Otara LT Cluster Meeting

Vanitha Govini

Phoebe Fabricius

Numeracy facilitators

Pirate Problem:Three pirates have some treasure to share. They decide to sleep and share it equally

in the morning. • One pirate got up at at 1.00am and took 1/3 of the treasure.

• The second pirate woke at 3.00am and took 1/3 of the treasure.

• The last pirate got up at 7.00am and took the rest of the treasure. Do they each get an equal share of the

treasure? If not, how much do they each get?

Game time!

FRACDICE!

• Learning intention• Key knowledge

Objectives:

• Explore common misconceptions with fractions in order to develop teachers content knowledge.

• Explore key ideas, equipment and activities used to teach fraction knowledge and strategy.

• Exploring the language demands of mathematics

• Scaffolding learning for English Language Learners (ELL)

The ‘big’ ideas??Brainstorm any key ideas that underpin

fractional understanding

• Comparing **What’s it got to do with division?

• Ordering• Improper **Finding a fraction of a set• Mixed• Equivalency **Representing a fraction• Unit fractions• Why does ‘the whole’ matter?

Write down all the misconceptions you think exist within Fractions.

The Problem with Language•Stress the meaning of the numerators and denominators.

Use words first before using the symbols

e.g. one fifth not 1/5

How do you explain the top and bottom numbers?

1

2

The number of parts chosenThe number of parts the whole has been divided into

Ideas for Fraction language and recording. Fraction as a Number• Do not assume children understand the symbols.

• Call the fraction by its Call the fraction by its name:name:

• It is wise to use words first not symbols e.g. record 1 half rather than 1/2.

See

Say

Do Doing some ‘Skemp’ activities to teach Fractions.

Skemp activitySTART ACTION RESULT NAME

Biscuit Leave this on as it is

(Put it here) The whole of a biscuit

Biscuit Make 2 equal parts

(Put it here) These are halves of a biscuit

Biscuit Make 3 equal parts

(Put it here) These are third-parts of a biscuit

Biscuit Make 4 equal parts

(Put it here) These are fourth-parts of a biscuit. Also called Quarters.

Biscuit Make 5 equal parts

(Put it here) These are fifth-parts of a biscuit.

What does Fractions mean?

€

3

73 over

73 : 7

3 out of 7 3 ÷ 7

3 sevenths

How would each of the views solve this problem?

€

3

7

3 over 7

3 : 7

3 out of 7

3 ÷ 7

3 sevenths

of

€

3

7of

€

1

3

42

€

3

73 over

73 : 7

3 out of 7 3 ÷ 7

3 sevenths1 seventh of 42 is 6 so 3 x 6 = 18Children need good Mult and Div basic facts7 is telling you how many equal parts and the 3 is telling how many of those parts have been selected

Fraction is actually tenths1/10 of 42 = 4 so 3 x 4 =, 12 with some left over3/7 is actually a ratio of 3:4Demonstrate with unifix cubes

Is possible but refers to both numerator and denominator as being whole numbers.Need to picture groups of 7 and then three of each of these. Encourages additive thinking

Refers to both numerator and denominator as being whole numbers. If you place three over seven are you actually working with tenths.

Difficult as you need to work out what 3 divided by 7 as a decimal (leads to needing calculator). Can be worked out by (3x42) / 7. Generates unnecessary multiplication.A calculator will give the answer, though this would be devoid of meaning

Draw 3 pictures to represent three quarters.Continuous model of fractions

Discrete model of fractions

Label your pictures of three quarter as either continuous (shape/region) or discrete (sets).

10

Whole to Part:

• Most fraction problems are about giving students the whole and asking them to find parts.

• Show me ¼ of this circle?

Part to Whole:

• We also need to give them part to whole problems, like:

• ¼ of a number is 5. What is the number?

Perception Check:Model Part - to -

WholeWhole- to - Part

Continuous(Region or length)

Discrete(sets)

Model Part - to - Whole

Whole- to - Part

Continuous(Region or length)

This is one quarter of a shape. What is the shape?

How many ways can you cut this shape into quarters?

Discrete(sets)

Hemi got two thirds of the lollies. How

many were there altogether?

Here are 12 lollies. If you eat one quarter of them, how many do you

get?

Perception Check:

Emphasise the ‘ths’ code

1 dog + 2 dogs = 3 dogs

1 fifth + 2 fifths = 3 fifths

+ =

3 fifths + = 1

1 - =

€

2

5

€

1

5

€

?

5

€

3

5

€

?

5

€

3

5

Scenario one• A group of students are investigating the

books they have in their homes. • Steve notices that of the books in his

house are fiction books, while • Andrew finds that of the books his family

owns are fiction.

• Steve states that his family has more fiction books than Andrew’s.

Consider….Is Steve necessarily correct?Why/Why not?What action, if any, do you take?

€

1

5

2

1

Key Idea:The size of the fraction depends on the

size of the whole.• Steve is not necessarily correct because the amount of books that each fraction represents is dependent on the number of books each family owns.

• For example: of 30 is less than of 100.

• Key is to always refer to the whole. This will be dependent on the problem!

2

1

€

1

5

Scenario Two

• You observe the following equation in Bill’s work:

Consider…..

• Is Bill correct?• What is the possible reasoning behind his answer?

• What, if any, is the key understanding he needs to develop in order to solve this problem?

Key Idea:To divide the number A by the number B is to find out

how many lots of B are in A. When dividing by some unit fractions the answer gets bigger!

• No he is not correct. The correct equation is

• Possible reasoning behind his answer:• 1/2 of 2 1/2 is 1 1/4.

– He is dividing by 2. – He is multiplying by 1/2. – He reasons that “division makes smaller” therefore the answer must be smaller

than 2 1/2.

Chocolate activity

Fraction starter: Fraction Three in a Row

Dotty Pairs Game: p. 22 Book 7

• One player is dots the other is crosses• Number line from 0 to 6 or 0 to 10• Roll 2 dice and form a fraction, place this on number line (use materials if necessary)

• Aim is to get 3 marks uninterrupted by your opponent’s marks on the number line.

• If a player chooses a fraction that is equivalent to a mark that is already there they lose a turn.

Three in a row:

0 1 2 3 4 5 6

e.g. Roll a 3 and a 5

Mark a cross on either 3 fifths or 5 thirds.

The winner is the first person to get three crosses in a row.

X X

Are fractions always less than 1?Using the material provided, can you show 5 halves? 5 halves

will always be 2 and a half as a number.How would you mark 5 halves on a

number line?

0 5

• Explore one activity from a strategy stage in your table groups.

• Focus on :• What key knowledge is required before beginning this stage.

• Highlight the important key ideas at this stage.• The learning intention of the activity.• Work through the teaching model (materials, imaging, number properties).

• Possible follow up practice activities.• The link to the planning units and Figure It Out support.

Book 7:

• Let’s have a look at another example.

• How much more of green is there than of blue?

• What knowledge do children need to be able to apply this understanding to realitity?

Ratios:

• What is the fraction of blue and green cubes?• Can you make another structure with the same

ratio? What would it look like?• What confusions may children have here?

Pipe Music Using Deci-pipes to teach decimals!

Objectives: •Identify and Order decimals

•Identify no. of tenths / hundredths in a number

•Add / Sub decimals Game: Zap!

Decimals

Winnie uses materials and claims 2•5 + 1•8 = 3•13. What error has Winnie probably made?

Problem solving

In a game of netball, Irene gets in 43 out of her 50 shots. Sarah takes 20 shots and gets in 17. Who is the better shot?

The Connection between Fractions and PercentagesWhat does % mean?

• In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, % For example, 45% (read as "forty-five percent") is equal to 45 hundredths or 0.45.

• What do we need to do to fractions so that it can be read as a percentage?

• What key mathematical knowledge do children need to be able to do this?

Teaching Percentages:

• Double Number lines:Students enter the information they have onto

a double number line, then extend the pattern to find the information they need. Students are encouraged to find relationships vertically and horizontally.

For example: 40% of 70 = ?

0

0 100%

4

410%

7

40%

28 707010 = 7

10010 = 10

Solve these Problems using the Double Number line:

• Emily’s team won the basket ball game 120-117. Emily shot 60% of the goals. How many goals did Emily get?

• John scored 104 runs in a one day cricket match, that was 40% of the teams total. How many runs did his team score altogether?

• In a bike race, 30% of cyclists drop out. 42 riders finish the race. How many cyclists started the race?

Summary of key ideas

• Fraction language - emphasise the “ths” code• Fraction symbols - use words and symbols with

caution• Continuous and discrete models - use both • Go from Part-to-Whole as well as Whole-to-Part• Fractions are numbers and operators• Fractions are a context for add/sub and mult/div

strategies• Fractions are always relative to the whole

Book 7:• Explore one activity from a strategy stage in your

table groups. • Focus on :• What key knowledge is required before beginning this

stage.• Highlight the important key ideas at this stage.• The learning intention of the activity.• Work through the teaching model (materials, imaging,

number properties).• Possible follow up practice activities.• The link to the planning units and Figure It Out

support.

The Connection between Fractions and PercentagesWhat does % mean?

• In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, % For example, 45% (read as "forty-five percent") is equal to 45 hundredths or 0.45.

• What do we need to do to fractions so that it can be read as a percentage?

• What key mathematical knowledge do children need to be able to do this?

Lemon activity

What are the key messages? Why?

Share it with a partner.