Fractions Workshop

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

Proportions and Ratios Workshop Session 1: 9:15-10:30 Warm-up, Police Test, What is Proportional Thinking?- Key Ideas Model “Fraction Circles” lesson. Play Dotty pairs. Session 2: 11:00-12:15 Stage 7 & 8 Equivalent Fractions. Play games/become familiar with activities. Session 3: 1:00-2:45 Stage 6+ Decimal Fractions, Percentages, Ratios and Rates Explore lessons from Book 7.

Proportions and Ratios Workshop “He has no sense of proportion”

Warm Up- I have… Who has…

Reviewing Multiplication and Division…(Stage 7 = whole numbers, Stage 8 = decimals) Tidy Numbers Place Value 7 x 0.29 7.2 ÷ 4 Proportional Adjustment Written form

A sample of numerical reasoning test questionsas 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

Proportional Thinking

To be proportional thinker you need to be able to think multiplicatively How do you describe the change from 2 to 10? Additive Thinking: views the change as an addition of 8 Multiplicative Thinking: Describes the change as multiplying by 5

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. What is proportional thinking?

Revision of Early Proportional Thinking (Level 1 - 3) STRATEGY KNOWLEDGE

Key Idea 1Introduce fractional language with care Use words first then introduce symbols. e.g. ‘1 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

2 3 1 2 3 5 8 6 2 3 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”!!!!! x 24 = 2 out of 3 multiplied by 24! = 8 out of 6 parts!

Key Idea 2 Connect different representations words - symbols – regions - sets - numberlines one quarter Sets (Discrete Models) Shapes/Regions(Continuous models)

Year 7 Student Responses 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 Key Idea 3 Fractions can be more than 1 Push over 1 whole to consolidate the meaning of fraction symbols and how 1 whole is created.

Sam had one half of a cake, Julie had one quarter of a cake, so Sam had most. True or False or Maybe Key Idea 4 Fractions are always relative to the whole. Halves are not always bigger than quarters, it depends on what the whole is. Sam Julie

Communicating this idea to students What is B? (what’s the part, what’s the whole?)

1/2 1/2 1/2 1/2 1/2 Key Idea 5 Division is the most common context for fractions i.e. quotients (Wafers, Book 7 p.16) 5 children share three chocolate bars evenly. How much chocolate does each child receive? What are these pieces called? 3 ÷ 5 2/12 !! What would have been an easier way to solve this? 1/2 +1/10 =

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

Choose your share of chocolate!

A B C D E F 0 1 2 3 Which letter represents 5 halves? Key Idea 6 Fractions are numbers as well as operators e.g. 3/5 is a number between 0 and 1 (number) e.g. Find 3/5of 100 (operator)

3 5 100 0 1 5 1 0 Communicating fractions as operators to students using double number lines x3 20 60 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 / double number line to find 3/5 of 100. (Fractions as an operator). How will you work it out?

3 5 1 4 1 0 Operating with fractions - Result Unknown The distance between Masterton and Wellington is 80 kilometres. Hemi has travelled 3/4 of the trip. How many kilometres is that? 80 20 20 20 20 x3 80 60 20 0

3 5 1 9 Tino has travelled 16km, which is four ninths of his journey. How much further does he need to travel? What is the number sentence for this problem? Operating with fractions - Change Unknown 4/9 of ? = 16, 16 is four ninths of what number? Draw a diagram to help to solve this problem. 16 4 4 4 4 4 ÷ 4 X 9 16 36 0 0 4 9 1

Key Idea Go from part-to-whole (change unknown) as well as whole-to-part (result unknown) with shapes and sets. Your turn…. If this is one quarter of a shape. What does the whole look like? 12 is two thirds of what number?

Lesson Modelling Fraction Circles (Book 7, p20) Play “Dotty Pairs” game

Session 2: Stages 7 & 8

Finding Fractions Throw 2 dice and make a fraction, e.g. 4 and 5 could be 4 fifths of 5 quarters. Try and make a true statement each time the dice is thrown. Throw dice 10 times, Miss a go if you cannot place a fraction.

Moving to Stage 7 and 8 (Level 4-5) STRATEGY KNOWLEDGE

Equivalent Fractions

Objectives • Teaching Fractional equivalence • In order to apply understanding to: • Adding and subtracting with fractions • Multiplying and dividing with fractions

What do students need to know about fractions before Stage 7? Any fraction equivalence??

Stage 7 (AM) Key Ideas (level 4) Fractions and Decimals • Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3 • Find equivalent fractions using multiplicative thinking, and order fractions using equivalence and benchmarks. e.g. 2/5 < 11/16 • Convert common fractions, to decimals and percentages and vice versa. • Add and subtract related fractions, e.g. 2/4 + 5/8 • Add and subtract decimals, e.g. 3.6 + 2.89 • Find fractions of whole numbers using multiplication and division e.g.2/3 of 36 and 2/3 of ? = 24 • Multiply fractions by other factions e.g.2/3 x ¼ • Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 =9/6 ÷ 1/6 =9 • Solve division problems expressing remainders as fractions or decimals e.g. 8 ÷ 3 = 22/3 or 2.66 Percentages • Estimate and solve percentage type problems such as ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10% and 5% Ratios and Rates • Find equivalent ratios using multiplication and express them as equivalent fractions, e.g. 16:8 as 8:4 as 4:2 as 2:1 = 2/3 • Begin to compare ratios by finding equivalent fractions, building equivalent ratios or mapping onto 1). • Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour is equivalent to 21 boxes in 1½ hours.

Stage 8 (AP) Key Ideas (level 5) Fractions and Decimals • Add and subtract fractions and mixed numbers with uncommon denominators, 2/3 + 14/8 • Multiply fractions, and divide whole numbers by fractions, recognising that division can result in a larger answer, e.g. 4 ÷ 2/3 = 12/3 ÷ 2/3 = 6 • Solve measurement problems with fractions like ¾ ÷ 2/3 by using equivalence and reunitising the whole • Multiply and divide decimals using place value estimation and conversion to known fractions, e.g. 0.4 × 2.8 = 1.12 (0.4< ½ ) , 8.1 ÷ 0.3 = 27 (81÷ 3 in tenths) • Find fractions between two given fractions using equivalence, conversion to decimals or percentages Percentages • Solve percentage change problems, e.g. The house price rises from $240,000 to $270,000. What percentage increase is this? • Estimate and find percentages of whole number and decimal amounts and calculate percentages from given amountse.g. Liam gets 35 out of 56 shots in. What percentage is that? Ratios • Combine and partition ratios, and express the resulting ratio using fractions and percentages, e.g. Tina has twice as many marbles as Ben. She has a ratio of 2 red to 5 blue. Ben’s ratio is 3:4.If they combine their collections what will the ratio be? i.e. 2:5 + 2:5 + 3:4 = 7:14 = 1:2, • Find equivalent ratios by identifying common whole number factors and express them as fractions and percentages, e.g. 16:48 is equivalent to 2:6 or 1:3 or ¼ or 25% Rates: • Solve rate problems using common whole number factors and convertion to unit rates, e.g. 490 km in 14 hours is an average speed of 35 k/h (dividing by 7 then 2). • Solve inverse rate problems, e.g. 4 people can paint a house in 9 days. How long will 3 people take to do it?

How could you communicate this idea of equivalence to students? Multiplicative thinking Fraction Circles Paper Folding x2 1/4 = ?/8 x2 Fraction Tiles / Strips

Equivalence Games/Activities • Fraction Frenzy, FIO Number Level 3, Book 3 • www.maths-games.org Click on “Fraction Games” (Fraction Booster, Fraction Monkeys, Melvin’s Make-a-Match, Fresh Baked Fractions) • Fraction circles/wall and dice game • Fraction bingo, pictures then words • The Equivalence Game: PR3-4+ p.18-19

Once you understand equivalence you can…… • Compare and order fractions • Add and Subtract fractions • Understand decimals, as decimals are special cases of equivalent fractions where the denominator is always a power of ten.

What did you do to order them? Circle the bigger fraction of each pair. B 6/4 or 3/5 7/8 or 9/7 7/3 or 4/6 C 7/16 or 3/8 2/3 or 5/9 5/4 or 3/2 D 7/10 or 6/8 7/8 or 6/9 5/7 or 7/9 Key IdeaOrdering using equivalence and benchmarks A ½ or ¼ 1/5 or 1/9 5/9 or 2/9 unit fractions More or less than 1 related fractions unrelated fractions Example of Stage 8 fraction knowledge 2/3 3/42/5 5/83/8

Which is bigger? (Order/compare fractions: Stage 7) 4/5 or 2/3 12/15 10/15

Find fractions between two fractions, using equivalence: Stage 8 Feeding Pets 3/4 2/3 What fractions come between these two?? Usefulness of decimal conversion and equivalent fraction methods? Both are equivalent fraction methods. When is one method easier than another? When the fractions are easier to convert to decimals fifths, tenths, halves, quarters, eighths or commonly known ones- eg. thirds).

Tri Fractions Game for comparing and ordering fractions FIO PR 3-4+

Add and Subtract related fractions (Stage 7) e. g ¼ + 5/8 Fraction circles / fraction wall tiles *Play create 3 (MM 7-9) • halves, quarters, eighths • halves, fifths, tenths • halves, thirds, sixths What could you use to help students understand this idea?

Add and Subtract fractions with uncommon denominators (Stage 8) e.g. 2/3 + 9/4 Using fraction circles / fraction wall tiles *Play “Fractis” • How?? • Find common denominators/equivalent fractions

Create 3 (MM7-9) Each player chooses a fraction to place their counter on Take turns to move your counter along the lines to another fraction Add the new fraction to your total. The first player to make exactly three wins. Go over three and you lose.

Multiplying Fractions (Stage 7) Using fraction circles / fraction wall tiles • 6 x ¼ Using paper folding / wall tiles / OHT fractions/drawing • ½ x ¼ Using multiplicative thinking, not additive

Fractions Workshop