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Proportional reasoning. Lead teachers Northland 2010. How could you describe this diagram?. 4 times as many red stars as there are yellow ones Ratio of red stars to yellow stars is 4:1 Ratio of yellow to red is 1:4 4 red stars to every yellow star 4/5 of the stars are red
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Proportional reasoning Lead teachers Northland 2010
How could you describe this diagram? • 4 times as many red stars as there are yellow ones • Ratio of red stars to yellow stars is 4:1 • Ratio of yellow to red is 1:4 • 4 red stars to every yellow star • 4/5 of the stars are red • 80% of the starts are red 20% of the stars are yellow. • 0.8 of the stars are red etc. • The proportion of red stars is greater than the yellow. • Proportion, Fraction, Ratio, Rate
What is proportional thinking? • Piaget describes it as the difference between concrete levels of thought and formal operational thought. • We use “Ratio” to describe the relationship. • We do not become proportional thinkers simply by getting older. We need help and lots of practice in a range of contexts. • Symbolic or mechanical means do not help develop proportional thinking. eg cross products.
Key concepts • A ratio is a comparison of any two quantities • Proportions involve comparing 2 quantities multiplicatively. . • A proportion expresses the relationship between two ratios. • To develop proportional thinking we need to involve the students in a wide variety of activities over a considerable period of time.
Where does it fit on the framework? • Stage 6 - equivalent fractions - Comparing fractions • Stage 7 - ratios as fractions - Simplifying ratios - Simple rates • Stage 8 - Equivalent ratios - Sharing amounts in a given ratio - Expressing ratios as %
What about National Standards? By end of year 7 Apply multiplicative strategies flexibly to whole numbers, ratios, and equivalent fractions By end of year 8 Apply multiplicative strategies flexibly to whole numbers, ratios, and equivalent fractions Ie the difference between the thinking at level three and level four.
What do they need to know? • Basic facts • How to simplify fractions • How to write equivalent fractions • How to order fractions • How to find the factors of a number
Ratios Comparing Same Types of Measures Comparing different types of measures Part/whole (fraction) Rate Part/part
Other examples of ratio • Pi or is the ratio of the circumference of a circle to the diameter • The slope of a line is the ratio of vertical to sideways movement. • The 3/4/5 or the 5/12/13 right angled triangles • The Golden ratio is found in many spirals • Proportions within the human body • Packets of soap powder and cornflakes. • What happens to area when you double the length of a side of a square?
Knowledge recap Fraction versus a Ratio • What fraction of the group is the pear? the lemon? • What fraction of the group are bananas?, apples? • How many bananas compared to the apples in the group? • What is the ratio of lemons to pears?, lemons to bananas?
Fruit Bowl Problems Apples and Oranges There are 3 oranges to every one apple in the bowl How many apples and how many oranges, if there are 40 pieces of fruit in the bowl?
Fruit Bowl Problems • 24 in the bowl? • 16 in the bowl? • 52 in the bowl? • What fraction (proportion) are apples? • What fraction (proportion) are oranges? • What is the ratio of oranges to apples?
Harder ratios Apples and Bananas 3 apples to every 2 bananas in the fruit bowl How many apples and how many bananas if • 40 in the bowl? • 25 in the bowl? • 60 in the bowl?
More Fruit Apples, bananas and oranges For every 4 apples in a box there are 3 bananas and 2 oranges How many of each fruit if • 45 in the box? • 180 in the box? • 72 in the box?
Make up Own Problems • Make up a problem that someone else can work out using three types of fruit, in a given ratio. • Challenge students to find a range of different amounts in the box. • Make up a problem using a different context that they can choose.
ImagingTransfer the model to other situations • In a school of 360 pupils there are 5 boys to every 4 girls. How many girls are there?
Proportional reasoning activities • These are informal and exploratory to start. Eg When I planted two little trees they were 80cm and 120 cm tall. Now they are 110cm and 150cm. • Which one grew the most?
Comparing ratios • Pizzas were ordered for the class There were 3 pizzas for every 5 girls and 2 pizzas for every 3 boys. Did the boys or the girls have more pizza to eat ?
Look alike rectangles • This activity links geometry with ratio. • Cut out the rectangles • Sort into 3 “families” • Arrange each family smallest to largest • What patterns do you see in each family. • Stack each family largest on the bottom sharing bottom left corner. • Now what do you notice? • Could you fit another rectangle into each family? • Fill in on the table by family length and width. • Look for patterns • Make the table into a series of ratios • Use the long rectangles and plot length and width on graph • What do you think this line means? • Use different colours for each family.
Division in a given ratio • Joe works 3 days and Sam works 4 days painting a roof. Altogether they get paid $150. How much should each get?
Multilock block models • Colour mixes • 18 yellow with 6 blue in a mix to make a green • What mix might go into smaller pots to make the same colour?
Proportional relationships-rate • It takes 20 bales of hay a day to feed 300 sheep. How many bales would you need each day to feed 120 sheep. • How did you work it out? • The number of dog biscuits to be feed to a dog depends on the weight of the dog. • If the packet recommends that an 18kg dog needs 12 biscuits, how many biscuits should you feed a 30kg dog, a 10kg dog? • How did you work it out?
