# Skills development for fifth and sixth classes

Through completing the strand units of the mathematics curriculum the child should be enabled to

##### Applying and problem-solving
• select appropriate materials, concepts and processes for particular tasks and applications
• apply concepts and processes in a variety of contexts
• analyse problems and plan an approach to solving them
• select and apply a variety of strategies to complete tasks and projects or solve problems
• reflect upon and evaluate solutions to problems
##### Communicating and expressing
• discuss and explain the processes used and the results of mathematical activities, problems and projects in an organised way
• listen to and discuss other children's mathematical descriptions and explanations
• discuss and record the processes and results of work using a variety of methods
• discuss problems and carry out analyses
##### Integrating and connecting
• connect informally acquired mathematical ideas and processes with formal mathematical ideas and processes
• recognise mathematics in the environment
• represent mathematical ideas and processes in different modes: verbal, pictorial, diagrammatic and symbolic
• understand the connections between mathematical procedures and the concepts he/she uses
• recognise and apply mathematical ideas and processes in other areas of the curriculum
##### Reasoning
• make hypotheses and carry out experiments to test them
• make informal deductions
• search for and investigate mathematical patterns and relationships
• reason systematically in a mathematical context
• justify processes and results of mathematical activities, problems and projects
##### Implementing
• devise and use mental strategies and procedures for carrying out mathematical tasks
• use appropriate manipulatives to carry out mathematical procedures
• execute standard procedures efficiently with a variety of tools
##### Understanding and recalling
• understand and recall facts, definitions and formulae.

## Strand: Number

### Strand unit: Place value

Content for fifth class

The child should be enabled to

• read, write and order whole numbers and decimals
extend previous conceptual and practical work to include larger numbers and decimals
• identify place value in whole numbers and decimals
extend previous conceptual and practical work to include larger numbers and decimals
• round whole numbers and round decimals
round whole numbers to nearest ten, hundred, thousand
round decimals to nearest whole number.

Content for sixth class

The child should be enabled to

• read, write and order whole numbers and decimals
• identify place value in whole numbers and decimals
• round decimals
round decimals to one, two or three decimal places.

### Strand unit: Operations

Content for fifth class

The child should be enabled to

• estimate sums, differences, products and quotients of whole numbers
use strategies for estimation, e.g. front-end estimation, rounding, clustering, special numbers
estimate calculations and compute answers with a
calculator
e.g. 450 x 9 = 4500 (estimation based on 450 x 10)
estimate first, then use calculator to get actual result
• add and subtract whole numbers and decimals (to three decimal places) without and with a calculator
develop and extend the use of existing algorithms
• multiply a decimal (up to three places) by a whole number, without and with a calculator
develop and extend the use of existing algorithms 8.125 x 9
• divide a three-digit number by a two-digit number, without and with a calculator
explore the concept of division with concrete materials
develop the long division algorithm from repeated
subtraction and multiples of repeated subtraction
• divide a decimal number by a whole number, without and with a calculator
explore the concept of division of decimals with concrete materials, money and measurement
extend the algorithm in conjunction with place value 75.6 divided by 4.

Content for sixth class

The child should be enabled to

• estimate sums, differences, products and quotients of decimals
use strategies for estimation
estimate calculations and compute answers with a calculator
• add and subtract whole numbers and decimals (to three decimal places) without and with a calculator
• multiply a decimal by a decimal, without and with a calculator
develop and extend the use of existing algorithms
7.25 x 1.5; 13.2 x 0.75
understand that multiplication does not always make larger
• divide a four-digit number by a two-digit number, without and with a calculator
develop and extend the use of existing algorithms 7852 divided by 26
• divide a decimal number by a decimal, without and with a calculator
explore the concept of division by decimals with
concrete materials, money and measurement
36.92 divided by 2.6; 27.6 divided by 0.2
understand that division does not always make smaller.

### Strand unit: Fractions

Content for fifth class

The child should be enabled to

• compare and order fractions and identify equivalent forms of fractions with denominators 2 - 12
explore, compare and record simple equivalence using concrete materials, paper folding, and fraction charts
• express improper fractions as mixed numbers and vice versa and position them on the number line
establish equivalence by using concrete materials explore, compare and record simple improper fractions and mixed numbers diagrammatically, numerically and on the number line
• add and subtract simple fractions and simple mixed numbers
use equivalent fractions to simplify calculations
• multiply a fraction by a whole number
develop concepts with concrete materials, paper folding and fraction charts
four x three quarters of a pizza is how many pizzas?
• express tenths, hundredths and thousandths in both fractional and decimal form
explore and compare using concrete materials
express as fractions and as decimals.

Content for sixth class

The child should be enabled to

• compare and order fractions and identify equivalent forms of fractions
order equivalent fractions on the number line and on fraction charts
• express improper fractions as mixed numbers and vice versa and position them on the number line
• add and subtract simple fractions and simple mixed numbers
commondenominator should be found by listing multiples
• multiply a fraction by a fraction
explore and develop concept by using concrete materials and the number line and by drawing diagrams to illustrate examples, leading to the development of analgorithm
• express tenths, hundredths and thousandths in both fractional and decimal form
• divide a whole number by a unit fraction
how many quarters in 2?
2 divided by one quarter 4 ; 15 divided by one fifth 5
• understand and use simple ratios
explore and record the relationship between the natural numbers and their multiples.

### Strand unit: Decimals and percentages

Content for fifth class

The child should be enabled to

• develop an understanding of simple percentages and relate them to fractions and decimals
express percentages as fractions and as decimals, and vice versa calculate simple percentages, e.g. 50%, 25% 10%
• compare and order fractions and decimals
explore, compare and record using concrete materials and money order diagrammatically or on the number line
• solve problems involving operations with whole numbers, fractions, decimals and simple percentages
use diagrams; estimate and compute answers with acalculator, include simple discount and increase examples 10% off all jeans, 20%extra free.

Content for sixth class

The child should be enabled to

• use percentages and relate them to fractions and decimals
express quantities as percentages
• compare and order percentages of numbers
• solve problems relating to profit and loss, discount, VAT, interest, increases, decreases.

Measures: Money
Integration
Geography: Human environments

The treatment of content as suggested in the exemplars is common to both classes.

### Strand unit: Number theory

Content for fifth class

The child should be enabled to

• identify simple prime and composite numbers
define a prime number, i.e. a number greater than 1 with exactly two divisors, itself and 1
identify simple prime numbers by trial and error, e.g. 2, 5, 7, 11
identify and record primes with Sieve of Eratosthenes
define a composite number, i.e. a number that has more than two divisors, e.g. 4, 6, 9
identify and record composite numbers using number facts and/or a calculator
investigate relationship with odd and even numbers
• identify square and rectangular numbers
construct diagrams on geoboards, pegboards and squared paper to illustrate simple square and rectangular numbers explore, compare and record these numbers
• identify factors and multiples
identify factors and multiples from basic multiplication facts.

Content for sixth class

The child should be enabled to

• identify simple prime and composite numbers
• identify and explore square numbers
16 = 4 x 4 = 4 to the power of 2
• explore and identify simple square roots
construct diagrams
record and relate to square numbers
• identify common factors and multiples
explore and record factors and multiples to identify common factors and multiples
• write whole numbers in exponential form
1000 = 10 x 10 x 10 = 10 to the power of 3
8 = 2 x 2 x 2 = 2 to the power of 3 .

## Strand: Algebra

### Strand unit: Directed Numbers

Content for fifth class

The child should be enabled to

• identify positive and negative numbers in context
examine and discuss money affairs, video counters and calculator displays, sports reports, golf scores, temperature, sea level and lifts, leading to the need to
distinguish between amounts above and below zero
refer to positive and negative numbers as 'positive seven' and 'negative three' record positive and negative numbers with + or -
signs raised e.g. + 7, - 3
rewind a video tape
pupils draw and label a thermometer, mark in temperatures, consult weather forecasts in newspapers.

Integration
Geography: Natural environments: weather, climate and atmosphereContent for sixth class

Content for sixth class

The child should be enabled to

• identify positive and negative numbers on the number line
walk the number line to experience positive and negative numbers that arise in discussion and/or in context, identify and mark positive and negative numbers on
personal and class number lines
• add simple positive and negative numbers on the number line
add simple positive and negative numbers by walking the number line and by counting on the class and personal number line
+5 + -7 = ? 9 + -3 = ?
-8 + +2 =
add positive and negative numbers that arise contextually, e.g. a golfer's score over four rounds was 6 under par, 2 over par, 3 under par, and 1 under par; what was her final score relative to par?

Integration
Geography: Natural environments: weather, climate and atmosphere

### Strand unit: Rules and properties

Content for fifth class

The child should be enabled to

• explore and discuss simple properties and rules about brackets and priority of operation
identify, discuss and compute expressions with brackets in a variety of positions
10 + (4 + 7) = _
(10 + 4) + 7 = _
(8 - 1) + 4 = _
8 - (1 + 4) = _
(3 x 4) + 5 = _
3 x (4 + 5) = _
8 divided by (2 + 2) = _
(8 divided by 2) + 2 = _
what is the significance of the positions of the brackets?
identify, discuss and compute expressions with brackets excluded
4 + 3 x 5 = _
12 x 6 + 3 = _
2.45 divided by 5 - 0.75 = _
96 divided by 8 - 12 = _
what is the significance of starting operations at different points?
e.g. 4 + 3 before 3 3 5 or vice versa in 4 + 3 3 5
establish the value of brackets, leading to the priority of multiplication and division over addition and subtraction
explore these properties and rules without and with a calculator
• identify relationships and record verbal and simple symbolic rules for number patterns
identify and discuss rules for simple number sequences 2.0, 3.5, 5.0, 6.5 ... i.e. sequence increases by adding 1.5
81, 27, 9 ... decreases by dividing by 3
1, 4, 9, 16, 25, 36 ...

Content for sixth class

The child should be enabled to

• know simple properties and rules about brackets and priority of operation
use the calculator in exercises to find missing numerals and missing operator
e.g. 37 ? 21 ? 23 = 800
27 ? (36 ? 11) = 675
• identify relationships and record symbolic rules for number patterns
deduce and record rules for given number patterns
2, 6, 12, 20, 30 ...
4:1, 8:2, 16:4 ...

The treatment of content as suggested in the exemplars is common to both classes.

### Strand unit: Variables

Content for sixth class

The child should be enabled to

• explore the concept of a variable in the context of simple patterns, tables and simple formulae and substitute values for variables
identify and discuss simple formulae from other strands
e.g. d = 2 x r; a = l x w
substitute values into formulae and into symbolic rules developed from number patterns.

### Strand unit: Equations

Content for fifth class

The child should be enabled to

• translate number sentences with a frame into word problems and vice versa
create number stories to describe a given number sentence
how many teams of four can the teacher make for relays from a class of twenty-eight children?
28 / 4 = _
a man has twenty-eight windows to clean; it takes him an hour to clean four; how long will it take him altogether?
construct number sentences to describe mathematically a given word problem
• solve one-step number sentences and equations
75 - 43 = _ 3.5 x _ - 14
25% of _ = 15.

Content for sixth class

The child should be enabled to

• translate word problems with a variable into number sentences
Peter cut a length of ribbon into five equal parts; each part was 30 cm long. How long was the ribbon before it was cut?
x / 5 - 30
• solve one-step number sentences and equations
-3 + +6 - _
-4 + _ -+1
10 x _ - 8 x 5.

The treatment of content as suggested in the exemplars is common to both classes.

## Strand: Shape and space

### Strand unit: 2-D shapes

Content for fifth class

The child should be enabled to

• make informal deductions about 2-D shapes and their properties
• use angle and line properties to classify and describe triangles and quadrilaterals
name, explore and compare a wide variety of three and four-sided figures in terms of size and number of angles, type and number of sides e.g. trapezium, scalene
triangle, regular hexagon
• identify the properties of the circle
explore and compare circles of various unit diameters
measure and identify the relationship of diameter to radius
examine area by counting square units
• construct a circle of given radius or diameter
draw using a compass
• tessellate combinations of 2-D shapes
• classify 2-D shapes according to their lines of symmetry
explore, compare and record lines of symmetry in 2-D shapes
• use 2-D shapes and properties to solve problems
make a specified shape with Tangram shapes.

Content for sixth class

The child should be enabled to

• make informal deductions about 2-D shapes and their properties
• use angle and line properties to classify and describe triangles and quadrilaterals
• construct triangles from given sides or angles
complete the construction of triangles, given two sides and the angle between them or given two angles and the line between them
• identify the properties of the circle
relate the diameter of a circle to its circumference by measurement
measure the circumference of a circle or object such as a rolling-pin or wheel e.g. use a piece of string
• construct a circle of given radius or diameter
• tessellate combinations of 2-D shapes
• construct a circle of given radius or diameter
• classify 2-D shapes according to their lines of symmetry
• plot simple co-ordinates and apply where appropriate
use geoboards and squared paper
• use 2-D shapes and properties to solve problems.

### Strand unit: 3-D shapes

Content for fifth class

The child should be enabled to

• identify and examine 3-D shapes and explore relationships, including tetrahedron (faces, edges and vertices)
explore, compare and record the number of faces of 3-D shapes
identify number of edges and vertices of 3-D shapes
name the shape of the faces deconstruct 3-D shapes into nets; examine and discuss
• draw the nets of simple 3-D shapes and construct the shapes
discuss and draw simple net including flaps where necessary
construct 3-D shapes from nets.

Integration
Visual arts: Construction

Content for sixth class

The child should be enabled to

• identify and examine 3-D shapes and explore relationships, including octahedron (faces, edges and vertices)
• draw the nets of simple 3-D shapes and construct the shapes.

Integration
Visual arts: Construction

The treatment of content as suggested in the exemplars is common to both classes.

### Strand unit: Lines and angles

Content for fifth class

The child should be enabled to

• recognise, classify and describe angles and relate angles to shape and the environment
explore and compare a wide variety of angles and shapes measure and record angles as acute, obtuse, reflex or right angles, and determine the number of such angles in relation to common regular shapes
• recognise angles in terms of a rotation
examine, measure and record the angles (including the reflex angle) formed by the hands of a clock at a variety of different times
extend by using manipulatives, e.g. straws, lollipop sticks, Meccano, string, 360¡ protractor, LOGO computer language if available
• estimate, measure and construct angles in degrees
measure and record a wide variety of angles using a protractor
construct angles of various sizes using a protractor
estimate angle sizes and check by measuring with a protractor
• explore the sum of the angles in a triangle
cut off the three corners of a paper triangle and put them together to make 180 degrees
measure the angles in a variety of triangles using a protractor
calculate and record their sum
examine and discuss results.

Content for sixth class

The child should be enabled to

• recognise, classify and describe angles and relate angles to shape
identify types of angles in the environment
• recognise angles in terms of a rotation
• estimate, measure and construct angles in degrees
• explore the sum of the angles in a quadrilateral
cut off the four corners of a paper quadrilateral and put them together to make 360 degrees
measure the angles in a variety of quadrilaterals and calculate their sums.

The treatment of content as suggested in the exemplars is common to both classes.

## Strand: Measures

### Strand unit: Length

Content for fifth class

The child should be enabled to

• select and use appropriate instruments of measurement
ruler for shorter objects
metre stick for longer objects or distances
trundle wheel for distances
• estimate and measure length using appropriate metric units
estimate and measure a large variety of objects and places, both outdoors and indoors: books, desks, corridors, driveways, playing-pitch sidelines
how far can you throw a ball? jump?
run in 20 seconds?
use appropriate measuring units
mm (shorter objects) cm (longer objects)
m (short distances) km (long distances)
• estimate and measure the perimeter of regular and irregular shapes.

Integration
Physical education: Athletics; Outdoor and adventure activities

Content for sixth class

The child should be enabled to

• select and use appropriate instruments of measurement
• rename measures of length
rename measurements of appropriate metric units; express results as fractions and decimal fractions of appropriate metric units
233 m = 0.233 km
1 m 11 cm = 1.11 m
• estimate and measure the perimeter of regular and irregular shapes
• use and interpret scales on maps and plans
identify given scale on a map or plan and draw items to a larger or smaller scale.

Integration
Geography: Natural environments
Physical education: Athletics; Outdoor and adventure activities

### Strand unit: Area

Content for fifth class

The child should be enabled to

• discover that the area of a rectangle is length by breadth
determine by repeated experiments using rectangles withsides measured in whole centimetres and square units of one square centimetre
• estimate and measure the area of regular and irregular 2-D shapes
measure a wide variety of regular and irregular shapes using square units of onesquare centimetre and one square metre
• calculate area using square centimetres and square metres
choose appropriate measuring units:
square centimetres (smaller objects)
square metres (large objects or rooms)
• compare visually square metres and square centimetres.

Content for sixth class

The child should be enabled to

• recognise that the length of the perimeter of a rectangular shape does not determine the area of the shape
construct rectangles of constant perimeter with varying areas
• calculate the area of regular and irregular 2-D shapes
estimate and calculate area of shapes, and check by measuring with square centimetre units circles: calculate by counting squares only
• measure the surface area of specified 3-D shapes
measure 3-D surfaces by measuring individual 2-D faces or by extending into nets
• calculate area using ares and hectares
fields, large playgrounds, car parks
• identify the relationship between square metres and square centimetres
explore and compare areas of one, four, twenty-five and one hundred square centimetres to establish relationships
• find the area of a room from a scale plan
measure and calculate area of rectangular shapes by partitioning into rectangles and combining individual areas
extend to finding area of room plans (rectangular)
extend to using scale to find area of rooms from plans.

The treatment of content as suggested in the exemplars is common to both classes.

### Strand unit: Weight

Content for fifth class

The child should be enabled to

• select and use appropriate instruments of measurement choose measurement instruments appropriate to given tasks, e.g. balance, kitchen scales, bathroom scales and spring balance
• estimate and measure weight using appropriate metric units
estimate and measure a large variety of objects use appropriate measuring units
grams (pencils and copybooks) kilograms (school bags and people).

Content for sixth class

The child should be enabled to

• select and use appropriate instruments of measurement
• rename measures of weight
rename measurements of appropriate metric units
express results as fractions or decimals of appropriate metric units
750 g = 0.75 kg
4 kg 45 g = 4.045 kg.

### Strand unit: Capacity

Content for fifth class

The child should be enabled to

• select and use appropriate instruments of measurement
choose measurement instruments appropriate to given tasks graduated jugs, litre containers or fractional litre containers
• estimate and measure capacity using appropriate metric units
estimate and measure a large variety of objects use appropriate measuring units
millilitres (cups), litres (watering-can).

Content for sixth class

The child should be enabled to

• select and use appropriate instruments of measurement
• rename measures of capacity
rename measurements of appropriate metric units
express results as fractions or decimals of appropriate metric unit
625 ml = 5 eighths of a litre = 0.625 l
8 l 253 ml = 8.253 l
• find the volume of a cuboid experimentally
fill a cuboid container with water and measure capacity in litres
fill a cuboid container with unit cubes and count.

### Strand unit: Time

Content for fifth class

The child should be enabled to

• read and interpret timetables and the 24-hour clock (digital and analogue)
bus, train, air, ship, films, theatre, school, class
• interpret and convert between times in 12-hour and 24-hour format
10:30 p.m. = 22:30 hours
07:50 hours = 7:50 a.m.

Content for sixth class

The child should be enabled to

• explore international time zones
identify and discuss the need for time zones calculate time differences between Ireland and other countries
• explore the relationship between time, distance and average speed
measure, using a stop-watch, the time taken for short journeys to be completed or short distances to be covered and compile database to examine averages.

Data: Representing and interpreting
Integration
Physical education: Athletics

The treatment of content as suggested in the exemplars is common to both classes.

### Strand unit: Money - euro

Content for fifth class

The child should be enabled to

• compare 'value for money' using unitary method
compare the cost of 6 apples costing 75 cents and 4 apples costing 50 cents
calculate pay, based on hourly or daily rate
calculate totals of shop bills.

Content for sixth class

The child should be enabled to

• explore value for money
calculate sale prices, e.g. 10% discount, 20% VAT added
• convert other currencies to euro and vice versa
identify and discuss exchange rates from newspaper
calculate major currency equivalents for basic sums of euro
convert sums of money in other currencies to euro equivalents.

Number: Operations, Decimals and percentages

The treatment of content as suggested in the exemplars is common to both classes.

## Strand: Data

### Strand unit: Representing and interpreting data

Content for fifth class

The child should be enabled to

• collect, organise and represent data using pictograms, single and multiple bar charts and simple pie charts
collect data from the environment in tabular form and represent in appropriate format
discuss and explore modes of representation
• read and interpret pictograms, single and multiple bar charts, and pie charts
examine and discuss class-based examples and interpret
charts from newspapers, magazines and computergenerated
charts

Number: Fractions
Shape and space: Angles
Integration
Geography: Human environments

• compile and use simple data sets
compile lists of statistics from children's experiences
e.g. personal data (height, age, hair colour) sports results (wins, losses, scores)
use data as source for representation, interpretation and setting problems

• explore and calculate averages of simple data sets
identify the most frequently occurring item in a data set
calculate average by adding all the values and dividingby the number of items (use a calculator)
• use data sets to solve problems
solve problems based on data sets and representations used in class
what were the most popular buns at a cake sale?

Content for sixth class

The child should be enabled to

• collect, organise and represent data using pie charts and trend graphs
sales or rainfall per month
• read and interpret trend graphs and pie charts
e.g. height or weight in relation to age

Number: Fractions
Shape and space: Angles
Integration
Geography: Human environments

• compile and use simple data sets
compile lists of statistics from children's experiences e.g. personal data (height, age, hair colour) sports results (wins, losses, scores)
use data as source for representation, interpretation and setting problems
• explore and calculate averages of simple data sets
identify the most frequently occurring item in a data set
compare calculated averages with the most frequently occurring items
• use data sets to solve problems.

### Strand unit: Chance

Content for fifth class

The child should be enabled to

• identify and list all possible outcomes of simple random processes
discuss and list all possible outcomes of:
rolling a die (1, 2, 3, 4, 5, 6)
drawing a cube from a bag containing blue, red and green cubes (blue cube, red cube, green cube)
• estimate the likelihood of occurrence of events
if we toss a coin, say, 100 times, how many heads would we expect to get? a head has 50 chances in 100, or 1 chance in 2, of appearing; heads and tails are equally likely to occur if we roll a die: how often would we expect to get a 2? (1 chance in 6);
each of the 6 outcomes is equally likely; this activity can be done in groups with
each child or group throwing the die (or coin) 20 times and pooling the results; discuss the fairness of board games
• construct and use frequency charts and tables
perform the experiment (toss a coin, roll a die, draw a cube from a bag containing 3 blue and 6 green cubes ...) a large number of times (50-100 times) this activity can be done in groups with each child or group throwing the die (or coin) 20 times and pooling the results
record the outcomes and use to construct a frequency table; for example, if drawing a cube from a bag as above, the table might be as follows:
colour number of times drawn
blue 36
green 64
we estimate the likelihood of a blue cube to be 36 in 100 and that of a green cube to be 64 in 100
discuss: is that what we expected?
data sets compiled from children's experiences (personal data, weather, sports) might be used; for example, a survey of favourite cereals might have produced the
following table:
cereal number of pupils who prefer it
corn flakes 19
porridge 4
crispies 9
muesli 3
the likelihood that a pupil picked at random prefers corn flakes is estimated to be 19 in 35.

Integration
Music:Improvising and creating

Content for sixth class

The child should be enabled to

• identify and list all possible outcomes of simple random processes
discuss and list all possible outcomes of:
rolling two dice and calculating the total
(2, 3, 4 ... 12)
selecting two numbers at random from the numbers
1, 2, 3, 4, 5 (ten possibilities)
• estimate the likelihood of occurrence of events; order on a scale from 0 to 100%, 0 to 1
when tossing a coin, a head has 1 chance in 2 of occurring; thus the likelihood of a head is 1 in 2, or 1-2 or 50%, similarly for a tail when rolling a die, each outcome has a 1 in 6 chance of occurring -- therefore the likelihood is 1-6 when drawing a cube from a bag containing 3 red and 6 blue cubes, a blue cube has 6 chances in 9 of occurring and thus has a probability of 6-9 or 2-3 ; the probability of drawing a red cube is 3-9 or 1-3 what if the bag contains 5 red, 5 blue and 5 green cubes? or 3 red, 6 blue and 6 green?
• construct and use frequency charts and tables
perform the experiment (toss two coins, draw a cube from a bag containing a number of different-coloured cubes) a large number of times; larger numbers of throws can be achieved by using group work
record the outcomes and use to construct a frequency table; for example, when tossing two coins, the table might look as follows:
outcome frequency
2 tails 28