Download 2.1 Adding and subtracting fractions and mixed numbers 2

2 2 Fractions
Specification
GCSE 2010
N a (part) Add, subtract… any number
N h Understand equivalent fractions,
simplifying a fraction by cancelling all
common factors
N i Add and subtract fractions
FS Process skills
Recognise that a situation has
aspects that can be represented using
mathematics
FS Performance
Level 2 Apply a range of mathematics to
find solutions
Resources
ActiveTeach resources
Simplifying fractions quiz
Adding fractions 2 interactive
2 2.1 Adding and subtracting fractions and
mixed numbers
Concepts and skills
•
•
•
Add, subtract … fractions.
Find equivalent fractions.
Add and subtract fractions.
Functional skills
•
L2 Carry out calculations with numbers of any size in practical contexts.
Prior key knowledge, skills and concepts
Students should already know how to
• add and subtract integers
•
•
•
•
find the LCM of two numbers
write a fraction in its simplest form (N h)
change from mixed numbers to improper fractions and vice versa (N h)
order fractions (N b).
Starter
•
Ask students to find the LCM of pairs of numbers, e.g. 3 and 4 (12), 2 and 6 (6), 4 and 6
(12), 8 and 10 (40).
•
Discuss this question with students. Can you tell me all the fractions that are equivalent
1
1
to 2 ? (No – there are an infinite number.) Ask them for other fractions equivalent to 2 .
Main teaching and learning
•
•
Tell students that they are going to learn how to add and subtract fractions.
•
Discuss how you could add 32 + 41 . Ask students for ideas. Drawing a diagram to illustrate
each fraction on a rectangular grid, using 3 columns and 4 rows, is one way to move
into a discussion of common denominators.
•
•
Explain the subtraction of fractions in the same way, using diagrams where necessary.
Explain that, to add fractions, the denominators must be the same. Diagrams are a
useful way to show that, for example, 25 + 51 = 53 .
Discuss adding and subtracting mixed numbers. Encourage students to deal with the
integer parts first and then the fraction parts.
Common misconceptions
•
When adding fractions you do not ‘add the top numbers’ and ‘add the bottom numbers’.
This is a very common error.
•
When adding fractions that have a common denominator add only the numerators; the
denominator remains the same.
Enrichment
•
Students could consider the circumstances under which the LCM of two numbers is not
the product of the two numbers (when they have factor(s) in common).
Plenary
•
68
equivalent fractions
7
Students could be asked to find different fraction sums that lead to an answer of 8
11++33, ,11––11, ,55++11,etc
etc
22 88
88 88 44
((
mixed number
))
2 2 Fractions
Specification
GCSE 2010
N a (part) … multiply… any number
N o (part) Interpret fractions … as
operators
FS Process skills
Recognise that a situation has
aspects that can be represented using
mathematics
Use appropriate mathematical
procedures
FS Performance
Level 2 Apply a range of mathematics to
find solutions
2 2.2 Multiplying fractions and mixed
numbers
Concepts and skills
•
•
•
… multiply… fractions….
Multiply… by any number between 0 and 1.
Find a fraction of a quantity.
Functional skills
•
L2 Carry out calculations with numbers of any size in practical contexts …
Prior key knowledge, skills and concepts
Students should already know how to
• multiply integers
•
change between mixed numbers and improper fractions.
Starter
Resources
ActiveTeach resources
Adding fractions quiz
Multiplying a fraction 2 animation
•
Ask students to change some mixed numbers to improper fractions and vice versa. For
23 3 7 1
2 10 , 3 2 3 .
example 2 53 135 , 4 29 389 , 10
() () ( ) ( )
Main teaching and learning
•
•
Tell students that they are going to learn to multiply both fractions and mixed numbers.
•
Discuss the fact that if the final answer needs simplifying, then it is likely that the
simplifying could have been done before the multiplication.
•
•
Ask students how they would multiply mixed numbers.
Explain that to multiply fractions you multiply the numerators and multiply the
denominators.
Explain that mixed numbers need to be changed to improper fractions before
multiplication can take place.
Common misconceptions
•
When multiplying fractions students often multiply the whole numbers together and
then multiply the fractions together.
Enrichment
•
More able students could practise multiplying three mixed numbers together.
Plenary
•
70
improper fraction
Practise multiplying simple fractions together mentally. For example, 71 × 25
(352 ).
2 2 Fractions
2 2.3 Dividing fractions and mixed numbers
Specification
GCSE 2010
N a (part) … divide any number
FS Process skills
Recognise that a situation has
aspects that can be represented using
mathematics
Use appropriate mathematical
procedures
FS Performance
Level 2 Apply a range of mathematics
to find solutions
Concepts and skills
•
•
… divide… fractions….
… divide by any number between 0 and 1.
Functional skills
•
L2 Carry out calculations with numbers of any size in practical contexts …
Prior key knowledge, skills and concepts
Students should already know how to
• multiply integers and fractions
•
change between mixed numbers and improper fractions.
Starter
•
Remind students that the reciprocal of a whole number is 1 divided by the number, so
1
the reciprocal of 4 is 4 , and to find the reciprocal of a fraction you turn the fraction
upside down.
•
Ask students to give you the reciprocal of various fractions and integers. For example
1
1 2 3 5 4
5, 5 2, 2 , 3 2 , 4 5 .
Resources
ActiveTeach resources
Dividing integers quiz
Dividing a fraction 2 animation
() () () ()
Main teaching and learning
•
Tell students that they are going to learn how to divide by a fraction and by mixed
numbers.
•
Discuss the connection between multiplication and division, e.g. multiplying by 2 (the
reciprocal of 2) is the same as dividing by 2.
•
Explain that to divide by a fraction you multiply by the reciprocal of the fraction.
Illustrate this by a worked example (e.g. Example 10).
•
Ask students how this could be extended to mixed numbers. First change the mixed
numbers to improper fractions and then proceed in the same way as for dividing by
fractions.
•
Encourage students to cancel before division and to give their final answers in their
simplest form, using mixed numbers where appropriate.
1
Common misconceptions
•
•
It is the second fraction that must be ‘turned upside down’, not the first one.
Fractions must be kept in the correct order and division is not commutative.
Enrichment
•
Students could be given problems that contain a mixture of division and multiplication.
For example, 121 × 2 23 ÷ 131 or calculations involving fractions and BIDMAS.
Plenary
•
72
inverted
Use a mixture of division and multiplication of fraction questions on the board to ensure
that students can differentiate between the two methods.
2 2 Fractions
2 2.4 Fraction problems
Specification
GCSE 2010
N a Add, subtract, multiply and divide
any number
N o Interpret fractions, decimals and
percentages as operators
FS Process skills
Recognise that a situation has
aspects that can be represented using
mathematics
Use appropriate mathematical
procedures
FS Performance
Level 2 Apply a range of mathematics to
find solutions
Concepts and skills
•
•
ActiveTeach resources
Multiplication and division quiz
Fraction and percentage finder
interactive
RP KC Fractions knowledge check
RP PS Fractions problem solving
Find a fraction of a quantity.
Functional skills
•
L2 Carry out calculations with numbers of any size in practical contexts …
Prior key knowledge, skills and concepts
Students should already be able to
• multiply and divide by an integer
•
•
add, subtract, multiply and divide fractions
solve word problems.
Starter
•
Resources
Add, subtract, multiply and divide… fractions….
Use a number of word problems using integer values to ensure that students know
when to use the different arithmetic operations.
Main teaching and learning
•
Ask students for examples in real life where fractions are used. For example, sales in
shops, interest rates.
•
•
Tell students that they are going to learn about the use of fractions in solving problems.
•
How would you find 14 of 20? (20 ÷ 4 = 5). How could you use this result to find
(5 × 3 = 15).
3
4
of 20?
Discuss the need to be able to work with fractions in problems. Use the examples in the
Student Book to illustrate problems using fractions.
Common misconceptions
•
When working with questions involving a mix of units students need to think carefully
about their final answer.
•
Students use the wrong arithmetical operation.
Plenary
•
74
Give students a mixture of simple fraction calculations to underpin the different
techniques needed for adding, subtracting, multiplying and dividing fractions and
mixed numbers.