Teach for Understanding: Fractions and decimals
Contents
Operations with Decimals are in Teach for Understanding: Whole numbers
1 Basic fraction concepts p3 [VELS 3]
1
2
3
4
5
6
7
8
9
10
11
Folding paper and naming fractions
Circle fractions1
Tangrams
Circle fractions2
Square fractions
Triangle fractions
Strip fractions
Number lines up to 1
Fractions of a dozen
Pattern block fractions
Rod fractions
2 Improper fractions & mixed numbers p15
[VELS 3]
1
2
3
4
5
6
7
8
Improper fractions (pizzas)
Improper fractions (halves)
Improper fractions (quarters)
Improper fractions (thirds)
Improper fractions (pattern blocks)
Improper fractions (rods)
Mixed numbers (number line)
Mixed numbers (dice game)
3 Fractions of whole numbers p23
[VELS 3]
1
2
3
4
5
6
Fractions of whole numbers (counters)
Guessing and checking fractions
Clock face fractions
Fractions of an hour
Fractions of pizzas
Fractions of rods
4 Comparing fractions p30 [VELS 3]
1
2
3
4
Comparing fractions (strips)
Comparing fractions (counters)
Comparing fractions (circles)
Rectangle shapes
5 Equal fractions p35 [VELS 3]
1
2
3
4
5
6
7
Equal fractions (circles)
Equal fractions (squares)
Equal fractions (triangles)
Equal fractions (strips)
Equal fractions (counters)
Equal fractions (clocks)
Equal fractions (graphs)
6 Fraction operations p43 [VELS 3]
1
2
3
4
5
6
7
Adding and subtracting (counters)
Adding and subtracting (strips)
Adding and subtracting (squares)
Adding and subtracting (triangles)
Multiplying fractions (counters)
Multiplying fractions (strips)
Multiplying fractions (squares)
7 Fractions and division p51 [VELS 4]
1
2
3
4
5
6
Fractions and dividing
Sharing pizzas evenly
Sharing money evenly
Steps
Remainders as fractions (counters)
Dividing by fractions
8 Decimals – place value p57 [VELS 3]
1
2
3
Decimals include tenths
Decimals include hundredths
Decimals include thousandths
9 Fractions & percentages p61 [VELS 4]
1
2
3
Percentage squares
Percentages and fractions
Bounce fractions and percentages
10 Fractions, decimals and percentages p65
[VELS 4]
1
2
3
4
5
6
7
Fractions and decimals (wall)
Fractions and decimals (ruler)
Fractions of a dollar
Guitar fractions and decimals
Decimals and percentages
Changing fractions to decimals
Changing decimals into fractions
11 Ratio p73 [VELS 4]
1
2
3
Whole number comparisons
Mixed number comparisons
Fraction comparisons & reciprocals
12 Proportion & percentage problems p77
1
2
3
4
5
Equal ratios
Recognising proportional situations
Solving proportion problems
Solving percentage problems
Percentage increases and decreases
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2
Red = MAV product hyperlink
Resources for learning
The curriculum described in this section does not use textbooks. Instead it calls on the wealth of high quality
learning resources that are available, mainly through MAV.
Lesson plans
Maths300
This is available from MAV. There is an annual subscription, for a user name and password for on-line access to
over 170 lesson plans, many with high quality associated software.
RIME
This collection of lesson plans is also available from MAV. There are three books in the series, also available on
CD, with extra spreadsheets. Choose RIME (Measurement, Space, Chance & data).
RIME 5&6: A set of RIME-style lessons written specifically for upper primary. From MAV.
Teaching advice
Continuum
Assessment for Common Misunderstandings
Scaffolding Numeracy in the Middle Years
Problem solving
Maths With Attitude
For each content dimension and for VELS levels 3, 4, 5 and 6, this is a repackaging of the best Maths300 lessons
and the best 20 problem solving tasks, with a useful guide. From MAV.
Mathematics Task Centre
This collection of problem solving tasks are available from Doug Williams www.blackdouglas.com.au/taskcentre,
or as part of the Maths With Attitude kits, from MAV.
Action Numeracy – Middle Primary
The following stimulus books include material on fractions: Digging deep, Expeditions, The future of forests
Action Numeracy – Upper Primary
The following stimulus books include material on fractions, decimals or percentages: Bikes, Exploring space, The
facts of living, Technology to the rescue, Water and food, Who wants to be a millionaire?
Worksheets
Active Learning (Number & Algebra)
This is a set of graded worksheets in a book from MAV. They are also available on a CD containing the contents
of all three books in the series, plus extra worksheets describing how to use the hundreds of spreadsheets also
on the CD.
Active Learning 2 (Number & Algebra): More of the same in a book or a CD.
Tuning in with task cards – lower, middle, upper primary
Each book is a set of 150 workcards to guide students into hands-on activities. From the Curriculum Corporation.
Computers
Interactive Learning
One CD from MAV, containing hundreds of spreadsheets requiring no knowledge of Excel, and covering all levels
and dimensions.Very useful for homework!
Learning Objects (FUSE)
Free high quality software from a large Federal Government project. Available to government schools from
www.education.vic.edu.au/fuse (via a password) or from your CEO or AISV office.
Resources from MAV
may be purchased with a credit card or school order number on-line,
using the MAV’s web site: www.mav.vic.edu.au/shop.
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3
Red = MAV product hyperlink
1 Basic fraction concepts [VELS 3]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
•
Fraction and Decimal classroom activities Cuisenaire rods and pattern blocks p3
•
People Count #19 Basic fractions
•
Fractions: pikelets and lamingtons
•
Guidelines in Number:
Fractions: p37-39, 59-61, p83-84
•
RIME: Number
•
Mathematics Task Centre:
•
•
19 Clock fractions, 20 Compass fractions
201 Rectangle Fractions, 202 Rod Mats, 203 Make The Whole, 211 Soft Drink Crates,
218 Guessing Colours Game
Spreadsheets from the Interactive Learning CD
Fraction bars and pies, Fractions up to 1, Fractions of an amount, Rod fractions,
Learning Objects (FUSE):
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
Free software:
http://nlvm.usu.edu/en/nav/
Fraction pieces, Fractions - visualizing, - parts of a whole, - naming, - comparing,
http://illuminations.nctm.org/Activities.aspx Equivalent fraction
standards.nctm.org/document/eexamples/index.htm Communicating about Mathematics Using Games:
Playing Fraction Track
MORE >
Teach Fractions and decimals for understanding
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1.1
4
Red = MAV product hyperlink
Folding paper and naming fractions
Teach for understanding
This introductory experience should be both enjoyable and successful. It will be successful if the students feel
they have learned something about fractions.
It is strongly recommended that you join the students in these introductory activities. As you go talk about the
basic ideas of the fractions.
•
Discuss what is the ‘whole’ (the strip, square or rectangle, of length of string).
•
Count the equal pieces into which it is split (the denominator, but you probably won’t use that word).
Make sure they know how we say the word for that fraction.
This is not so obvious before sixth, seventh, etc. From then on we just add ‘th’ to the word for the number of parts
(tenth). But a half is not a ‘twoth; a third is not a ‘threeth’ and a quarter is not a ‘fourth’ though that is sometimes
used. Remember that some children will need help with the special words we use in mathematics.
Folding strips
Cut strips of paper from blank A4 sheets, the long way (‘landscape’). Children fold these into halves, quarters,
eighths, and label the parts when the strip is unfolded.
Folding squares
Use more blank paper; some teachers prefer to use brightly coloured red or yellow kindergarten squares.
(Indigenous colours are preferred where a choice exists.)
Fold a rectangle diagonally as shown and cut off the extra. This makes a square.
•
By folding in two perpendicular directions these can be folded into rectangles or squares.
•
If they are folded at 45° the result is triangles. When it is opened out, students can colour in 4 etc.
3
Halving string
How far is half way across the room? Students use a length of string that equals the width of the room. Fold it in
half once to get half and check.
Repeat with quarters or eighths of the initial length.
In this way they mark the position of each fraction along the wall. (Some equal fractions will appear.)
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction and Decimal classroom activities Partitioning geoboards p9
•
Spreadsheets from the Interactive Learning CD: Fraction bars and pies
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
1.2
5
Red = MAV product hyperlink
Circle fractions 1
Teach for understanding
It is strongly recommended that you join the students in these introductory activities. As you go talk about the
basic ideas of the fractions.
•
Discuss what is the ‘whole’ (the strip, square or rectangle, of length of string).
•
Count the equal pieces into which it is split (the ‘denominator’, but you probably won’t use that word).
•
Make sure they know how we say the word for that fraction.
This is not so obvious before sixth, seventh, etc. From then on we just add ‘th’ to the word for the number of
parts (tenth). But a half is not a‘twoth; a third is not a ‘threeth’ and a quarter is not a ‘fourth’ though that is
sometimes used. Remember that some children will need help with the special words we use in
mathematics.
Suggested activities
Children need more than one variation on this theme in order to abstract from them all the essential idea of the
fraction. In activity 1 we folded strips and squares. In activity 2 we subdivide circles by folding.
It is recommended that you actually use circles, at this stage. Make flat circular shapes two for each child, and
provide scissors to cut them. Please actually do the activity with the children. It makes all the difference.
Cutting halves, quarters and eighths
Fold and cut your circles into halves first. Discuss the meaning.
Then fold and cut each half into two parts and show that there are now four equal parts, called quarters. Discuss
one quarter, two quarters and three quarters. Show that four quarters is the whole circle.
Finally fold and cut each quarter to make eighths, and discuss this in the same way.
Cutting thirds, sixths and twelfths
Follow the same procedure with thirds. (It is not so easy! Think clock: 12, 4 and 8 o’clock.) Discuss one, two and
three thirds. Note that it doesn’t matter which you take.
Divide each third to make sixths. Discuss similarly.
Divide each sixth to make twelfths. You might note the similarity to a clock face.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Fraction and Decimal classroom activities Draw the spinner p29
•
From the Interactive Learning CD: Fraction bars and pies
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
1.3
6
Red = MAV product hyperlink
Tangrams
Teach for understanding
The tangram is a famous spatial puzzle. It is claimed that it comes from China.
The large square (10 cm by 10 cm) is cut into seven separate pieces, with these relative sizes.
Suggested activities
•
Students should first cut up the square puzzle into seven pieces.
•
Find which ones cover the others. For example, the two small triangles cover the middle triangle, the square
and also the parallelogram.
•
Determine the fraction of the total for each part of the puzzle.
•
Make a kangaroo, by putting together the pieces to form the same shape as the black silhouette.
Here is the solution.
1
8
middle
triangle
1
8
parallel
-ogram
1
8
square
1
16
1
4
1
16
1
4
large
triangle
large
triangle
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction and Decimal classroom activities Tangram parts p7
•
Spreadsheets from the Interactive Learning CD: Fraction bars and pies
•
Free software:
http://nlvm.usu.edu/en/nav/ Tangrams
Teach Fractions and decimals for understanding
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1.4
7
Red = MAV product hyperlink
Circle fractions 2
Teach for understanding
You will need circle pieces, all using the same radius – 2 halves, 3 thirds, 4 quarters, 6 sixths and 8 eighths.
We are still developing the basic concept of a fraction. You should not be surprised if some children still do not
understand yet. In the first three activities we were mainly concerned only with the denominator – the number of
pieces into which the ‘whole’ was split. We divided strips, squares and circles into halves, quarters etc.
Suggested activities
In the next few activities we are concerned equally with both numerator (that tells us how many of the given size
there are) and the denominator (that tells us the size of each piece).
•
Have the children sort the shapes into similar piles.
•
Then have them put them together to form complete circles.
•
They should name the fractions used to make up each circle: halves, thirds, quarters, sixths and eighths.
•
Some equal fractions
It is quite possible that children will notice that some different fraction names seem to show the same
2
3
4
1
amount of the circles. For example 4 and 6 and 8 are all equal to 2 .
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Fraction and Decimal classroom activities Draw the spinner p29
•
Spreadsheets from the Interactive Learning CD: Fraction bars and pies
•
Active Learning (Number & Algebra) F1 Circles
Teach Fractions and decimals for understanding
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1.5
8
Red = MAV product hyperlink
Square fractions
Teach for understanding
You will need square pieces – 2 halves, 3 thirds, 4 quarters (squares), 4 quarters (triangles), 6 sixths (rectangles),
all using the same size of square, such as 10 cm square. This is clearly similar to the previous activity but uses
the basic shape of a square. Here we are concerned with both numerator (that tells us how many of the given
size there are) and the denominator (that tells us the size of each piece).
Suggested activities
•
Have the children sort the squares into similar piles.
•
Then have them put them together to form complete squares of the same sized pieces.
•
They should name the fractions used to make up each square: halves, thirds, quarters, sixths and eighths.
•
Then have them put them together to form complete squares of differently sized pieces. There are many
ways to do this. They should name the fractions used to make up each square: for example 1 half and 2
quarters.
•
Using this approach they should soon be able to recognise equal fractions (that is, fraction names that have
the same value).
•
Children should make up outlines of fractions made from plastic pieces, and challenge others to name the
fraction shown
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction and Decimal classroom activities Partitioning geoboards p9
•
Spreadsheets from the Interactive Learning CD Fraction bars and pies
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
1.6
9
Red = MAV product hyperlink
Triangle fractions
Teach for understanding
You will need plastic triangle pieces – 2 halves, 3 thirds, 4 quarters (equilateral triangles), 6 sixths, 8 eighths. It is
clear from the designs which ones belong together. They are cut from this design.
This is clearly similar to the previous activities but uses the basic shape of an equilateral triangle. Here we are
concerned with both numerator (that tells us how many of the given size there are) and the denominator (that tells
us the size of each piece).
Suggested activities
•
Have the children sort the triangles into similar piles.
•
Then have them put them together to form complete triangles of the same sized pieces.
•
They should name the fractions used to make up each triangles: halves, thirds, quarters, sixths and eighths.
•
Then have them put them together to form complete triangles of differently sized pieces. There are many
ways to do this. They should name the fractions used to make up each triangles: for example 1 half and 2
quarters.
•
Using this approach they should soon be able to recognise equal fractions (that is, fraction names that have
the same value).
•
Children should make up outlines of fractions made from plastic pieces, and challenge others to name the
fraction shown
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction triangles
•
Spreadsheets from the Interactive Learning CD: Fraction bars and pies
Teach Fractions and decimals for understanding
10
Blue = free hyperlink, black = book, no hyperlink
1.7
Red = MAV product hyperlink
Strip fractions
Teach for understanding
A set of strips (fraction wall – see next page) shows the fractions of halves, thirds, quarters, fifths, sixths, eighths,
ninths, tenths and twelfths. These are used at this stage to reinforce the basic idea and to allow simple
comparison of size by means of comparing length.
Suggested activities
3
5
•
Students should use the strips to find and compare pairs of fractions. e.g. 4 and 6 .
•
Many students have the misconception that fractions with larger denominators must be bigger, when exactly
2
2
2
2
the opposite is the case. Explore this using comparisons such as these: 3 and 4 and also 5 and 6 .
There are very many chances for students to find equal fractions. This is not the main aim of this activity,
but it is a major goal later, so progress towards that at this stage is welcome.
•
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with fraction strips
•
Spreadsheets from the Interactive Learning CD Fraction bars and pies, Fractions up to 1
•
Fraction and Decimal classroom activities Colour in fractions p11
1
-2
1
-2
1
4
1
-4
1
8
1
8
1
-4
1
8
1
-8
1
4
1
8
1
8
1
8
1
8
1 whole strip
1
3
1
3
1
6
1
12
1
9
-----
1
6
1
12
1
12
-----
1
10
1
-9
1
10
-----
1
12
1
9
-----
1
-5
-----
1
-6
1
12
1
9
1
-5
-----
1
10
-----
1
3
-----
1
10
-----
1
-6
1
12
1
12
-----
----1
9
1
-5
1
10
-----
1
10
-----
1
6
1
12
1
9
1
12
1
9
1
--5
-----
1
12
-----
1
10
-----
1
-6
1
12
-----
1
12
1
-9
-----
-----
1
9
1
--5
1
10
-----
1
10
-----
1
10
-----
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
11
Red = MAV product hyperlink
1.8 Number lines up to 1
Teach for understanding
For many students putting fractions on a number line is the first step to recognising fractions as numbers. Clearly
the number line is closely related to the fraction wall above, however collapsing the strips onto a line is not
conceptually an easy thing.
Suggested activities
•
Start with simple fractions: halves, quarters, eighths, and name them all.
•
Work with ‘separate families’, such as [thirds, sixths, twelfths], [fifths, tenths, twentieths]
•
On the same line, combine the simpler fractions from different ‘families’, e.g. halves, thirds, sixths, fifths.
This naturally leads to observations about equal fractions (e.g. 3 sixths is one half) or comparing (e.g. 2
fifths is less than one half).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction and Decimal classroom activities Sticky numbers p24, Estimating fractions p42
•
Spreadsheets from the Interactive Learning CD Fractions up to 1
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
12
Red = MAV product hyperlink
1.9 Fractions of a dozen
Teach for understanding
You will need a lot of counters. It might be also useful to have some loops of string.
We are still working on the basic idea: denominator tells us how many equal parts to split the whole thing into,
and the numerator tells us how many of those equal parts to look at. This time we take a natural unit of number —
the dozen — which happens to have many simple fractions. For this reason 12 was the basis for imperial
measurement: for money (12 pennies made 1 shilling) and measurement (12 inches made 1 foot).
Suggested activities
•
Talk about dozens. Eggs will certainly come to mind, but we will use something more Australian, and less
breakable — counters.
•
Give each child a lot of counters and a loop of string. Ask them to put one dozen (12) counters into the loop.
Now get them to split the counters inside the loop into two parts. How many counters make one half? If
necessary, demonstrate this.
•
Continue in this way with many other fractions, getting the child to say the fractions. Use examples such as
two thirds, three quarters, five sixths, etc. Include equal fractions, such as four sixths and two thirds.
•
Ask the children to try to make one fifth of the dozen. Discuss with them why it is not possible. (Because 12
cannot be split evenly into a whole number of 5s; there is always 2 over.)
•
Deal with comparisons, which should be based on the number of counters coloured. So, for example, 6 is
2
2
2
2
less than 4 because 6 has 4 counters coloured, but 4 has 6 counters coloured.
•
Ask for those fractions that are equal, based only on the fact that they have the same number of counters
3
2
coloured. For example, 6 and 4 both have 6 counters coloured.
•
If time, extend to 20 counters in the ring. Find halves, quarters, fifths this time, and tenths. It will not be
possible to make thirds or sixths.
•
Use the 20 counter rings to compare these fractions. Please help students to actually do it.
2
5
5
10
•
1
or 2
3
or 4
1
5
or 4
1
5
10
or 5
2
3
4
or 5
2
5
or 4
3
2
3
5
or 10
7
4
5
or 4
3
Write at least one other fraction equal to this one using the 20 counters only.
1
2
1
4
2
5
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with a dozen counters
•
People Count #20 Fractions of amounts
•
Active Learning (Number & Algebra) F2 Fractions of things
•
Spreadsheets from the Interactive Learning CD
Fraction bars and pies, Fractions of an amount.
6
10
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
13
Red = MAV product hyperlink
1.10 Pattern block fractions
Teach for understanding
This activity and the next (using cuisenaire rods) demonstrate the flexible use of materials. Mathematically the
major idea is that the value of a fraction depends on what it is that represents the 1.
is quite different from one half of
.
So one half of
You will need a set of Pattern Blocks. For the purposes of learning fractions it is best to remove the red squares
and the tan (narrow) rhombuses (diamonds).
You may need to borrow them from one of the junior levels of the school where they are frequently used for
making geometric patterns. Their potential for representing fractions is less well known.
It is best if each students has a small set, but that several students at the same table can share and compare.
Suggested activities
•
Let them make patterns first. This is good, because: (a) you won’t be able to stop them anyway! and (b) it
gets them familiar with the very relationships that will be needed to explore the fraction potential of the
material.
•
Ask: What shapes are half of other shapes? Find as many as you can and show them. For each one, show
why it is half.
•
What shapes are one-third of other shapes? Show two-thirds.
•
What shapes are one-sixth of other shapes? Show two-sixths, three-sixths, four-sixths, five sixths.
•
We can make shapes to show 1 that involve more than one piece.
For example:
•
Find all the fractions you can, based on this.
•
Find all the fractions you can, based on each of these.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with pattern blocks
•
Fraction and Decimal classroom activities Pattern blocks p6 (Note rhombus should be called parallelogram)
•
Spreadsheets from the Interactive Learning CD Fraction bars and pies, Fractions up to 1
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
14
Red = MAV product hyperlink
1.11 Rod fractions
Teach for understanding
For this activity your students must use cuisenaire rods.
As with the previous activity, this shows the importance of knowing what represents the 1 when you are finding a
fraction.
Suggested activities
•
Let the children ‘play’ with the rods for a while. They will discover relationships as they do it, and can be
guaranteed to build a staircase!
•
Ask: What rods are half of other rods? Find as many as you can and show them. For each one, show why it
is half. (There are five: ‘1’ and ‘2’, ‘2’ and ‘4’, ‘3’ and ‘6’, ‘4’ and ‘8’, ‘5’ and ‘10’.)
•
What rods are one-third of other rods? (There are three: ‘1’ and ‘3’, ‘2’ and ‘6’, ‘3’ and ‘9’.)
•
Show two-thirds. (There are three: ‘2’ and ‘3’, ‘4’ and ‘6’, ‘6’ and ‘9’.)
•
What rods are one-quarter of other rods? (There are two: ‘1’ and ‘4’, ‘2’ and ‘8’.)
•
Show two-quarters (‘12’ and ‘4’, ‘4’ and ‘8’). Show three-quarters (‘3’ and ‘4’, ‘6’ and ‘8’).
•
Students search for pairs of rods which have other fraction relationships.
1
1 1
There are many unit fractions, such as 10 , 9 , 8 , etc, using the white rod and each of the others for the
larger.
They will find all the thirds, quarters, fifths, sixths, sevenths, eighths, ninths and tenths.
2
3
Some of these appear several times, which suggests equal fractions. (4 = 6 etc.)
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using Cuisenaire rods
•
Fraction and Decimal classroom activities Cuisenaire rods and pattern blocks p4
•
Spreadsheets from the Interactive Learning CD Rod fractions
Teach Fractions and decimals for understanding
15
Blue = free hyperlink, black = book, no hyperlink
2
Red = MAV product hyperlink
Improper fractions and mixed numbers [VELS 3]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
People Count #19 Improper fractions and mixed numbers
•
Fractions: pikelets and lamingtons
•
Guidelines in Number: Fractions: p37-39, 59-61, p83-84
•
RIME: Number
19 Clock fractions, 20 Compass fractions, 21 Estimation with fractions
•
Spreadsheets from the Interactive Learning CD
Mixed numbers, Improper fractions, Skip count fractions, Missing fractions
2.1 Improper fractions (pizzas)
Teach for understanding
Improper fractions, and their equivalent mixed numbers, are important ideas that are not well understood.
Relating improper fractions and mixed numbers is not an easy idea for many children. There is a great emphasis
placed on a fraction being less than the ‘whole thing’. So the idea of fractions greater than the ‘whole thing’ is
rather a surprise, at least at first.
This deals with that idea by defining the ‘whole thing’ as a pizza. Children are familiar with having many pizzas,
and fractions of pizzas are familiar. ‘Mixed numbers’ combine whole numbers and a fraction.
Suggested activities
•
Use paper or card circles for pizzas. Initially at least this is better than drawing. Deal with halves.
1
,
2
•
1
1
1
1, 12 , 2, 22 , 3, 32 etc.
1
Convert any mixed number (with halves) into a number of halves by cutting the circles into halves. So 32
7
will have 7 halves, written as 2 .
1
•
Repeat with quarters, e.g. 24 is 9 quarters. Repeat with other denominators (thirds, sixths, etc.)
•
So that they all understand the idea, do not try to teach any kind of rule at this stage. If a student figures it
out for themselves get them to tell you in their own words.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD Mixed numbers, Improper fractions,
Teach Fractions and decimals for understanding
16
Blue = free hyperlink, black = book, no hyperlink
Red = MAV product hyperlink
2.2 Improper fractions (halves)
Teach for understanding
This deals with that idea by defining the ‘whole thing’ as the number of counters that will fit into an outline. In this
activity we use pairs, so the ‘unit’ is 2, so each counter is one-half. This is all that is needed to see the improper
1
7
fraction. For example, if there are 7 counters, each worth 2 , there are 7 halves, which we can write as 2 .
Students can put the counters into the outlines, and there will possibly be some left over. In the case of 7
counters, there will be three outlines filled and one tile over. Students will easily see that this represents 3 whole
units, but calling the one left over is not always so obvious.
3 12 = 27
1
This is where the idea of one thing as a fraction of another comes in. The one counter left over is 2 of the pair of
7
1
counters that we called the ‘whole thing’ in the outline. So 2 is the same as 32 .
You will need a collection of large counters. Instead of the paper outlines you could cut an egg carton into six
pieces so that you have natural sets of two to define the unit.
Suggested activities
•
Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold two
counters and so defines the 1.
•
Go through the process of adding one counter at a time, saying the improper fraction, putting them into the
outlines, and naming the mixed number (whole number part and fraction part).
1
,
2
•
1
1
1
1, 12 , 2, 22 , 3, 32 etc.
So that they all understand the idea, do not try to teach any kind of rule at this stage. If a student figures it
out for themselves get them to tell you in their own words.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD Improper fractions, Mixed numbers
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2.3
17
Red = MAV product hyperlink
Improper fractions (quarters)
Teach for understanding
1
This time we deal with quarters. For example if the outline holds 4, each counter will be 2 . Then students can put
the counters into the outlines and there will be some which will not fill an outline. These will be the fraction part of
1
the mixed number (e.g. 2 ). Students can also say how many quarters they used, in this case 15.
3 34 = 15
4
You will need a collection of large counters. (An alternative is plastic one-inch squares.)
Instead of the paper outlines you could cut an egg carton into three pieces so that you have natural sets of four to
define the unit.
Suggested activities
•
Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold four
counters and so defines the 1.
•
Go through the process of adding one counter at a time, saying the improper fraction, putting them into the
outlines, and naming the mixed number (whole number part and fraction part).
•
Some students will realise that two-quarters is the same as one-half. This is to be encouraged. Have a
student who realises this explain it to the others.
1 2
,
4 4
•
1 3
1
2
1
3
1
2
1
3
or 2 , 4 , 1, 14 , 14 or 12 , 14 , 2, 24 , 24 or 22 , 24 , 3 etc.
So that they all understand the idea, do not try to teach any kind of rule at this stage. If a student figures it
out for themselves get them to tell you in their own words.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
Spreadsheets from the Interactive Learning CD Improper fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2.4
18
Red = MAV product hyperlink
Improper fractions (thirds)
Teach for understanding
Use counters and outlines. (Outlines could be parts of an egg carton cut into four equal parts, so there are three
spots to each part.) The outline will hold three counters and so defines the 1.
Suggested activities
•
Use counters and outlines. Demonstrate that the outline (on paper or with the egg carton) will hold three
counters and so defines the 1.
•
Go through the process of adding one counter at a time, saying the improper fraction, putting them into the
outlines, and naming the mixed number (whole number part and fraction part).
1 2
, ,
3 3
•
1
2
1
2
1, 13 , 13 , 2, 23 , 23 , 3 etc.
So that they all understand the idea, do not try to teach any kind of rule at this stage. If a student figures it
out for themselves get them to tell you in their own words.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD Improper fractions, Mixed numbers
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2.5
19
Red = MAV product hyperlink
Improper fractions with pattern blocks
Teach for understanding
This activity continues the idea of naming improper fractions as mixed numbers.
Suggested activities
Use the rhombus as the 1. One at a time, add each small triangle, getting the students to name the
improper fraction and the mixed number.
•
1
,
2
1
1
1
1, 12 , 2, 22 , 3, 32 etc.
Use the hexagon as the 1. One at a time, add each trapezium (half hexagon), getting the students to name
the improper fraction and the mixed number.
•
1
,
2
1
1
1
1, 12 , 2, 22 , 3, 32 etc.
Use the hexagon as the 1. One at a time, add each rhombus (diamond), getting the students to name the
improper fraction and the mixed number.
•
1 2
, ,
3 3
1
2
1
2
1, 13 , 13 , 2, 23 , 23 , 3 etc.
Use the hexagon as the 1. One at a time, add small triangle, getting the students to name the improper
fraction and the mixed number.
•
1 2
,
6 6
1 3
1 4
2 5
1
22
1
3
1
4
2
5
or 3 , 6 or 2 , 6 or 3 , 6 , 1, 16 , 1 6 or 13 , 16 or 12 , 16 or 13 , 16 , 2, etc.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with pattern blocks
•
Spreadsheets from the Interactive Learning CD Improper fractions, Skip count fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2.6
20
Red = MAV product hyperlink
Improper fractions with rods
Teach for understanding
This activity continues to develop the ideas of improper fractions and mixed numbers.
You need enough rods for each child at the table to be able to work alone, comparing results with others.
Suggested activities
Choose the small red rod as the unit. In this case, some answers will include halves.
Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed
numbers
•
1
1
1
1
The answers will be 2 , 1, 12 , 2, 22 , 3, 32 etc.
Choose the small light green rod as the unit. In this case, some answers will include thirds.
Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed
numbers.
•
1 2
1
2
1
2
1
The answers will be 3 , 3 , 1, 13 , 13 , 2, 23 , 23 , 3, 33 etc.
Choose the small pink rod as the unit. In this case, some answers will include quarters (and maybe halves).
Add many white (1) cuisenaire rods, one at a time. Say the name as both improper fractions and mixed
numbers.
1 2
1 3
1
2
1
3
1
2
1
3
The answers will be 4 , 4 or 2 , 4 , 1, 14 , 14 or 12 , 14 , 2, 24 , 24 or 22 , 24 , 3 etc.
For an extra challenge, students should choose any two different rods. They make the smaller one the unit
(1). Then they express the larger one as a mixed number. (Some students may need to replace the larger
rod by small white ones to see the link to the previous work.),
•
2
1
For example, if you choose black (8) and dark green (6), the black is 16 (and also 13 ).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using Cuisenaire rods
•
Spreadsheets from the Interactive Learning CD: Improper fractions, Skip count fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
2.7
21
Red = MAV product hyperlink
Mixed numbers on number lines
Teach for understanding
For a proper understanding of fractions, and mixed numbers in particular, it is most important that students spend
time working with number lines. The aim of this activity is to get familiar and comfortable with the positions of
mixed numbers on number lines, and introducing ‘skip counting’.
For this activity it would be good to have many spare copies of the number lines. Suggested problems for you to
use are given below.
Suggested activities
•
Hand out copies of number lines: one marked in halves, one in thirds, one in quarters etc. Discuss the
pattern with students.
1
1 The first line has whole numbers every two marks, to show halves. Talk about finding 12 etc.
1
2 The second line has whole numbers every three marks, to show thirds. Talk about finding 13 etc.
The pattern continues. Make sure students can tell you about it.
1
3 On the first line, put a dot at the first mark after 0 and write its name. (It is 2 .)
1
1
1
4 Continue to write the names of the positions every two marks along. They are 12 , 22 , 32 etc.
1
5 On the second line, put a dot at the first mark after 0 and write its name. (It is 3 .)
Continue to write the names of the positions every two marks along.
2
1
2
1
They are 1, 13 , 23 , 3, 33 , 43 , 5, etc
6 Continue in this way for the other lines.
7 Repeat for every third mark on each of the number lines.
•
After some fluency begins, students could be asked to speak the names of the mixed numbers obtained by
‘skip-counting’ in this way.
•
Mark a large-scale number line outdoors. The marks can be one step-length apart. Children should choose
the fraction they are using (e.g. quarters) and write the whole numbers into position on the line with chalk.
They then walk up and down the line saying the position they are in at each step.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD:
Improper fractions, Skip count fractions, Missing fractions
Teach Fractions and decimals for understanding
22
Blue = free hyperlink, black = book, no hyperlink
2.8
Red = MAV product hyperlink
Mixed numbers – dice game
Teach for understanding
This game is designed to give children some greater appreciation of the positions of fractions and mixed numbers
on the number line.
You will need one die. Each two or three children need one copy of a sheet with parallel number lines from 0 to 3.
They are marked in halves, thirds, quarters, fifths and sixths. Each child needs a marker which is small enough to
show a position on the number line.
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
0
1
2
3
Suggested activities
•
In turn each child rolls the die twice. The first roll is the numerator, and the second is the denominator. (If
the second roll is a 1, the child should roll again.) The child then finds that position on the appropriate
number line on the sheet. If the numerator is larger than the denominator, the child will have to convert the
3
1
fraction to a mixed number, for example 2 becomes 12 .
•
When all the children have rolled the die and placed their markers, the child with the largest number wins.
This is easily found by seeing which is further from 0 on the number lines. (There are some equal fractions,
1
3
such as 2 and 6 . If two children have equal fractions and they are both winners, then those children only roll
the die twice again, and choose the winner from that ‘play-off’.)
•
Extension: Each child rolls the die three times, and adds the first two rolls to make the numerator. This will
produce larger mixed numbers. Some of these will fall on the unlabeled part of the line, and children will
have to work out the position for themselves, with the others checking! Others will be so large that they do
11
1
not fit onto the page. For example rolls of 6, 5 and 2 gives 2 and therefore the mixed number is 52 . This
will mean that children will have to find their own methods of comparing and proving to the others that they
have won!
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD:
Improper fractions, Skip count fractions, Missing fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3
23
Red = MAV product hyperlink
Fractions of whole numbers [VELS 3]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
•
Fraction and Decimal classroom activities
•
People Count #20 Fractions of amounts
•
Guidelines in Number: Fractions: p83-84
•
RIME (Number)
•
Fractions: pikelets and lamingtons
•
Active Learning (Number and Algebra):
F1 Fraction circles, F2 Fractions of things, F3 Fraction rally: a fraction dice game
•
Mathematics Task Centre:
•
Spreadsheets from the Interactive Learning CD
•
Learning Objects (FUSE):
•
Free software:
19 Clock fractions, 20 Compass fractions, 21 Estimation with fractions, 22 Fractions of whole numbers
201 Rectangle Fractions, 202 Rod Mats, 203 Make The Whole, 211 Soft Drink Crates, 218 Guessing
Colours Game
Fractions of an amount, Fractions of dollars, Fractions of time, Fractions of numbers, Fractions of money
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
http://nlvm.usu.edu/en/nav/ Fraction bars, Fraction pieces, Fraction parts of a whole, Fractions - visualizing,
Number line bars - fractions
http://illuminations.nctm.org/Activities.aspx Equivalent fraction
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.1
24
Red = MAV product hyperlink
Fractions of whole numbers
Teach for understanding
The most common use of fractions is finding fractions of whole numbers. It is at this point that you will start to find
out who really does understand the ideas and who still needs at lot of help.
Each student needs a set of counters.
Suggested activities
•
Start with 6 counters. Ask the child to split them into thirds. (There may be some children who will react by
splitting the 6 counters into 3s. Be patient!)
•
Discuss how to show one-third (2 counters), and two-thirds (4 counters). Discuss the meaning of threethirds (all 6 counters).
•
The activity asks students to use 12 counters. It is quite important for most children that they actually use
objects, e.g. counters. Students colour in the circles to show the selected counters.
There are two ways to conceptualise this question:
2
- For example, for 3 of 12 counters share into three equal parts, and then choose (colour) two of those
parts;
2
- For example, for 3 of 12 cookies organize them into groups of 3, and choose (colour) 2 out of every 3
circles.
•
They find all the halves, thirds, quarters, sixths and twelfths. This also reveals that there are many equal
fractions. Hopefully they will discover this for themselves.
•
Use array models. For example for 6 counters, use two rows of 3 to represent one unit. Then finding 2
1
1
2
should be easy and finding 3 or 3 should be easy too.
•
For some children you might be able to ask if they can see a short cut. All children should find their way to
this eventually, but do not rush them; it is best if it is understood, or even discovered by the child.
•
When they split 12 counters into 6 equal parts (for sixths) they are dividing 12 by 6. Then when they select 5
of those parts, they are multiplying by 5.
•
Discuss that you can ‘multiply first then divide’ or ‘divide first then multiply’.
5
(For example, 6 of 12 can be 12 x 5 ÷ 6 = 60 ÷ 6 = 10 or 12 ÷ 6 x 5 = 2 x 5 = 10.)
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with a dozen counters
•
Spreadsheets from the Interactive Learning CD: Fractions of dollars
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.2
25
Red = MAV product hyperlink
Guessing and checking fractions
Teach for understanding
This set of activities aim to develop further the ideas of finding fractions of whole numbers. This is done by
dividing a length into a number of equal parts and then choosing some of those parts.
There are two ways of doing this. One is estimation. This means that the child looks at the line, mentally divides it
into equal parts, and then chooses a number of those parts. This then gives the position of the fraction of the
2
whole line. For example, to find 3 of a length, the child imagines it split into three equal parts, and then puts a
mark at the end of two of those thirds. Estimation also motivates the calculation part for the next step.
This is to measure the length of the line (in centimetres) and divide this number into so many equal parts, and
then multiply to get the number of centimetres in the required fraction. For example, the child measures a line 60
2
2
cm long. Then to find 3 of 60 cm, the child divides 60 by 3 (20 cm for each third) and multiplies by 2 (40 cm for 3 .)
2
Then a ruler or tape is used to find the correct position of the 3 and compare it with the estimation.
You will need a strip of blank tape or length of string (or heavy cord). A good length is 120 cm, since this can be
divided by 2, 3, 4, 5, 6, 8 or 10. It is useful to have a few clothes pegs to act as markers. It would also be useful to
have a measuring tape (longer than a 30 cm ruler) such as a 150 cm sewing tape.
Suggested activities
2
•
Two students hold the tape tight, and a third student places a clothes peg at the point they think is 3 from
one end. They then measure the tape, calculate the fraction of its length and compare the position of the
peg with the ‘correct’ answer. (The process is discussed above.)
•
Repeat the activity with a wide variety of fractions, such as one-half, all the thirds, all the quarters, all the
fifths, all the sixths, all the eighths and all the tenths. (You can go even further if you also let the children
use calculators to divide.)
•
You can extend the activity easily by asking child to estimate, say, two thirds of the distance across the
room. After measuring the distance, the child will need to use a calculator to divide by 3 and multiply by 2.
This provides excellent conceptual development (through the estimation) and links it to the computation
process (through the use of the calculator).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction and Decimal classroom activities Estimating fractions p42
•
Maths300: Fraction Estimation
•
Spreadsheets from the Interactive Learning CD: Fractions of time
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.3
26
Red = MAV product hyperlink
Clock face fractions
Teach for understanding
This is another chance to practise finding fractions of whole numbers. In this case the whole number is always 12,
and the application is a very real and useful one – fractions of one clock face of 12 hours.
It would be useful if you had a real clock face or one designed for teaching time.
Suggested activities
•
Use the page of clock faces (above). Students find the number of hours that match certain fractions of the
clock face. Start with halves and quarters; these will be familiar because of common usage.
•
Thirds are more difficult. Divide the clock face into three equal parts, each 4 hours. So 3 is 8 hours. Try to
get the children to see that they can divide 12 by 3 then multiply by 2.
•
Continue with sixths, eighths and twelfths, by getting the fraction of 12.
2
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Spreadsheets from the Interactive Learning CD: Fractions of time – 12 hours.
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.4
27
Red = MAV product hyperlink
Fractions of an hour
Teach for understanding
This is another chance to practise finding fractions of whole numbers. In this case the whole number is always 12,
and the application is a very real and useful one – fractions of one clock face of 12 hours.
It would be useful if you had a real clock face or one designed for teaching time.
Suggested activities
•
Use the page of clock faces (below). Students find the number of minutes that match certain fractions of one
hour. Start with halves and quarters; these will be easy because of common usage.
•
Thirds are more difficult. Divide the 60 minutes into three equal parts, each 20 minutes. So 3 is 40 minutes.
Try to get the children to see that they can divide 60 by 3 then multiply by 2.
•
Continue with sixths, fifths and tenths, by getting the fraction of 60.
2
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Spreadsheets from the Interactive Learning CD Fractions of time – 60 minutes.
(This presents the problem and the diagram already drawn. This is visual confirmation of the answer to be
obtained by calculation.)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.5
28
Red = MAV product hyperlink
Fractions of circles
Teach for understanding
Finding fractions of whole numbers is relatively easy when the denominator divides exactly into the whole
2
number, such as 3 of 6. It is much more difficult when the denominator does not divide exactly into the whole
2
number, such as 3 of 5. This activity works with this idea using small numbers of circles. (Some students may be
familiar with the ways that pizzas are cut. This will help.)
Each student should have access to at least three sets of circles – divided into thirds, quarters, sixths, eighths
and twelfths (see below). Organise these into complete sets, so they may be used for the problems.
Suggested activities
•
Have students find fractions of two plastic circles where the answer is a fraction or mixed number.
3
Start with 4 of 2. Because of the quarters we choose two circles divided into four parts.
3
1
There are 8 quarters, so 4 of 2 circles will be of 8 quarters = 6 quarters. This is 12 circles.
•
There is another way to find this: get three quarters of each circle and add the results.
3
1
Either way you should get 4 of 2 is 6 quarters or 12 .
•
2
Continue with 3 of 2. Because of the thirds, we choose two circles divided into three parts.
2
2
There are six thirds, so 3 of 6 thirds is 4 thirds. (Alternative: get 3 of each circle and add.)
1
Either way, this is 13 .
•
5
Continue with 8 of 2. Because of the eighths, we choose two circles divided into eight parts.
5
10
2
1
There are 16 eighths, so 8 of 16 eighths is 8 which is 18 or 14 .
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Fraction and Decimal classroom activities Boys, girls and pizza p39
•
Spreadsheets from the Interactive Learning CD: Comparing fractions
•
Active Learning (Number & Algebra) F1 Circles
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
3.6
29
Red = MAV product hyperlink
Fractions of rods
Teach for understanding
A similar activity can be done with number lines. A number of units is selected and we get a fraction of it. It is
necessary to choose a line divided into suitable parts.
1
2
Here is an example, using a 15 cm line. 3 of 5 is 13 .
0
1 23
1
2
1 third
3
3 13
4
1 third
5
1 third
Suggested activities
•
Use Cuisenaire rods as units to make this clearer. They are useful because they have width and therefore
unit lengths of 1 cm. Demonstrate
•
For 3 of 5 we can choose light green rods of length 3 units. We choose this length because it may be split
1
1
into 3 equal parts. We need five of them. Here they are for 3 of 5, placed end to end along a ruler – to a
2
length of 15 cm. You can see that one third of the total length is 13 .
0
1
1 23
2
1 third
3
3 13
4
5
1 third
1 third
Instead of a ruler you may replace the light green rods with white rods (cubes of length 1) to show how the
•
2
splits occur. Again, one third of the total length is 13 .
0
1
1 23
1 third
2
3
3 13
1 third
4
5
1 third
The general idea is to choose a rod for the unit whose length may be divided by the denominator. Students
divide the total length of a number of rods (laid out end to end) by the denominator and multiply by the
numerator.
Discuss another example. It uses the yellow rods (5), because we are making fifths.
•
4
5
4
2
of 3 splits into 15 parts. So it is 5 of 15 fifths, which is 12 fifths or 25 .
1 fifth
Resources for learning
1 fifth
1 fifth
1 fifth
1 fifth
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using Cuisenaire rods
•
Spreadsheets from the Interactive Learning CD:
Fractions of numbers (Use this to practise this skill mentally.)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
4
30
Red = MAV product hyperlink
Comparing fractions [VELS 3]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
•
People Count #22 Comparing fractions
•
Guidelines in Number: Fractions: p83-84
•
Fractions: pikelets and lamingtons
•
RIME: Number
N23 Fractions by comparing lengths,
N24 Order of fractions,
N25 Equal fractions
•
Active Learning (Number and Algebra):
F1 Fraction circles, F2 Fractions of things, F3 Fraction rally: a fraction dice game
•
•
Mathematics Task Centre
201 Rectangle Fractions, 202 Rod Mats, 203 Make The Whole, 211 Soft Drink Crates,
218 Guessing Colours Game
Spreadsheets from the Interactive Learning CD
Comparing fractions, Fraction triangles, Equal circle fractions, Equal square fractions,
Equal line fractions, Equal block fractions, Clock fractions, Graph fractions
•
Learning Objects (FUSE):
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
•
Free software:
http://nlvm.usu.edu/en/nav/ Pattern blocks (under Geometry), Fraction pieces, Fractions - visualizing, parts of a whole, - naming, - comparing,
http://illuminations.nctm.org/Activities.aspx Equivalent fractions
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
4.1
31
Red = MAV product hyperlink
Comparing fractions (strips)
Teach for understanding
An important part of understanding fractions is to be able to compare their sizes. This also improves
understanding of the roles of denominator (the number of equal parts of the whole) and the numerator (how many
of those equal parts are used in the fraction).
Students can use fraction strips to compare fractions. This activity has three different comparisons. It is important
that students understand, particularly the second type.
• Type 1: with the same denominator. Clearly the larger numerator shows the bigger fraction.
• Type 2: with the same numerator. Not so clearly, the smaller denominator shows the bigger fraction.
This is a surprise to many students, as they focus on the larger number (in the denominator) and think at first
that this makes the fraction larger.
In fact, a larger denominator means that the whole is split into more equal parts, so each part is smaller.
2
2
For example 3 is bigger than 4 . The larger the denominator, the smaller the fraction, if the numerators are the
same.
• Type 3: when one fraction has both numerator and denominator increased by the same number to form a
second fraction, the larger numbers show the bigger fraction.
2
1
3
2
For example, 3 is bigger than 2 , and 4 is bigger than 3 . The increased fraction is always bigger.
The fraction strips are an ideal way for the students to learn this. They all have the same length but are clearly
split into equal parts.
Suggested activities
•
Find and compare the sixths. 5 sixths is longer than 4 sixths, is longer than 3 sixths etc.
•
Now compare one sixth and one quarter. If you write the fractions (6 and 4 ) you will find some students who
1
1
1
think 6 is bigger, thinking ‘6 is bigger than 4’. Let them use the strips to compare.
1
4
1
6
1
1
•
Discuss why 4 is actually larger than 6 .
•
Try one with a numerator of 2: compare 3 and 5 . Discuss why 3 is larger than 5 . Try to get a general
understanding expressed by the children. Try other examples of two fractions with the same numerator to
help them generalise the rule.
•
The last step is to compare fractions in which both numerator and denominator of one are increased by 1 to
form the other fraction. The fraction with the increased numbers will be larger, providing that the original
fraction is less than 1.
2
2
2
2
In fact this applies to increasing numerator and denominator by any (same) number.
2
4
4
For example, if 3 is changed to 5 then the increased fraction (5 ) will be greater, providing that the original
fraction is less than 1.
Note: If the original fraction is greater than 1, the increased fraction is the lesser. A more general rule is that the
increased fraction will always be closer to 1.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with fraction strips
•
Spreadsheets from the Interactive Learning CD Comparing fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
4.2
32
Red = MAV product hyperlink
Comparing fractions (counters)
Teach for understanding
In this activity students compare two fractions by finding fractions of the same larger whole number and
3
2
comparing the results. For example, by using 12, 4 of 12 (9) is bigger than 3 of 12 (8).
This leads directly to using the same denominator as a method for comparing, adding and subtracting:
3
9
2
8
So 4 = 12 and 3 = 12 . It will also have the advantage of reviewing ‘fractions of whole numbers’ which was done
earlier. Any children who were absent, or who didn’t quite understand earlier should be given extra help at this
stage.
Suggested activities
•
Each child has a small pile of counters.
Put 4 counters aside and call them 1. (Make sure it is understood that the set of four is the unit, 1.)
1
So each counter shows 4 .
1
3
1
3
Now ask one child to make 2 and another child to make 4 . Ask: Which is bigger, 2 or 4 ?
•
1
Now make 6 counters represent 1. So each counter shows 6 .
1
2
Now ask one child to make 2 and another child to make 3 .
1
2
5
2
Ask: Which is bigger, 2 or 3 ? Which is bigger, 6 or 3 ?
•
1
2
1
1
Which is bigger, 2 or 5 ?
Why is this not possible with 6 counters? (Because we cannot make fifths.)
How many counters should represent 1? Why? (10, because we can make both half and fifths.)
1
1
Which is bigger, 2 or 2 ? Which is bigger, 2 or 2 ?
•
1
1
How many counters should represent 1if we want to work out which is bigger, 2 or 2 ?
(We use 12, so we can make quarters and thirds.)
3
2
3
2
Which is bigger, 4 or 3 ? Which is bigger, 4 or 3 ?
•
3
5
How many counters should represent 1 if we want to work out which is bigger, 4 or 6 ?
(We use 12, so we can make quarters and sixths. Note that 24 will also work, but 12 is easier.)
3
5
1
1
Which is bigger, 4 or 6 ? Which is bigger, 4 or 6 ?
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above.
•
Understanding fractions with a dozen counters
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
4.3
33
Red = MAV product hyperlink
Comparing fractions (circles)
Teach for understanding
This model of fractions may be used to make comparisons. You will need circle pieces, one set for each student.
1 1 1
1
1
A complete set has a circle made from each of these fractions: 4 , 6 , 8 , 10 , 12 .
Suggested activities
1
3
Each child makes a circle using quarters. Now ask one child to make 2 and another child to make 4 . Ask:
•
1
3
Which is bigger, 2 or 4 ?
Each child makes a circle using sixths.
•
1
2
1
2
Now ask one child to make 2 and another child to make 3 . Ask: Which is bigger, 2 or 3 ?
5
2
Another question: Which is bigger, 6 or 3 ?
1
2
1
2
Which is bigger, 2 or 5 ?
Why is this not possible with a circle in sixths? (Because we cannot make fifths.)
Which circle should we use? Why? (10 because we can make both half and fifths.)
•
7
4
Which is bigger, 2 or 5 ? Which is bigger, 10 or 5 ?
3
2
Which circle should we use if we want to work out which is bigger, 4 or 3 ?
•
(We use the circle in twelfths, so we can make quarters and thirds.)
3
2
3
11
Which is bigger, 4 or 3 ? Which is bigger, 4 or 12 ?
3
5
Which circle should we use if we want to work out which is bigger, 4 or 6 ?
(We use the circle in twelfths, so we can make quarters and sixths. Note that 24 will also work, but 12 is
easier.)
•
3
5
1
1
Which is bigger, 4 or 6 ? Which is bigger, 4 or 6 ?
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Spreadsheets from the Interactive Learning CD Comparing fractions
Teach Fractions and decimals for understanding
34
Blue = free hyperlink, black = book, no hyperlink
4.4
Red = MAV product hyperlink
Rectangle shapes
Teach for understanding
This activity is about the shapes of rectangles. Although they are all rectangles (even the squares!) some are tall
and thin, others are short and fat, and so on. Rectangles that are ‘closer to squares’ have heights more nearly
equal to their widths.
height
Rectangles that are the same shape will have equal fractions formed with width .
height
The fraction width is closer to 1 the closer the rectangle is to a square.
Here is an example. For these rectangles, height is less than width.
3
2
The rectangle on the right is closer to a square, so fraction 5 is bigger than the fraction 4 .
2 cm
4 cm
Fraction 2
4
3 cm
Fraction 3
5
5 cm
Each rectangle has one diagonal drawn as well.
Rectangles that have the same shape will have diagonals with the same slope.
3
Larger fractions have steeper slopes for their diagonals. For example the diagonal for 5 is steeper than the
1
diagonal for 2 .
Suggested activities
•
Explore rectangles wherever you can find them.
•
The activity with rectangles should now be extended to real rectangles around the room: doors, cupboards,
blackboard, paper, posters, books, tables, floor, walls, etc.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with rectangles
•
Spreadsheets from the Interactive Learning CD: Fraction triangles
•
Maths300 Rectangle fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5
35
Red = MAV product hyperlink
Equal fractions [VELS 3]
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
•
Fraction and Decimal classroom activities Find me a partner p14
•
People Count #23 Equal (equivalent) fractions
•
Guidelines in Number: Fractions: p83-84
•
Fractions, pikelets and lamingtons
•
Maths300: 33 Estimating Fractions, 144 Rod Mats
•
RIME (Number):
•
Spreadsheets from the Interactive Learning CD
•
Learning Objects (FUSE):
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
•
Free software:
N19 Clock fractions, N24 Order of fractions, N25 Equal fractions
Equal circle fractions, Equal square fractions, Equal line fractions, Equal block fractions, Clock fractions,
Graph fractions
http://nlvm.usu.edu/en/nav/ Fractions - equivalent, Fractions- comparing,
http://illuminations.nctm.org/Activities.aspx
Equivalent fractions, Fraction game, Fraction model (I, II, III)
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.1
36
Red = MAV product hyperlink
Equal fractions (circles)
Teach for understanding
This is the first of seven activities on equal fractions. Confidence with this topic is very useful, both for further work
with fractions, such as adding and subtracting, and with real life situations, where equal fractions may be used to
solve ratio problems.
Equal fractions have been met before on many occasions. These activities formalise the idea, through the
comparison of several models: circles, squares, triangles, strips, whole numbers, clocks and rectangles.
Each student will need circle pieces, to make complete circles in halves, thirds, quarters, fifths, sixths, eighths,
tenths and twelfths.
Suggested activities
1
Ask someone to show you 6 of a complete circle. Write the fraction. Now find two equal pieces that can
•
1
2
take its place. This shows that 6 = 12 . Write this out. Ask the students to look for patterns.
5
Ask someone to show you 6 of a complete circle. Write the fraction. Now find equal pieces that can take its
•
5
10
place. This shows that 6 = 12 . Write this out. Tell the students to look for patterns.
1
Ask someone to show you 5 of a complete circle. Write the fraction. Now find two equal pieces that can take
•
1
2
its place. This shows that 5 = 10 . Write this out. Remind the students to look for patterns.
After working through the problems, the pattern should be fairly obvious.
•
Have kids tell you how to create a fraction equal to another one.
Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’. For example, make the
•
6
fraction 8 . Now find a way to make the same amount of circle with fewer equal pieces. You can use three
6
3
quarter pieces, so 8 = 4 .
Use equal fractions to compare pairs of fractions.
•
5
3
10
9
For example, 6 is bigger than 4 , because 12 is bigger than 12 . Check with the circles.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Fraction and Decimal classroom activities Find me a partner p14
•
Spreadsheets from the Interactive Learning CD: Equal circle fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.2
37
Red = MAV product hyperlink
Equal fractions (squares)
Teach for understanding
Each student will need square and rectangle pieces, to make complete squares in halves, thirds, quarters, fifths,
sixths, eighths, tenths and twelfths.
Suggested activities
•
1
Ask someone to show you 6 of a complete square. Write the fraction.
1
2
Now find two equal pieces that can take its place. This shows that 6 = 12 . Write this out.
Tell the students to look for patterns.
•
4
Ask someone to show you 6 of a complete square. Write the fraction.
4
2
Now find fewer equal pieces that can take its place. This shows that 6 = 3 . Write this out.
Tell the students to look for patterns.
•
After working through some problems, a pattern should be fairly obvious.
Have kids tell you how to create a fraction equal to another one.
Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’.
6
For example, make the fraction 8 .
Now find a way to make the same amount of square with fewer equal pieces. You can use three quarter
6
3
pieces, so 8 = 4 .
•
Use equal fractions to compare pairs of fractions.
5
3
10
9
For example, 6 is bigger than 4 , because 12 is bigger than 12 . Check with the square pieces.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with rectangles
•
Fraction and Decimal classroom activities Find me a partner p14
•
Spreadsheets from the Interactive Learning CD: Equal square fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.3
38
Red = MAV product hyperlink
Equal fractions with equilateral triangles
Teach for understanding
Each student may need the triangle pieces, to make complete triangles in halves, thirds (two ways), quarters,
sixths and eighths. The diagram uses sixths, eighths and twenty fourths.
Suggested activities
•
1
Ask someone to show you 4 of a complete triangle. Write the fraction.
1
2
Now find two equal pieces that can take its place. This shows that 4 = 8 . Write this out.
Tell the students to look for patterns.
•
1
Ask someone to show you 3 of a complete triangle. Write the fraction.
2
1
Now find equal pieces that can take its place. This shows that 6 = 3 . Write this out.
Tell the students to look for patterns.
•
After working through some problems, the pattern should be fairly obvious. Have kids tell you how to create
a fraction equal to another one.
•
Discuss the idea of reducing fractions to the lowest numbers: ‘simplest terms’.
6
For example, make the fraction 8 . Now find a way to make the same amount of square with fewer equal
6
3
pieces. You can use three quarter pieces, so 8 = 4 .
•
Use equal fractions to compare pairs of fractions.
5
3
10
9
For example, 6 is bigger than 4 , because 12 is bigger than 12 . Check with the triangle pieces.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
See above. Particularly Fraction triangles
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.4
39
Red = MAV product hyperlink
Equal fractions with strips
Teach for understanding
Use the fraction strips. These have the same length split into these numbers of equal parts: 2, 3, 4, 5, 6, 8, 9, 10,
12. They therefore display very many equal fractions. The focus in this activity is more on the ‘families of equal
fractions’. The members of the same family are all equal to each other, so we can recognise equality if we can
6
10
2
reduce them to the simplest fraction in the family. For example, 9 and 15 both belong to the family 3 .
Suggested activities
1
•
Find all the fractions the same length as 2 , and so on.
•
Compare the fractions 8 and 12 . Putting the eighths and twelfths side by side will make it obvious that they
are the same length.
2
3
----12
2
-----12
1
-----12
4
----12
7
----12
6
----12
5
-----12
7
--8
6
-8
5
-8
4
--8
3
--8
2
-8
1
-8
•
3
8
-----12
9
----12
10
----12
1
Why are they the same? Let the children discuss it until they realise that both are equal to 4 .
They are ‘in the same family’.
•
4
6
6
9
Why are 8 and 12 equal? (Which family are they in?)
Why are 8 and 12 equal?
•
11
-----12
4
6
Use the sixths and ninths to ask: Why are 6 and 9 equal?
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using strips
•
Fraction and Decimal classroom activities Find me a partner p14
•
Spreadsheets from the Interactive Learning CD Equal line fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.5
40
Red = MAV product hyperlink
Equal fractions with counters
Teach for understanding
Equal fractions have been met before on many occasions. These activities formalise the idea, through the
comparison of several models: circles, squares, triangles, strips, whole numbers, clocks and rectangles.
The emphasis in this activity is firmly on reducing a fraction to ‘lowest terms’ and possibly using this to then find
another fraction from the same family that is equal to the first (and to the second).
Each student should have access to a large number of counters (or equivalent counters). This enables students
to find equal fractions using fractions of whole numbers.
Suggested activities
4
•
Make a pile of ten counters. Find 10 of it (4). What simpler fraction is equal to this one?
•
(The counters can be grouped into 2s, showing that 4 counters is 5 of 10.)
•
Make a set of 18 counters. Make a small pile from 9 of them (12).
2
6
What simpler fraction is equal to this one?
2
(The counters can be grouped into 6s, showing that 12 counters is 3 of 18.)
4
2
Is the 6 of 18 the same amount? (Yes.) What simpler fraction is equal to this one? (Also 3 , obtained by
•
grouping in 3s.)
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with a dozen counters
•
Spreadsheets from the Interactive Learning CD Equal block fractions
Teach Fractions and decimals for understanding
41
Blue = free hyperlink, black = book, no hyperlink
5.6
Red = MAV product hyperlink
Equal fractions (clocks)
Teach for understanding
This is the fifth of seven activities on equal fractions. Confidence with this topic is very useful, both for further work
with fractions, such as adding and subtracting, and with real life situations, where equal fractions may be used to
solve ratio problems.
As well as providing further reinforcement for the basic idea of equal fractions, this activity offers practice in
fractions of 60, and in many fractions of an hour.
Suggested activities
•
Use the clocks and find equal fractions by finding minutes for many fractions. Use the spreadsheet.
•
Here is one example. 40 minutes is 3 of 60 and also 6 of 60. It is also equal to 12 of 60.
2
4
8
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Spreadsheets from the Interactive Learning CD Clock fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
5.7
42
Red = MAV product hyperlink
Equal fractions (graphs)
Teach for understanding
This is the last of the activities on equal fractions. Students who get this far are expected to be ready for an
extended challenge.
The spreadsheet introduces the idea of using graph coordinates to represent a fraction.
The numerator is the distance up (‘rise’)
The denominator is the distance right (‘run’)
numerator
rise
Then the slope of the line is the size of the fraction: denominator , sometimes called run .
Finding equal fractions then becomes a matter of looking for points that lie along a line from the bottom left
corner.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Spreadsheets from the Interactive Learning CD Graph fractions
•
Active Learning (Number and Algebra) F4 Fractions on a graph
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6
43
Red = MAV product hyperlink
Fraction operations [VELS 3]
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum:
2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
3.25 Multiples (of fractions) and fractions of fractions
•
People Count
#26 Addition or subtraction of fractions,
#27 Multiplication of fractions
•
Fractions: pikelets and lamingtons
•
Guidelines in Number: Fractions: p37-39, 59-61, p83-84
•
RIME Number
•
RIME 5&6: Newspaper fractions
•
Active Learning (Number & algebra):
•
Mathematics Task Centre: 201 Rectangle Fractions, 202 Rod Mats, 203 Make The Whole
•
Spreadsheets from the Interactive Learning CD
N27 Adding and subtracting fractions,
N28 Multiplying fractions
F1 Fraction circles, F2 Fractions of things, F3 Fraction rally: a fraction dice game
Add circle fractions, Add strip fractions, Add fractions, Subtract fractions,
Fractions of (circles), Fractions of (strips), Fractions of (rectangle)
•
Learning Objects (FUSE):
•
Free software:
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
http://nlvm.usu.edu/en/nav/ Fractions – adding, – comparing, – rectangle multiplication
http://illuminations.nctm.org/Activities.aspx Equivalent fractions
standards.nctm.org/document/eexamples/index.htm
Communicating about Mathematics Using Games: Playing Fraction Track
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.1
44
Red = MAV product hyperlink
Adding and subtracting (pizzas)
Teach for understanding
This activity provides a gentle introduction to adding and subtracting, by way of stories about children eating
pizzas. The adding comes from finding ‘how much pizza is eaten’. The subtracting comes from finding ‘how much
more one person’s share is than another’s’ and ‘how much is left from the whole pizzas’. At this stage it is strongly
recommended that children do all problems using concrete materials, or by colouring diagrams.
Do not press for short cuts, or mention common denominators, or make other moves towards a final algorithm.
Great damage can be done by pushing too early for these pen and paper methods when a bit of common sense
and thinking will help children understand what is going on.
Many fraction problems can be grasped and solved mentally if a good mental image is used. Concentrate on
building this image of what it all means. In this activity we use circles.
You will need several sets of the plastic circles.
Suggested activities
1
1
Make up a story about two of them eating pizzas. One eats 4 , one eats 2 .
•
Make the fractions with pizza pieces in quarters. There are three questions:
Who ate the most?
How much more did this person eat?
How much did they eat in total?
1
1
1
They only bought one pizza. Note that after taking away the 4 , the 2 means half of the whole pizza, not 2
of what was left! How much was left at the end?
3
1
Two other children were given three pizzas to share. One child ate 4 , one ate 12 .
•
Make the fractions with pizza pieces in quarters.
Who ate the most? How much more did this person eat? How much did they eat in total?
How much was left from the three pizzas?
Show two other fractions. Ask the children to make up stories, and answer the same three questions.
(These problems will need circles in sixths. Equal fractions (halves or thirds to sixths) may be required.
These should occur easily at this stage after the extensive work on this concept.)
•
2
3
5
and 6 from two pizzas.
1
5
13 and 6 from three pizzas.
2
3
1
and 16 from two pizzas.
Let the children try the activity when they are ready to succeed.
•
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Spreadsheets from the Interactive Learning CD: Add circle fractions
•
Active Learning 2 (Number & Algebra) N5 Music and fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.2
45
Red = MAV product hyperlink
Adding and subtracting (strips)
Teach for understanding
Students will need more than one way of thinking about it if they are to create their own mental abstraction of the
process. For this work, students should have their own set of fraction strips. There are important ideas here.
Please make sure they understand what is going on through the activity.
Suggested activities
First, use common denominators. For example, using tenths, try these additions and subtractions.
•
3
10
1
7
10
+ 10
1
3
10
+ 10
1
– 10
7
10
1
– 10
For adding, count along the strip, for example 3 tenths, and then 1 more.
The answer is 4 tenths.
Now we look for another fraction of the same length; 4 tenths is the same as 3 fifths. (Place them side-byside to check.)
For subtracting, count along the strip for the first number then count backwards.
1
1
Try 2 + 4 . Discuss the fact that the first fraction is about a strip divided into 2 parts, but the second fraction
is about a strip divided into 4 parts. What can we do?
•
1
2
We change 2 into 4 . These are equal fractions. (Put the strips side-by-side to show this.)
1
1
2
1
3
Then 2 + 4 becomes 4 + 4 , which is 4 .
5
1
Try 9 – 3 . Discuss the fact that the first fraction is about a strip divided into 9 parts, but the second fraction
is about a strip divided into 3 parts. What can we do?
•
1
3
We change 3 into 9 . These are equal fractions. (Put the strips side-by-side to show this.)
5
1
5
3
2
Then 9 – 3 becomes 9 – 9 , which is 9 .
Provide more examples if needed.
•
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with fraction strips
•
Spreadsheets from the Interactive Learning CD: Add strip fractions.
•
Active Learning (Number & Algebra) F3 Fraction rally
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.3
46
Red = MAV product hyperlink
Adding and subtracting (squares)
Teach for understanding
Students will need more than one way of thinking about it if they are to create their own mental abstraction of the
process. For this work, students should have their own set of plastic squares.
There are important ideas here. Make sure they understand what is going on before you let them try the activity.
At this stage we go directly to the problems with mixed denominators. You should use several examples to make
sure children have the ideas.
Suggested activities
•
1
1
Try to add 2 and 3 by putting them next to one another.
The problem is not in the adding, but in finding the proper name for the fractions they produce. The picture
1
3
1
2
shows that you can swap 2 for 6 and 3 for 6 . This makes naming the total easy.
•
5
3
5
3
Try to subtract 12 from 4 by putting the 12 on top of 4 and trying to name the rest.
The problem is not in the subtracting, but in finding the proper name for the fraction left. The picture shows
3
9
that you can swap 4 for 12 . This means we are taking 5 twelfths from 9 twelfths, and makes the fraction left
4 twelfths.
4
1
This time we can see that 12 is the same size as 3 .
•
At this stage we are focusing entirely on the understanding. The ideas about common denominators and
equal fractions are what is essential. Please do not attempt to teach a routine method for solving this kind of
problem to any child until you are absolutely sure that the children will clearly understand the reasons
behind each step.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with rectangles
•
Spreadsheets from the Interactive Learning CD: Adding fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.4
47
Red = MAV product hyperlink
Adding and subtracting (triangles)
Teach for understanding
This is the fourth activity on adding and subtracting. Students will need more than one way of thinking about it if
they are to create their own mental abstraction of the process.
For this work, students should have their own set of plastic triangles.
There are important ideas here. Make sure they understand what is going on before you let them try the activity.
Suggested activities
•
The first two questions use common denominators. Try a few of these with triangles that you make up at the
time. For example, using twelfths, try these additions and subtractions.
5
12
+ 12
1
7
12
+ 12
1
5
12
– 12
1
7
12
– 12
1
1 2 1
1
Show how each of these answers can be replaced by a simpler fraction: 2 , 3 , 3 and 2 .
•
1
1
Try 2 + 4 . What can we do?
1
2
We change 2 into 4 . These are equal fractions. (Two quarter-triangles fit on top of the half-triangle.) Then
1
2
•
1
2
1
3
+ 4 becomes 4 + 4 , which is 4 .
1
1
Try 2 – 3 . What can we do?
1
1
3
2
We change 2 and 3 into 6 and 6 . These are equal fractions. (Show these.)
1
1
3
2
1
Then 2 – 3 becomes 6 – 6 , which is 6 .
•
Provide more examples if needed.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction triangles
•
Spreadsheets from the Interactive Learning CD: Subtracting fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.5
48
Red = MAV product hyperlink
Multiplying fractions (pizzas)
Teach for understanding
Although the activity is called ‘Multiplying’ we are concentrating on the meaning of ‘x’, which is ‘of’.
1
2
• The meaning of 2 of 3 is simple.
2
1
We make the 3 first, then find 2 of it. In this case the fraction was easy to find, without using equal
1
fractions. It is 3 .
2
1
• The meaning of 3 of 2 is also simple.
1
2
We make the 2 first, then find 3 of it. This time we need to convert to sixths (equal fractions) to name the
1
answer. It is also 3 .
1
2
2
1
• These two examples also show another idea: 2 of 3 is the same as 3 of 2 .
The questions give the same answer in either order.
The students you are teaching should have their own plastic circles to help them work out the answers.
Remember to focus entirely on the main idea, which is in three parts.
• Make the second fraction.
• Find the first fraction of it.
• Name the answer.
You should start with some very simple examples that focus clearly on the idea of ‘of’.
Suggested activities
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
1
2
1
•
Show 2 . Find a piece that is half of it. (It is 4 .) This shows that 2 of 2 is 4 . This is how it is written.
•
Show 4 . Find a piece that is half of it. (It is 8 .) This shows that 2 of 4 is 8 .
•
Show 3 . Find a piece that is half of it. (It is 6 .) This shows that 2 of 3 is 6 .
•
Show 3 . Find a piece that is half of it. (It is 3 .) This shows that 2 of 3 is 3 .
•
Show 2 . Find three-quarters of it. Change the 2 for 8 . Then 4 of 2 is 8 .
•
Show 2 . Find two-thirds of it. Change the 2 for 6 . Then 3 of 2 is 6 or 3 .
•
Students try more problems of this type, using their plastic circles. Make sure they can be done using the
circles.
1
1
1
1
4
3
2
3
1
3
1
2
1
The idea that the answer is the same when the fractions are reversed might be left as a discovery!
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions using pizzas
•
Spreadsheets from the Interactive Learning CD: Fractions of (circles)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.6
49
Red = MAV product hyperlink
Multiplying fractions (strips)
Teach for understanding
This is the second activity on the meaning of ‘x’, which is ‘of’.
1
2
2
1
• The meaning of 2 of 3 is simple. We make the 3 first, then find 2 of it.
whole strip
1
In this case the fraction was found without using equal fractions. It is 3 .
2
1
• The meaning of 3 of 2 is also simple.
whole strip
1
2
We make the 2 first, then find 3 of it. This time we need to convert to sixths (equal fractions) to name the
1
answer. It is also 3 .
1
2
2
1
Note that 2 of 3 is the same as 3 of 2 . The fractions give the same answer in either order.
The students you are teaching should have their own plastic strips to help them work out the answers. Remember
to focus entirely on the main idea, which is in three parts.
• Make the second fraction.
• Find the first fraction of it. • Name the answer.
You should start with some very simple examples that focus clearly on the idea of ‘of’.
Suggested activities
1
1
1
1
1
1
•
Show 2 . Use the quarters-strip to find half of it. So 2 of 2 is 4 . This is how it is written.
•
Show 4 . Use the eighths-strip to find half of it. (It is 8 .) This shows that 2 of 4 is 8 .
•
Show 3 . Use the thirds-strip to find half of it. (It is 3 .) This shows that 2 of 3 is 3 .
•
Show 2 . Find a piece that is three-quarters of it. Change the 2 for 8 . Then 4 of 2 is 8 .
•
Show 2 . Find a piece that is two-thirds of it. Change the 2 for 6 . Then 3 of 2 is 6 .
2
1
1
1
1
1
1
1
4
3
2
1
1
2
1
3
1
1
2
3
1
(This can also be simplified to 3 , but this is not essential at this stage.)
The idea that the answer is the same when the fractions are reversed might be left as a discovery!
•
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with fraction strips
•
MApps: Maths investigations in the real world Guitar
•
Spreadsheets from the Interactive Learning CD: Fractions of (strips)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
6.7
50
Red = MAV product hyperlink
Multiplying fractions (squares)
Teach for understanding
This is the third activity on the meaning of ‘x’, which is ‘of’.
1
2
2
1
• The meaning of 2 of 3 is simple. We make the 3 first, then find 2 of it.
1
In this case the fraction was found without using equal fractions. It is 3 .
2
1
1
2
• The meaning of 3 of 2 is also simple. We make the 2 first, then find 3 of it.
1
This time we need to convert to sixths (equal fractions) to name the answer. It is also 3 .
1
2
2
1
• Note that 2 of 3 is the same as 3 of 2 . The fractions give the same answer in either order.
The students you are teaching should have their own plastic squares to help them work out the answers.
Remember to focus entirely on the main idea, which is in three parts.
• Make the second fraction.
• Find the first fraction of it.
• Name the answer.
You should start with some very simple examples that focus clearly on the idea of ‘of’.
Suggested activities
1
Show 2 on a square. Show half of it. What fraction of the square is this?
•
1
1
1
(It is 2 of 2 is 4 . This is how it is written.)
1
Show 4 on a square. Show half of it. What fraction of the square is this?
•
1
1
1
1
(It is 8 .) This shows that 2 of 4 is 8 .
2
Show 3 on a square. Show half of it. What fraction of the square is this?
•
1
1
2
1
(It is 3 .) This shows that 2 of 3 is 3 .
1
1
4
3
1
3
2
3
•
Show 2 . Find three-quarters of it. Change the 2 for 8 . Then 4 of 2 is 8 .
•
Show 2 . Find two-thirds of it. Change the 2 for 6 . Then 3 of 6 is 6 .
1
1
3
2
1
(This can also be simplified to 3 , but this is not essential at this stage.)
The idea that the answer is the same when the fractions are reversed might be left as a discovery!
•
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions with rectangles
•
Spreadsheets from the Interactive Learning CD: Fractions of (rectangles)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
51
Red = MAV product hyperlink
7 Fractions and division [VELS 4]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 3.5 Fraction as a number
•
People Count #25 Fractions, division and remainders
•
Fractions: pikelets and lamingtons
•
Guidelines in Number: Fractions: p37-39, 59-61, p83-84
•
RIME: Number N26 Card thickness
•
Mathematics Task Centre: 201 Rectangle Fractions, 202 Rod Mats
•
Spreadsheets from the Interactive Learning CD
Dividing strips, Sharing pizzas, Dividing money, Steps, Remainders, Dividing with remainders
MORE >
Teach Fractions and decimals for understanding
52
Blue = free hyperlink, black = book, no hyperlink
7.1
Red = MAV product hyperlink
Fractions and dividing
Teach for understanding
This is the first activity aimed at developing an understanding of fractions as the result of dividing. This uses the
linear model.
Normally we relate fractions to parts of one whole thing. However, fractions also arise from dividing a number of
things (the numerator) by another number (the denominator). Below are examples.
You need many strips of paper and some sticky tape. Cut A4 sheets into strips 2 cm wide. This gives 10 strips per
sheet. Go through this carefully with the students before they tackle the sheet.
Suggested activities
•
Lay out three strips in a row. Tape them together.
1
1
Fold to divide into 2 parts. 3 ÷ 2 = 12 . The crease is 12 strip lengths from the end.
strip 1
•
strip 2
crease
strip 3
Now fold the same three strips to divide into four parts.
3
3
So 3 ÷ 4 = 4 ; the creases are 4 of a strip length apart.
strip 1
strip 2
crease
crease
•
strip 3
crease
Now fold the same three strips to divide into six parts.
3
1
1
So 3 ÷ 6 = 6 = 2 ; the creases are 2 of a strip length apart.
strip 1
crease
•
strip 2
crease
crease
strip 3
crease
crease
Now fold the same three strips to divide into five parts. (This may require a bit of trial and error.)
3
3
So 3 ÷ 5 = 5 ; the creases are 5 of a strip length apart.
strip 1
strip 2
crease
strip 3
crease
crease
crease
For this last one is not so easy to guess the fraction. It can be checked by folding strip 1 into five equal
parts.
2
1
1
•
Use 2 metres of string. Fold it into quarters. Each part is 4 = 2 of a metre (50 cm.) So 2 ÷ 4 = 2 .
•
Use 2 metres of string. Fold it into sixths. Each part is 6 = 3 of a metre (333 cm.) So 2 ÷ 6 = 3 .
•
Use 2 metres of string. Fold it into fifths. Each part is 5 of a metre (40 cm.) So 2 ÷ 5 = 5 .
2
1
2
1
1
2
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Dividing strips
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
7.2
53
Red = MAV product hyperlink
Sharing circles evenly
Teach for understanding
This aims to develop an understanding of fractions as the result of dividing using the circle model.
Normally we relate fractions to parts of one whole thing. However, fractions also arise from dividing a number of
things (the numerator) by another number (the denominator). Below are examples.
Each student needs one set of circles: in thirds, quarters, sixths, eighths and twelfths. Work together in sets, since
you will need several similar circles.
Suggested activities
Lay out three circles in quarters. “Here are three circles. They are to be shared by two people. How much
•
1
does each get?” (Each gets 6 quarters, which is 12 .)
A A
A
B
B B
A
A
B
C
C D
A A
A
B
B B
A B
B
C
D D
3
•
The three circles are to be shared by four people. How much does each get?” (Each gets 4 .)
•
The three circles are to be shared by six people. How much does each get?” (Each gets 6 = 2 .)
3
A A
C
C
E
E
B B
D
D
F
F
C
C
A A
B B
A
B
C
F
D D
E E
D
E
F
H
F G
HH
1
G
G
•
The three circles are to be shared by eight people. How much does each get?”
(Now we need circles cut in eighths; see above right. Each gets Error!.)
•
Note that we needed 24 part circles in order to share three circles among 8 people. How can you tell how
many parts you will need? They may need to work together to have enough circles.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Understanding fractions uaing pizzas
•
Spreadsheets from the Interactive Learning CD: Sharing pizzas
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
7.3
54
Red = MAV product hyperlink
Sharing money evenly
Teach for understanding
We all need to be able to handle money. It is a good thing to use to practise with fractions. Sharing whole
numbers of dollars evenly is a good way to get fractions, and the answers will be expressed as decimals also,
reinforcing decimal and fraction equivalences.
If at all possible this should be done with real coins, at least for the demonstration and discussion with a small
group.
Suggested activities
•
“$4 is shared between 5 people.” Here it is split into 20 cent coins. There are 20 coins.
Each of the five people gets 4 coins. So each gets 80 cents = $ = $0.80.
•
“$3 is shared between 4 people.” Here it is split into 20 cents, then into 20, 10 and 5 cent coins.
•
Each of the 4 people gets 75 cents = $ = $0.75.
Make sure they can write the answer as both a fraction and as a decimal (that is, as dollars).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Dividing money
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
7.4
55
Red = MAV product hyperlink
Steps
Teach for understanding
A great emphasis placed on the idea that a fraction is found by dividing a 1 by the denominator then multiplying
2
by the numerator. Therefore it surprises many students that 2 ÷ 3 is the same as 3 .
This uses the steps model. Make sure students count the steps, not the footprints.
Draw these diagrams, preferably on large paper, and discuss them with the children.
Suggested activities
•
When you walk along a concrete footpath you will take a number of steps to cover a number of concrete
slabs. Three examples are drawn below.
•
A
B
4
2
6
5
3
1
C
1
4
2
5
3
8
6
7
A shows 1 slab covered in 2 steps, therefore 2 in 4 steps and 3 in 6 steps.
1
1
Thus 3 ÷ 6 = 2 ÷ 4 = 1 ÷ 2 = 2 , Each step is 2 a slab.
2
2
3
3
B shows 2 slabs covered in 3 steps, hence 4 in 6 steps. Each step is 3 of a slab. Thus 2 ÷ 3 = 3 .
C shows 3 slabs covered in 4 steps, hence 6 in 8 steps. Each step is 4 of a slab. Thus 3 ÷ 4 = 4 .
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Steps
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
7.5
56
Red = MAV product hyperlink
Remainders as fractions (counters)
Teach for understanding
When we divide any remainder can be treated in at least three ways, depending on the problem.
For example, this problem needs the remainder to be rounded upwards.
A: “Cars will take 4 children each. There are 11 children; how many cars will we need?” (3 cars)
This problem needs a whole number remainder.
3
B: “A ribbon is 11 cm long. We want to cut 4 cm pieces from it. How much is left at the end?” (3 cm of 4 of a
ribbon.)
This problem needs a fraction, either as a common fraction or a decimal.
3
C: “We have $11 to pay for fabric at $4 for each metre. How much can we buy?” (2.75 m, or 2 4 m)
Here is the last problem in pictures. The circles show $1 coins.
When we divide and there is a remainder it is the fraction part of a mixed number.
11 ÷ 4 means
‘How many 4s in 11?’
There are two groups of 4 and 3 over.
The 3 is 3 of a group of 4.
4
So the answer is 2 3
4
Note: There are two different meanings for division.
“How many 4s in 11?” is a different question from “Share $11 in four equal parts.”
It is good for children to see both ways to think of a division.
Suggested activities
•
Use a set of counters. Make a group of 7. “How many groups of 2 can you make from the 7?”
The group size is 2. The remainder is 1; it is half of the group size, so the answer is 3.
•
Here is the same problem expressed as a ‘sharing’ problem. “Share $7 equally between 2 people.” The
answer is clearly $3.50, or $3.
•
Use the problem above, in the teacher’s explanation. $11 ÷ $4 = 2 people, preferably with real money.
There is $3 left over.
•
Here is the same problem expressed as a ‘sharing’ problem. “Share $11 equally between 4 people.” The
3
answer is clearly $2.75, or $24 .
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Remainders, Division with remainders
Teach Fractions and decimals for understanding
57
Blue = free hyperlink, black = book, no hyperlink
7.6
Red = MAV product hyperlink
Dividing by fractions
Teach for understanding
The rule ‘invert and multiply’ is usually based on very little understanding. There is another way to tackle dividing
by fractions that makes much more sense. It uses intuitive ideas of ratio.
Since we are dividing one quantity by another, we cannot use the ‘sharing’ idea of division. Instead we can ask
‘how many times does the second fraction fit into the first?’. There is a graded set of problems we can use.
a
Problems that can be easily solved by drawing or visualising with models. At this stage, the answer is always
a whole number.
1
1
For example 1 ÷ 4 = 4. How many 4 ‘s in 1?
b
2
2
or 2 ÷ 3 = 3. How many 3 ‘s in 2?
Problems where the answer is a mixed number. Again drawing (or models) is the way to develop
understanding. In a ÷ b, the second number (b) has to be thought of as 1. Because the answer is not a whole
number the issue is how to deal with the ‘remainder’. It has to be expressed as a fraction of the second
number (b).
3
1
1
1
2
3
For example 4 ÷ 2 = 12
or 2 ÷ 4 = 23
3
3
How many 2 ‘s fit into 4 ?
How many 4 ‘s fit into 2?
1
The one quarter over is half of the 2 .
2
3
The two parts of the ‘remainder’ makes 3 of the 4 .
If both numbers are expressed with the same denominator it becomes clear that the answer is the ratio of
the two numerators.
3
1
becomes
3
4
÷4
which is
1
3
or 1
2
2
3
For example 4 ÷ 2
c
or 2 ÷ 4 .
8
2
3
becomes 4 ÷ 4
2
8
which is 3 or 23
Problems where the answer is a fractional number. Again drawing (or models) is the way to develop
understanding. Because the second number is larger than the first, the answer is a fraction.
1
3
2
or 3 ÷ 4 = 9
3
1
“What fraction of 4 is 3 ?”
For example 2 ÷ 4 = 3
“What fraction of 4 is 2 ?”
2
3
8
3
2
If both numbers are expressed with the same denominator the answer is the ratio of the two numerators, as
before.
1
3
2
3
For example 2 ÷ 4
becomes 4 ÷ 4
which is
2
3
2
3
8
9
or 3 ÷ 4
becomes 12 ÷ 12
8
which is 9
Suggested activities
•
Let each group of students use a set of 12 counters. Pose the problems above and discuss the problems
with the students. Lead them to see that the division is comparing the size of two groups (using ratio), and
this can be done by comparing their numerators.
•
Let each group of students use a set of fraction strips and pose and discuss above.
•
Let them use whatever other shape they wish to try the same problems, and come to the same conclusions.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Spreadsheets from the Interactive Learning CD: Dividing by fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
58
Red = MAV product hyperlink
8 Decimals – place value [VELS 3]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum: 2.5a Early fraction ideas with models, 2.5b Early fraction ideas with models
•
Fraction and Decimal classroom activities p46, p48, p49
•
People Count #13 Addition and subtraction with decimals
•
Guidelines in Number: Decimals: p83, 95, 108-113,
•
Mathematics Task Centre: 75 What's It Worth?
•
Spreadsheets from the Interactive Learning CD
•
Learning Objects (FUSE): Wishball (XR7DMG), Decimaster (2SRAVC)
•
Free software:
Skip count decimals, Money by 10 or 100, 10 or 100 times what? Reading a scale,
http://nlvm.usu.edu/en/nav/vlibrary.html Base blocks Decimals
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
8.1
59
Red = MAV product hyperlink
Decimals include tenths
Teach for understanding
The problems that many students have with decimals arise from lack of understanding, not lack of skills. The
basic idea is place value, but now the places are extended to the right and the values divided by ten each time.
It is convenient that we have familiar examples of decimals: money and metric units of length. Using these
applications, and renaming of the base 10 materials, we can make sure students grasp the main ideas.
Suggested activities
•
Work with ten-cent coins, and write as dollar amounts; ten of them make $1.00.
•
Note that added zeros do not change the value: $0.10 is the same as $0.1.
•
Count by tenths: (0.1, 0.2, 0.3 ...), by 0.2, by 0.3 etc.
•
Name the places on a number line. For example:
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
•
Find the decimals above on a metric ruler or tape.
•
Measure a length in millimetres, and write it in centimetres. (Example: 35 mm = 3.5 cm)
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Some ideas about teaching decimals
•
Spreadsheets from the Interactive Learning CD: Skip count decimals
1.5
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
8.2
60
Red = MAV product hyperlink
Decimals include hundredths
Teach for understanding
The most common practical application of decimals with two places is money. However many children will also
know metric length, using centimetres. It is wise to make frequent links to these practical applications.
Suggested activities
0
•
Work out the numbers halfway between the tenths. Use money (halfway from 60 c to 70 c), a number line
and a tape. For example halfway from 60 c to 70 c is 65 c.
•
Mark the ‘5-cent’ positions on the number line. Name positions as above.
0.05
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45 1.55
•
Find the centimetre positions on a metric ruler or tape. Name them.
•
Consider order. Which is the larger number: 0.4 or 0.15?
•
Compare how much money they represent and their positions on a number line or tape.
•
Recognise that the number of tenths is more important than the number of hundredths, just as with whole
numbers tens are more important than ones. The left-most place has more value.
•
Use the number chart. Each row is one tenth of the chart, so 0.23 means two rows and 3 more numbers.
•
Measure a length in centimetres, and write it in metres. (Example: 35 cm = 0.35 m, 153 cm = 1.53 m)
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Some ideas about teaching decimals
•
Spreadsheets from the Interactive Learning CD: Skip count decimals
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
8.3
61
Red = MAV product hyperlink
Decimals include thousandths
Teach for understanding
The most common application of decimals to three places is metric length (e.g.1.234 m, or 1.234 km), and many
children will have experienced other metric measures in this way: mass (1.234 kg) or capacity (1.234 L). It is wise
to make frequent links to these practical applications.
Suggested activities
•
0
We have the model of millimetres for this. Students refer to a ruler to help complete gaps in number lines.
0.01 0.02
0.03 0.04
0.05 0.06 0.07
0.08 0.09
0.1
0.11 0.12
0.13 0.14
0.15
0.005 0.015 0.025 0.035 0.045 0.055 0.065 0.075 0.085 0.095 0.105 0.115 0.125 0.135 0.145
•
Use Base 10 blocks (if you have them) and name the biggest block as the 1.
Then a flat shows 0.1, a long shows 0.01 and a mini shows 0.001.
Make models of different decimal numbers, such as 0.105, 0.025 and 0.3 and decide which has the most
wood (and shows the largest number).
•
Measure a length in millimetres, and write it in metres. (435 mm = 0.435 m, 1504 mm = 1.504 m)
•
Measure a capacity (eg. water) in millilitres, and write it in litres.
(435 mL = 0.435 L, 1504 mL = 1.504 L)
•
Use flashcards for metric conversion and for scale readings.
Resources for learning
–––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Some ideas about teaching decimals
•
Fraction and Decimal classroom activities p48, p49
•
Spreadsheets from the Interactive Learning CD:
10 or 100 times what? Estimating decimals, Ordering decimals , Reading a scale,
Skip count decimals
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
62
Red = MAV product hyperlink
9 Fractions and percentages [VELS 4]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
People Count
#14 Percentages, #21 Percentages and fractions
•
Guidelines in Number: Fractions: p183-184
•
Active Learning (Number & Algebra) F13 Estimating percentages
•
RIME 5&6: This goes with this
•
Spreadsheets from the Interactive Learning CD
Percentage square, Percentage pie, Percentage line, Percentage fraction, Percentage full,
Percentage rectangle
•
Mathematics Task Centre
•
Free software:
201 Rectangle fractions, 202 Rod mats, 203 Make the whole, 211 Soft drink crates,
218 Guessing colours game
http://nlvm.usu.edu/en/nav/ Percentages
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
9.1
63
Red = MAV product hyperlink
Percentage squares
Teach for understanding
This is the first of a number of activities designed to introduce the concept of percentage – “out of 100”.
Suggested activities
•
A square made from small squares (10 by 10) has some of the small squares coloured. Students estimate
the percentage of the large square that is coloured. They are then count to check their guesses.
•
The activity is extended with activities inviting some creativity – colouring in a pattern but colouring the right
number of small squares.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Active Learning (Number & Algebra) F13 Estimating percentages
•
Spreadsheets from the Interactive Learning CD: Percentage square, Percentage pie
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
9.2
64
Red = MAV product hyperlink
Percentages and fractions
Teach for understanding
This activity involves reading the percentage and fraction scales. However there are a number of good estimation
activities which should be done first.
You will need a metre ruler, which is blank on one side, with centimetres marked on the other side.
Suggested activities
Put a chalk mark or otherwise indicate a spot on the blank side of the ruler. (A rubber band around the ruler
is useful as it also shows the position on the centimetre scale.)
•
Ask the students to guess the percentage from the zero end.
Then turn the ruler over and show them the answer.
40 cm = 40%
Students mark where they think a given percentage of the total width of the blackboard will be.
•
For example, guess where 20% from the left end of the board will be.
They each put a mark on the board with their initials.
They measure the width of the blackboard (e.g. 452 cm).
They then turn the percentage into a decimal. (For example, 20% = 0.2)
They calculate the distance by multiplying the decimal by the width of the board on a calculator. (For
example, 0.2 x 452 cm = 90.4 cm.) If this is too difficult, do it for them.
They measure the correct distance from the left of the board and compare their answers with it.
Measure correct distance
Measure width
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Active Learning (Number & Algebra) F13 Estimating percentages
•
•
RIME N21 Estimation with fractions
Spreadsheets from the Interactive Learning CD: Percentage fraction
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
9.3
65
Red = MAV product hyperlink
Bounce fractions and percentages
metric ruler or tape
Teach for understanding
This tries to answer this question: Do balls always bounce up to the
same percentage of their previous height?
Use a real experiment. You need a ball, preferably a ‘super ball’.
You also need a metre ruler or sewing tape fixed to a wall.
drop height
bounce height
Suggested activities
•
Students drop the ball from exactly 1 metre. The bottom of the ball should be level with the 1.00 m. They get
their eyes level with the height to which the ball rises. In this way they estimate the height to which the
bottom rises on the first bounce.
•
They work out the percentage (bounce height ÷ drop height x 100).
•
Repeat 1 and 2, dropping the ball from 90 cm. Find the percentage.
•
Repeat from several other heights. The percentages should be similar.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Active Learning (Number & Algebra) F13 Estimating percentages
•
Spreadsheets from the Interactive Learning CD: Percentage full
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
66
Red = MAV product hyperlink
10 Fractions, decimals and percentages [VELS 4]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Fraction and Decimal classroom activities Convert & compare: fractions to decimals p34
•
People Count #21 Percentages and fractions, #24 Fractions and decimals
•
Guidelines in Number: Fractions: p83-84, 118
•
Active Learning (Number & Algebra) F13 Estimating percentages
•
RIME: Number N26 Card thickness
•
Maths300: 11 This goes with this
•
Active Learning (Number & Algebra): F6 Reciprocals and dividing
•
Spreadsheets from the Interactive Learning CD
•
Learning Objects (FUSE):
Design a school (4HJMYK) Fraction fiddle (U73M4Q), Cassowary fractions (ABX3BL)
Estimating decimals, Fractions of dollars, Guitar fractions, Decimals to fractions, Fraction to decimal,
Ten clocks, Repeating decimals
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
67
Red = MAV product hyperlink
10.1 Fractions and decimals (wall)
Teach for understanding
This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of
the Whole numbers activities, which stresses their place value properties.)
Students will come to see that decimals are equal to other fractions, so that they are just another way to write
them. Decimals are very convenient for some things, and very awkward for others.
The attached strip ‘wall’ includes a ‘ruler’ marked in decimals and percentages. Students can use a ruler vertically
to measure the length of a particular fraction as a decimal; and as a percentage. They will also discover that the
percentages are just the same as the first two places of the decimals.
0.0 0.05
0.1
0.15
0.2
0.25
0.3
0.4
0.35
1
-----20
2
----10
2
----20
4
----20
3
-----20
0.5
0.6
0.55
3
-----10
5
-----20
4
----10
5
-----10
8
-----20
7
-----20
6
----20
0.65
0.7
0.75
9
----20
10
-----20
6
----10
7
-----10
14
-----20
13
----20
12
-----20
11
----20
0.8
0.85
0.9
0.95 1.00
4
-5
3
-5
2
--5
1
-5
1
-----10
0.45
8
----10
15
-----20
9
-----10
18
----20
17
-----20
16
----20
19
-----20
1
--2
2
-----12
1
-----12
3
----12
5
----12
4
-----12
8
----12
7
-----12
6
----12
1
-3
1
--8
2
--8
9
-----12
11
----12
10
----12
5
-6
4
-6
3
--6
2
--6
1
--6
3
--4
2
-4
1
-4
2
--3
3
--8
1
5
--8
4
--8
5
6
-8
7
-8
3
For example, the line above shows that 4 is the same as 20 , 12 etc. and the same as 0.25 and also 25%.
The same result can be obtained using a calculator. You just divide the numerator by the denominator, for
example 1 ÷ 4 = 0.25, and 5 ÷ 20 = 0.25 also. This becomes a very quick and useful way to decide whether two
fractions are equal, or which one is the greater.
Suggested activities
•
Show students how to find some fractions that are equal, using a ruler or other straight edge.
•
Explore the tenths, as multiples of 0.1 and as common fractions. This should stress the equivalence.
•
After finding equal fractions on the ‘wall’, students may use their calculators to check the answers.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Fraction-decimal wall
•
Fraction and Decimal classroom activities Convert and compare: fractions to decimals p34
•
Spreadsheets from the Interactive Learning CD: Estimating decimals
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
10.2 Fractions and decimals
(ruler)
Teach for understanding
Use a metre ruler or a sewing tape, usually
150 cm long.
The strips on the right of this page can be
used. This is a rich experience
Cut out each of the four strips, and stick them
together. with the short marks at the end lined
up.
Each strip is 25 cm, so the combination will
make one metre.
The marks show:
• quarters (at the ends of the
separate strips),
• fifths (20 cm intervals),
• tenths (10 cm intervals),
• twentieths (5 cm intervals),
• 25 ths (at 4 cm intervals)
• 50 ths (at 2 cm intervals).
Suggested activities
•
Measure some distances as fractions of
a metre, and give the answers as
decimals by reading the number of
centimetres from the tape: for example,
40 cm = 0.40 m.
•
With the students count up and label the
marks as shown above.
•
Choose a number of the fractions and
convert them to decimals by measuring
their distance from 0 with the measuring
tape or ruler.
•
For example, 5 measures 80 cm, so 5 =
4
4
0.8.
•
Discuss the fact that we can put in, or
leave out the zero after the 8. (0.8 is the
same as 0.80.)
•
Let the students work through some
questions using a tape and the strip as a
reference. Remember the main aim is to
develop understanding of decimals.
However a review of the basic ideas of
fractions will certainly take place as well.
Resources for learning ––––––––––––
See above.
68
Red = MAV product hyperlink
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
69
Red = MAV product hyperlink
10.3 Fractions of a dollar
Teach for understanding
This is the third activity that uses decimals to represent fractions of something. The first one used fractions of the
‘wall’. The second used real metric lengths, using a metre as the unit.
For this one it is important to use real money. It shows we are dealing with the real world.
You need at least two 50 c coins, ten 10 c coins, five 20 c coins, and a few 5 c coins.
Suggested activities
•
Have children find fractions of a dollar, giving the answers as decimals:
for example, 40 cents = $0.40.
•
Start with 10 cents, and discuss that it is written as $0.10, but could be written as $0.1.
1
How many lots of $0.10 do you need to make $1.00? (10). This means that $0.10 is 10 of a dollar.
2
1
•
Extend this to $0.20 (both 10 and 5 ). Extend it to $0.30, and to $0.40, and to $0.50.
•
Discuss what amount you need to have four times to make $1.00, that is, a quarter of a dollar. Some
children have trouble with this because it is not a single coin. (Unless you’re American!)
3
Extend this to 4 of a dollar.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Fractions of dollars
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
70
Red = MAV product hyperlink
10.4 Guitar fractions and decimals
Teach for understanding
This is another real-world example of decimals linked to fractions. It will benefit greatly if you demonstrate with a
real guitar. Playing is not needed.
The frets on a guitar are very close to exact fractions of the total free string length.
Musical Note
E D# D C# C B A# A G# G
Bridge
F#
F
E
D#
D
C#
C
B
A#
A
String
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
The frets are shown by the vertical lines above. On a real guitar, the frets allow the player to shorten the length of
string that vibrates. There are many interesting relationships involving fractions. They were first discovered by
Pythagoras (yes, the right-angled triangle chap!) over two millennia ago. Pythagoras did not use decimals.
1
The first major discovery is that 2 of the open string length always creates a note exactly an octave higher. So on
the finger-board above the A in the middle is exactly half of the total length, from the bridge to the A on the right
end. For the same reason the distance to B in the middle is half the distance to B near the right end, and so on.
This explains why the frets get closer together as you move to the left.
The answers from a real guitar make the other fractions clear enough.
Note
Decimal
A
0.53
Fraction
1
2
G#
0.56
G
0.6
F#
0.63
5
9
3
5
F
0.67
E
0.71
2
3
D#
0.75
D
0.8
C#
0.83
C
0.89
B
0.94
3
4
4
5
5
6
8
9
A#
1.0
A
0.5
1
Suggested activities
•
Students measure in millimetres. They first measure the open string length. All other lengths will be fractions
of that.
•
Students measure the lengths from the bridge to each fret. They divide the length to each fret by the open
string length.
•
They experiment to find the simple fraction closest to the value obtained.
Note for teachers only
The more accurately we measure the less the Pythagoras’ fractions hold up. There are two reasons.
1 The string is stretched when it is played. So the fret positions must take account of this.
2 The scale used on a modern guitar uses tuning that Pythagoras never heard. It is called ‘equal temperament’.
It means, for example, that E is at a fraction of 0.6674 and not at 0.6666.The C# is not at 0.8000, but at
0.7837, and so on. We are used to having most notes slightly ‘out of tune’, so we can play in all keys.
Pythagoras worked with a ‘purer’ form of tuning, used in musical instruments (such as organs) until the
Renaissance or even later.
Note: The values quoted in the note above use an exponential formula, above our students.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Guitar fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
71
Red = MAV product hyperlink
10.5 Decimals and percentages
Teach for understanding
There are several levels of complexity in the relationship between these two forms of fractions. Of course they all
use place value.
•
The basic idea is that percentages (“out of 100”) are just hundredths. So 0.23 is 23%, 1.23 is 123%, and
0.234 is 23.4%. Note that whole numbers are always hundreds of percents (e.g. 2 = 200%).
•
The conversions also go the other way: 35% = 0.35, 135% = 1.35, 3000% = 30.
•
It gets more complex when fractions are also included in the percentage. These are not uncommon, e.g.
1
1
122 % is one eighth, and is often expressed as a percentage. This tell us that the decimal has 122
hundredths, so it must be 0.125, the 0.005 being 5 thousandths or half of 1 hundredth, since that is 10
thousandths.
Examples with repeating decimals will be covered in 10.6.
•
Suggested activities
Use a place value chart
This should clearly show the hundredths, and also the percentages.
•
NUMBER
PERCENTS
tens
ones
•
tenths
1
2
3
1000s
100s
tens
hundredths thousands
4
ones
5
•
tenths
•
Discuss the examples above.
•
Discuss how to use the % key on a simple calculator. Most of them multiply the decimal by 100.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
•
•
Paper abacus (decimals)
Spreadsheets from the Interactive Learning CD:
Fracs-dec-% on line, Fractions to percentages
Active Learning 2 (Number & Algebra)
N5 Fractions
N9 Fraction applications
N13 Converting forms of fractions
N14 Maths @ work: Using fractions
N15 Maths@work: bank on it
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
72
Red = MAV product hyperlink
10.6 Changing fractions to terminating or repeating decimals
Teach for understanding
This is a complex idea with various levels of complexity.
Some decimal numbers have their exact value after a small number of decimal places. (0.56, 0.25, 0.875 etc.)
Some decimal numbers can never be written out in full to have their exact value because there is always a little
bit more. These decimals are of two kinds: those that repeat a pattern of digits over and over (repeating or
recurring decimals), and those that never repat but keep on varying the sequence.
The easiest way to distinguish between the first two types is to look at their fraction form, in ‘simplest form’.
o If the denominator is a power of 5 or 2 (or both) the fraction will eventually terminate.
o If the denominator includes as a factor a prime number (other than 2 or 5) then it will repeat.
The third type of numbers is irrational numbers, because they cannot be expressed as a ratio (i.e. fraction).
Helping students to understand all this requires time and a set of guided explorations. The best tool is a
computer, so the spreadsheets below are recommended. A calculator frequently does not show enough digits for
a pattern to become clear.
Suggested activities
•
Fractions are changed into decimals by dividing the denominator into the numerator. For convenience it is
best to start by comparing the simple 1 and 2 digit denominators.
1 ÷ 2 is easy, but the fundamental issue appears with 1 ÷ 3. There is always 1 remainder, so the same
division (10 ÷ 3) is always going to be repeated. The decimal can never be written out in full.
Demonstrate with a short division problem.
0.33333333333…
3 ) 1.00000000000…
•
Compare 2 ÷ 3, 4 ÷ 3, 5 ÷ 3 to see the pattern.
•
Dividing by 4 and 5 are easy and terminate quickly, but 1 ÷ 6 show up a different pattern again. This time
the first non-zero digit is 1 and then 6 is repeated. Compare 0.1666… to half of one third 0.333….
•
Explore the pattern for sevenths. There is a cycle of 6 digits that appear in the same order, but with a
different starting number.
•
Eighths are easy. Explore ninths. Note that nine ninths is 0.999…, and this must be 1.
•
Once the idea is clear, there are many fascinating explorations with repeating decimals. For example, how
can you predict how long the repeating cycle will be? The pattern shown in the ‘Ten clocks’ spreadsheet
reveals hidden symmetry in these numbers.
•
An additional challenge is converting percentages such as 333 % to a decimal. The 33% shows that the first
1
1
1
two digits are 0.33, and the 3 % shows that the remaining digits continue the pattern in 3s. Similarly 86 % will
become 0.0816666…
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD:
Fraction to decimal, Ten clocks, Repeating decimals
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
73
Red = MAV product hyperlink
10.7 Changing decimals into fractions
Teach for understanding
There are many occasions when it is convenient to be able to recognise a decimal (repeating or terminating) and
convert it into a fraction. Because these situations generally come as part of something else, it becomes valuable
for a keen student to memorise as many of these as possible.
Suggested activities
•
Memorise the common terminating decimals with denominators: 2, 4, 8, 16, 5, 10, 20, 25, 40, 50, 80.
•
Memorise the simple repeating patterns with these denominators: 3, 6, 9, 11, 12, 15, 18
•
Understand and learn to perform the process to convert a repeating decimal to a fraction.
Example:
0.714285 714285 … is a repeating decimal with a cycle of 6 repeating digits. Call it F.
Then 1000 000 times F is 714285.714285 714285 … and the difference between these (subtracting) is just
714285.
This is equal to 999 999 times F. If you divide 714285 by 142857 and 999 999 by 142857 you will see that
this is the same as five sevenths!
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Spreadsheets from the Interactive Learning CD: Decimals to Fractions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
74
Red = MAV product hyperlink
11 Ratio [VELS 4, 5]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum 5.5 Ratio and proportion questions
•
Assessment for Common Misunderstandings level 5 Proportional reasoning
•
Scaffolding Numeracy in the Middle Years Zone 6 Strategy extending
•
Ratio CD:
Hands-on Ratio, Hands-on Scale
•
RIME:
Number N29 Sharing by ratio, N30 Finding scale factors N34 Garbage bag kite, N35 Spirograph
•
People Count #22 Comparing fractions and ratios
•
Active Learning (Space): S37 Photos and scale
•
Spreadsheets from the Interactive Learning CD:
Estimating ratios, Estimating reciprocals, Guitar fractions, Orange cordial, Ratio on a number line,
Right-angled triangles, Sharing by ratio, Simple interest, Travel
Free software
•
Learning Objects (FUSE): Biscuit factory (2KSQQL), Squirt (DJ78M5)
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
75
Red = MAV product hyperlink
11.1 Whole number comparisons
Teach for understanding
A ratio is a measure of how many times bigger (or smaller) one thing is than another.
It may be a single number (e.g. 2), but is sometimes expressed as a two-number expression using a colon (e.g.
2:1). In either case, the numbers may be whole numbers, fractions or decimals. Often the single number is a
percentage. So 50% means one-half, or 0.5 or 1:2.
Research and teaching experience indicate that students find whole number comparisons much easier than
others. This occurs when the answer to the question: “B is how many times A?” is a whole number.
Suggested activities
•
Coin ratios
Compare the value of common coins – from 5 c to $2. (Avoid 20c to 50c).
•
Body ratios (rounded)
Students use tapes to compare wrist circumference with neck or head, finger span with arm length etc. and
look for (rounded) whole number ratios.
•
Informal measures
Height of a wall (using the number of rows of bricks), mass of wall, words in a book (using words per line,
lines per page, pages per book)
• Simple rates
These are compared to numbers of seconds: Pulse rates, Speeds,
• Comparing units
How many millimetres in a centimetre? etc. How many inches in a foot? feet in a yard?
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Ratio CD: Hands-on Ratio p5-8
•
Spreadsheets from the Interactive Learning CD:
Estimating ratios, Orange cordial, Ratio on a number line, Sharing by ratio
Free software
•
Learning Objects (FUSE): Biscuit factory (2KSQQL), Squirt (DJ78M5)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
11.2
76
Red = MAV product hyperlink
Mixed number comparisons
Teach for understanding
This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of
the Whole numbers activities, which stresses their place value properties.)
Suggested activities
Coin ratios
Compare the values of common coins – from 5 c to $2. (Include 20c to 50c).
•
• Rates
Students measure: each others speeds for walking and running, bike riding or car speeds.
Explore usage rates for water, electricity, phone calls
Compare surface area to volume for different 3D shapes (using unit cubes) and look at biological
applications.
Simple interest is a rate (say 10%, which compares the interest paid with the principal, per year –another
rate)
• Olympic and other sporting records
Students calculate the sped of runners, swimmers, etc.
Rectangle aspect ratios
The most common use of this is on TV screens; some pictures don’t fit your screen.
The aspect ratio of A4 paper is a comparison of height to width. It is 1.414.
•
• Comparing units
Compare metric to British, using rounded values. How many:
• centimetres to the inch?
• feet to the metre?
• kilometres to the mile
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Ratio CD: Hands-on Ratio p9-13
•
Spreadsheets from the Interactive Learning CD:
Estimating ratios, Orange cordial, Ratio on a number line, Right-angled triangles, Sharing by ratio,
Free software
•
Learning Objects (FUSE): Biscuit factory (2KSQQL), Squirt (DJ78M5)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
77
Red = MAV product hyperlink
11.3 Fraction comparisons and reciprocals
Teach for understanding
This activity introduces decimals as one particular way to write a fraction. (Decimals are also introduced as part of
the Whole numbers activities, which stresses their place value properties.)
Suggested activities
•
Coin ratios
Compare the value of common coins – from 5 c to $2 by asking “What fraction is the smaller (e.g. 5 c) of the
larger (e.g. $1)?”
•
Reciprocals
Notice that most people find what fraction a shorter length is than a longer one by working out how many
times the shorter fits into the longer, and then getting the reciprocal.
For example, what fraction is the shorter than the longer line?
• Simple rates
Find your average step length (in metres).
Find the thickness of a sheet of paper by measuring many, and dividing.
Look up currency conversions, and convert both ways.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
• Ratio CD: Hands-on Ratio p14-18
•
Spreadsheets from the Interactive Learning CD:
Estimating ratios, Estimating reciprocals, Guitar fractions, Orange cordial, Ratio on a number line,
Right-angled triangles, Sharing by ratio, Simple interest, Travel
Free software
•
Learning Objects (FUSE): Biscuit factory (2KSQQL), Squirt (DJ78M5)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
78
Red = MAV product hyperlink
12 Proportion and percentage problems [VELS 5, 6]
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
•
Continuum 5.5 Ratio and proportion questions, 5.5 Solving percentage problems
•
Assessment for Common Misunderstandings level 5 Proportional reasoning
•
Scaffolding Numeracy in the Middle Years Zone 6 Strategy extending, Zone 7 Connecting
•
Ratio CD: Hands-on Proportion and percentages
•
People Count #28 Proportions and rates
•
RIME:
Number N31 Same-shape rectangles, N33 Slopes
•
Active Learning (Number & Algebra): N33 Proportion concept test
•
Active Learning 2 (Number & Algebra): N16 Who uses proportion? N17 Understanding proportions
•
MApps: Maths investigations in the real world Food energy, Plastic money
•
Spreadsheets from the Interactive Learning CD
Distance-time-speed, Find the whole, Graph fractions, Multiplying any numbers, Proportion problems,
Proportion triangles, Proportions, River width, Simple interest, Spirograph
Free software
•
Learning Objects (FUSE):
In proportion: rates and scales (EZ95P5), : ratios (23DCDD), : variables in ratios (P7NJJ9)
MORE >
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
79
Red = MAV product hyperlink
12.1 Proportions show equal ratios
Teach for understanding
Ratios compare two numbers, such as 4 compared to 2. Such a ratio is often written as a singe number (2)
showing how many times bigger one is than the other.
3
3
But there are other pairs of numbers where one is 2 times the other: 6 to 3, 15 to 7.5, to and so on. (Notice
8
4
that any kinds of numbers can be part of a ratio.)
A proportion is an equation that says that two ratios are equal. It is often written so that it looks like equivalent
3
4 10
15
4
fractions. For example, =
, or
= . Clearly the last case explains the need for the colon notation, so 15 :
2
5
7.5 3
8
3 3
7.5 = : .
4 8
The concept of equal ratios is a very useful one, with very many real-life applications, such as scales on maps
and plans, pricing, etc. Even percentages can be seen as an example of a proportion; for example the ratio 3 to 4
3
is equal to the ratio 75 to 100, so = 75%.
4
Suggested activities
•
Percentages as proportions
5
3
) into percentages under 100. Show that ratios over 1 (e.g )
4
4
become percentages of 100 (e.g. 125%) and discuss what this means.
Convert ratios less than 1 (such as
•
Aspect ratio of rectangles
The aspect ratio is height ÷ width. For a rectangle in ‘portrait’ orientation this is over 1, but in ‘landscape’
orientation it is under 1. ‘Similar’ rectangles will present the same-shape rectangle at different scales and
provide examples of proportions.
•
Step length
Assuming constant step length, you can use the length of 10 steps to find the distance for other number of
steps, such as 20, 30, 40 and 25 steps etc .
•
Speeds
Assuming constant speed, you can use the time to walk 10 m to find the time for other number of steps,
such as 20 m, 30 m, 40 m and 25 m etc. This allows you to find distances by measuring times.
•
A-series of paper
An interesting extension of the aspect ratio is to consider the aspect ratio for A4 paper. It will be close to
1.414, and it is possible to show that it must be √2.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Assessment for Common Misunderstandings level 5 Proportional reasoning
•
Scaffolding Numeracy in the Middle Years Zone 6 Strategy extending, Zone 7 Connecting
•
Ratio CD: Hands-on Proportion and percentages p40-46
•
People Count #28 Proportions and rates
•
RIME: Number
N31 Same-shape rectangles, N33 Slopes
•
Active Learning 2 (Number & Algebra): N16 Who uses proportion?
•
Spreadsheets from the Interactive Learning CD:
Distance-time-speed, Proportion triangles, Proportions
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
80
Red = MAV product hyperlink
12.2 Recognising proportional situations
Teach for understanding
It is sometimes assumed that all a teacher needs to do to teach proportion is to provide a simple algorithm, or
procedure, so that all problems may be solved. Research suggests that the matter is far more complex than this.
The first step is for the student to be able to identify when a proportion situation exists – and when it does not.
This requires some understanding of the ‘multiplicative’ character of the relationship between the variables.
Suggested activities
•
Simple multiplication tables
Provide a simple table based on multiplication only, such as this one.
n
1
p
4
2
5
9
2.5
7.5
There are very many patterns to be found in such a table.
•
Conversion tables
Provide a more complex table based on multiplication only, such as this one (rounded values).
inches
2
centimetres
5
4
6
3
9
12
There are very many patterns to be found in such a table. Note that this assumes 1 inch = 2.5 cm.
•
Using linear graphs – through the origin
A graph of either of the tables above, or any similar ones, will go through the origin. Any two pairs of
numbers on the graph will form a proportion. For example, for y = 4x, we have (8,2) and (20,5). The gradient
(4) is the ratio of any y-value to any x-value.
•
Methods of solving proportion problems
The method used by a student to solve a proportion problem can indicate the level of development of
‘multiplicative thinking’ of the student. Some students will use entirely additive methods, some will use
methods that mix adding with dividing and multiplying, and others will use purely ‘multiplicative thinking’
(dividing and multiplying). The important step is to use a problem that is more complex than whole number
comparisons (e.g. 6 to 4 = ? to 10), and to ask the student to explain their reasoning.
•
The ‘double – double’ property
One way to recognize a proportion situation is to check that whenever one variable doubles, the other also
doubles. It should apply to any change using multiplication (x3) or division (÷2).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Continuum 5.5 Ratio and proportion questions
•
Assessment for Common Misunderstandings level 5 Proportional reasoning
•
Ratio CD: Hands-on Proportion and percentages p45-49
•
Active Learning (Number & Algebra): N33 Proportion concept test
•
Spreadsheets from the Interactive Learning CD:
Distance-time-speed, Graph fractions, Proportion triangles, Proportions,
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
81
Red = MAV product hyperlink
12.3 Solving proportion problems
Teach for understanding
The ability to solve direct proportion problems is extremely useful, both in real life (prices etc) and in sciences
such as chemistry and physics.
Suggested activities
•
Get the relationships clear
Solving a proportion problem means finding a missing number. Three numbers will be known, and for one
variable you will have two values. It is useful to carefully set out the four numbers in a grid, with the known
pairing first. Here are some examples:
a 6 items for $10
b 6 items cost $10
9 items for $? (two values of item numbers)
? items cost $25 (Two values of prices)
•
Look for simple ratios to use, where you can
Many proportion problems can be solved mentally, using the simple ratios involved. For example,
- 6 items cost $10;
? items cost $30? (Just multiply by 30/10 = 3).
- 6 items cost $10;
30 items cost $? (Just multiply by 30/6 = 5).
How some ratios might not be so simple:
- 6 items cost $10;
? items cost $25? (Multiply by 25/10 = 2.5. Some people might just double and then add on half the
number you can get for $10.)
•
For more complex problems, use the unitary method
A common method is to use an intermediate step to find out the rate: how much for one. Which rate you
choose will depend on what you have to find.
In these examples the intermediate step is ‘how much money for 1 item?’
- 6 items cost $24
7 items cost $? (Each item costs $4, so 7 items will cost $4 x 7 = $28).
- 4 items cost $15
7 items cost $? (Each item costs $15/4; the rate is $3.75 per item. So you pay $3.75 x 7).
In these examples the intermediate step is ‘how many items for $1?’
- 8 items cost $5
? items cost $30? (For each $1 you get 8/5 = 1.6 items. So the answer is 1.6 x 30 items = 48.)
- 6 items cost $4
? items cost $9? (For each $1 you get 6/4 = 1.5 items. So the answer is 1.5 x 9 items = 13.5. You can buy
13.)
•
Estimate proportions
It is important to be able to estimate approximate answers, as real-life applications - such as supermarket
shopping - require quick estimates.
•
Some common applications
- At the same time of day the length of a shadow is proportional to the height of an object.
- For a given material the mass is proportional to the volume; the ratio ‘mass/volume’ is called density.
- Many problems involving similar figures, including trigonometry (using similar right-angled triangles).
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Continuum 5.5 Solving percentage problems
•
Assessment for Common Misunderstandings level 5 Proportional reasoning
•
Assessment for common misunderstandings level 5 Orange juice task
•
Ratio CD: Hands-on Proportion and percentages p40-41
•
People Count #28 Proportions and rates
•
Active Learning 2 (Number & Algebra): N17 Understanding proportions
•
Spreadsheets from the Interactive Learning CD
Find the whole, Proportion problems, Proportion triangles, River width, Spirograph
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
82
Red = MAV product hyperlink
12.4 Solving percentage problems
Teach for understanding
There are basically three different types of percentage problems.
• Finding an amount that is a percentage of a quantity.
For example, find the amount that is 25% of $60.
• Find what percentage an amount is of another quantity
For example, find the percentage that $15 is of $60.
• Given the amount and the percentage, find the original quantity (100%).
For example, given that $15 is 25% of a quantity, find that quantity.
These problems are very similar to proportion questions, but instead of the fourth number being able to change, it
is always 100. The suggestions below build on understanding, not rules like ‘multiply by 100 over 1’.
Suggested activities
•
Use diagrams to get the relationships clear
Solving a percentage problem means finding a missing number. Two numbers will be known, and 100 is
one of the four numbers you need.
Diagrams will probably help. Draw the 100% and the part of it. Put in the numbers you know.
For example, what is 25% of $60.
The $60 is the 100%, and we need to find s smaller part of it, labeled 25%.
MONEY
$??
$60
PERCENTAGES
25%
100%
Using simple proportion, you divide the 100 by 4 to get 25, so you divide the $60 by 4 to get $15.
Some people find a dual number line useful – money (or other) on one side and percentages on the other.
•
Build understanding by using simple numbers first
Here are the other two types of percentage problems formulated in this way.
For example, what percentage is $15 of $60.
The $60 is the 100%, and we need to find s smaller part of it, labeled 25%.
MONEY
$15
PERCENTAGES
??%
$60
100%
Using simple proportion, you divide the $60 by 4 to get $15, so you divide the 100% by 4 to get 25%.
For example, given that $15 is 25% of a quantity, find that quantity.
The unknown is the 100%, and we know that 25% is $15 .
MONEY
$15
PERCENTAGES
25%
$??
100%
Using simple proportion, you multiply the 25% by 4 to get 100%, so you multiply the $15 by 4 to get $60.
•
For more tricky numbers, simplify the numbers to be sure of the procedure first.
This is particularly needed for the third kind of problem, finding the value for 100%.
For example, given that $30 is 15% of a quantity, find that quantity.
MONEY
$30
$??
PERCENTAGES
15%
100%
The problem is that it is not intuitively easy to see how many times the 15% is the 100%.
If the 15% were 10%, then you divide the 10 into the 100 to get the answer.
100
. So this is the multiplier.
So this is what you do with the 15%. So 100% ÷ 15% = 100 ÷ 15 =
15
100
100
Using simple proportion, you multiply the 15% by
to get 100%, so you multiply the $30 by
15
15
to get $200.
Teach Fractions and decimals for understanding
83
Red = MAV product hyperlink
Include problems for which the ‘part’ percentage is over 100%.
Sometimes students think that percentages must always be under or equal to 100, because “You can’t get
more than 100% on a test.” But of course a percentage is just another name for a fraction, so the fraction
1.5 means the same as 150%.
Diagrams will probably help. Draw the 100% and the part of it. Put in the numbers you know.
For example, what percentage is $60 of $25.
Establish first that it is more, so the answer is more than 100%.
The $25 is the 100%, and we need to find the percentage corresponding to $60.
Blue = free hyperlink, black = book, no hyperlink
•
•
MONEY
$25
$60
PERCENTAGES
100%
??%
Using simple proportion, you multiply the 25 by 4 to get 100, so you multiply the $60 by 4 to get 240%.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Continuum 5.5 Solving percentage problems, 5.25 Adding and taking off a percentage
•
Ratio CD: Hands-on Proportion and percentages p53-55
•
People Count #29 Finding 100%
•
Active Learning (Number & Algebra):
F14 Percentages in your head, N34 Finding the whole thing
•
MApps: Maths investigations in the real world Food energy, Plastic money
•
Spreadsheets from the Interactive Learning CD
Annuity, Find 1 or 100%, Mortgage, Percentage converter, Percentages of money
Free software
•
Learning Objects (FUSE):
Design a school (T755RN), Design a school - 10 by 10 grid assessment (HYT8HB), Playground (85SW6N)
Teach Fractions and decimals for understanding
Blue = free hyperlink, black = book, no hyperlink
84
Red = MAV product hyperlink
12.5 Percentage increases and decreases
Teach for understanding
Percentage increases include price increases, profits, and increases of other kinds, including interest.
There are two types of problems: finding the percentage that corresponds to the increase, or finding the increase
that corresponds to the percentage.
There are two ways to solve such problems.
• Subtract to find the actual increase, and find the increase as a percentage of the smaller (original)
amount.
For example, a price increases from $25 to $35. What is the percentage increase?
Subtract to find the increase itself: $35 - $25 = $10.
Find the percentage that $10 is of the smaller original amount: $10 is 40% of $25.
• Find the final amount as a percentage of the smaller (original) amount, then subtract 100%.
For example, a price increases from $25 to $35. What is the percentage increase?
$35 is 140% of $25. So the increase ($10) must be 40%.
Percentage decreases include price decreases, discounts, and decreases of other kinds including depreciation.
There are two types of problems: finding the percentage that corresponds to the decrease, or finding the
decrease that corresponds to the percentage.
There are two ways to solve such problems.
• Subtract to find the actual decrease, and find the decrease as a percentage of the larger (original)
amount.
For example, a price decreases from $25 to $20. What is the percentage decrease?
Subtract to find the decrease itself: $25 - $25 = $10.
Find the percentage that $10 is of the larger original amount: $10 is 20% of $25.
• Find the final amount as a percentage of the larger (original) amount, then subtract from 100%.
For example, a price decreases from $25 to $20. What is the percentage decrease?
$20 is 80% of $25. So the decrease ($10) must be 20%.
Suggested activities
Use diagrams to get the relationships clear
Again it is really just a matter of working out what is what, and a diagram always helps.
The original amount is always 100%.
INCREASE
MONEY
$25
$10 increase
$35
PERCENTAGES
100%
?% increase
?%
Use the $25 and 100% and one other pair of numbers.
DECREASE
MONEY
$20
$10 decrease $25
PERCENTAGES
?% ??% decrease 100%
Use the $25 and 100% and one other pair of numbers.
Resources for learning –––––––––––––––––––––––––––––––––––––––––––––––––––––
See above, particularly
•
Continuum 5.25 Adding and taking off a percentage
•
Ratio CD: Hands-on Proportion and percentages p57-60
•
•
Active Learning (Number & Algebra):
N35 Percentage increases and decreases
Spreadsheets from the Interactive Learning CD
Commission, Credit card, Discount, Hire purchase, PAYG tax, Personal loan, Used car