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To be a proportional thinker you need to be able to think multiplicatively

To be a proportional thinker you need to be able to think multiplicatively. How do you describe the change from 2 to 10?. Additive Thinking : Views the change as an addition of 8 Multiplicative Thinking: Views the change as multiplying by 5. Proportional Thinking.

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To be a proportional thinker you need to be able to think multiplicatively

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  1. To be a proportional thinker you need to be able to think multiplicatively How do you describe the change from 2 to 10? Additive Thinking: Views the change as an addition of 8 Multiplicative Thinking: Views the change as multiplying by 5

  2. Proportional Thinking A sample of numerical reasoning test questions as used for the NZ Police recruitment

  3. ½ is to 0.5 as 1/5 is to a. 0.15 b. 0.1 c. 0.2 d. 0.5

  4. 1.24 is to 0.62 as 0.54 is to a. 1.08 b. 1.8 c. 0.27 d. 0.48

  5. Travelling constantly at 20kmph, how long will it take to travel 50 kilometres? a. 1 hour 30 mins b. 2 hours c. 2 hours 30 mins d. 3 hours

  6. If a man weighing 80kg increased his weight by 20%, what would his weight be now? a. 96kg b. 89kg c. 88kg d. 100kg

  7. Developing Proportional thinking Fewer than half the adult population can be viewed as proportional thinkers And unfortunately…. We do not acquire the habits and skills of proportional reasoning simply by getting older. What is proportional thinking?

  8. Pre-Stage 7 • What fraction and decimal ideas should you already know about before moving to Stage 7

  9. Fractional Key Ideas Pre-Stage 7

  10. Fractions Teaching Key Ideas • Use sets as well as regions from early on and connect different representations 1 quarter Sets (Discrete Models) Shapes/Regions(Continuous models)

  11. Fractions Key Ideas • Use sets as well as regions from early on and connect different representations. • Use words first then introduce symbols with care. • How do you explain the top and bottom numbers? 1 2 The number of parts chosen The number of parts the whole has been divided into

  12. 2 3 1 2 3 5 8 6 2 3 The problem with “out of” + = “I ate 1 out of my 2 sandwiches, Kate ate 2 out of her 3 sandwiches so together we ate 3 out of the 5 sandwiches”!!!!! x 24 = 2 out of 3 multiplied by 24! = 8 out of 6 parts!

  13. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. Fractions Key Ideas 6 is one third of what number? This is one quarter of a shape. What does the whole look like?

  14. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. • Division is the most common context for fractions when units of one are not accurate enough for measuring and sharing problems. • Initially this is done by halving and halving again. • e.g. 3 ÷ 5 Fractions Key Ideas

  15. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. • Division is the most common context for fractions, e.g. 3 ÷ 5 • Fractions are not always less than 1. Push over 1 early to consolidate the understanding of the top and bottom numbers. Fractions Key Ideas Y7 responses: What is this fraction? 5/2 2 fifths, five lots of halves, tenth, five twoths

  16. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. • Division is the most common context for fractions, e.g. 3 ÷ 5 • Fractions are not always less than 1. Push over 1 early to consolidate understanding. • Fractions are numbers as well as operators Fractions Key Ideas 3/4 is a number between 0 and 1 (number) Find three quarters of 80(operator)

  17. 20 60 3 5 100 0 1 5 1 0 x3 Place 3/5 on the number line. (number) Find 3/5 of 100. (operator). Using a double number line or bead string

  18. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. • Division is the most common context for fractions, • Fractions are not always less than 1. Go over 1. • Fractions are numbers as well as operators. • Fractions are always relative to the whole. Fractions Key Ideas Sam had one half of a cake, Julie had one quarter of a cake, so Sam had most. True or False or maybe

  19. What is B? What is the whole? (Trains Book 7, p32)

  20. Use sets as well as regions from early on and connect different representations. • Use words first & introduce symbols with care. • Go from part-to-whole as well as whole-to-part with both shapes and sets. • Division is the most common context for fractions, e.g. 3÷5 • Fractions are not always less than 1. Go over 1. • Fractions are numbers as well as operators • Fractions are always relative to the whole. • Fractions are really a context for applying add/sub and mult/div strategies. Fractional Key Ideas – Pre Stage 7

  21. Stage 7 Decimals and Percentages Decimals are special cases of equivalent fractions where the denominator is always a power of ten.

  22. Stage 7 (AM) Level 4 Key Ideas Fractions • Rename improper fractions as mixed numbers, e.g. 17/3 = 52/3 • Find equivalent fractions using multiplicative thinking,, e.g. 2/6 = how many twelfths? • Order fractions using equivalence and benchmarks like 1 half , e.g. 2/5 < 11/16 • Add and subtract related fractions, e.g. 2/4 + 5/8 • Find fractions of whole numbers using mult’n and div’n e.g.2/3 of 36 and 2/3 of ? = 24 • Multiply fractions by other factions e.g.2/3 x ¼ • Solve measurement problems with related fractions, e.g. 1½ ÷ 1/6 =9/6 ÷ 1/6 =9 Decimals • Order decimals to 3dp • Round whole numbers and decimals to the nearest whole or tenth • Solve division problems expressing remainders as decimals, e.g. 8 ÷ 3 = 22/3 or 2.66 • Convert common fractions, i.e. halves, quarters etc. to decimals and percentages • Add and subtract decimals, e.g. 3.6 + 2.89 Percentages • Estimate and solve percentage type problems like ‘What % is 35 out of 60?’, and ‘What is 46% of 90?’ using benchmark amounts like 10% & 5% Ratios and Rates • Find equivalent ratios using multiplication and express them as equivalent fractions, e.g. 16:8 as 8:4 as 4:2 as 2:1 = 2/3 • Begin to compare ratios by finding equivalent fractions, building equivalent ratios or mapping onto 1). • Solve simple rate problems using multiplication, e.g. Picking 7 boxes of apples in ½ hour is equivalent to 21 boxes in 1½ hours.

  23. Misconceptions with Decimal Place Value:How do these children view decimals? • Bernie says that 0.657 is bigger than 0.7 • Sam thinks that 0.27 is bigger than 0.395 • James thinks that 0 is bigger than 0.5 • Adey thinks that 0.2 is bigger than 0.4 • Claire thinks that 10 x 4.5 is 4.50

  24. Developing understanding of decimal tenths and hundredths place value The CANON law in our place value system is that 1 unit must be split into TEN of the next smallest unit AND NO OTHER!

  25. Developing Decimal Place Value Understanding • Use decipipes, candy bars, or decimats to understand how tenths and hundredths arise and what decimal numbers ‘look like’ • Make and compare decimal numbers, e.g. Which is bigger? 0.6 and 0.47 • How much more make.. e.g. 0.47 + ? = 0.6

  26. Using Decipipes • establish the whole, half, quarter rods then tenths • 1 half = ? tenths • 1 quarter = ? tenths + • 1 eighth = ? tenths? + View children’s response to this task Now compare: 0.4 0.38 0.275

  27. Using candy bars 3 ÷ 5 3 chocolate bars shared between 5 children. 30 tenths ÷ 5 = 0 wholes + 6 tenths each = 0.6

  28. Using decimats and arrow cards

  29. 2. Make and compare decimals • Which is bigger: 0.6 or 0.43? • How much more make…

  30. Add and subtract decimals • Rank these questions in order of difficulty. • 0.8 + 0.3, • b) 0.6 + 0.23 • c) 0.06 + 0.23, Exchanging ten for 1 Mixed decimal values Same decimal values

  31. Add and subtract decimals (Stage 7) using decipipes or candy bars Place Value Tidy Numbers 1.6 - 0.98 Reversibility Equal Additions Standard written form (algorithm)

  32. When you multiply the answer always gets bigger. True False 0.4 x 0.3Which is the correct answer?0.12 1.2 0.012

  33. Decimals multiplied by a whole number(Stage 8) Using candy bars or decipipes Tidy Numbers Place Value 7 x 0.2 Written form Proportional Adjustment

  34. Decimal Multiplied by a Decimal (Stage 8) • Convert to a fraction, i.e. • 0.25 x 0.8 is the same as 1 quarter of 0.6 • 2. Use Arrays e.g. 0.4 x 0.3 Ww w

  35. Using Arrays 0.4 x 0.3 = 0.12 0 0.3 1 0.4 Ww w 1

  36. Division of decimals by a decimal Sue had 2.5 kg of fruit, if it takes 0.5 kg of fruit to make 1 jar of jam, how many jars can Sue make? 2.5 ÷ 0.5 Division of decimals by a whole number 4.2 metres of string is cut int 7 equal lengths, how long is each length? 4.2 ÷ 7

  37. ‘Target Time’ (from FIO Number L3 Book 2 page 16) Target Number is 6 + = • Roll a dice and place the number thrown. • Try and make the number sentence as close to the target number as possible. • Score = the difference between your total and the target number.

  38. Decimal Keyboard

  39. Decimal Games and Activities • Digital learning Objects: http://digistore.tki.org.nz/ec/viewMetadata.action?id=L1079 • Decimal Sort • First to the Draw • Four in a Row Decimals • Beat the Basics • Decimal Keyboard • FIO N3–4:2 Fraction Distraction

  40. Thought for the day A DECIMAL POINT When you rearrange the letters becomes I'M A DOT IN PLACE

  41. = Why calculate percentages? It is a method of comparing fractions by giving both fractions a common denominator i.e. hundredths. So it is useful to view percentages as hundredths.

  42. Applying Percentages Types of Percentage Calculations at Level 4 (stage 7) • Estimate and find percentages of amounts, • e.g. 25% of $80 • Expressing quantities as a percentage • (Using equivalence – Jo’s workshop) • e.g. What percent is 18 out of 24?

  43. Estimate and find percentages of whole number amounts. 25% of $80 Using common conversions halves, thirds, quarters, fifths, tenths Book 8:21 (MM4-28) , Decimats. Bead strings, slavonic abacus Practising instant recall of conversions Bingo, Memory, I have, Who has, Dominoes, 35% of $80 Using benchmarks like 10%, and ratio tables FIO: Pondering Percentages NS&AT 3-4.1(p12-13)

  44. Find __________ (using benchmarks and ratio tables)

  45. Find 35% of $80 $80

  46. Find 35% of $80 $80

  47. Find 35% of $80

  48. 10% $8 30% $24 5% $4 $4 $8 $8 $8 Find 35% of $80 35% $28

  49. Now try this… 46% of $90

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