mathematics leadership community matamata

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Mathematics Leadership CommunityMatamata

Term 1 2013

Honor Ronowicz honorr@waikato.ac.nz

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I OREA TE TUATARA KA PUTA KI WAHO

A problem is solved by continuing to find solutions

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Today’s Agenda

• Cluster Data• Leadership in Maths• Maths Time• Updates and ideas

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Dotty 63

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5

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2Green wins!

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Data 2012

• What do you notice?• Celebrations?• Concerns?• What else may we need to know?• What Professional Development do we need in

2013?

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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?

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Decimal DiveYou 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

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Maths Lead teacher?

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Proportions and ratios

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• www.nzmaths.co.nz• Fractions tutorial

• Key ideas about fractions• How to communicate these to students

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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

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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.

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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…

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Number of books

Fraction of books that are fiction

Number of fiction books

Steve’s family 30 15

Andrew’s family 100 20

Number of books

Fraction of books that are fiction

Number of fiction books

Steve’s family 40 20

Andrew’s family 40 8

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

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

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Key Idea: The size of the fractional amount depends on the size of the whole.

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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?

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Scenario Two

•You observe the following equation in Emma’s work:

• + =

•Is Emma correct?

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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?

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•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.

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Key Idea 1:

When working with fractions, the whole needs to be clearly identified.

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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?

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Key Idea 2:

When adding fractions, the units need to be the same because the answer can only have one denominator.

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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:

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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 ?

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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?

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• 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.

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Key Idea:

The key to proportional thinking is being able to see combinations of factors within numbers.

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Use ratio tables to identify the multiplicative relationships between the numbers involved.

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• Use double-number lines to help visualise the relationships between the numbers.

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Scenario Four

•Anna says is not possible as a fraction.

Consider…..

•Is possible as a fraction?

•What action, if any, do you take?

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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.

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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

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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?

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• 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.

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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

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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:

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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?

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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?

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Key Idea:

Equivalent fractions have the same value.

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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

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• 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.

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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?

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Key Idea:

The more pieces a whole is divided into, the smaller each piece will be.

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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?

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•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.11

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Benchmarking

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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.

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• 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.

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The Big Ideas

• Use lots of equipment. • Allow explorations, investigations and

discussions.• Don’t rush to teaching rules!

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Updates and Ideas

• Nzmaths• Studyladder• Multiplication.com• Nrich

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Ten Principles of Effective Mathematics Teaching

1. Ethic of care2. Arranging for learning3. Building on student’s thinking4. Worthwhile mathematical tasks5. Making connections6. Assessment for learning7. Mathematical communication8. Mathematical language9. Tools and representations10. Teacher knowledge

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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.

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Communicating with whānau

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

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Noho ora mai