Pickups 2010 Fractional Thinking. Lisa Heap, Jill Smythe & Alison Howard Numeracy Facilitators. Pirate Problem. While you are waiting… Three pirates have some treasure to share. They decide to sleep and share it equally in the morning.
Pickups 2010Fractional Thinking Lisa Heap, Jill Smythe & Alison Howard Numeracy Facilitators
Pirate Problem While you are waiting… • 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, proportions and ratios. • Further develop teacher’s confidence and content knowledge of fractions. • Explore key ideas, equipment and activities used to teach fraction knowledge and strategy.
The 4 Stages of the P.D 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 Reacting to the Student Interpret and respond to what the student is doing
Developing Proportional Thinking: A chance to recap what needs to be taught at the different stages. • Decide which strategy stage fits each scenario. • Use the number framework to help you. • 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 from 1-5 • 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
Morning Tea • After morning tea we will split again; Stages 1, 2, 3 – with Alison Stages 4 – 8 – with Jill and Lisa
Ratios: • In the rectangle below, what is the ratio of green to blue cubes? • 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?
More on Ratios…. • Divide a rectangle up so that the ratio of its blue to green parts is 7:3. • Think of other ways that you can do it. • What is the fraction of each colour? • If I had 60 cubes how many of them will be of each colour?
A Ratio Problem to Solve: • There are 27 pieces of fruit. The ratio of fruit that I get to the fruit that you get is 2:7. How many pieces do I get? • How many pieces would there have to be for me to get 8 pieces of fruit? • What key mathematical knowledge is required here?
What about this? • Two students are measuring the height of the plants their class is growing. • Plant A is 6 counters high. • Plant B is 9 counters high. • When they measure the plants using paper clips they find that Plant A is 4 paper clips high. • What is the height of Plant B in paper clips ? Consider….. Scott thinks Plant B is 7 paper clips high. Wendy thinks Plant B is 6 paper clips high. • Who is correct? • What is the possible reasoning behind each of their answers? • How would you further support Scott’s thinking?
Key Idea: The key to proportional thinking is to be able to see combinations of factors within numbers. • Wendy is correct, Plant B is 6 paper clips high. • Scott’s reasoning: To find Plant B’s height you add 3 to the height of Plant A; 4 + 3 = 7. • Wendy’s reasoning: • Plant B is one and a half times taller than Plant A; 4 x 1.5 = 6. • The ratio of heights will remain constant. 6:9 is equivalent to 4:6. • 3 counters are the same height as 2 paper clips. There are 3 lots of 3 counters in plant B, therefore 3 x 2 = 6 paper clips.
Exploring Book 7: Stage 4-5: Fraction Circles (page 20) Stage 5-6: Birthday Cakes (page 26) Stage 6-7 Hot Shots (page 46)
Views of Fractions: • What 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 equal parts the whole has been divided into
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 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?
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: • Fiona says the following: ¼ + ¼ + ¼ = 3/12 • Why do you think Fiona 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. • Key is to always refer to the whole. This will be dependent on the problem!
Misconceptions with Fractions: • Heather says is not possible as a fraction. Consider….. • Is possible as a fraction? • Why does Heather 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?
What could be the misconception here? • 2 chocolate bars shared amongst 5 students: • What does each student get?
Reason: • Because the divisor is 5 the natural denominator is fifths. Each bar is broken into five equal pieces. • One way of solving the problem is to give each student one piece from each bar. Each will have 2 pieces. Compared with one bar each student has 2 fifths of a bar. • The common error here is for students to think the answer is 2/10 because they think the answer is 2 out of 10.
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.
⅓ Misconceptions with Fractions • When you multiply by some fractions the answer gets smaller • 1/4 x 1/3 = 1/12 • This is ⅓ of one whole strip. • If it is cut into quarters, four equivalent pieces, what will each new piece be called?
Fractions Video: • What was the key purpose of the lesson? • What key mathematical language was being developed? • How did materials/equipment support the children’s learning? What may have happened if the equipment was not present? • Why did the teacher use the example 101/4 in the lesson? • In terms of the teaching model, where do you think the children are at? • What would be you next step with this group of children?
Summary of key ideas: • Fraction language - emphasise the “ths” code • Fraction symbols – use symbols with caution, start with words • 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 and the whole can be bigger than one.
Thought for the day: • Smart people believe only half of what they hear. • Smarter people know which half to believe.