Intermediate Effective Numeracy Fractions Workshop Have a go at the Fraction Hunt on your table while you are waiting! 4 out of 3 people have trouble with fractions

Objectives Understand the progressive strategy stages of fractions, proportions and ratios Understand common misconceptions and key ideas when teaching fractions and decimals. Explore equipment and activities used to teach fraction knowledge and strategy

Choose your share of chocolate!

4 Stages of the PD Journey Organisation Organising routines, resources etc. Focus on Content Familiarisation with books, teaching model etc. Focus on the Student Move away from what you are doing to noticing what the student is doing Talk to the ‘journey’ where might they be? Reacting to the Student Interpret and respond to what the student is doing

Assess Yourself – Fraction Strategy and Knowledge

Use of Addition and known facts e.g. Stage 1 Stage 2-4 (AC) Stage 5 (EA) Unequal Sharing Equal Sharing Use of Addition and known facts e.g. 5 + 5 + 5 = 15

Stage 6 (AA) Using multiplication Stage 7 (AM) Using division

Stage 8 (AP)

What misconceptions may children have with fractions? Misconceptions about finding one half when beginning fractions: Share without any attention to equality Share appropriate to their perception of size, age etc. Measure once halved but ignore any remainder So what do we need to teach to move to equal sharing? Introduce the vocabulary of equal / fair shares with both regions and sets for halves and then quarters, then other fractions.

Draw two pictures of one quarter Key Idea 1 Draw two pictures of one quarter

Shapes/Regions(Continous models) one quarter Connect different representations words - symbols - drawings - number lines Sets (Discrete Models) Shapes/Regions(Continous models)

Work with shapes, lengths and sets of fractions from early on. Recapping Key Idea 1 Work with shapes, lengths and sets of fractions from early on.

Key Idea 2 3 sevenths 3 out of 7 7/3 7 thirds

5 views of fractions 3 ÷ 7 3 out of 7 3 : 7 3 over 7 3 sevenths

The problem with “out of” “I ate 1 out of my 2 sandwiches, Kate ate 2 out of her 3 sandwiches so together we ate 3 out of the 5 sandwiches”!!!!! 1 2 3 5 + = 2 3 x 24 = 2 out of 3 multiplied by 24! 8 6 = 8 out of 6 parts!

Fraction Language Use words first then introduce symbols with care. e.g. ‘one fifth’ not 1/5 How do you explain the top and bottom numbers? 1 2 The number of parts chosen The number of parts the whole has been divided into

Fractions In 2001 42% of year 7 & 8 students who sat the initial NUMPA could not name these symbols

Fractional vocabulary One half One third One quarter Don’t know

Emphasise the ‘ths’ code 1 dog + 2 dogs = 3 dogs 1 fifth + 2 fifths = 3 fifths 1/5 + 2/5 = 3/5 3 fifths + ?/5 = 1 1 - ?/5 = 3/5

Fraction language is confusing. Emphasise the ‘ths’ code. Recapping Key Idea 2 Fraction language is confusing. Emphasise the ‘ths’ code. Use words before symbols. Introduce symbols with care. The bottom number tells how many parts the whole has been split into, the top number tells how many of those parts have been chosen.

6 is one third of what number? Key Idea 3 6 is one third of what number? This is one quarter of a shape. What does the whole look like?

Recapping Key Idea 3 Go from part-to-whole as well as whole-to-part with both shapes and sets. Children need experience in both reconstructing the whole as well as dividing a whole.

Perception check on two key ideas Where in the table does this question fit? Hemi got two thirds of the lollies. How many were there altogether? Part-to-Whole Whole-to-Part Continuous (region or length) Discrete (sets)

Key Idea 4 : Anticipate the result of equal sharing by grouping (using addition or skip counting) to solve fraction problems rather than equal sharing by ones(Early Additive Stage 5)

The Strategy Teaching Model Using Number Properties Using Imaging Using Materials New Knowledge & Strategies Existing Knowledge & Strategies

Developing the use of addition facts (to find one quarter of 12) ? + ? + ? + ? = 12 3 ? ? ? ?

Find one quarter of 12 (imaging) Key idea: quarters means you need 4 equal groups. One quarter is the number in one of those groups. 3

Using Number Properties Find one quarter of 40, 400, 4000

Hungry Birds (Book 7, page 22) Four birds found a worm in the ground 20 smarties long. What proportion of the worm do they each get? How many smarties will each bird get? 5 + 5 + 5 + 5 = 20

3 ÷ 5 Division 1/5+1/5+1/5 =3/5 Y7 response: “3 fifteenths!” Why?

Key Idea 5 Division is the most common context for fractions when units of one are not accurate enough for measuring and sharing problems. Initially this is done by halving and halving again but harder examples require more sophisticated methods Cubes is in Book 5 and really establishing PV e.g. 3 ÷ 5 = 3 fifths

Order these fractions: ¼ 6 quarters, 3/4 nine quarters, 2/4 Key Idea 6 Order these fractions:   ¼ 6 quarters, 3/4 nine quarters, 2/4

Key Idea 6 Fractions are not always less than 1 Key Idea 6 Fractions are not always less than 1. Push over 1 early to consolidate the understanding of the top and bottom numbers. 1 5 12/4 6 quarters

Year 7 student responses (decile 10) What is this fraction? 5/2 2 fifths, five lots of halves, tenth, five twoths How do I write 3 halves? 3 1/2 1/3

0 1 half 2 halves 3 halves 4 halves Using fraction number lines to consolidate understanding of denominator and numerator Push over 1 0 1 half 2 halves 3 halves 4 halves 0 1/2 2/2 3/2 4/2 0 1/2 1 11/2 2

Three in a row (use two dice or numeral cards) A game to practice using improper fractions as numbers 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

Perception Check – Discuss these key ideas. (Stage 4 and 5). Use sets as well as shapes/regions from early on Fractions are a context for add/sub and mult/div strategies Fraction Language - use words first and introduce symbols carefully Go from Part-to-Whole as well as Whole-to-Part. 5. Division is the main context for using fractions 6. Fractions are not always less than 1, push over 1 early to consolidate meaning of fraction symbols.

Watch Vince in action

Moving to Stage 6 (AA) What key knowledge is required before moving on to Stage 6? Identify symbols for halves, quarters, thirds and fifths and tenths Order unit fractions (¼ ½,) and fractions with like denominators (3/4, 1/4 ) Identifies symbols for improper fractions, e.g. 5/4 Which number games/activities could you adapt to practice fraction knowledge?

Fraction Snakes and Chances Memory/ Happy Families Fraction / Number Mat Fraction Bingo Fraction Dominoes Fraction Snakes and Chances Fraction Memory/ Happy Families Fraction I have, Who Has

Key Idea 6. Which letter shows 5 halves as a number? 1 2 3 A B C D E F

Key Idea 6 (Stage 6) Fractions are numbers as well as operators 3/4 is a number between 0 and 1 (number) Find three quarters of 80 (operator)

3/4 of 80 20 20 20 3/4 of 80 = 3 x 20 = 60 Key Idea 7 (Stage 6 AA) The distance between Masterton and Wellington is 80 kilometres. Hemi has travelled 3/4 of the trip. How many kilometres is that? 3/4 of 80 80 20 20 20 3/4 of 80 = 3 x 20 = 60

Recapping Key Idea 7 (Stage 6) Apply understanding of links between addition and multiplication to solve fraction problems. (Birthday Cakes, Book 7 ,p.26)

Using Double Number Lines 100 1 5 20 60 3 5 1 Put a peg on where you think 3/5 will be. (Fractions as a number). How will you work it out? Use a bead string and double number line to find 3/5 of 100. (Fractions as an operator). How will you work it out?

Key Idea 8 (stage 6) Sam had one half of a cake, Julie had one quarter of a cake, so Sam had most. True or False or Maybe Julie Sam

What is the whole? (Trains Book 7, p32) What is B? What is the whole? (Trains Book 7, p32) A B C D cuisenaire

Recapping Key Idea 8 (stage 6) Fractions are always relative to the whole. Halves are not always bigger than quarters, it depends on what the whole is.

Key Idea 9 (stage 6) - Ratios! 1:1 Write 1/2 as a ratio 3: 4 is the ratio of red to blue beans. What fraction of the beans are red? 3/7 Think of some real life contexts when ratios are used.

Key Idea 9 (Stage 6) Ratios What is the link between ratios and fractions? Ratios describe a part-to-part relationship e.g. 2 parts blue paint : 3 parts red paint But fractions compare the relationships of one of the parts with the whole, e.g. The paint mixture above is 2/5 blue

Ratios and Rates What is the difference between a ratio and a rate? Both are multiplicative relationships. A ratio is a relationship between two things that are measured by the same unit, e.g. 4 shovels of sand to 1 shovel of cement. A rate involves different measurement units, e.g. 60 kilometres in 1 hour (60 km/hr)

Summary of Fractions Key Ideas Use sets as well as shapes/regions from early on Fraction Language - use words first and introduce symbols carefully. Go from Part-to-Whole as well as Whole-to-Part Division is the most common context for fractions. Fractions are not always less than 1, push over 1 early. Fractions are numbers as well as operators. Fractions are always relative to the whole. Consider the relationship between ratios and fractions Use addition/skip counting to find fractions of sets then develop and apply multiplicative thinking – Fractions are really a context for add/sub and mult/div strategies

Moving to Stage 7 AM Matiu started with a whole box of chocolates and ate fife-ninths. That left only 16 chocolates. How many chocolates were in the box to begin with? What is the number sentence for this problem? 4/9 of ? = 16 Draw a diagram to help to solve this problem.

4/9 of ? = 16 16 is four ninths of what number? 36 16 8 4 4 4 4 4 4 At Stage 7, students should be using a range of multiplication and division strategies to solve problems with fractions, proportions and ratios.

Developing Proportional thinking Fewer than half the adult population can be viewed as proportional thinkers And unfortunately…. We do not acquire the habits and skills of proportional reasoning simply by getting older.

Summary of Stage 7 Ideas Find simple equivalent fractions using multiplicative thinking. Convert common fractions as decimal fractions and percentages and vice versa. Add/subtract related fractions, e.g. 2/4+5/8 and decimals, e.g.3.6+ 2.89 (multiplication and division of fractions and decimals is developed further at Stage 8) Solve division problems expressing remainders as fractions or decimals e.g. 8 ÷ 3 = 2 2/3 or 2.66 Estimate and solve percentage type problems such as What % is 35 out of 60, and 46% of 90 Find equivalent ratios and compare by converting to equivalent fractions, equalising the total number of parts or mapping onto 1. Solve simple rate problems

Equivalence halves, thirds, quarters, fifths, tenths A key idea at this stage is equivalence. Which equivalent fractions should students be able to recall and order at this stage? halves, thirds, quarters, fifths, tenths Order these fractions: 2/3 3/4 2/5 5/8 3/8 What did you do to help order them?

How could you communicate this idea of equivalence to students? Paper Folding Fraction Tiles / Strips/ Fraction circles

Differentiating whole class warm ups Circle the biggest fraction B 6/4 or 3/5 7/8 or 9/7 C 4/5 or 3/8 7/8 or 6/9 A ½ or ¼ 1/5 or 1/9 D Adding and Subtracting fractions with related denominators

Renaming fractions as decimals You need to understand equivalent fractions before understanding decimals, as decimals are special cases of equivalent fractions where the denominator is always a power of ten. What is 3/8 as a decimal? (decipipes)

A Fractional Thought for the day Smart people believe only half of what they hear. Smarter people know which half to believe.

A sample of numerical reasoning test questions as used for the NZ Police recruitment

½ is to 0.5 as 1/5 is to a. 0.15 b. 0.1 c. 0.2 d. 0.5

1.24 is to 0.62 as 0.54 is to a. 1.08 b. 1.8 c. 0.27 d. 0.48

Travelling constantly at 20kmph, how long will it take to travel 50 kilometres? a. 1 hour 30 mins b. 2 hours c. 2 hours 30 mins d. 3 hours

If a man weighing 80kg increased his weight by 20%, what would his weight be now? a. 96kg b. 89kg c. 88kg d. 100kg

Misconceptions with Decimal Place Value What number is three tenths less than 2? Why might children have get this wrong? Does the student understand that 1 whole has an infinite number of fraction names. In this case, it would be most sensible to assume ten tenths equals one whole.

Misconceptions with Decimal Place Value: Bernie says that 0.657 is bigger than 0.7 Why do you think he thinks that? Does the student mistakenly regard the digits following the decimal point as representing whole numbers? May take out if not enough time as already covered this earlier.

Misconceptions with Decimal Place Value: Sam thinks that 0.27 is bigger than 0.395 Why do you think he thinks that? Does the student mistakenly think the more decimal places a number has, the smaller the number is because the last place value digit is very small. May take out if not enough time as already covered this earlier.

3.9 plus 4 tenths Jane thinks it is 3.13. Bill thinks it is 7.9. Melinda thinks it is 4.3. Who is correct? What misconceptions do they have? How would you respond to this?

Misconceptions with Decimal Place Value: Martika and Karl are having an argument. Martika says 1.5  10 = 1.50, while Karl says its 10.50 Why do you think he thinks that? What would you do to respond to this? May take out if not enough time as already covered this earlier.

Decimal Misconceptions Summary Decimals are two independent sets of whole numbers separated by a decimal point, e.g. 3.71 is bigger than 3.8 and 1.8 + 2.4 = 3.12 The more decimal places a number has, the smaller the number is because the last place value digit is very small. E.g. 2.765 is smaller than 2.4 Decimals are negative numbers. 1/2 is 0.2 and 1/4 is 0.4, so therefore 0.4 is smaller than 0.2 When you multiply decimals the number always gets bigger. When you multiply a decimal number by 10, just add a zero, e.g. 1.5 x 10 = 1.50

The moral of the story… Decimals need explicit teaching – don’t assume students understand them. Introduce them once students understand how whole numbers work, and have the concept of a fraction. Equipment demonstration – how I introduce decimals Spend time on decimal place value “See, Say, Make”, e.g.0.65

Teaching Decimal Place Value Decimal Fraction Mats Book 4:8-9, (MM 4-21) Decimal Number Lines Book 4:15 (MM 4-31) Candy Bars (Book7)

Operating with decimals Understanding tenths and hundredths using candy bars: Pose division problems using the equipment to find the number of wholes, tenths and hundredths; e.g. 6 ÷ 5, 4 ÷ 5, then 5 ÷ 4, 3 ÷ 4, 13÷ 4 Operate with the decimals using addition/subtraction and multiplication to consolidate understanding requiring exchanging across the decimal point, e.g. 3.6 - 1.95, 3.4 + 1.8, 4.3 - 2.7, 7 x 0.4, 1.25 x 6

Your Choice! – Further PD with… Stage 7 Decimals (Add/sub, Mult/Div) Percentages Ratios and rates Stages 1 – 5 Beginning Fractions

What misconceptions may young children have when beginning fractions? Misconceptions about finding one half when beginning fractions: Share without any attention to equality Share appropriate to their perception of size, age etc. Measure once halved but ignore any remainder So what do we need to teach to move to equal sharing? Introduce the vocabulary of equal / fair shares with both regions and sets for halves and then quarters.

Plan a lesson from Book 7 Focus on the key ideas we have discussed today

What now? Use your data from IKAN and GloSS (Re-GloSS fractions if necessary) to identify class needs. Use long-term planning units for Fractions Teach fraction knowledge and proportions & ratios strategies with your groups/whole class. In-Class Modelling visit

Evaluation

Little League Video Clip

When you rearrange the letters becomes Thought for the day A DECIMAL POINT   When you rearrange the letters becomes I'M A DOT IN PLACE

Thought for the day There are three things to remember when teaching; Know your stuff, Know whom you are stuffing, And stuff them elegantly. Lola May