Transcript Ratio and proportion : teachers version

Problem-Solving
Teachers’ Version
Bridging the Gap
Aim of resource
The overall aim is to support pupils to move from Level 3 to Level 4 in the area of ratio and
proportion.
This slideshow provides detailed teaching support for the introduction or revision of the basic
concepts of ratio, proportion and fractions, including:
•The terms ‘ratio’, ‘proportion’ and ‘fractions’.
•The language associated with each one.
•The connection between the three.
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)
Teachers’ Note 2: Objectives
Objectives from the Yearly Teaching Programmes
(Page references are to the supplement of examples for Years 4, 5 and 6 of the Primary Framework)
Key Objectives
Y6:
1) Solve simple problems involving ratio and proportion. (pp. 26-27)
2) Identify and use the appropriate operations (including combinations
of operations). (pp. 82-89)
Y5:
1) Know by heart all multiplication facts up to 10 x 10. (pp. 58-59)
2) Derive quickly division facts corresponding to 10 x 10. (pp. 58-59)
Other Objectives
Y5/Y6: 1) Use doubling or halving, starting from known facts (see Recipes). (pp. 60-61)
2) Check the results of calculations with an equivalent calculation. (pp. 72-73)
(Eg: Calculate an ingredient using proportion, but check using ratio)
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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
First Thoughts …
Form 6A has 30 pupils:
Form 6B has 18 pupils:
10 Rovers fans
20 United fans
9 Rovers fans
9 United fans
You’re a Rovers fan.
Which class would you rather be in?
Discuss and take a class vote.
Did You Pick 6A?
6A: 10
+ 20
= 30
6B: 9
+ 9
= 18
–
Rani, a mad Rovers fan, picked 6A:
“10 is bigger than 9,
so there’s more Rovers fans
in 6A than 6B.
Yey!”
Did You Pick 6B?
6A: 10
+ 20
= 30
6B: 9
+ 9
= 18
2 to 1fans picked 6B:
- But other red Rovers
is a ratio MARTIN SAYS
third and half
are fractions
BECKY SAYS:
“Only one third
of 6A Rovers
fans. It’s one
half in 6B.
“I’m not going to 6A - I’d
be outnumbered 2 United
to every 1 Rovers fan!”
1 out of 3
is a
proportion
HANNAH SAYS:
“6A only has 10 out
of 30 Rovers fans.
That’s 1 out of 3.”
WhoButis- 3right?
All answers are correct!
different sorts of answer.
Discuss
take
a class
vote!
Whichand
do you
prefer?
Why?
Recap – 3 Ways to Compare Numbers
“6A only has 10 out
of 30 Rovers fans.
That’s 1 out of 3.”
“I’m not going to 6A.
I’d be outnumbered 2
to every 1.”
“Only one third of 6A
Rovers fans.
It’s one half in 6B.”
Which question
is a …
Fraction?
Ratio?
Proportion?
What’s the Difference - Summing Up: .
3 different ways to
3 different ways to
say the same thing
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
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
Eg: Rovers : United = 1 to every 2 = 1:2
…You can compare part with whole using
PROPORTIONS
or FRACTIONS
Eg: 1 out of 3 are Rovers fans Eg: 1/3 are Rovers fans
Ratio, Proportion or Fraction?
two out of five
two fifths
This is a … proportion This is a … fraction
four tenths
four to every ten
This is a … fraction
This
is
a
…
ratio
ten to every four
four out of ten
This is a … ratio
This is a … proportion
4/10
This is a … fraction
4:10
This is a … ratio
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
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
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
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
PART 2: Simplest Ratios and
Proportions
Part 2 offers teaching support in:
•the development of the concept of equivalent ratios and proportion,
using mental pictures and detailed examples.
4 girls to every 8 boys
The dinner queue
Ratio of girls
to boys?
girls : boys = 4 : 8
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
8 boys to every 4 girls
The dinner queue
Ratio of boys
to girls?
boys : girls = 8 : 4
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
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 ?
The dinner queue
What
proportion
is girls?
4 girls out of 12
Simplest proportion of girls
is still 1 out of 3
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
The dinner queue
What
proportion
is boys?
Proportion
of of
boys?
8 boys out
12
Simplestproportion
proportionof
of
Simplest
boys?
boys is still
2 out of 3
6 boys out
of 9
Proportion
of boys?
Simplest
Simplestproportion
proportionof
of
boys is still
2 out of 3
boys?
4 boys out
of 6
Proportion
of boys?
Simplest
Simplestproportion
proportionofof
boys is still
2 out of 3
boys?
Proportion
of boys?
2 boys out
of 3
Simplest proportion of
boys?
This is the simplest
proportion of boys
= 2 out of 3
The dinner queue
What
fraction is
girls?
What fraction
4 / 12is girls?
Simplest
of of
girls
Simplestfraction
proportion
girls?1 / 3
is still
3 /is9 girls?
What fraction
Simplest
fractionofof
girls
Simplest fraction
girls?
is still 1 / 3
2 /is6 girls?
What fraction
Simplest fraction of girls is
Simplest fraction of girls?
still 1 / 3
1 /is3 girls?
What fraction
This is the simplest
Simplest fraction of girls?
fraction of girls = 1 / 3
The dinner queue
What
fraction is
boys?
What fraction
8 / 12is boys?
Simplestfraction
proportion
of is
Simplest
of boys
boys?
still
2/3
6 /is9 boys?
What fraction
Simplest
fraction
of boys
Simplest
fraction
of is
still
2/3
boys?
4 /is6 boys?
What fraction
Simplest
fraction
of boys
Simplest
fraction
of is
still
2/3
boys?
2 /is3 boys?
What fraction
This
Simplest
is thefraction
simplest
of
fraction of
boys?
boys = 2 / 3
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
Click to check your answer
girls : boys = 2 : 1 means 2 girls to every boy
ORDER MATTERS!
Be careful what you write!
True or Not 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 ..
PART 3: Recipes and Proportion
How to scale a recipe for 10 people down to 4 people
without disaster!
Part 2 offers teaching support in:
• the development of the concept of equivalent ratios and proportion,
using mental pictures and detailed examples.
• the concept of scaling up and down in recipes, keeping the ingredients
in proportion
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 …
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
6
+
? tsp of butter = 3 omelettes
3
? eggs
8
+
? tsp of butter = 4 omelettes
4
? eggs
10
eggs
++
? tsp of butter = 5 omelettes
5
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
10 eggs
+
+
2? tsp of butter = 2? omelettes
5? tsp of butter = 5? omelettes
? eggs
20
eggs
++ 10
10tsp
tspofofbutter
butter==10
? omelettes
omelettes
? eggs
22
eggs
++ 11
11tsp
tspofofbutter
butter==11
? omelettes
omelettes
STEP 1: CHOOSE
the maths!
RECAP
x or
?
How did
you÷work
these out?
BIGGER AMOUNTS mean X
SMALLER AMOUNTS mean ÷
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
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.
10 Mouth-watering Shortbread Biscuits!
This is
recipe
for 20
STEP
1: the
Choose
the maths!
x or ÷ ?
BIGGER means X
STEP the
3: CHECK
biscuits. Can you scale down
recipethe
formaths!
10?
- using ratio.
Eg: The weight of the butter
STEP
2:
Do
the
maths!
Remember to keep your recipe in proportion – ifisyou
halve
the the
sugar,
always
2 times
weight
SMALLER
means
÷
DO
THE
SAME
X
or
÷
to
of the
sugar.
remember to halve the rest as well or it won’t taste
good!
ALL ingredients
400g
+ 400g
+ 200g
flour
butter
sugar
÷?2
÷?2
+
+ 200g
+ 100g
flour
butter
sugar
=
semolina
÷?2
200g
200g
+
÷?2
100g
semolina
20
biscuits
÷?2
=
10
biscuits
STEP 1: CHOOSE the maths!
CHOOSE FROM x or ÷
RECAP
… BIGGER means X
… SMALLER means ÷
3 Steps for Scaling
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…
… Nirmal makes 2 shortbread biscuits
Nirmal
Mistake in STEP 1: Choose the maths!
scaled
down
the+ recipe
for 2 biscuits,
CHOOSE
FROM
- his
x ÷ working
Check
out but
Did
BIGGER means X
SMALLER means ÷
heBUT…
scale
down
andofkeep
Nirmal
didcorrectly
X 10 instead
÷ 10 the
+ 400g
+ 200g
flour
butter
sugar
÷ 10
?
x10
flour
recipe in proportion?
Then click to see if you’re right
400g
40 gg
4000
they don’t taste right!
÷10
10
x?
+
÷10
10
x?
200g
semolina
÷10
10
x?
20 gg + 2000
20 gg
40 gg + 2000
+ 4000
butter
sugar
=
semolina
=
20
20
biscuits
biscuits
÷ 10
x?10
2
WHAT
200
DIFFERENCE
biscuits
BISCUITS
DID THE
INSTEAD OF
MISTAKE
2!
MAKE?
STEP 1: CHOOSE the maths!
CHOOSE FROM x or ÷
RECAP
… BIGGER means X
… SMALLER means ÷
3 Steps for Scaling
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…
… Liana makes 5 shortbread biscuits
Check
her2:for
working
outbut they don’t taste right!
Mistake
STEP
Do5 the
maths!
Liana scaled down
theinrecipe
biscuits,
DO THE SAME X or ÷ to ALL ingredients
BUT …Liana did not do ÷ 2 to the sugar.
Did she scale
keep theright
recipe in proportion?
Thendown
clickcorrectly
to seeand
if you’re
400g
+ 400g
+ 200g
flour
butter
sugar
+
200g
=
semolina
20
biscuits
÷4
÷4
÷4
÷2
100 g
+ 100 g
50 gg
+ 100
flour
butter
sugar
+
÷4
50 g
semolina
÷?4
=
5
WHAT
THE
DIFFERENCE
biscuits
BISCUITS
DID THE
WERE TOO
MISTAKE
SWEET!
MAKE?
STEP 1: CHOOSE the maths!
CHOOSE FROM x or ÷
RECAP
… BIGGER means X
… SMALLER means ÷
3 Steps for Scaling
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…
Further Practical Examples
Recipe 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.
STEP 1: Choose the maths!
x or ÷ ?
BIGGER means X
STEP 3: CHECK the maths!
- using ratio.
Eg: The bananas’ weight is
ALWAYS twice the melons’
weight.
Do the maths!amounts?
Can you work outSTEP
the2:missing
SMALLER means
÷
DO THE SAME X or ÷ to
ALL ingredients
=
Melon
Merenga for 8
? g g + 200
? g g + 100
? gg =
300
Melon
Merenga for 4
600 g +
400 g +
?
? gg +
150g g + 100
200 g
? gg
50
=
? gg
+ 25
=
? g g + 300
? gg + 150
? gg
450
=
? gg +
75
? gg
50
General Method:
1) Work out the
recipe for 1.
2) Multiply by the
Melon
number you need
Merenga for 2
Melon
Shortcut:
You can work this out
Merenga for 1 1)from
Work
out
from the
the
quantities
quantities
2 …
for
1 person,for
BUT
people.
Can you see a
Melon
2) X 3shortcut?
Merenga for 6
(easier than X6)
Further Practical Examples
Recipe 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
STEP 1: Choose the maths!
x or ÷ ?
BIGGER means X
STEP 3: CHECK the maths!
- using ratio.
Eg: The raspberries’ weight
is ALWAYS twice the melons’
weight.
Do the maths!amounts?
Can you work outSTEP
the2:missing
SMALLER means
÷
DO THE SAME X or ÷ to
ALL ingredients
150 g
=
Raspberry
Fruitloop for 10
+
?75 gg
=
+
?15 g
=
Raspberry
Fruitloop for 5 General Method:
1) Work out the
recipe for 1.
Raspberry
2) Multiply by the
Fruitloop for 1
number you need
500 g +
250 g +
? gg +
250
? gg
125
?25 g
?50g g
+
?
200gg +
? gg
100
+
?60 gg
=
?
400gg +
? gg
200
+
? gg
120
=
Shortcut:
Raspberry
1)
Work
out this
fromout
You
can work
Fruitloop for 4 from
the the
quantities
quantities
for
1 person,
BUT …
for
4 people.
Can you
see any
2) Just
double
it!
Raspberry
shortcuts?
(Easier
than X8)
Fruitloop for 8
Further Practical Examples
Recipe 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
STEP 1: Choose the maths!
x or ÷ ?
BIGGER means X
STEP 3: CHECK the maths!
- using ratio.
Eg: The weight of bananas is
always 5 times the weight of
the melon.
Do the maths!amounts?
Can you work outSTEP
the2:missing
SMALLER means
120 g +
?
60 gg
+
÷
600 g +
? gg
300
+
120 g
=
Banana Boat
for 12
?
60 gg
=
Banana Boat
for 6
+
?
40 g
g
=
Banana Boat
for 4
?
50 g
g
+
?
10 g
g
=
Banana Boat
for 1
? gg
+ 400
+
?
g
80 g
=
Banana Boat
for 8
+
? gg
200
?10 gg
+
?80 gg
?40gg
DO THE SAME X or ÷ to
ALL ingredients
General Method:
1) Work out the
recipe for 1.
2) Multiply by the
number you need
Shortcut:
You can work this out
1) Work out from
from the quantities
quantities
for 1the
person,
BUT …
foryou
4 people.
Can
see a
2) Just
double it!
shortcut?
(Easier than X8)
Still not had enough?
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