9 are 810, 828, 846, 864, 882 and 900.
Number divisible by 2, 4 and 8 are 808, 816, 824, 832, 840, 848, 856, 864, 872,
880, 888 and 896.
Using Maths Tracks, Stage 3B, Unit 14
2
Supervisor notes
Activity 3
Looking at the number 2356.
It is divisible by 2 as it is an even number.
It is not divisible by 3 and 6 as its digits add up to 16.
It is divisible by 4 as the last two digits are divisible by 4.
It is not divisible by 5 as the last digit is not a 5 or a 0.
7145 ÷ 67 = 7000 ÷ 70 = 100 (and 106.64 with a calculator)
8445 ÷ 38 = 8400 ÷ 40 = 210 (and 222.24 with a calculator)
Activity 4
Looking at the number 2356.
It is not divisible by 7 as 235 – 12 = 223 and 223 is not divisible by 7
It is not divisible by 8 as the last three digits are not divisible by 8.
It is not divisible by 9 as its digits add up to 16.
It is not divisible by 10 as the last digit is not a zero.
$488.00 was divided evenly among a group of people. Each person received $61.00.
How many people were in the group?
$488 ÷ __ = $61
Estimate first: $480 ÷ 60 = 8
The last two digits are divisible by 8 so the answer could be 8 people.
Check: $61.00 x 8 = $488.00
The money was evenly divided among 8 people.
Maths Tracks Student Book Stage 3B, page 36.
2
4
Write an estimate for each division problem. Use a calculator to check your
estimates.
a
7215 ÷ 68 = 7000 ÷ 70 = 100 (Check: 106.1)
b
8147 ÷ 39 = 8000 ÷ 40 = 200 (Check: 208.9)
c
5173 ÷ 47 = 5000 ÷ 50 = 100 (Check: 110.06)
d
5614 ÷ 52 = 6000 ÷ 50 = 120 (Check: 107.96)
Write the division algorithm for each problem and estimate the answer. Use a
calculator to check your answers.
a
276 ÷ 6 = 300 ÷ 6 = 50 pages
b
1026 ÷ 19 = 1000 ÷ 20 = 50 is his average score
c
2989 ÷ 61 = 3000 ÷ 60 = 50 wagons
d
2385 ÷ 9 = 2000 ÷ 10 = 200 passengers
Using Maths Tracks, Stage 3B, Unit 14
3
Supervisor notes
Activity 5
Number
2
3
4
5
6
8
9
10
4590
6300
6401
4500
4560
7200
The secret number on the piece of paper is 7200.
Reflection
Suggested recording answers:
•
Discuss and explain how the divisibility rules can help you to estimate and check
division answers.
Divisibility rules help you decide if a number can be evenly divided by
another number. They also help you to round numbers to a closer estimate
or an algorithm that is easier to use.
•
What must be added to 2353 to make it divisible by 2, 4 and 8? Explain to your
teacher how you worked it out.
7 needs to be added to 2353 to make it a number divisible by 2, 4 and 8.
Adding 7 takes the number to 2360. The last two digits must be divisible
by 4 (60 ÷ 4 = 15); the last three digits must be divisible by 8 (360 ÷ 8 =
45); it is also an even number so is divisible by 2.
•
Change one digit in 3840 so that the number is divisible by 3, 6 and 9.
3840 is divisible by 3 and 6 but not 9. Change one digit to 6840.
6840 is divisible by 3 as all digits added together equal 18 which is
divisible by 3.
6840 is divisible by 6 as it is an even number and divisible by 3.
6840 is divisible by 9 as its digits add up to 18.
Using Maths Tracks, Stage 3B, Unit 14
4
Supervisor notes
Feedback
Supervisor
The feedback you provide will help the teacher assess your student’s progress and plan
future learning experiences. Please mark the scale and comment on the activities that
your student completed.
Student’s name
Date
•
use divisibility rules to divide 3-digit
or 4-digit numbers by 1-digit or
2-digit numbers (NS3.3)
•
make reasonable estimations and
check to justify solutions
(WMS3.4)
Using Maths Tracks, Stage 3B, Unit 14
Activity
with
difficulty
(Tick along line)
with
independently
help
All
Introduction,
1, 3, 4
Adapted from: Mathematics K–6 Syllabus, © Board of Studies NSW 2002.
Did your student:
5
Supervisor notes
Feedback
Student
In this unit I learnt about
Help your student
to give feedback
on their learning
for completed
activities.
______________________________________________________________
________________________________________________________________________________________.
Divisibility rules can help you to
________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________.
I now know the divisibility rules for the following numbers:
_____________________________
________________________________________________________________________________________.
My favourite activity for this unit was
because
___________________________________________________
_______________________________________________________________________________.
I had to work hard at
__________________________________________________________________.
Using Maths Tracks, Stage 3B, Unit 14
6
Supervisor notes
Student's name:
Checking up
Make sure your
student completes
this work
independently
for return to the
teacher.
Using Maths Tracks, Stage 3B – Unit 14
Number: Multiplication and Division
Entry 2: Division Strategies
2
3
4
Use your knowledge of divisibility rules to write the numbers between 2 and 10 that
divide equally into these numbers:
a
3636
b
2445
c
4860
d
5292
Circle the numbers that are divisible by:
a
Both 3 and 4.
88
93
192
564
1384
b
Both 3 and 5.
85
75
450
861
2565
c
Both 4 and 6.
96
132
672
4622
6348
d
Both 6 and 9.
63
366
576
8379
6372
Use your estimation skills to identify the closest answer to these problems.
a
464 ÷ 5 =
80
149
90
b
687 ÷ 7 =
100
90
190
c
3272 ÷ 4 =
1 000
400
800
d
63 339 ÷ 3 =
21 000
2100
210
Write a word problem to match this division algorithm.
Estimate the answer, then use a calculator to check.
6 4852
Using Maths Tracks, Stage 3B, Unit 14
7
Supervisor notes
Adapted from: Maths Tracks Teacher’s Resource Book Stage 3B © Harcourt Education, 2004.
1
Using Maths Tracks, Stage 3B, Unit 14
8
Supervisor notes
Stage 3B – Unit 14
Student and supervisor guide
Unit contents
About this unit ....................................................................................... ii
What you’ll do ................................................................................... ii
What you need ................................................................................ iii
Words you need to know ............................................................ iii
Icons ....................................................................................................iii
Using this guide .............................................................................. iv
Returns................................................................................................ iv
Introduction .............................................................................................1
Activity 1 ....................................................................................................3
Activity 2 ....................................................................................................5
Activity 3 ....................................................................................................7
Activity 4 ................................................................................................. 10
Activity 5 ................................................................................................. 12
Activity 6 ................................................................................................. 14
Reflection ................................................................................................ 15
Checking up .......................................................................................... 16
Student sheets ................................................................................... 17
Using Maths Tracks, Stage 3B, Unit 14
i
Student and supervisor guide
About this unit
What you’ll do
√
Introduction
•
discuss divisibility rules
•
write a 3-digit or 4-digit number and use rules to check its
divisibility
•
revise rules for rounding
Activity 1
•
choose numbers that can be divided equally by 2, 3, 4, 6, 8, 9 and
10 on a 100 chart
•
round numbers to the nearest 10 in a division algorithm
Activity 2
•
create number charts and colour counting patterns for 3, 6, 9 and
2, 4, 8
•
use the divisibility rules for these counting patterns
Activity 3
•
review the divisibility rules
•
review rounding numbers to make estimation of division problems
easier
•
match statements with a divisibility rule
•
choose the best estimate for division problems
•
recognise numbers that divide equally into larger numbers
Activity 4
•
use the divisibility rules
•
review underlining relevant information in problems
•
use the divisibility rules to find missing divisors
•
choose the best estimate for division problems
Activity 5
•
find a 4-digit number that is equally divisible by 2, 3, 4, 5, 6, 8, 9
and 10
Activity 6
•
play a game to match the numbers with divisors and identify the
divisibility rule
√
Reflection
•
explain how the divisibility rules can help you to estimate and
check division answers
•
add a number to a 4-digit to make it divisible by 2, 4 and 8
•
change one digit in a 4-digit number so that the number is divisible
by 3, 6 and 9
√
Checking up
•
select a divisibility rule and write a 4-digit number that the rule
can be applied to
•
explain how the divisibility rules can help you to solve problems
•
complete the assessment task.
Using Maths Tracks, Stage 3B, Unit 14
ii
Student and supervisor guide
What you need
Introduction
•
calculator
Activity 1
•
coloured pencils
Activity 2
•
Microsoft word and colour printer
Activity 3
•
Maths Tracks Student Book Stage 3B, page 35
•
calculator
Activity 4
•
Maths Tracks Student Book Stage 3B, page 36
Words you need to know
division
divide
divisor
divisible
divisibility rule
rounding
estimate
counting pattern
Icons
Record this for the teacher.
Return this to the teacher.
Use the page in the Maths Tracks Student Book.
Page x
Use a computer for this activity.
Using Maths Tracks, Stage 3B, Unit 14
iii
Student and supervisor guide
Using this guide
The boxes on the right-hand side of pages in the
Student and supervisor guide contain information
and suggestions for the supervisor.
After each activity, circle the face that shows
how you feel about your work and talk
about it with your supervisor.
Returns
in g
d
n
u
o
r
Student sheet 3 – Divisibility Rules – Activity 2
Student sheet 4 – Problem-solving – Activity 5
Checking up sheet
personal tape or recording – Reflection and Checking up
Supervisor and student Feedback sheets
es
this guide (if the teacher asks for it)
division
ate
ule
r
y
t
i
l
i
divisib
countin
divide
g
pa
tte
rn
divisor
tim
divisible
Using Maths Tracks, Stage 3B, Unit 14
iv
Student and supervisor guide
Introduction
In this unit you will be looking at the rules for
divisibility and revising the rules for rounding
numbers for estimation.
Divisibility rules
Divisibility rules can tell you whether a number
is divisible by another number (leaving no
remainder) without actually doing the division.
Can you remember any divisibility rules?
If so, write one of them in the box below.
You will need to check your
student’s divisibility rule.
Find Student sheet 1.
On that sheet you will find the divisibility rules
for the numbers 2, 3, 4, 5, 6, 7, 8, 9 and 10.
(Keep this sheet handy as it will be useful in the future.)
Work your way down the sheet and use a
calculator to check each example.
Write a 3-digit or 4-digit number in the box
below. Use the divisibility rules to find out what
your number is divisible by.
Show your calculating in the box.
Use a calculator to check your answer.
Using Maths Tracks, Stage 3B, Unit 14
1
You will need to check your
student’s working out.
Student and supervisor guide
Revising rules for rounding
Remember
that we use rounding
to give a quick
estimation.
Look at 236 ÷ 8 = ?
Round 236 to the nearest 10 and then
divide by 8.
240 ÷ 8 = 30
Think: 24 ÷ 8 = 3, then
write the zero to get 30
Use rounding in the following division algorithms.
538 ÷ 9 =
782 ÷ 3 =
1223 ÷ 6 =
6485 ÷ 5 =
Check your answers with a calculator.
Refer to the Answer guide
in the Supervisor notes.
Feedback:
lots of
help
Using Maths Tracks, Stage 3B, Unit 14
2
some
help
no
help
Student and supervisor guide
Activity 1
30
0
2
Using divisibility rules
Find Student sheet 2.
On the 100 chart put a cross on the numbers
that can be divided equally by 10.
What is the same about all of those numbers?
Count by 5s, starting at 5, and put a green cross
on all the numbers in the 5s counting pattern.
Can you identify the rule?
Repeat, using different coloured crosses, for the
numbers contained in the 2, 3, 4, 6, 8 and 9
counting patterns.
0
5
Your student should say
they all end in 0.
10
Refer to the Answer guide in
the Supervisor notes.
Refer back to the divisibility rules on Student
sheet 1 and check that the rules match their
coloured numbers on the chart.
Using rounding to the nearest 10 to
get a close estimate
40
Remember that when you round
numbers you are changing the
exact value of a number by giving
that number a more convenient
value, usually for the purpose of
estimating.
0
6 70
It is important to understand counting patterns
of divisible numbers to help you recognise the
easiest and closest estimate.
Using Maths Tracks, Stage 3B, Unit 14
3
Student and supervisor guide
Look at 119 ÷ 3 = ?
Round 119 and rewrite the problem.
Remember you can divide
12 by 3 and then add a
zero so 120 ÷ 3 = 40.
That’s easy!
o
r
Your student should write
120 ÷ 3.
divide
d
i
n
ng
u
Use rounding and the divisibility rules
to estimate the answers to the following
algorithms.
386 ÷ 5 =
6032 ÷ 2 =
Refer to the Answer guide
in the Supervisor notes.
divisibility
Feedback:
lots of
help
Using Maths Tracks, Stage 3B, Unit 14
4
some
help
no
help
Student and supervisor guide
Activity 2
Using divisibility rules and number
patterns
It takes practice before you can use the
divisibility rules effectively.
Recognition of number patterns is helpful.
Open a new Word document and insert a table,
10 columns by 10 rows.
Your student may need help
to create a table.
Once you have a 10 x 10
table, highlight it and select
‘centre’ alignment and
double spacing. (Do this
from the Format menu.)
Create a Number chart by numbering the cells
from 801 to 900. Here is one that has been
started for you.
801
811
821
802
812
822
803
813
823
804
814
824
805
815
825
806
816
826
807
817
827
801
930
900
808
818
828
809
819
829
810
820
830
3
Make a copy of your chart as you will need two.
Encourage your student to
use the divisibility rule for 3.
On the first chart, find the first number that
can be divided equally by 3 and change the cell
colour for that number’s cell.
Using Maths Tracks, Stage 3B, Unit 14
5
Student and supervisor guide
6
Continue changing the cell colour on all the
numbers that can be divided equally by 3.
3
Repeat, using different cell colours for the 6 and
9 counting patterns.
Print out your document.
Draw a circle around the numbers that can be
divided by 3, 6 and 9.
Under the table, write the numbers that can be
divided by 3, 6 and 9.
On the second table, do the same thing using
different colours for the numbers that can be
divided by 2, 4 and 8.
Your student needs to be
aware that not all numbers
that are divisible by 3 will
be divisible by 6 and 9.
Your student needs to be
aware that not all numbers
divisible by 2 will be
divisible by 4 and 8.
Refer to the Answer guide
in the Supervisor notes.
Print out the chart and circle and write the
numbers that are divisible by all three numbers.
Find Student sheet 3 and paste both tables onto
the sheet with your answers.
12
9
18
6
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6
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Student and supervisor guide
2
3
Activity 3
Divisibility rules
Look at the number 2356.
Use the divisibility rules to find out if the
numbers 2, 3, 4, 5 and 6 can be divided equally
into 2356.
Show your working out in the box below.
Rounding numbers to make
estimation easier
5
Refer to the Answer guide
in the Supervisor notes.
6
4
2356
Rounding numbers makes estimation of division
problems easier.
Using a combination of rounding and the
divisibility rules, you are able to make a quick
estimate of a division algorithm.
Using Maths Tracks, Stage 3B, Unit 14
7
Student and supervisor guide
For example, in the following problem,
724 ÷ 91 = ?
you would round the divisor and the dividend to
make the algorithm easier.
The closest estimate is
720 ÷ 90 = ? Cross out
the zeros. 72 ÷ 9 = 8,
so 720 ÷ 90 = 8. So
724 ÷ 91 is around 8
Now look at the division algorithm
7314 ÷ 67 = ?
If you round the divisor and the dividend, you
will get a closer estimate of the quotient.
Rounding makes it 7000 ÷ 70 = 100
So 7314 ÷ 67 will be around 100.
In fact, it is actually 109.16
Use rounding and the divisibility rules to
estimate the following division algorithms.
Try to use the closest estimate.
7145 ÷ 67 =
Your student can round
8445 to 8400 and 38 to
40 so algorithm becomes
8400 ÷ 40 =
Your student should
recognise that 40 divides
into 8400 as 400 is divisible
by 4.
Refer to the Answer guide
in the Supervisor notes.
8445 ÷ 38 =
Check your answers with a calculator.
Using Maths Tracks, Stage 3B, Unit 14
8
Student and supervisor guide
Find page 35 in the Maths Tracks Student Book.
Page 5
1
Use the divisibility rule for 6 to find the
numbers that can be divided by 6 to find a
path through the maze.
2
You can refer to the divisibility rules on
the Student sheet 1 if you need to, but
otherwise see if you can complete the
activity from memory.
3
When dividing a larger number by a 2-digit
or 3-digit number you may want to round
both numbers.
4
Use your divisibility rules to find the
numbers that divide equally into the
number in the circle.
div
is
ib
it
il
Mark your answers for this page at the back of
the Maths Tracks Student Book. Have another
try if you went off the track.
y
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lots of
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Using Maths Tracks, Stage 3B, Unit 14
9
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Student and supervisor guide
no
help
Activity 4
Look at the number 2356.
Use the divisibility rules to find out if the
numbers 7, 8, 9 and 10 can be divided equally
into the number.
Show your working out in the box below.
Underline relevant information in the following
problem.
$488.00 was divided evenly among a group of
people. Each person received $61.00.
How many people were in the group?
Refer to the Answer guide
in the Supervisor notes.
Your student should go
through each number on
Student sheet 1 until they
find a number that fits the
algorithm.
Estimate first and then use your knowledge of
divisibility rules to find the missing amount.
Show your working out in the box below.
Think: $488 ÷ __ = $61
Using Maths Tracks, Stage 3B, Unit 14
10
Student and supervisor guide
Find page 36 in the Maths Tracks Student Book.
Page 36
1
If you need to, you can refer to Student
sheet 1. Otherwise use your knowledge of
divisibility rules to tick the numbers that are
exactly divisible by the number at the top of
each column.
2
Use rounding to write an estimate for each
division problem.
3
Go through each number on Student
sheet 1 until you find a number that fits
the algorithm. It might be a good idea to
estimate the answer first.
4
Underline important information in the
problems. Use rounding to help you
estimate. Just estimate; do not worry
about the actual calculations.
Refer to the Answer guide
in the Supervisor notes.
Refer to the Answer guide
in the Supervisor notes.
r
Mark your answers for this page at the back of
the Maths Tracks Student Book. Have another
try if you went off the track.
u
o
n
i
d
g
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Student and supervisor guide
Activity 5
Problem-solving
Find the secret code
Find Student sheet 4 and show your working out
for the following problem.
A bank manager has forgotten the numbers in
the code to open up the bank safe.
On a piece of paper she has several possible
numbers.
The number she is looking for is divisible by 2,
3, 4, 5, 6, 8, 9 and 10.
Using Maths Tracks, Stage 3B, Unit 14
12
Student and supervisor guide
The bank manager has hired you to help her find
the secret code.
Refer to the Answer guide
in the Supervisor notes.
Use your knowledge of divisibility rules to find
the number that is divisible by 2, 3, 4, 5, 6, 8, 9
and 10 so she can open the safe.
3
9
6
10
5
8
4 2
Feedback:
lots of
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Using Maths Tracks, Stage 3B, Unit 14
13
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no
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Student and supervisor guide
Activity 6
Divisibility rules
Find Student sheet 5 and cut out all the little
game cards carefully. Then play the game
‘Divisibility rules’ with your supervisor.
Aim
•
To match the numbers with the divisors,
and identify the divisibility rule.
How to play
•
Shuffle the cards and deal them all out so
that each player has half the cards.
•
Leave the cards face-down and both players
turn over their top card.
•
If one card is a number and the other card
is an operator that divides that number
equally call out “Divisibility!”
•
The first player to identify the match and
state why the divisibility rule can be applied
keeps the cards. For example, 2470 can
be divided evenly by 5 because the number
ends with a 0.
•
If the cards don’t make a pair, or if they are
both number cards or both divisor cards,
return them face-down to the bottom of the
piles.
•
Continue until all cards have been matched.
Divisibility rule for 5
If the number ends in a 0 or 5,
it is divisible by 5.
For example, 235 ÷ 5
Feedback:
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Student and supervisor guide
Reflection
Record this talk and your answers to the
following questions for your teacher.
•
Discuss and explain how the divisibility rules
can help you to estimate and check division
answers.
•
What must be added to 2353 to make it
divisible by 2, 4 and 8? Explain to your
teacher how you worked it out.
•
Change one digit in 3840 so that the
number is divisible by 3, 6 and 9.
Stop the recording now.
2353
Using Maths Tracks, Stage 3B, Unit 14
15
Student and supervisor guide
Checking up
Record this for your teacher.
•
Think of a 4-digit number and then select a
divisibility rule that can be applied to that
number. Discuss the number and divisibility
rule with your teacher.
•
How does knowing about the divisibility
rules help you solve problems?
It is expected that your
voice may be heard
prompting and praising your
student.
Stop the recording now.
Complete the Checking up sheet without any
help from your Supervisor.
After you have finished the Checking up sheet,
fill in the student side of the Feedback sheet.
You may need to look back at the smiley faces
you circled to remind yourself how you felt
about each activity.
Using Maths Tracks, Stage 3B, Unit 14
16
The Checking up sheet and
Feedback sheet are near the
back of the Supervisor notes
for this unit.
Make sure your student
works on this assessment
task independently with
your assistance to read and
interpret the instructions.
Return the Checking
up sheet to the teacher
unmarked.
Student and supervisor guide
Divisibility Rules
Introduction
Divisibility rule for 2
If the last number is even, or a zero, it is divisible by 2.
For example, 348 ÷ 2.
Divisibility rule for 3
If the sum of the digits is divisible by 3, the number is divisible by 3.
For example, 261 ÷ 3; 2 + 6 + 1 = 9; 9 ÷ 3.
Divisibility rule for 4
If the last two digits of the number are divisible by 4, the number is
divisible by 4.
For example, 436 ÷ 4; 36 ÷ 4
Divisibility rule for 5
If the number ends in zero or 5, it is divisible by 5.
For example, 235 ÷ 5
Divisibility rule for 7
Take the last digit, double it and subtract it from the rest of the number;
if the answer is divisible by 7, then the number is divisible by 7.
For example, 595 ÷ 7; 59 – 10 = 49; 49 ÷ 7
Divisibility rule for 8
If the last two digits of a three-digit number, or the last three digits of a
four-digit number (and so on), are divisible by 8, the number is divisible
by 8.
For example, 732 ÷ 8; 32 ÷ 8 and 5816 ÷ 8; 816 ÷ 8
Divisibility rule for 9
If the sum of the digits is divisible by 9, the number is divisible by 9.
For example, 873 ÷ 9; 8 + 7 + 3 = 18; 18 ÷ 9
Divisibility rule for 10
If the number ends in zero, it is divisible by 10.
For example, 91 250 ÷ 10
Using Maths Tracks, Stage 3B, Unit 14
17
Student sheet 1
Adapted from: Maths Tracks Teacher’s Resource Book Stage 3B © Harcourt Education, 2004.
Divisibility rule for 6
If the number is even and the sum of the digits is divisible by 3, the
number is divisible by 6.
For example, 648 ÷ 6; 648 is even and 6 + 4 + 8 = 18; 18 ÷ 3
Using Maths Tracks, Stage 3B, Unit 14
18
Student sheet 1
Activity 1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99 100
Using Maths Tracks, Stage 3B, Unit 14
19
Student sheet 2
Adapted from: Maths Tracks Teacher’s Resource Book Stage 3B © Harcourt Education, 2004.
100 Chart
Using Maths Tracks, Stage 3B, Unit 14
20
Student sheet 2
Name:
Using divisibility rules and
number patterns
Activity 2
Glue your tables onto this sheet.
Using Maths Tracks, Stage 3B, Unit 14
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Student sheet 3
Using Maths Tracks, Stage 3B, Unit 14
22
Student sheet 3
Name:
Problem-solving
Activity 5
Find the code
The secret number is divisible by 2, 3, 4, 5, 6, 8, 9 and 10.
This table is one way of showing your working out. If you would like to
solve this problem differently show your working out in the box below the
table. Tick in each box if number is divisible.
Number
2
3
4
5
6
8
9
10
4590
6300
6401
4500
4560
7200
Other working out.
Did you find the secret code?
_________________
What was the secret number?
Using Maths Tracks, Stage 3B, Unit 14
_________________
23
Student sheet 4
Using Maths Tracks, Stage 3B, Unit 14
24
Student sheet 4
Divisibility rules
Activity 5
Divide Divide Divide Divide Divide Divide Divide Divide
by 3
by 4
by 5 by 6
by 8
by 9
by 2 by 10
Divide Divide Divide Divide Divide Divide Divide Divide
by 3
by 4
by 5 by 6
by 8
by 9
by 2 by 10
3284
2646
9831
4815
1756
7436
5544
2470
8436
1584
6320
1452
5672
3255
7641
9054
8935
6993
5721
4347
7293
2620
9459
2783
Using Maths Tracks, Stage 3B, Unit 14
25
Student sheet 5
Adapted from: Maths Tracks Teacher’s Resource Book Stage 3B © Harcourt Education, 2004.
Divide Divide Divide Divide Divide Divide Divide Divide
by 3
by 4
by 5 by 6
by 8
by 9
by 2 by 10
Using Maths Tracks, Stage 3B, Unit 14
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Student sheet 5
Centre for Learning Innovation
NSW Department of Education and Training
51 Wentworth Road
Strathfield NSW 2135