Mathematics Leadership Community Matamata Term 1 2013 Honor Ronowicz honorr@waikato.ac.nz

I orea te tuatara ka puta ki waho            A problem is solved by continuing to find solutions This whakatauaki has a literal meaning of using a spear to probe a tuatara from its hole – by being persistent you will succeed. Note, some versions use I orea te tuatara ka patu ki waho with patu being ‘to hit’ I orea te tuatara ka puta ki waho  

Today’s Agenda Cluster Data Leadership in Maths Maths Time Updates and ideas

Dotty 6 3 4 2 1 5 Green wins! With thanks to nzmaths Robyn to feedback after the game Green wins! With thanks to nzmaths

Data 2012 What do you notice? Celebrations? Concerns? What else may we need to know? What Professional Development do we need in 2013? Photocopy for all Where is Year 7 and 8? Gloss moderation at a more “natural time” at the next meeting perhaps? Print National Expectations from NZ maths so they can compare. Trajectory for Proportions and ratios domain falls off and less are achieving at Year 6 than are achieving. Ethic groups? Gender?

School Targets What are yours? What is in place to achieve them? How has your school aligned Teaching as Inquiry cycles with School targets? What does your long term plan look like in regards to your school targets? Buddy share or small groups? Feedback to all? Depending on how much they wan to share Find out about Curriculum Plans at this time also

Refer to this if needed

Decimal Dive You need: 2 dice Goal: To be the first player to reach zero. How to play: Each player starts at 21. Take turns to throw both dice. Choose which digit is ones and which digit is tenths. Subtract this number from 21. Continue subtracting from your previous score, when ‘1’ is reached, use one dice. The winner is the first player to reach zero exactly. Acknowledgements to Margot Nielsen Brainstorm before moving on What is your role as a lead teacher?

Maths Lead teacher? Then show the slide.

Proportions and ratios With thanks to nzmaths

Key ideas about fractions How to communicate these to students www.nzmaths.co.nz Fractions tutorial Key ideas about fractions How to communicate these to students This tutorial is available on nzmaths, Homework. To do at school before the next meeting. With thanks to nzmaths

In your groups Look at each scenario and consider…. Are they right or wrong? What is the thinking behind their answer? What is the key idea the student needs to develop in order to solve this problem? What will you do in your teaching now? Consider equipment/ representations you could use? What knowledge may be required? Hand out scenarios Work in groups Feed back at end of session. 15 mins to go Vivids and paper on desks With thanks to nzmaths

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.

Scenario 1 -Summary 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… Make the point it depends on how much they own With thanks to nzmaths

Fraction of books that are fiction Number of fiction books 1 Number of books Fraction of books that are fiction Number of fiction books Steve’s family 30 15 Andrew’s family 100 20 Andrew’s family has more fiction books than Steve’s. Number of books Fraction of books that are fiction Number of fiction books Steve’s family 40 20 Andrew’s family 8 Steve’s family has more fiction books than Andrew’s.

1 Key Idea: The size of the fractional amount depends on the size of the whole. This is so often a problem. Students really do not see what we mean by the whole! What can we so about it With thanks to nzmaths

1 What can we do? Demonstrate with clear examples, as in the previous tables. Use materials or diagrams to represent the numbers involved (if appropriate). Question the student about the size of one whole: Is one half always more than one fifth? What is the number of books we are finding one fifth of? How many books is that? What is the number of books we are finding one half of? How many books is that? When working with students on fractions it’s important to define what the “whole” is in all situations. The best way to do this will depend on the student and the problem. With thanks to nzmaths

Scenario Two You observe the following equation in Emma’s work: + = + = Is Emma correct?

but wait…. You question Emma about her understanding and she explains: “I ate 1 out of the 2 sandwiches in my lunchbox, Kate ate 2 out of the 3 sandwiches in her lunchbox, so together we ate 3 out of the 5 sandwiches we had.” What, if any, is the key understanding Emma needs to develop in order to solve this problem? With thanks to nzmaths

Emma needs to know that the relates to a different whole than the 2 Emma needs to know that the relates to a different whole than the If it is clarified that both lunchboxes together represent one whole, then the correct recording is: + = Emma also needs to know that she has written an incorrect equation to show the addition of fractions. 2 key ideas her

2 Key Idea 1: When working with fractions, the whole needs to be clearly identified. With thanks to nzmaths

2 What can we do? Use materials or diagrams to represent the situation. For example: Question the student about their understanding. The one out of two sandwiches refers to whose lunchbox? Whose lunchbox does the two out of three sandwiches represent? Whose lunchbox does the three out of five sandwiches represent? Be careful about the language “out of” Ways to communicate this could be counting fractions and going over the 1 Use bsicuits With thanks to nzmaths

2 Key Idea 2: When adding fractions, the units need to be the same because the answer can only have one denominator. Counting fractions will help this use tiles to demonstrate With thanks to nzmaths

2 What can we do? Use a diagram or materials to demonstrate that fractions with different denominators cannot be added together unless the units are changed. For example: You could relate this to the idea of apples and oranges: If we add 2 apples and 3 oranges together, the answer is neither 5 apples or 5 oranges but 5 pieces of fruit. In order to add apples and oranges together we need to change the counting unit. The equation is better described as 2 pieces of fruit, plus 3 pieces of fruit equals 5 pieces of fruit. With thanks to nzmaths

Scenario Three 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 ? With thanks to nzmaths

Consider… Scott thinks Plant B is 7 paper clips high. 3 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? With thanks to nzmaths

Wendy is correct, Plant B is 6 paper clips high. Scott’s reasoning: 3 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. Additive thinking With thanks to nzmaths

3 Key Idea: The key to proportional thinking is being able to see combinations of factors within numbers. With thanks to nzmaths

3 Use ratio tables to identify the multiplicative relationships between the numbers involved. Note there are other relationships: The difference between the height in toothpicks and the height in counters of each plant is x 1.5 The difference between Plant B and Plant A’s height (in counters or toothpicks) is x two-thirds. Multi link cubes With thanks to nzmaths

3 Use double-number lines to help visualise the relationships between the numbers. With thanks to nzmaths

Scenario Four Consider….. Anna says is not possible as a fraction. Is possible as a fraction? What action, if any, do you take?

Key Idea: A fraction can be more than one whole. 4 Key Idea: A fraction can be more than one whole. The denominator tells the number of equal parts into which a whole is divided. The numerator specifies the number of these parts being counted. Compare this to the view of fractions, where one third is seen as “one out of three.” This “out of” approach can be limiting. With thanks to nzmaths

What can you do… Use materials and diagrams to illustrate. 4 What can you do… Use materials and diagrams to illustrate. 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? Let’s try Go over the whole Count fractions Question their understanding of top and bottom numbers Out of thinking can be limiting for them. With thanks to nzmaths

Scenario Five Consider….. 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? With thanks to nzmaths

No he is not correct. The correct equation is 5 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. Note that it is a common misconception that “division makes smaller” because division involving natural numbers results in a quotient smaller than the dividend. (dividend divided by divisor equals quotient). For example in the equation 10 divided by 5 equals 2, 2 is less than 10. This is not always the case with division (or multiplication) involving fractions. Dividing by a number less than one makes the number larger. For example 2 divided by ¼ equals eight, eight is more than 2. With thanks to nzmaths

5 Key Idea To divide the number A by the number B is to find out how many lots of B are in A. so Division is the opposite of multiplication. The relationship between multiplication and division can be used to help simplify the solution to problems involving the division of fractions. For example: There are 4 lots of 2 in 8 There are 5 lots of 1/2 in 2 1/2 Sometimes referred to as the “measurement” model of division With thanks to nzmaths

5 What can you do? Use meaningful representations for the problem. For example: I am making hats. If each hat takes 1/2 a metre of material, how many hats can I make from 2 1/2 metres? Use materials or diagrams to show there are 5 lots of 1/2 in 2 1/2: With thanks to nzmaths

Or….. Use contexts that make use of the inverse operation: 5 Or….. Use contexts that make use of the inverse operation: A rectangular vegetable garden is 2.5 m2. If one side of the garden is 1/2 a metre long, what is the length of the other side? Half of a skipping rope is 2.5 metres long. How long is the skipping rope? The “partitive” and “product and factors” models of division make use of the inverse properties of division and multiplication. The vegetable garden context is an example of the “product and factors” model of division and the skipping rope context is an example of the “partitive” model of division. The learning object found at http://www.nzmaths.co.nz/learningobjects/fractions/index.swf will also help develop this idea. With thanks to nzmaths

Scenario Six Consider: Which shape has of its area shaded? Sarah insists that none of the shapes have of their area shaded. Consider: Do any of the shapes have of their area shaded? What action, if any, do you take? With thanks to nzmaths

Key Idea: Equivalent fractions have the same value. 6 Key Idea: Equivalent fractions have the same value. With thanks to nzmaths

What can you do… Use diagrams or materials to show equivalence. 6 Paper folding Cut up pieces of fruit to show, for example, that one half is equivalent to two quarters. Fraction tiles Book 7 paper folding FIO Allow exploration of equipment to discover stuff With thanks to nzmaths

6 Question students about their understanding. For example, using the fraction tiles you could ask: How many twelfths take up the same amount of space as two sixths? How many sixths take up the same amount of space as one third? Can you see any other equivalent fractions in this wall? Record the equivalent fractions as they are identified. With thanks to nzmaths

Scenario Seven Consider: You observe the following equation in Bruce’s work: Consider: Is he correct? After checking that Bruce understands what the “>” symbol means, what action, if any, do you take? With thanks to nzmaths

Key Idea: The more pieces a whole is divided into, the smaller each piece will be. Kims story Equipment discover stuff With thanks to nzmaths

What can you do? Demonstrate the relative size of fractions with materials or diagrams. Question students about the relative size of each fractional piece: If we had 2 pizzas and we cut one pizza into six pieces and the other into 4 pieces, which pieces would be bigger? The use of one half as a reference point can be supported by the idea that “half” can be found by dividing any denominator in 2. In the example above, half of 11 is five and a half, so 8/11 is larger than one half. This reasoning can also be extended to use one quarter and three quarters as reference points. With thanks to nzmaths

Benchmarking The use of reference points 0, 1/2 and 1 can be useful for ordering fractions larger than unit fractions. For example: Which is larger is larger than one half and is less than one half.

Key Ideas about Fractions The size of the fractional amount depends on the size of the whole. When working with fractions, the whole needs to be clearly identified. A fraction can represent more than one whole. When adding fractions, the units need to be the same because the answer can only have one denominator. Equivalent fractions have the same value. The key to proportional thinking is being able to see combinations of factors within numbers. With thanks to nzmaths

Division is the opposite of multiplication. The denominator tells the number of equal parts into which a whole is divided. The numerator specifies the number of these parts being counted. How many of what! Division is the opposite of multiplication. The relationship between multiplication and division can be used to help simplify the solution to problems involving the division of fractions. The more pieces a whole is divided into, the smaller each piece will be. With thanks to nzmaths

The Big Ideas Use lots of equipment. Allow explorations, investigations and discussions. Don’t rush to teaching rules!

Updates and Ideas Nzmaths Studyladder Multiplication.com Nrich

Click on Professional Development

Ten Principles of Effective Mathematics Teaching Ethic of care Arranging for learning Building on student’s thinking Worthwhile mathematical tasks Making connections Assessment for learning Mathematical communication Mathematical language Tools and representations Teacher knowledge Go back to ‘professional development’

New module 5a – Making judgements in aspect of mathematics Under Professional development – Illustrations and the national standards

Studyladder Awarded ‘Best Website for Teaching and Learning 2012’ Aligned with the National Standards Problem Solving Now includes thousands of real-life questions Sharing Ideas Teachers will have the ability to share ideas and resources with others If every class at your school signs up to use Studyladder in 2013, they will give all students extended homework access at no cost. Contact support@studyladder.co.nz for more details. Address issues in media.

Communicating with whānau http://nzcurriculum.tki.org.nz/National-Standards/Supporting-parents-and-whanau/Resources#snapshot_pdfs

Noho ora mai