Number Concepts/
Number and Relationship
Operations
General Curriculum Outcome A:
Students will demonstrate number sense and
apply number-theory concepts.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
i) demonstrate an understanding
of number meanings with
respect to whole numbers,
fractions, and decimals
SCO: By the end of grade 4,
students will be expected to
A1 identify and model
fractions and mixed
numbers
Many pairs of fractions can be
compared without using a formal
algorithm, such as finding a
common denominator or changing
each fraction to a decimal.
Children need informal ordering
schemes to estimate fractions quickly
or to judge the reasonableness of
answers. They can be led to
discover these relationships if they
have had experiences in constructing
mental images of fractions. (NCTM
1989 Yearbook, p.162)
4-2
A1 It is important that students develop visual images for fractions
and be able to tell about how much a particular fraction represents.
For a broader understanding of fractions, students should model
fractions and mixed numbers using a variety of materials such as
- fraction circles
- fraction squares and/or rectangles
green
- pattern blocks
red
- geoboards and grid paper
2
Students should understand that fractions can describe
- a part of a whole
- a part of a group
Invite students to determine what fraction of the letters in
their names are vowels (e.g., TARA EDAM ).
To strengthen the fraction concept, it is recommended that the
size of the whole be changed regularly. For example, show the
students the yellow pattern-block hexagon and say, "If this
represents one, what is this (blue rhombus)?" Continue by
showing the red trapezoid and asking, "If this is one, show me
.
If this (trapezoid) is one, what is the hexagon?"
Break egg cartons into sections (1 through 11), and use complete
cartons as well. Distribute at least one of the sections to each of
the students and say, “If this (whole carton) is one, what is
? If this (9 section piece) is one whole, show me one third. If
this (2 sections) is one, show me 2 ," etc. Students should
realize that any one section can have many different names
depending on the size of the whole. It is also beneficial for
students to frame these types of questions for their classmates.
In their work with fractions, students should recognize that some
models will show fractional parts that make a whole; others will show
fractional parts that make more than one whole. For example, 3 thirds
( ) makes a whole and 6 thirds ( ) makes 2 wholes. It is quite
appropriate to use these fractions. This can then be extended to
mixed numbers. For example, seven-thirds
( ), equalling 2 , can be modelled as
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A1.1 Ask the students to flip a coin 10 times and name the fraction
which describes the frequency of “tails.”
A1
A1.2 Have the students “shake and spill” a number of two-coloured
counters and ask them to name the fraction that represents the red
counters.
A1.3 Provide pattern blocks for the students. Ask them to make and
describe a design that shows 4 . Have them model and provide
another name to describe it.
Teacher’s Guide and Journal
Unit 6:
Activities 1-8 (pp. 219-250)
Culminating Work (pp. 251-254)
Indiv. Perf. Assessment (p. 256)
Paper and Pencil
A1.4 Ask the student to divide the cake below into thirds in 2 different
ways:
A1.5 Provide the student with a shape. Ask: If this shape represents a
whole, draw one that would show 2 .
Interview
A1.6 Ask the student to explain how both diagrams below show two
thirds.
A1.7 Tell the student that Martin said that the green pattern-block
triangle represented . Stephen said that Martin must be wrong because
he knew the blue rhombus represented . Ask the student to provide an
explanation for the difference of opinion.
Portfolio
A1.8 Ask the students to design flags which can be described using
thirds.
A1.9 Ask the students to decide which they would rather have, of a
pizza or of a pizza. Have them explain their choice in writing.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-3
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
i) demonstrate an understanding
of number meanings with
respect to whole numbers,
fractions and decimals
SCO: By the end of grade 4,
students will be expected to
A2 interpret and model
decimal tenths and
hundredths
A2 Because decimals are fractional parts, it is essential that the
relationship between decimals and fractions be regularly addressed.
Students should use a variety of materials to model and interpret
decimal tenths and hundredths. Models could include:
- base-ten materials
- grids or decimal squares
- metre sticks
- 10 x 10 geoboards
s
- hundredths circles or disks
Models must be used at all grade
levels to develop fraction concepts
adequately. Further,... children
should have experiences with a
wide assortment of models.
(Elementary School Mathematics,
pp. 222-23)
4-4
- money
= 0.23
Most students may see the relationship between 0.01, 0.1 and 1.0
better if analogies are made to real-life objects which are sized
proportionally.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A2.1 Ask the student to show 0.2 if
whole;
A2
represents one
Teacher’s Guide and Journal
Unit 6:
if
represents one whole; if
represents one whole.
Activities 7 and 8 (pp. 243-250)
Culminating Work (pp. 251-254)
A2.2 Ask: About where would you place 1.76 on the number line?
Explain your choice.
Indiv. Perf. Assessment (p. 256)
Unit 11:
Paper and Pencil
A2.3 Have the students use a hundred grid to show a capital “T”
that takes up more than 0.20 of the grid and one that takes less than
0.20 of the grid. Ask them what decimal part would make the task
very difficult.
Activity 3 (pp. 427-430)
Interview
A2.4 Ask the student to select two different models with which to
show 0.38.
A2.5 Ask the student to use a model of choice to explain why 0.40 is
the same as 0.4.
A2.6 Have the student estimate 0.36 of a circle. Provide an acetate
overlay of a hundredths circle/disk so that the estimate can be
checked.
A2.7 Ask the student: How long is 0.25 metres? How do you
know?
A2.8 Tell the student that 0.53 could be read as 53 hundredths or
5 tenths and 3 hundredths. Ask the student which he/she prefers
and why. Also ask when such a number might be used.
A2.9 Ask the student to give the number that is 0.01 more than, or
less than, 3.24.
A2.10 Ask the student to identify a situation in which 0.36 can refer
to a large amount.
Portfolio
A2.11 Ask each student to use a hundred grid to draw an “E” which
covers more than 0.3, but less than 0.5, of the grid.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-5
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
iii) read and write whole
numbers and decimals and
demonstrate an understanding
of place value (to millions and
to thousandths)
SCO: By the end of grade 4,
students will be expected to
A3 model and record numbers
to 99 999
A3 Students should recognize the value represented by each digit in
a number, as well as what the number means as a whole. Include
situations in which students use
• base-ten materials (e.g., to model 10 000 have the class make a long
rod with 10 big cubes. It will be a 10 thousand rod.)
• money (e.g., How many $100 bills are there in $12 347?)
• place value charts
Thousands
Students should have opportunities to
• model numbers containing zeroes.
For example, 1003 means
1 thousand
3 ones
• read numbers several ways. For example, 12 347 is read 12
thousand, three hundred forty-seven but might also be expressed as
12 thousands, 34 tens, 7 ones; 12 thousands, 33 tens, 17 ones;
123 hundreds, 47 ones; etc.
• record numbers. For example, ask students to write twenty-eight
thousand sixty; a number which is eighty less than ninety thousand; etc.
Invite students to investigate the length of a line comprising
10 000 pennies.
Encourage students to share the various strategies they used to
investigate this problem. It is also important to have them share
strategies that they considered, but rejected.
Provide 10-sided dice, prepared cards marked 0 to 9, or playing
cards ace (1) to 9 (with the joker as the zero). Shuffle the cards.
Have students select 5 cards each (or toss a die 5 times) and make
the greatest (least) possible number. Have students lay out the
cards (leaving a space after the thousands) and read their numbers
to their groups. Have some students write their numbers on the
board and read them. Ask: Who thinks they might have the
greatest number? How far from the greatest possible number is
yours? Would it be possible for someone to have the greatest and
the least possible numbers when the cards are rearranged? What
digits would you want in order to have the greatest difference
between your greatest and your least numbers?
4-6
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A3.1 Provide a stack of 4 sets of shuffled cards numbered 0 - 9. Ask
the students to select 5 cards and arrange them to make the greatest
possible number. Ask them to record and read the number and to
rearrange the cards to make the least possible number. Have this
number recorded under the larger number. As an extension, have the
students estimate the difference between the two numbers. This is
an ideal opportunity activity for students to practise front-end
subtraction.
A3
Teacher’s Guide and Journal
Unit 2:
Activities 1-6 (pp. 47-70)
Culminating Work (pp. 71-74)
Indiv. Perf. Assessment (p. 76)
A3.2 Ask the student to use base-ten materials to model 2046 in
three different ways. Have him/her explain the models.
Paper and Pencil
A3.3 Tell the students that a number is represented by using 10
large base-ten cubes and some flats. Ask: What might it be?
A3.4 Tell the students that a number has 5 digits. The digit in the
ten thousands place is greater than the digit in the tens place. Ask:
What is the greatest number this could be? the least?
A3.5 Ask the students to record a series of numbers that have been
read to them (such as forty-six thousand eighty-two, ninety thousand
five). Include examples such as the greatest 5-digit number or a number
one hundred less than the greatest 5-digit number.
Interview
A3.6 Ask the student to imagine flats placed on top of each other to
form a tower. Ask: How many hundred flats would be required to
construct a tower representing 75 000? How high would this be?
A3.7 Ask: For how many $100 bills could $15 000 be fairly
exchanged?
A3.8 Ask the student to describe the base-ten blocks that he/she
would need to show 75 089.
A3.9 Tell the student that the number 13 420 has 134 hundreds
and 2 tens. Ask him/her to describe 3 other ways to express it.
A3.10 Tell the student that a car costs $16 135. Ask: If one were to
pay for it in $10 bills, how many would be needed?
A3.11 Ask: How are 1003 and 10 003 different? similar?
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-7
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
iv) order whole numbers,
fractions and decimals and
represent them in multiple
ways
SCO: By the end of grade 4,
students will be expected to
A4 compare and order whole
numbers
A4 Students should investigate meaningful contexts to compare and
order two or more numbers, both with and without models. It is
expected that they be able to explain why one number is greater or
less than another. For example, 2542 < 3653 because 2542 is less
than 3 thousands while 3653 is more than 3 thousands.
Provide the students with opportunities to practise comparing
numbers such as 32 998 and 33 010 and ask them to explain
their reasoning.
Prepare cards for students to order from least to greatest.
For example:
Assign pairs of students the task of making challenging number cards
for their classmates to order.
Number sense is the ability to
understand and use numbers and
operations on numbers in
computation, measurement, and
estimation situations. This ability
takes many years to develop and is
well worth the investment; it is
valuable to both children and
adults as they encounter
mathematical situations in and
out of school. When children have
experiences that encourage them to
model and describe numbers in a
variety of settings, they will readily
learn to apply mathematical
understandings in appropriate and
efficient ways. (Curriculum and
Evaluation Standards, Addenda
Series, Fourth-Grade Book, p. 9)
4-8
Provide situations in which students
- name numbers which are greater than or less than a given
number (Note: In some cases the amount greater or less could
be specified, such as 29 more or 3000 less, etc.)
- name numbers which are between given numbers
Invite students to decide which is worth more
11 356 quarters
27 462 dimes or
99 999 pennies
Have the students predict first, then use calculators to help solve
the problem.
Display a 5-digit number on an overhead calculator (or on a card, or
on the board). Have students enter on their calculators a number
which differs by 1 digit. Have them read their numbers and ask
others to determine if they are greater than or less than the number
on the overhead. Collect five of their numbers and ask the students
to order them. Ask for an explanation of the order.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A4.1 Have the students use a reference book to find the populations
of 2 communities. Then ask them to find another population that is
greater than that of one of the communities, but less than the other.
A4
A4.2 Give the students some number cards and ask them to order
them from greatest to least.
Paper and Pencil
A4.3 Ask the students to find 3 ways to fill in the boxes to make the
statement true:
2 245 > 15 8 4
Teacher’s Guide and Journal
Unit 2:
Activities 2 (pp. 51-54),
5 and 6 (pp. 63-70)
A4.4 Ask the student to record two numbers: the first has 3 in the
thousands place, but is less than the second which has 3 in the
hundreds place.
A4.5 Ask the students to each write a number that has 5210 tens.
A4.6 Ask the students to each write a number that would fall about half
way between 95 987 and 100 000.
A4.7 Tell the students that you are thinking of a 5-digit number that has
4 thousands, a greater number of tens, and an even greater number of
ones. Ask them to give three possibilities.
A4.8 Have the students create a number with the same digits as
98 950, but which is greater. Ask: How many of these numbers are
there?
Interview
A4.9 Tell the student that Jodi’s number had 9 hundreds, but Fran’s had
only 6 hundreds. Fran’s number was greater. Ask: How was this
possible?
A4.10 Ask: Which number below must be greater? Explain why.
4
2
9
3
A4.11 Ask the student how many whole numbers are greater than
8000, but less than 8750.
A4.12 Ask the student how he/she might advise a younger student
to determine which of 2 numbers is greater.
Portfolio
A4.13 Have the students find large numbers from newspapers and
magazines. Ask them to create a collage that would illustrate the
order of the numbers from least to greatest.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-9
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
iv) order whole numbers,
fractions and decimals and
represent them in multiple
ways
SCO: By the end of grade 4,
students will be expected to
A5 compare and order
fractions
A5 Students should recognize that there are various ways to compare
fractions. It is important that these early experiences be embedded in
investigations with various models. It is through these investigations
that students will develop a visual image of fractions which is
essential for fraction number sense.
Provide situations in which students will explore and compare
fractions using
• area models (part of a whole area)
>
• length models (part of a length measurement)
• set models (part of a set of like objects)
Provide regular opportunities to compare fractions
• having the same denominator. For example, < because if an item
is cut into 6 equal pieces, 2 of those pieces are less than 5 of them.
• having the same numerator. For example, > because if 3 people
share 1 item, they will each get more than if 4 people share the same
item.
Many pairs of fractions can be
compared without using a formal
algorithm, such as finding a
common denominator or changing
each fraction to a decimal.
Children need informal ordering
schemes to estimate fractions quickly
or to judge the reasonableness of
answers. They can be led to
discover these relationships if they
have had experiences in constructing
mental images of fractions.
(NCTM 1989 Yearbook, p. 162)
4-10
• using reference points ( , 1, etc.). For example,
is almost 1, while is not even .
<
because
Encourage students to use various comparison methods, depending
on the situation.
Invite students to create problems for others to solve. For
example: Ted ate of a cheese pizza, and Lee ate of a
mushroom pizza. Each ate 2 pieces. Who ate more? (It is
important that they recognize that this can be solved only if
they know the size of the pizzas.)
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A5.1 Ask the student to estimate where the following fractions
would be on a number line marked only with a 0 and a 1:
A5
,
,
Teacher’s Guide and Journal
Unit 6:
, and
Activity 3 (pp. 227-230)
A5.2 Give pairs of students cards with the following fractions on them:
Ask them to arrange the cards in order from least to greatest and to
give reasons for their arrangement.
Paper and Pencil
A5.3 Ask the students to list 3 fractions between
and 1.
A5.4 Ask: What possible numerators could be used in the statement
below if both fractions are less than 1?
Interview
A5.5 Ask the student to explain why
A5.6 Ask: Why is it easy to compare
is greater than
and
.
?
Presentation
A5.7 Ask pairs of students to work together to provide the class with
an explanation as to how they know that
is closer to 1 than .
Portfolio
A5.8 Have the students explain in writing how it is possible for
one pizza to be less than
of another.
of
A5.9 Ask the student to write a letter to a younger child explaining
why having
of a number of dimes is better than having
of the
same number of dimes.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-11
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
iv) order whole numbers,
fractions and decimals and
represent them in multiple
ways
SCO: By the end of grade 4,
students will be expected to
A6 rename fractions with and
without the use of models
A6 Students should be able to find equivalent fractions
(e.g., =
= ).
To develop conceptual understanding of equivalency, it is important that
models be used to generate the different representations of a fraction.
The “rules” for finding equivalent fractions should not be taught or used
until the students understand the reason for applying these rules. They
must understand why a fraction can have another name (e.g., = )
and still have the same value. Encourage the students to look for a
pattern in equivalency. It is recommended that the term “reduce” be
avoided since it suggests to the students that the size actually changes.
Students must develop number sense for fractions. The use of visuals is a
key component in this development.
In this example, if each third is cut in half, there will be twice as
many pieces and twice the number will be coloured.
Pattern blocks work well to show equivalence. For instance, if
is one whole unit, then
Invite students to make designs with pattern blocks and give a
value to them.
If
= 1, then
=3 .
Students may present their designs for others to rename by
covering them with other combinations of pieces.
For example:
=3
The use of manipulatives is crucial
in developing students' understanding
of fraction ideas. Manipulatives
help students construct mental
referents that enable them to
perform fraction tasks meaningfully.
(NCTM 1989 Yearbook, p. 158)
4-12
Paper folding is also an effective way of showing equivalency.
Ask the students to fold a piece of paper in half, open and shade in
one part. Have them refold the paper and once again fold in half.
When opened, the folds now provide the coloured visual for = .
Continue to fold to show
.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A6.1 Ask the student to use a rectangle on a geoboard to determine
other names for .
A6
A6.2 Ask the student to use a model of choice to show that
Teacher’s Guide and Journal
=
.
Unit 6:
Activities 2 (pp. 223-226),
A6.3 Ask the students to fold pieces of paper to show that
=
.
6-8 (pp. 239-250)
Paper and Pencil
A6.4 Have the students use a diagram to show that
=
.
A6.5 Ask: What equivalent fractions does this diagram show? List all
the pairs you can.
Interview
A6.6 Tell the student that Sally ate
of a large pizza. Billy ate
of
a medium-sized pizza. Sally said that they ate the same amount
because she had learned that = . Ask the student to respond to
Sally’s comment.
A6.7 Ask: Why is there always another name for a fraction?
Presentation
A6.8 Ask students to work in pairs to provide an explanation for
their classmates as to why 10 cannot be another name for .
Portfolio
A6.9 Ask the students to write what they know about equivalent
fractions.
A6.10 Have the students draw a visual representation and explain in
writing how they know that 3 = .
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-13
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
KSCO: By the end of grade 6,
students will have achieved the
outcomes for entry-grade 3 and
will also be expected to
iv) order whole numbers,
fractions and decimals and
represent them in multiple
ways
SCO: By the end of grade 4,
students will be expected to
A7 compare and order
decimals with and without
models
A7 Students should compare and order decimals (with and without
using models) while investigating relevant situations.
Examples:
- sports events (measurement of times, distances, and scores)
- capacity of various containers (25 mL, 0.5 L, 500 mL)
Invite students to make a
table of distances (expressed
in metres) that represents how
far each person can kick a
tissue, flip a coin, etc. They
might then, for example, list
each set of distances from least
to greatest.
The number line and metre stick both provide good models for decimals.
Have students work together to locate 0.5 m and 0.6 m on a metre
stick or measuring tape and to discover how they would name the
points between the two.
It is important that work with decimals not be distinct from work
with fractions; their relatedness must be emphasized. Decimals are
fractional parts, and the connection between the two should be a
major part of exploration.
Hundredths disks and base-ten materials are helpful when making
this connection.
4-14
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Worthwhile Tasks for Instruction and/or Assessment
Suggested Resources
Performance
A7.1 Ask the students to use base-ten materials to explain why
1.02 < 1.2.
Paper and Pencil
A7.2 Have the students record three decimal amounts between 0.4
and 0.5.
A7.3 Have the students arrange the digits 4, 2, 9, and 0 in the boxes
below to make the greatest and least amounts.
.
A7.4 Ask the students to find at least three ways to make the following
statement true:
1.3
< 1.
2
Ask: Can the statement be true if the first ‘square’ contains a 2? If the
second contains a 3? Explain.
Interview
A7.5 Provide the times for four sprinters in a 100-metre race. Ask
the student to determine the first-, second-, and third-place winners.
A7.6 Ask the student to order the following from least to greatest
and to provide an explanation for the order.
1.24 m, 2.01 m, 0.97 m, 2.20 m, 3 m, and 108 m
A7.7 Ask the student to name a situation in which a contest result of
0.23 might actually beat 0.26.
Portfolio
A7.8 Have the students use each of the digits 0 - 9 once to fill in the
ten spaces and make the statements true.
Ask: How many different ways can be found?
A7.9 Ask the students to write about when they would prefer to use
fractions and when decimals would be their choice.
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE
4-15
SPECIFIC CURRICULUM OUTCOMES, GRADE 4
GCO A: Students will demonstrate number sense and apply number-theory concepts.
Elaboration - Instructional Strategies/Suggestions
4-16
ATLANTIC CANADA MATHEMATICS CURRICULUM GUIDE