National 5 Numeracy: Fractions

Learning Objective

I can perform operations and combinations of operations on fractions including mixed numbers.

Previous learning

Third Level Benchmarks

  • Converts any fraction, decimal fraction or percentage into an equivalent fraction, decimal fraction or percentage.
  • Adds and subtracts commonly used fractions including when changing a denominator.
  • Converts between whole or mixed numbers, improper fractions and decimal fractions.
  • Uses knowledge of fractions, decimal fractions and percentages to carry out calculations with or without a calculator.

Fourth Level Benchmarks

  • Chooses the most efficient form of fractions, decimal fractions or percentages when making calculations and justifies the methods used.
  • Uses calculations to support comparisons, decisions and choices and justifies the method used.
  • Applies addition, subtraction and multiplication skills to solve problems involving fractions and mixed numbers.

National 4

  • Find simple fractions of shapes and quantities e.g.½, ⅓, ¼, 1/10, 1/5.
  • Convert equivalences between common fractions, decimal fractions and percentages.

General Guidance

Although fractions are covered extensively in both the Third and Fourth levels of the BGE, learners who have come from National 4 will not have been assessed on carrying out the four operations or working with mixed numbers. This makes the need for a diagnostic assessment more pertinent.

Finding out what learners really know

The following questions could be used to diagnose where learners are with their learning or identify common misconceptions.

Learning Intentions

Learning Experiences - Concrete, Pictorial, Abstract Models

Equivalent Fractions

Ask learners to use fraction tiles to create as many fractions as possible that are equivalent to or etc. When they are comfortable with this challenge them to extend to fractions they don’t have tiles to represent. They may choose to use a pictorial model or they may go straight to abstract models.

When learners are comfortable with finding equivalent fractions they can be challenged to compare fractions with different denominators.

Mixed Numbers and Improper Fractions

Have learners represent a mixed number using the fraction tiles and then create an equivalent improper fraction and vice versa.

This can be progressed quite naturally to a pictorial model. In the pictorial model allow learners to decide how many boxes they will use to represent one whole unit. Discussion around this can highlight that you are breaking the whole unit into fifths.

Once learners are confident in converting fractions using a pictorial model, they can naturally progress onto abstract models.

Addition and Subtraction

When learners are completely comfortable with equivalent fractions we can use a concrete model to show adding fractions.

Creating pictorial models offers an opportunity for discussion around obtaining a common denominator. For example when asking learners to draw a decision needs to be made as to how many boxes will be used to represent one whole unit such that it can easily be divided into thirds and quarters.

This could be further expanded upon by choosing denominators such as 6 and 8 to prompt discussion around lowest common multiple.

When progressing to an abstract model a side by side pictorial model may help learners understand the step of finding equivalent fractions rather than viewing it as a process.

Multiplication and Division

It can be difficult to use the fraction tiles to represent fraction multiplication as the dimensions of the tiles do not match up exactly when you create the arrays. For that reason, it is perhaps preferable to start with a pictorial model as shown below.

Having obtained the results of several multiplication problems using an array learners might be asked to identify a strategy that eliminates the need for drawing an array. This would lead into a purely abstract model.

Consideration should be given to whether or not learners are encouraged to simplify before multiplying or carry out the multiplication and then simplify.

Key Vocabulary

  • Reciprocals
  • Factors
  • Lowest Common Multiple
  • Divisor
  • Dividend
  • Quotient
  • Remainder
  • Numerator
  • Denominator
  • Product

Interleaving

  • Link to percentages and decimals.
  • Order of Operations.
  • Can link multiplication and division of fractions with linear scale factors.
  • Link to area, perimeter and volume.
  • Link to solving equations involving fractions.
  • Link to probability calculations.
  • Although learners haven’t yet been exposed to algebraic fractions their understanding could be probed by exposing them to simple questions involving algebraic fractions i.e. .

Real-life Contexts

  • Measuring length using inches and fractions of inches then calculating area and/or perimeter.

Resources

Concrete Resources

  • fraction tiles
  • cuisenaire rods
  • fraction, decimal, percentage walls
  • empty number lines
  • counting sticks
  • grid paper
  • number line (with and without numbers)
  • hundred square (with and without numbers)

Online Resources

- Fraction Wall (Nrich Maths) Can be used to explore equivalence

- Equivalence (Kangaroo Maths)

-> Addition (Don Steward)

->Subtraction (Don Steward)

Addition and Subtraction

- Addition and Subtraction (Kangaroo Maths)

-> Multiplication (Don Steward)

-Multiplying Fractions

- Multiplication and Division (Kangaroo Maths)

->Videos, Practice Questions and Worksheets. (All Operations) (Corbett Maths)

- Four Operations (Kangaroo Maths)