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Bridging the Gap

L3. L4. to. Problem-Solving Pupils’ Version. Bridging the Gap. Revising Ratio and Proportion. What Will You Learn ?. Never heard of ratio before? How exactly do you keep numbers in proportion ? What’s the difference between ratio , proportion and fractions ?

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Bridging the Gap

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  1. L3 L4 to Problem-SolvingPupils’ Version Bridging the Gap Revising Ratio and Proportion .

  2. What Will You Learn ? Never heard of ratio before? How exactly do you keep numbersin proportion? What’s the difference between ratio, proportion and fractions? If you work through this slideshow, you will find out that: • Ratio and proportion are EASY! • Fractions will make more sense too • You can work out most of the maths in your head!

  3. How to use this resource You can control how fast or slow you go using: FORWARD: OR OR Enter OR Left-hand mouse BACK: OR OR Back Space TO START POWERPOINT:F5 ORSlide Show > View Show TO RETURN TO MENU:Escape

  4. Contents Slides 5 – 16: Part 1: Ratio, Proportion & Fractions What is ratio? What is proportion? What’s the connection? Where do fractions fit in? Slides 17 – 26 Part 2: Simplest Ratios and Proportions Equivalent ratios. Simplest ratios. Equivalent proportion. Simplest proportions. Slides 27 – 44 Part 3: Recipes & Proportion How to scale a recipe for 10 people down to 4 people without disaster. (TO GO TO LINK, HOLD DOWN CONTROL AND CLICK ON YOUR CHOICE)

  5. DO’s and DON’Ts Working Through DO go as fast or as slow as you like! You can use the mouse or arrow keys to control the speed of the slideshow. DON’T try to do it all in one go – there’s loads of stuff. Working Out You can work a lot of the questions out in your head, -but keep a pen and paper handy.! To START, press F5 To STOP, press ESC

  6. First Thoughts … There’s lots of differences between the classes in your school – … Some have more boys than girls … Less United fans than Rovers fans … More cat-lovers than dog-lovers … Less pizza-munchers than chicken-dippers

  7. Form 6A has 30 pupils: 10 Rovers fans 20 United fans Form 6B has 18 pupils: 9 Rovers fans 9 United fans First Thoughts … You’re a Rovers fan. Which class would you rather be in? Discuss and take a class vote.

  8. 6A: 10 +20 = 30 6B: 9 +9 = 18 – Rani, a mad Rovers fan, picked 6A: Did You Pick 6A? “10 is bigger than 9, so there’s more Rovers fans in 6A than 6B. Yey!”

  9. 6A: 10 +20 = 30 6B: 9 +9 = 18 - But other red Rovers fans picked 6B: Did You Pick 6B? 2 to 1 is a ratio 1 out of 3 is a proportion third and half are fractions MARTIN SAYS “I’m not going to 6A - I’d be outnumbered 2 United to every 1 Rovers fan!” BECKY SAYS: “Only one third of 6A Rovers fans. It’s one half in 6B. HANNAH SAYS: “6A only has 10 out of 30 Rovers fans. That’s 1 out of 3.” Who is right? Discuss and take a class vote! All answers are correct! But - 3 different sorts of answer. Which do you prefer? Why?

  10. Recap – 3 Ways to Compare Numbers Which question is a … “6A only has 10 out of 30 Rovers fans. That’s 1 out of 3.” Fraction? “I’m not going to 6A. I’d be outnumbered 2 to every 1.” Ratio? Proportion? “Only one third of 6A Rovers fans. It’s one half in 6B.”

  11. What’s the Difference - Summing Up:. 3 different ways to say the same thing 3 different ways to compare numbers “1 to every 2” is a Ratio “1 Rovers fan to every 2 United fans” “The ratio of Rovers to United fans is 1 to 2” “The ratio of United to Rovers fans is 2 to 1” “1 out of 3” is a Proportion “1 out of every 3 fans is a Rovers fan” “The proportion of Rover fans is 1 out of 3” “One third” is a Fraction “One third of all fans are Rovers fans “1/3 of all fans are Rovers fans” Ratios compare PART WITH PART Proportions compare PART WITH WHOLE Fractions compare PART WITH WHOLE using shorthand such as 1/3

  12. So comparing totals isn’t the only way to look at these numbers… Form 6A has 30 pupils: 10 Rovers fans 20 United fans Form 6B has 18 pupils: 9 Rovers fans 9 United fans …You can compare part with part using RATIO …You can compare part with whole using Eg: Rovers : United = 1 to every 2 = 1:2 PROPORTIONS or FRACTIONS Eg: 1 out of 3 are Rovers fans Eg: 1/3 are Rovers fans

  13. Ratio, Proportion or Fraction? two out of five This is a … two fifths This is a … proportion fraction four tenths This is a … four to every ten This is a … fraction ratio ten to every four This is a … four out of ten This is a … ratio proportion 4/10 This is a … 4:10 This is a … fraction ratio

  14. Ratio, Proportion or Fraction? 3 United fans to every 2 Rangers fans This is a … ratio 9 girls out of 10 use soap This is a … proportion 3 boys out of 10 use deodorant This is a … proportion

  15. Ratio, Proportion or Fraction? 3 out of 4 pizza-munchers love olives This is a … proportion Half of all girls in 6D love History This is a … fraction One third of all road accidents involve drinking This is a … fraction

  16. Ratio, Proportion or Fraction? 3 out of 4 drivers speed at some time This is a … proportion Three quarters of drivers speed This is a … fraction 3 drivers speed to every 1 which does not This is a … ratio

  17. Ratio and Proportion Ratio and Proportion: Different Words! Ratio: “to every” 1 Rovers fan to every 2 United fans Proportion: “out of” 1 out of 3 football fans is a Rovers fan

  18. PART 2: Simplest Ratios and Proportions

  19. The dinner queue 4 girls to every 8 boys girls : boys = 4 : 8 Ratio of girls to boys? But the simplest ratio is still 1 : 2 3 girls to every 6 boys girls : boys =3 : 6 But the simplest ratio is still 1 : 2 2 girls to every 4 boys girls : boys =2 : 4 But the simplest ratio is still 1 : 2 1 girl to every 2 boys girls : boys =1 : 2 This is the simplest ratio is 1 : 2

  20. The dinner queue 8 boys to every 4 girls boys : girls = 8 : 4 Ratio of boys to girls? But the simplest ratio is still 2 : 1 6 boys to every 3 girls boys : girls = 6 : 3 But the simplest ratio is still 2 : 1 4 boys to every 2 girls boys : girls = 4 : 2 But the simplest ratio is still 2 : 1 2 boys to every 1 girl boys : girls = 2 : 1 This is the simplest ratio is 2 : 1

  21. Simplest ratios 2 : 1 is a simpler ratio than 4 : 2 but They both mean the same 2 : 1 = 4 : 2 Can you explain why 2 : 1 = 6 : 3 ? 2 : 1 = 8 : 4 ? 2 : 1 = 100 : 50 ?

  22. The dinner queue 4 girls out of 12 Simplest proportion of girls is still 1 out of 3 What proportion is girls? 3 girls out of 9 Simplest proportion of girls is still 1 out of 3 2 girls out of 6 Simplest proportion of girls is still 1 out of 3 1 girl out of 3 This is the simplest proportion of girls = 1 out of 3

  23. The dinner queue Proportion of boys? 8 boys out of 12 Simplest proportion of boys? Simplest proportion of boys is still 2 out of 3 What proportion is boys? 6 boys out of 9 Proportion of boys? Simplest proportion of boys is still 2 out of 3 Simplest proportion of boys? 4 boys out of 6 Proportion of boys? Simplest proportion of boys is still 2 out of 3 Simplest proportion of boys? Proportion of boys? 2 boys out of 3 Simplest proportion of boys? This is the simplest proportion of boys = 2 out of 3

  24. What fraction is girls? 4 / 12 The dinner queue Simplest fraction of girls is still 1 / 3 Simplest proportion of girls? What fraction is girls? 3 / 9 What fraction is girls? Simplest fraction of girls is still 1 / 3 Simplest fraction of girls? 2 / 6 What fraction is girls? Simplest fraction of girls is still 1 / 3 Simplest fraction of girls? 1 / 3 What fraction is girls? This is the simplest fraction of girls = 1 / 3 Simplest fraction of girls?

  25. The dinner queue What fraction is boys? 8 / 12 Simplest proportion of boys? Simplest fraction of boys is still 2 / 3 What fraction is boys? 6 / 9 What fraction is boys? Simplest fraction of boys is still 2 / 3 Simplest fraction of boys? 4 / 6 What fraction is boys? Simplest fraction of boys is still 2 / 3 Simplest fraction of boys? 2 / 3 What fraction is boys? This is the simplest fraction of boys = 2 / 3 Simplest fraction of boys?

  26. Does order matter? There are 20 boys and 10 girls in 6D. Which of these are correct? b) girls : boys = 2 : 1 a) boys : girls = 2 : 1 c) boys : girls = 1 : 2 boys : girls = 2 : 1 means 2 boys to every girl girls : boys = 2 : 1 means 2 girls to every boy Click to check your answer ORDER MATTERS! Be careful what you write!

  27. GUESS! True or Not True? TRUE! Ratio of teachers to pupils = 30 : 1 Ratio of fans to players = 1 : 1000 This would mean all the players would be sitting in the stadium! Ratio of weekdays to weekend days = 2 : 5 You wish! Ratio of porridge lovers to pizza guzzlers = 100 : 1 - Not until they invent Porridge Takeaways .. TRUE! TRUE! TRUE!

  28. PART 3: Recipes and Proportion How to scale a recipe for 10 people down to 4 people without disaster!

  29. Whoops ! Biscuits and Bananas Skins Enjoy cooking? Check out these fantastic recipes! Some of them need changing to suit the number of people, - But remember to keep them in proportion, - And watch out for the or else …

  30. It’s supper time! You make a simple omelette like this: + = 2 eggs + 1 teaspoon of butter = 1 omelette Your omelette tastes amazing! - So of course your mates start turning up. Can you scale up your ingredients to feed them all? 4 eggs + 2 tsp of butter = 2 omelettes ? eggs + ? tsp of butter = 2 omelettes ? eggs + ? tsp of butter = 3 omelettes 6 eggs + 3 tsp of butter = 3 omelettes ? eggs + ? tsp of butter = 4 omelettes 8 eggs + 4 tsp of butter = 4 omelettes ? eggs + ? tsp of butter = 5 omelettes 10 eggs + 5 tsp of butter = 5 omelettes

  31. And the next week … + = 2 eggs + 1 teaspoon of butter = 1 omelette … word’s getting around and more mates turn up the next day. Can you scale up your ingredients to feed them? 4 eggs + ? tsp of butter = ? omelettes 4 eggs + 2 tsp of butter = 2 omelettes 10 eggs + ? tsp of butter = ? omelettes 10 eggs + 5 tsp of butter = 5 omelettes ? eggs + 10 tsp of butter = ? omelettes 20 eggs + 10 tsp of butter = 10 omelettes ? eggs + 11 tsp of butter = ? omelettes 22 eggs + 11 tsp of butter = 11 omelettes

  32. STEP 1: CHOOSE the maths! x or ÷ ? BIGGER AMOUNTS mean X SMALLER AMOUNTS mean ÷ RECAP How did you work these out? STEP 2: DO the maths! DO THE SAME X or ÷ to ALL ingredients (This is called ‘keeping it in proportion’) STEP 3: CHECK the maths! - using ratio ! Eg: If there are 2 eggs to every 1 tsp butter, the ‘eggs number’ is ALWAYS twice the ‘butter number’. Hungry for more? Try these recipes … REMEMBER THE 3 STEPS

  33. Recipe for Shortbread Biscuits Makes 20 shortbread biscuits Ingredients 200g butter 200g plain flour 100g golden caster sugar 100g fine semolina Pre-heat the oven to gas mark 2, 300°F (150°C). You will also need an 8 in (20 cm) diameter fluted flan tin, 1¼ in (3 cm) deep with a loose base.

  34. 10 Mouth-watering Shortbread Biscuits! This is the recipe for 20 biscuits. Can you scale down the recipe for 10?Remember to keep your recipe in proportion – if you halve the sugar, remember to halve the rest as well or it won’t taste good! STEP 1: Choose the maths! x or ÷ ? BIGGER means X SMALLER means ÷ STEP 3: CHECK the maths! - using ratio. Eg: The weight of the butter is always 2 times the weight of the sugar. STEP 2: Do the maths! DO THE SAME X or ÷ to ALL ingredients 20 biscuits 400g flour 200g sugar 200g semolina + + + = 400g butter ? ? ? ÷ 2 ÷ 2 ÷ 2 ? ? ÷ 2 ÷ 2 10 biscuits 200g flour 100g sugar 100g semolina + + + = 200g butter

  35. 3 Steps for Scaling RECAP STEP 1: CHOOSE the maths! CHOOSE FROM x or ÷ … BIGGER means X … SMALLER means ÷ STEP 2: DO the maths! DO THE SAME X or ÷ to ALL ingredients STEP 3: CHECK the maths! - using ratio. Eg: If the weight of the butter is always 2 times the weight of the sugar…

  36. 20 biscuits 400g flour 200g sugar 200g semolina + + + = 400g butter Whoops ! … Nirmal makes 2 shortbread biscuits Mistake in STEP 1: Choose the maths! CHOOSE FROM + - x ÷ BIGGER means X SMALLER means ÷ BUT… Nirmal did X 10 instead of ÷10 Nirmal scaled down the recipe for 2 biscuits, but they don’t taste right!Did he scale down correctly and keep the recipe in proportion? Check his working out Then click to see if you’re right 20 biscuits ÷ 10 ÷ 10 ÷ 10 ÷ 10 x 10 ? ? ? x10 x 10 x 10 ? ? x 10 ÷ 10 200 BISCUITS INSTEAD OF 2! WHAT DIFFERENCE DID THE MISTAKE MAKE? 2 biscuits 40 g flour 20 g sugar 4000 g flour 2000 g sugar 20 g semolina 2000 g semolina 40 g butter + + + = 4000 g butter

  37. 3 Steps for Scaling RECAP STEP 1: CHOOSE the maths! CHOOSE FROM x or ÷ … BIGGER means X … SMALLER means ÷ STEP 2: DO the maths! DO THE SAME X or ÷ to ALL ingredients STEP 3: CHECK the maths! - using ratio. Eg: If the weight of the butter is always 2 times the weight of the sugar…

  38. 20 biscuits 400g flour 200g sugar 200g semolina + + + = 400g butter Whoops ! … Liana makes 5 shortbread biscuits Mistake in STEP 2: Do the maths! DO THE SAME X or ÷ to ALL ingredients BUT …Liana did not do ÷ 2 to the sugar. Check her working out Then click to see if you’re right Liana scaled down the recipe for 5 biscuits, but they don’t taste right!Did she scale down correctly and keep the recipe in proportion? ÷ 4 ÷ 2 ÷ 4 ÷ 4 ? ÷ 4 ÷ 4 THE BISCUITS WERE TOO SWEET! WHAT DIFFERENCE DID THE MISTAKE MAKE? 5 biscuits 100 g flour 50 g semolina 100 g butter 100 g sugar 50 g sugar + + + =

  39. 3 Steps for Scaling RECAP STEP 1: CHOOSE the maths! CHOOSE FROM x or ÷ … BIGGER means X … SMALLER means ÷ STEP 2: DO the maths! DO THE SAME X or ÷ to ALL ingredients STEP 3: CHECK the maths! - using ratio. Eg: If the weight of the butter is always 2 times the weight of the sugar…

  40. Further Practical ExamplesRecipe No.1 Melon Merenga Serves 8 people Ingredients 300 g raspberries 200 g bananas 100 g melon Method Place ingredients in a juicer and switch on power for 30 seconds. Pour and serve with ice or ice-cream.

  41. STEP 1: Choose the maths! x or ÷ ? BIGGER means X SMALLER means ÷ STEP 3: CHECK the maths! - using ratio. Eg: The bananas’ weight is ALWAYS twice the melons’ weight. Can you work out the missing amounts? STEP 2: Do the maths! DO THE SAME X or ÷ to ALL ingredients Melon Merenga for 8 + + = 600 g 400 g 200 g MelonMerenga for 4 + + = ? g ? g ? g General Method: 1) Work out the recipe for 1. 2) Multiply by the number you need Shortcut: 1) Work out from the quantities for 2 people. 2) X 3 (easier than X6) 300 g 200 g 100 g MelonMerenga for 2 + + = ? g ? g ? g 150 g 100 g 50 g You can work this out from the quantities for 1 person, BUT … Can you see a shortcut? MelonMerenga for 1 + + = ? g ? g ? g 75 g 50 g 25 g MelonMerenga for 6 + + = ? g ? g ? g 450 g 300 g 150 g

  42. Further Practical ExamplesRecipe No.2 Raspberry Fruitloop(Same ingredients. Different amounts) Serves 10 people Ingredients 500 g raspberries 250 g bananas 150 g melon Method Place ingredients in a juicer and switch on power for 30 seconds. Pour and serve with fresh raspberries

  43. STEP 1: Choose the maths! x or ÷ ? BIGGER means X SMALLER means ÷ STEP 3: CHECK the maths! - using ratio. Eg: The raspberries’ weight is ALWAYS twice the melons’ weight. Can you work out the missing amounts? STEP 2: Do the maths! DO THE SAME X or ÷ to ALL ingredients Raspberry Fruitloop for 10 + + = 500 g 250 g 150 g Raspberry Fruitloop for 5 General Method: 1) Work out the recipe for 1. 2) Multiply by the number you need Shortcut: 1) Work out from the quantities for 4 people. 2) Just double it! (Easier than X8) + + = ? g ? g ? g 250g 125g 75g Raspberry Fruitloop for 1 + + = ? g ? g ? g 50g 25g 15g You can work this out from the quantities for 1 person, BUT … Can you see any shortcuts? Raspberry Fruitloop for 4 + + = ? g ? g ? g 200g 100g 60g Raspberry Fruitloop for 8 + + = ? g ? g ? g 400g 200g 120g

  44. Further Practical ExamplesRecipe No.3 Bombastic Banana Boat(Same ingredients. Different amounts) Serves 12 people Ingredients 120 g raspberries 600 g bananas 120 g melon Method Place ingredients in a juicer and switch on power for 30 seconds. Pour into a scooped out melon half . Serve with fresh raspberries

  45. STEP 1: Choose the maths! x or ÷ ? BIGGER means X SMALLER means ÷ STEP 3: CHECK the maths! - using ratio. Eg: The weight of bananas is always 5 times the weight of the melon. Can you work out the missing amounts? STEP 2: Do the maths! DO THE SAME X or ÷ to ALL ingredients Banana Boat for 12 + + = 120 g 600 g 120 g Banana Boat for 6 + + = ? g ? g ? g 60 g 300 g 60 g General Method: 1) Work out the recipe for 1. 2) Multiply by the number you need Shortcut: 1) Work out from the quantities for 4 people. 2) Just double it! (Easier than X8) Banana Boat for 4 + + = ? g ? g ? g 40 g 200 g 40 g You can work this out from the quantities for 1 person, BUT … Can you see a shortcut? Banana Boat for 1 + + = ? g ? g ? g 10 g 50 g 10 g Banana Boat for 8 + + = ? g ? g ? g 80 g 400 g 80 g

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