SCITT ProgrammeMathematics Training

Session ThreeCompleting Calculations!!!

Fractions, Decimals and Percentages

Liz HillsMathematics Subject Leader St Mary’s Catholic Primary School SLE Newick Teaching School

lhills@stmarysrc.e-sussex.sch.uk

Multiplication & Division

MULTIPLCIATION

Lattice Method

Grid Method

FORMAL WRITTEN METHOD OF SHORT MULTIPLICATION

FORMAL WRITTEN METHOD OF LONG MULTIPLICATION

FORMAL WRITTEN METHOD OF LONG MULTIPLICATION

+Remember to hold the place!

MULTIPLICATIONAdd a zero when completing long multiplication to hold the place.

Not adding in the carry over.

Too many digits –caused by not crossing out.

Think about how it is laid

out.

Use estimation to support accurate

calculation.MONSTER MISTAKES

DIVISION

Chunking

FORMAL WRITTEN METHOD OF SHORT DIVISION

FORMAL WRITTEN METHOD OF LONG DIVISION

FORMAL WRITTEN METHOD OF LONG DIVISION

DIVISION Bus stop method!!!

All operations in use.

Lots of opportunities

for error.

Need to know how to generate

multiplication facts.

Use of multiplication to check.

Lack of understanding of

place value.

MONSTER MISTAKES

TEACHING THROUGH MISCONCEPTIONS

Misconceptions with the Key Objectives

NCETM

Part Three: What is a Fraction?

1)Fractions are different from natural numbers in that they are relative rather than a fixed amount - the same fraction might refer to different quantities and different fractions may be equivalent (Nunes, 2006). Would you rather have one quarter of £20 or half of £5? The fact that a half is the bigger fraction does not necessarily mean that the amount you end up with will be bigger.

The question should always be, 'fraction of what?'; 'what is the whole?'.

These are all halves. All the halves are

different sizes!

2) Fractions can refer to objects, quantities or shapes, thus extending their complexity.

Half an Hour

Half of the same shape can look very different.

3) Fractions offer students their first lesson that many properties that are

true of whole numbers are not true for all numbers. For example, with

fractions, multiplication does not always lead to an answer larger than the multiplicands, division does not

always lead to an answer smaller than the dividend, and numbers do not have

unique successors.

4) Understanding that there are an infinite number of numbers between

any two fractions is challenging.

5) Fractions represent many different things: part/whole; division; proportions; ratios; numbers with a magnitude; measures; operators.

6) Fractions can also be represented as percentages and decimals.

6) Fractions can also be represented as mixed numbers, improper fractions, proper fractions, complex fractions and common fractions.

Less than one.

Value of one or more

These are known as complex fractions.

6) Fractions can also be represented as equivalent

fractions.

The idea of equivalence is a key one to introduce as children's understanding of fractions develops.

Consider whether some difficulties children experience are caused by what they understand as ‘equal’.It means ‘equal to’ not ‘identical to’.

Remember not only are learners encouraged to find families of fractions that are equivalent, but also to recognise decimal equivalents of fractions.

Vinculum: This stems from the Latin vincire meaning ‘binding’. It’s also used when writing a square root.

Solidus: This is Latin for compact. The symbol is sometimes used in fractions largely because in the early days of computers, there was no way to write a fraction. Hence, the solidus compensated for the vinculum.

Numerator: This tells us how many equal parts of a whole we have. Numerator stems from the Latin numerare meaning ‘to count’.

Denominator: This tells us how many equal parts our whole set is divided into. Denominator stems from the Latin denominare meaning ‘to name’. In other words we are naming (identifying) what we are dividing by.

What if you don’t understand the

language?

Term fraction means ‘ a piece broken off something’.

The word fraction actually comes from the Latin "fractio“ which means to break.

WHAT IS A FRACTION?

A FRACTION IS A PART OF A WHOLE.

And how often do we vary the

'whole'?Is it always 'one'?What about sharing two pizzas between

three?

Decimal numbers:

We could use decimal numbers instead of rational numbers but unfortunately not all rational numbers are easy to write as decimals.

Best example – 1/3 = o.3 recurring.

Decimal notation is generally considered to be less accurate than fractional equivalents (in pure maths) but in the real world where we might be measuring physical objects, a decimal format may be preferred.

Some decimal equivalents are quite easy to write eg. ½ = 0.5 others are harder eg. 1/107 has a 53 digit recurring decimal!!! Numbers such as this will not be accurately portrayed on a calculator.

HOW CAN WE SUPPORT GOOD UNDERSTANDING OF

FRACTIONS OVER THE COURSE OF THE PRIMARY

PHASE?

How can we ensure outstanding teaching

and learning of fractions?

To teach fractions well, teachers themselves must have a thorough understanding of fraction concepts and operations.

Researchers have found that students’ achievement in maths is positively correlated with the mathematical knowledge of their teacher.

Teachers’ Subject Knowledge and Understanding

Teachers’ Knowledge of the Curriculum

It is important for teachers to know not only the fraction concepts that are taught in their year group, but also the concepts that come before and after.

By understanding what students have been taught earlier, teachers can build upon what the students already know and better identify the sources of students’ misconceptions.

Understanding the material that will be taught later helps teachers to know what the students need to learn during the current year to provide a solid base for subsequent instruction.

Conceptual Understanding V Procedural Knowledge

Developing and improving students’ conceptual understanding of fractions is vital.

We define conceptual knowledge of fractions as knowledge of what fractions mean, for example their magnitudes and relations to physical quantities, an understanding of why arithmetic procedures with fractions are mathematically justified and why they yield the answers they do.

Such conceptual knowledge can be contrasted with procedural knowledge— the ability to execute a series of steps to solve a problem.

Concrete, Representational and Abstract

Teachers should focus on developing conceptual understanding along with procedural fluency.

One way to improve conceptual understanding is to use manipulatives and visual representations of fractions.

Studies that have taught fraction arithmetic using visual representations of fractions have shown positive effects on students’ computational skills.

In order to be able to develop their understanding and then generalise about fractions, children need to explore many representations and uses over a significant period of time.

Different representations are more or less appropriate for different situations. For example, diagrams can be useful in explaining sharing scenarios (e.g. splitting three cookies into five equal pieces) and in understanding fraction division. Number lines can help students to focus on the magnitudes of fractions.Area models (rectangles or circles that are partially shaded) can illustrate part/whole representation of fractions.

Intuitive KnowledgeYoung children understand the concept of equal sharing. Four-year olds can distribute a set of objects equally among a small number of recipients. By the age of 5, children can share a single object among multiple recipients. In addition, young children have an early understanding of proportional relations. For example, by the age of 6, children can match equivalent proportions represented by different geometric figures or everyday shapes.This early knowledge can be used to introduce the concept of fractions, connecting students’ intuitive knowledge to formal fraction concepts.

MisconceptionsMany students’ errors with fraction arithmetic can be avoided if they estimate their answers before attempting to use a formal algorithm. Estimation of fractions, however, does not come easily to many students. By practising estimation, students improve both their knowledge of fraction magnitudes and their understanding of fraction arithmetic. Estimation forces students to think out their answers and focuses students on what it means to add or multiply fractions, rather than just following a memorized rule without understanding.

Teaching and Learning Through TalkProviding children with opportunities to take part in meaningful mathematical discussions enables them to share their understanding, in addition to offering opportunities for teachers and children to challenge misconceptions.

It is important that children build up a vocabulary with which to talk about fractions. Of course this is a gradual process, but it is greatly helped by teachers modelling appropriate language wherever possible and drawing attention to good use of specific vocabulary by children themselves.

Instead of independently explaining these concepts as abstract formulae or steps, subject matter experts suggest that we

introduce the relation between the three. It helps to bring more clarity to students and as a result they don’t see them separately.

We encounter ideas related to fractions, decimals, ratios and percents on a daily basis.

Confident and flexible understanding of these ideas are key to everyday estimation and mental calculation in contexts as diverse as shopping and budgeting, diluting mixtures, understanding scales on

maps, interpreting probabilities and odds, and converting among metric units.

Not only is understanding these ideas essential to being numerate they are also foundational to more sophisticated mathematical ideas.

If you understand fraction, decimals, ratios and percents you have the basis for understanding topics such as similarity, trigonometry,

coordinate geometry, and fancy algebra.

Children may correctly answer a problem in a real-world context, and yet compute an incorrect answer when the problem is written in fraction notation. For example, a child may know that two halves make a whole, but say that 1/2 + 1/2 = 2/4.

Teachers should emphasize the connection between real-world problems and the fraction notation used to represent the problem.

Students can practice creating real-world contexts for problems presented in fraction notation and practice translating real-world problems into fraction notation.

This type of activity will help connect students’ intuitive knowledge of fractions with the formal notation.

USE A REAL WORLD CONTEXT

TO SUPPORT UNDERSTANDING

OF FRACTION NOTATION.

USE A NUMBER LINE TO HELP

DEVELOP UNDERSTANDING

OF THE RELATIONSHIP

BETWEEN FRACTIONS

DECIMALS AND PERCENTAGES.

Use a number line to help children connect the concepts of fractions, decimals and percents. Students can place a variety of numbers such as 3/5, 0.25 and 33% on the same 0 to 1 number line. In order to emphasize that fractions and decimals can represent the same magnitude, students could place fractions and their decimal equivalents on the same number line.

GIVE CHILDREN OPPORTUNITIES

WHICH ENCOURAGE

THEM TO MOVE FLUIDLY

BETWEEN FRACTIONS,

DECIMALS, AND PERCENTS.

Why do we need to be able to do this? Money involves decimals and

percents all the time. Prices are expressed with decimal numbers. We use percents to figure sales tax, tips, discounted prices and

interest. Knowing how to estimate and compute with decimals and

percents, whether with or without a calculator, is a critical skill for an

adult in our society.

The relationship between fractions, decimals and percents is clear when you look at the fraction

1/2.

When you divide the numerator by the common denominator you get a decimal.

That is seen in the following: 1 divided by 2 equals 0.5.

This relates to a percentage in the sense that 0.5 is a half and half is represented in percents as

50%.

So, if 0.5 = 50% and 1/2= 0.5 then 1/2= 50%.

Making a fraction into a percent is easy as you simply place the percentage over the whole number

100 and then reduce it.

So, 50% becomes 50/100 which is reduced to 1/2.

50% becomes a decimal by then dividing the whole number by a common denominator.

1/2 becomes 2 divided by 1 which equals 0.5

So, the relationship between fractions, decimals and percents

is that they are simply different numerical expressions of the

same value.

1 ÷ 2 = 0.5

1/2 = 50/100 = 50%

50% = 50/100 = 1/2

50/100 = 1/2 = 0.5

Primary Curriculum Mapping DocumentsThe documents on this page contain everything you need to include problem-solving activities in your planning, as they link up the National Curriculum statements with some of our favourite activities.

Part Four: Calculating with Fractions

The current National Curriculum in England, which became statutory in September 2014,

contains more content related to fractions than the previous curriculum. To support children in getting to grips with the concept of fractions it

is important to make sure they have lots of practical and varied experience using objects, shapes and quantities. This, combined with an

experience of rich tasks that stimulate and challenge their thinking; the opportunity to talk

and the chance to use models and images rather than 'tricks', will all support their

growing understanding of fractions.

Fractions are a deceptively tricky concept to teach.

Used by most people in daily lives – their interchangeable meaning and the mysterious rules used to manipulate them (teaching procedure rather than understanding) makes them a complicated concept.

Teachers must work hard to include a variety of classroom activities and teaching strategies, that focus on improving students’ conceptual understanding of fractions.

We define conceptual knowledge of fractions as knowledge of what fractions mean, for example their magnitudes and relations to physical quantities, an understanding of why arithmetic procedures with fractions are mathematically justified and why they yield the answers they do.

Such conceptual knowledge can be contrasted with procedural knowledge— the ability to execute a series of steps to solve a problem.

Teachers should focus on developing conceptual understanding along with procedural fluency.

One way to improve conceptual understanding is to use manipulatives and visual representations of fractions. Studies that have taught fraction arithmetic using visual representations of fractions have shown positive effects on students’ computational skills.

Many students’ errors with fraction arithmetic can be avoided if they estimate their answers before attempting to use a formal algorithm. Estimation of fractions, however, does not come easily to many students. By practising estimation, students improve both their knowledge of fraction magnitudes and their understanding of fraction arithmetic. Estimation forces students to think out their answers and focuses students on what it means to add or multiply fractions, rather than just following a memorized rule without understanding. When students are solving fraction arithmetic problems, they can be asked to estimate the answer and explain their reasoning before computing the answer. By checking if their computed answers are reasonable, students can recognize when they either used an incorrect computational procedure or made a mistake executing a correct procedure.

THE IMPORTANCE OF ESTIMATING WHEN CALCULATING WITH FRACTIONS

One estimation strategy is to use benchmarks. Benchmarks should be fractions that students are comfortable with such as 0, 1/2 and 1. Students can then decide whether any given fraction between 0 and 1 is closest to 0, 1/2 or 1. For example, if asked to add 6/7 and 5/8, a student might reason that 6/7 is close to 1 and 5/8 is close to 1/2, so the answer should be close to 1 ½.

Treating fractions’ numerators and denominators as separate whole numbers is a common error. When asked to subtract fractions, students often subtract the numerators and then subtract the denominators.These students are failing to treat the fraction as a unified number and instead treat the numerator and denominator as separate whole numbers. Teachers can help students to overcome the misconception that this is an acceptable procedure by presenting meaningful problems in the classroom.

THE IMPORTANCE OF ADDRESSING MISCONCEPTIONS

Leaving the denominator unchanged in fraction multiplication problems is another error. When multiplying fractions with equal denominators, students often leave the denominator unchanged.This error may occur because students are presented with more fraction addition problems than fraction multiplication problems, leading students to incorrectly generalize to multiplication the addition procedure for addends with equal denominators. Teachers can correct this error by reminding students that the problem can be reframed as “4/5 of 1/5”. Because the problem is asking for a part of 1/5, the answer cannot be larger than 1/5.

Students often have difficulties solving problems with mixed numbers. Some students ignore the fractional parts and instead only focus on the whole number (e.g. 4 2/3 – 1 2/5 = 3). Others decide that the whole numbers in the problem must have the same denominator as the fractions (e.g. 3 – 2/5 = 3/5 – 2/5 = 1/5). A related misconception is adding the whole number to the numerator of the fractional part (e.g. 2 2/5 x 5/6 = 4/5 x 5/6 = 20/30). All of these errors reflect a fundamental misunderstanding of what mixed numbers are and of the magnitudes they represent. Teachers should be sure to use proper fractions, improper fractions and mixed numbers in the classroom, and to translate often between mixed numbers and improper fractions.

Simplifying Fractions

We reduce a fraction to lowest terms by finding an equivalent fraction in which the numerator and denominator are as small as possible.This means that there is no number, except 1, that can be divided evenly into both the numerator and the denominator.To reduce a fraction to lowest terms, divide the numerator and denominator by their Greatest Common Factor (GCF). This is also called simplifying the fraction.

Reducing fractions before or in the middle

of a calculation can make the numbers

smaller and easier to handle.

Converting

Adding and Subtracting Fractions

Adding and Subtracting

Fractions with the Same

Denominator

Please note:Fractions are often taught using the idea that fractions represent a part of a whole. For example, one-fourth is one part of a whole

that has been split into four parts. This interpretation is important, but it fails to convey the vital information that fractions

are numbers with magnitudes. As such, fractions can be ordered from smallest to largest or be equivalent in value (1/2 =

2/4 = 3/6...). Children who only understand the part/whole approach to fractions often make errors, such as saying that 4/3 is not a number because a person cannot be given four parts of an object that is divided into three parts. The common error of trying to add fractions by first adding the numerators and then adding the denominators stems in part from not understanding that fractions are numbers with magnitudes. Solely relying on a

part/whole understanding of fractions often leaves children confused as to the meaning of fractions larger than 1 and to the

meaning of negative fractions.

Adding and Subtracting Proper Fractions with the

Same Denominator and Mixed Numbers

Adding and Subtracting Fractions with Different

Denominators

Visual representations can be used to help illustrate the need for common

denominators when adding and subtracting fractions. For example, a teacher can demonstrate addition by

using fractions of an object (e.g. 1/2 of a rectangle and 2/5 of a rectangle). By

placing the 1/2 of the rectangle and the 2/5 of the rectangle together above a

third rectangle, the teacher can show the approximate sum. She can then show

that 1/2 of the rectangle equals 5/10, and that 2/5 equals 4/10, and that the sum is exactly 9/10 of the rectangle. This type of

concrete demonstration can help students to understand why common denominators are necessary when adding and subtracting fractions.

How would it work for subtraction?

Do we always need to convert all the wholes into fractions to create an improper fractions?

Getting children to consider the size of the fraction helps.(This requires an understanding of the relationship between the numerator and the denominator)

THREE BOX METHOD

Understanding this method relies on children having a

secure understanding of common

denominators and equivalent fractions.

What happens when there are

more than 2 addends?

Need to return to understanding of common denominators and equivalent fractions and

addition as commutative.

Multiplying and Dividing Fractions

Multiplying Fractions

Multiplication pictorial representations can help students understand how multiplying fractions involves finding a fraction of a fraction. For example, to illustrate ¼ times 2/3, a student can start with a rectangle, divide it into thirds vertically, and then shade 2/3 of the rectangle with vertical lines. She would then divide the rectangle into fourths/quarters with three horizontal lines, and shade ¼ of the already shaded area with horizontal lines. At the end, two of the twelve small rectangles would be shaded both horizontally and vertically. Visual representations of fractions help develop conceptual understanding of computational procedures.

A BIT OF A BITA PART OF A PARTA FRACTION OF A FRACTION

What happens to the product when the

multiplicands involve fractions?

Fractions offer students their first lesson that many properties that are true of whole numbers are not true for all numbers. For

example, with fractions, multiplication does not always lead to an answer larger than

the multiplicands, division does not always lead to an answer smaller than the

dividend, and numbers do not have unique successors.

3/4 x 2/3

3/7 x 4/5

2/8 x 3/6

Multiplying fractions

Simple actionsMultiply topMultiply bot

3/4 x 2/3

3/7 x 4/5

2/8 x 3/6

Multiplying a Fraction and a Whole Number

Children need to understand multiplication as repeated

addition

As repeated addition children will see that they do not need to multiply the

denominators. (Identity property of 1)

SO…7 x 4

5 = 285 = 5 3

57 lots of 4/5

Dividing Fractions

HOW MANY HALVES IN EIGHT?

WHAT IS HALF SHARED BETWEEN EIGHT? Remember … division is

not commutative!DIVISION AS SHARING

DIVISION AS GROUPING

3 ÷ 9/17 = 3/1 x 17/9 = 3 x 17 / 1 x 9 =

51/9 = 5 6/9 = 5 2/3

1/171/17

1/17 1/171/171/17

1/171/171/17

DIVISION AS SHARING

DIVISION AS GROUPING