Fractions, Decimal Fractions and Percentages - a closer look. Jill Smythe Numeracy Facilitator (Based on work by Peter Hughes)
Pirate Problem (While you are waiting - there is a prize for whoever gets the answer first!) • 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?
Objectives: • Identify the progressive strategy stages of fractions, decimal fractions, percentages and ratios. • Further develop teacher’s confidence and content knowledge of fractions, decimal fractions, percentages and ratios. • Explore key ideas, equipment and activities used to teach knowledge and strategy of the above.
Developing Proportional Thinking: • Order the scenarios. • How well did you do? Check by using The Number Framework: Book 1 (Pg 15). • Highlight all the fractional knowledge across the stages (Pg18-22).
Fraction Knowledge Test: • Draw 2 pictures: (a) one half (b) one eighth • Mark 5 halves on a number line • 12 is three fifths of what number? • What is 3 ÷ 5? • Draw a picture of 7 thirds • Write one half as a ratio. • The ratio of kidney beans to green is 3:4. What fraction of the beans are green? • Order these fractions: 2/4, 3/4, 2/5, 7/16, 2/3, 6/49 • Now include these % and decimals into your order 30%, 75%, 0.38, 0.5
Group discussion Fraction questions Why do we have fractions? Shape on board - divided in 3. Are they thirds?
Views of Fractions: • What does this fraction mean?
Views of Fractions: • Does this fraction mean? 3 ÷ 7 3 out of 7 3 : 7 3 over 7 3 sevenths
The Problem with Language: Use words first before using the symbols: e.g. one half not 1 out or 2 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
Models of Fractions: • Show 3 quarters using any of the materials on your table.
0 1 Continuous Model: • Models where the object can be divided in any way that is chosen. e.g. ¾ of this line and this square are blue.
Discrete Model: • Discrete: Made up of individual objects. e.g. ¾ of this set is blue
Whole to Part: • Most fraction problems are about giving students the whole and asking them to find parts. • Show me one quarter of this circle?
Part to Whole: • We also need to give them part to whole problems, like: • 5 is a quarter of this number. What is the number?
Teaching Fractions: • What do you see as some of the confusions associated with the teaching and understanding of fractions?
Misconceptions with Fractions: • Charlotte believes that one eighth is bigger than one half. 1/2 1/3 1/4 1/8 • Why do you think Charlotte has this misunderstanding? • How would you address this misconception? • What equipment would you use?
Misconceptions with Fractions: • Jenna says the following: ¼ + ¼ + ¼ = 3/12 • Why do you think Jenna has this misunderstanding? • How would you address this misconception? • What equipment would you use?
Misconceptions with Fractions: • 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?
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.
Misconceptions with Fractions: • Anna says is not possible as a fraction. Consider….. • Is possible as a fraction? • Why does Anna say this? • What action, if any, do you take?
Key Idea: A fraction can represent more than one whole. Can be illustrated through the use of materials and diagrams. Question students to develop understanding: • Show me 2 thirds, 3, thirds, 4 thirds… • How many thirds in one whole? two wholes? • How many wholes can we make with 7 thirds?
Misconceptions with Fractions: • 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.
Fractional Knowledge: Having secure fractional knowledge is the key to operating successfully with fractions… Stage 4: Identifies and writes common fractions. Stage 5: Orders unit fractions. Stage 6: Identifies any fraction including improper fractions. Stage 7: Knows equivalent fractions for halves, thirds, quarters, fifths and tenths. Stage 8: Orders fractions, decimals and percentages. (Taken from IKAN assessment)
Exploring Book 7 & FIO: Folding Fractions Figure it Out Number Lev.2 Pg.18 Stage 5-6: Birthday Cakes Page 26 Focus on the following : • 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. • Link to the planning units and Figure It Out.
Fractions as Operators Sione has 35 marbles. At the end of play he only has 3 sevenths of his marbles left. At the end of lunch he has only two fifths of the marbles he started lunchtime with. How many marbles does he have left? Can you draw a grid to show your thinking?
Fractions as Operators So we are finding out a fraction of a fraction. 2/5 of 3/7
Fractions as Operators So we are finding out a fraction of a fraction. 2/5 of 3/7 We start with the 3/7 of 35 Which is 15.
Fractions as Operators So we are finding out a fraction of a fraction. 2/5 of 3/7 We start with the 3/7 of 35 Which is 15. Now we need to work out 2/5 of 15 Which is 6.
Multiplying with Fractions • When you multiply by some fractions the answer gets smaller • This is ⅓ of one whole strip. • If it is cut into quarters, four equivalent pieces, what will each new piece be called? 1/3 1/12
Summary of key ideas: • Fractional language - emphasise the “ths” code. • Fraction symbols - use symbols with caution. Combine language, symbols and a visual representation to consolidate understanding. • Continuous and discrete models - use both. • Go from Part-to-Whole as well as Whole-to-Part. • Fractions are always relative to the whole. • Begin to teach fraction when children are at stage 2!! • Use the teaching model to develop conceptual understanding.
Introducing Decimal Fractions: Present the class with unifix cubes that have been wrapped in tens. Each represents one bar of chocolate with 10 pieces. How many pieces/tenths would each person get? 2 ÷ 5 = ?
Discuss: • Why should we introduce decimal fractions by division of whole numbers rather than telling students that that the places after the decimal point represent tenths, hundredths, thousandths and so on?
Reason: • The whole must be broken into ten parts – canon of place value (ten for one) • Teaching division of whole numbers leads to fractional answers E.g 3 ÷ 7 = 3 sevenths of one whole • This is an essential precursor to decimal fractions
Solve these using Materials: • 7 ÷ 2 = ? Wholes and ? Tenths • 8 ÷ 5 = ? Wholes and ? Tenths • 6 ÷ 4 = ? Wholes and ? Tenths • 5 ÷ 2 = ? Wholes and ? Tenths • 2 ÷ 4 = ? Wholes and ? Tenths
Why have we used words? • Words emphasise what fractions really are. • Stops students thinking of fractions as division, out of, over or ratios. • Develops a deeper understanding of what fractions represent. • The number 3.5 should be read as 3 and 5 tenths rather than 3 point 5.
Solve this: • 4.5 ÷ 3 = • 4 wholes ÷ 3 = 1 whole for each share with 1 whole left over. This 1 whole is split into ten tenths – so there are now fifteen tenths altogether. 15 tenths ÷ 3 = 5 tenths for each share. Each share is 1 whole and 5 tenths (1.5)
I Whole - take one large piece of paper • Make 1 tenth • Make 1 hundredth
Becoming familiar with the equipment: • Decimal Fraction Mats (book 4, page 8 and book 7 pg 45). Prove that 1.7 is bigger than 1.68 using one of the above materials.
Misconceptions with Decimal Fractions: • Jill believes that 0.45 is less than 0 • Why do you think Jill has this misunderstanding? • How would you address this misconception? • What equipment would you use?
Misconceptions with Decimal Fractions: • John says the following: “one half is the same as 0.2” • Why do you think John has this misunderstanding? • How would you address this misconception? • What equipment would you use?
Key Idea: Decimals are special cases of equivalent fractions in that they always involve tenths, hundredths, or thousandths, etc. • John is not correct. ½ is 0.5 ( 5 tenths) and 0.2 is two tenths • Practise in needed in order to convert fractions to equivalent decimals
Misconceptions with Decimal Fractions: • Mary has to order the following fractions from smallest to largest:- • 0.67 0.8 0.532 • She orders them as follows:- • 0.8 0.67 0.532 • Consider…. • Is Mary correct? • Why/Why not? • What are the misconceptions? • What action, if any, do you take?
Misconceptions with Decimal Fractions: • Jacob added the following decimals together:- 3.4 + 1.8 = 4.12 • Why do you think Jacob has this misunderstanding? • How would you address this misconception? • What equipment would you use?
The Connection between Fractions and Percentages: What 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?
Hot Shots • Book 7 P 47 - 49 • Extending Hot Shots P 56 - 60
% Problems 20% of 150 is 20% of is 30 % of 150 is 30
Question (in context) The local dairy farmer is selling 20% of his herd of 150 cows. How many is he selling? Rewrite in maths language 20% of 150 is 150 0% 20% 100%