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) Part/part Rate Number of red marbles to marbles in the bag 6/24 Number of boys to girls 12 to 14 Number of km per litre 60 to 1 Number of maths rooms to number of rooms in the school 4/24 Number of footballs to number of soccer balls 5 to 7 Number of teachers per class 2 to 1 Number of answers correct to total score 6/34 Number of answers correct to number of answers not correct 4 to 6 Number of lambs per ewe 2 to 1

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?

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? Fruit Bowl Problems

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.

Imaging Transfer 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?

Ratios with whole numbers