Developing a Whole School Approach to Problem Solving and Word Problems Day 1 Dr Paul Swan and David Dunstan Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 1
Proficiency and Content Strands Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 2
What is a problem? • A problem is a situation where there is no obvious method of solution. • There is no basic routine to follow in order to solve the problem. • Students need to think about it. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 3
Four Stages of Problem Solving • Understand the problem • (See) • Devise a plan • (Plan) • Carry out the plan • (Do) • Look back • (Check) Polya It is better to solve one problem many ways, than many problems one way. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 4
Download the 16 page KenKen booklet from www.drpaulswan.com.au Go to the Resources section Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 5
Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 6
A 4 by 4 grid. The digits 1, 2, 3 and 4 are used. The operation is Addition. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 7
Always fill the ‘Freebie’ cells. The ‘Target Number (TN)’ is given. For example, in the bottom LHC, the TN is 1. 3 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 8
Look at the Green horizontal 2-cell Cage. 2 A Cage is defined by the heavy outline and includes the TN and generally, the operation. Why must the top LHC be ‘2’? Each row and column must use the digits 1, 2, 3 and 4, with NO REPEATS. 3 For a sum of 3, only 1 and 2 are used. In the first column, 1 cannot be repeated. 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 9
Another Strategy! 2 Why must the top LHC be ‘2’? Each row and column add to 10. So in the first column, 1 + 7 = 8 and 2 more makes 10 (or, 10 – 8 = 2). 3 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 10
In the RED cage, the sum of 7 can only be made with 3 and 4, 2 1 or 4 and 3. Due to the NO REPEAT rule, in row 3, the red cell must be a 4 . 3 4 3 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 11
Why is the bottom RHC = 2? 2 1 2 = 10 – (5 + 3) [each row and column adds to 10], or For two digits to make a sum of 5 (PINK cage), 3 the possible number sets are (1,4), (4,1), (2,3) and (3,2). Thus the 2 must go in the bottom RHC. 4 3 4 2 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 12
Remember, No Repeats; 2 4 1 so ‘4’ goes in the Top RHC (1 cannot go there). 3 1 4 3 4 2 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 13
Complete the top and bottom rows. 2 3 4 1 3 1 4 3 3 4 2 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 14
Complete the remaining cells. 2 3 4 1 Kengratulations! 3 4 2 1 4 2 1 3 3 4 2 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 15
A 4 by 4 Grid. The digits 1, 2, 3 and 4 are used. The operation is Addition. No Freebies. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 16
To make a sum of 3, use 1 and 2. 3 1 To make a sum of 4, use 1 and 3. So why must 3 be in the top LHC? Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 17
Why MUST ‘4’ go in the bottom LHC? 3 1 First column, 10 – (3 + 3) = 4 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 18
Complete the PINK cage . 3 2 1 1 2 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 19
Complete the RED cage. 3 2 1 Remember, No Repeats of ‘1’ in the second row. 2 1 1 2 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 20
Complete the YELLOW cage. 3 2 4 1 2 1 3 1 2 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 21
Complete the GREY cage. 3 2 4 1 2 1 3 1 2 3 1 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 22
Complete the remaining cages. 3 2 4 1 2 4 1 3 1 3 4 2 2 3 1 4 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 23
Expe pert! Wher ere e is a s a st strateg egic st starting c cel ell? www.drpaulswan.com.au |
“Thinking Clouds” can be used to assess student reasoning. Students can also add in their own “Thinking Clouds” to demonstrate their reasoning ability. Dating these records will assist teachers to monitor student progress. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 25
Partitions of 10 How many different ways can you partition (split) 10? 3 and 7 3 and 3 and 2 and 1 and 1 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 26
42 Partitions of 10 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 27
Decide what digits go in the boxes? • Can you find a solution? • How many solutions do you think there might be? • How will you know when you have found them all? Hurst, C. (2011). Connecting with the Australian Curriculum: Integrating learning through the proficiency strands. In J. Clark, B. Kissane, J. Mousley, T. Spencer, & S. Thornton (Eds.). Mathematics: Traditions and [New] Practices (Proceedings of the 34th annual conference of the Mathematics Education Research Group of Australasia and the Australian Association of Mathematics Teachers), pp. 973-980. Adelaide: AAMT and MERGA. Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 28
Questions • What sort of a number sentence do we have? • How big are the numbers? • What do we know about the numbers? • What digits could be in the ones place? • What numbers can we multiply together to make a number ending in 6? Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 29
Hints • 1, 6 what about 6, 1? • 2, 3 • 6, 6 Can you use a digit more than once? • 2, 8 • 7, 8 • 4, 4 • 4, 9 Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 30
Open – Ended Tasks Every school should buy at least one copy Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 31
One example • I have three coins in my hand, How much money might I have? • From Open – Ended Maths Activities, Sullivan and Lilburn Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 32
I have three coins in my hand. How much money do I have? Some Prompts: What is the lowest total? What is the highest total? Can any totals be made in more than one way? What are all the possible totals? How will you know when you have all of the solutions? (Parallel Problems) Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 33
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Solve a simpler – related problem: 2 Coins Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 35
Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 36
Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 37
Is it the Maths or Literacies that is the Problem? Newman Analysis 5 Pr Prom ompts 1. Please read the question to me. If you don't know a word, leave it out (Decoding) 2. Tell me what the question is asking you to do (Comprehending) 3. Tell me how you are going to find the answer (Transforming) Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 38
Newman Analysis cont 4. Show me what to do to get the answer. "Talk aloud" as you do it, so that I can understand how you are thinking (Applying) 5. Now, write down your answer to the question (Encoding into required format) Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 39
Word Problems • “70% of errors made by students on standard word problems could be attributed to lack of comprehension” (Clements, 2004) • However, “There is significant research evidence that reveals that recognising words in isolation does not necessarily mean they will be recognised as quickly in connected text.” (Ehri, 1997) Dr Paul Swan and David Dunstan Developing a Whole School Approach www.drpaulswan.com.au | 40
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