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Mathematics Leadership Community Matamata

Mathematics Leadership Community Matamata. Term 1 2013. Honor Ronowicz honorr@waikato.ac.nz. A problem is solved by continuing to find solutions. I orea te tuatara ka puta ki waho. Today’s Agenda. Cluster Data Leadership in Maths Maths Time Updates and ideas. 3 4 2 1 5 3 5 1

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Mathematics Leadership Community Matamata

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

  2. A problem is solved by continuing to find solutions I orea te tuatara ka puta ki waho

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

  4. 3 4 2 1 5 3 5 1 2 Dotty 6 Green wins! With thanks to nzmaths

  5. Data 2012 • What do you notice? • Celebrations? • Concerns? • What else may we need to know? • What Professional Development do we need in 2013?

  6. 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?

  7. 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

  8. Maths Lead teacher?

  9. Proportions and ratios With thanks to nzmaths

  10. www.nzmaths.co.nz • Fractions tutorial • Key ideas about fractions • How to communicate these to students With thanks to nzmaths

  11. In your groupsLook 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? With thanks to nzmaths

  12. 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.

  13. 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… With thanks to nzmaths

  14. 1 Andrew’s family has more fiction books than Steve’s. Steve’s family has more fiction books than Andrew’s.

  15. 1 Key Idea:The size of the fractional amount depends on the size of the whole. With thanks to nzmaths

  16. 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? With thanks to nzmaths

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

  18. 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

  19. 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.

  20. 2 Key Idea 1: When working with fractions, the whole needs to be clearly identified. With thanks to nzmaths

  21. 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? With thanks to nzmaths

  22. 2 Key Idea 2: When adding fractions, the units need to be the same because the answer can only have one denominator. With thanks to nzmaths

  23. 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: With thanks to nzmaths

  24. 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

  25. 3 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? With thanks to nzmaths

  26. 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. 3 With thanks to nzmaths

  27. 3 Key Idea: The key to proportional thinking is being able to see combinations of factors within numbers. With thanks to nzmaths

  28. Use ratio tables to identify the multiplicative relationships between the numbers involved. 3 With thanks to nzmaths

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

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

  31. 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. With thanks to nzmaths

  32. 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 With thanks to nzmaths

  33. Scenario Five • 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

  34. 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. 5 With thanks to nzmaths

  35. 5 Key IdeaTo divide the number A by the number B is to find out how many lots of B are in A.soDivision 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 With thanks to nzmaths

  36. 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

  37. 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? With thanks to nzmaths

  38. Scenario Six • 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

  39. 6 Key Idea: Equivalent fractions have the same value. With thanks to nzmaths

  40. 6 What can you do… • Use diagrams or materials to show equivalence. • Paper folding • Cut up pieces of fruit to show, for example, that one half is equivalent to two quarters. • Fraction tiles With thanks to nzmaths

  41. 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

  42. Scenario Seven • 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

  43. Key Idea: The more pieces a whole is divided into, the smaller each piece will be. With thanks to nzmaths

  44. 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? With thanks to nzmaths

  45. 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.

  46. 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

  47. 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

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

  49. Updates and Ideas • Nzmaths • Studyladder • Multiplication.com • Nrich

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