Perspectives on Proportions and Ratios Presented by Lyn Jones and Katreena Daniels
To get you thinking • Two students are measuring the height of the plants their class is growing. Plant A is 6 counters high. Plant B is 9 counters high. • When they measure the plants using paper clips they find that Plant A is 4 paper clips high. • What is the height of Plant B in paper clips ? With thanks to nzmaths
Consider… • Scott thinks Plant B is 7 paper clips high. • Wendy thinks Plant B is 6 paper clips high. • Who is correct? • What is the possible reasoning behind each of their answers? With thanks to nzmaths
Wendy is correct, Plant B is 6 paper clips high. Scott’s reasoning: To find Plant B’s height you add 3 to the height of Plant A: 4 + 3 = 7. Wendy’s reasoning: Plant B is one and a half times taller than Plant A: 4 x 1.5 = 6. The ratio of heights will remain constant. 6:9 is equivalent to 4:6. 3 counters are the same height as 2 paper clips. There are 3 lots of 3 counters in plant B, therefore 3 x 2 = 6 paper clips. With thanks to nzmaths
Use ratio tables to identify the multiplicative relationships between the numbers involved. With thanks to nzmaths
Use double-number lines to help visualise the relationships between the numbers. Tips on teaching with the dual number line: All the relationships are multiply or divide (not add or subtract). The same operations are done on both sides. The directions of the arrows are important. Acknowledge that there are several different correct ways to solve ratio problems. With thanks to nzmaths
Key Idea The key to proportional thinking is being able to see combinations of factors within numbers.
What’s the Difference? The ratio focuses on the relationship of the split of the whole into parts whilst the proportion shows the 'gap'- or 'space between' which needs to be maintained - almost like it provides the track or parallel lines the figures must travel along to maintain their relationship.
How full is the jug? How many different ways can you express the mathematical concepts depicted in this picture? Who is right?
How full is the jug? • How many different ways can you express the mathematical concepts depicted in this picture? • A fraction=1/2 • A percentage=50% • A decimal= .5 • A ratio? • Why/ Why not? • Who was right?
Key Ideas- Ratios A ratio is a way of writing how much of something there is compared to another thing. Ratios can compare a part to the whole.It can also be expressed as a fraction. Ratios can compare parts to parts. Part-to-whole and part-to-part ratios compare two quantities of the same thing. 50 % or .5 compares part to whole while 50:50 compares part-to-part where there are two quantities of the same thing Ratios can also be a rate, where a comparison is made between quantities of different things e.g. New minimum wage $13.75 per hour. A proportion expresses the relationship between two ratios. A proportion is a statement of the equality between two rations.
Common Misconceptions – Do you see what I see? Teacher • Teaching fractions of a circle- cutting / shapes referring back to the whole. Give a part and ask to draw the whole exercises. • Teaching numerator and denominator. Key=equal parts and irregular fractions. Give a part and ask to draw the whole exercises. • Ordering fractions- smallest to biggest and vise versa. • Always presenting equal parts as models – PAT mathsgliches • Reading, Reading, Reading- FIO-Unpack the reading. Rewrite the problem into user friendly words. Use materials. • A percentage is expressed as ?/100 – use of variety of visuals and materials. Student perspective: • Fractions are part of a circle. Stumped when drawing equal parts of rectangles, squares, diamonds, many other shapes • How may parts are coloured in. Stumped with irregular fractions. • Two is always the biggest number. • I remember this … it goes backwards • I just have to use visual clues to sort this out. • I am not a Level 26 reader yet? • Its out of a 100
Common Misconceptions – Do you see what I see? Teacher • Decimals add up to ten. Use of Cuisenaire rods to measure and portray decimals. • Proportions taught with a bias to percentages only. Cards to show the relationship between fractions, percentages and decimals. Student therefore understands the value of quantity of the part changes not the proportion of the whole and the inter relationships between the expressions of the proportions i.e. ½= 50%= .5 of the whole. Student perspective: • Its out of a 10. Stumped when adding and subtracting decimals. • Always expressed as percentages.
Progressions Reproduced thanks to Charlotte Wilkinson
Level One Unpacking & Attacking • Provide materials (any level) • Remind there are multiple possible answers and strategies to solve the problem • Encourage looking for patterns Possible Challenges • Colour perceptions (colour blindness) • Reading abilities • Reluctance to look for more than one solution • Once a colour has been used as a fraction e.g. half, then a block to seeing it as a third of a larger whole (different colour) With thanks Charlotte Wilkinson, Wilkie Way
Level Two Unpacking & Attacking • Recap on finding ½ and ¼ of sets mentally • Provide materials (any level), e.g. complete a hexagon using: poi = trapezium (1/2), rhombus = rakau (1/3), triangle = waiata (1/6) • Check understanding of relationship between number of students and materials mentioned in problem Possible Challenges • Reading abilities & language/experience barriers • Remind there are multiple possible strategies to solve the problem • Confusion or missing relationship between materials and number of students • Balking at larger numbers and not seeing relationship between them. 60 being double 30 or breaking down 204 into 60+60+60+24. With thanks Figure It Out, Number Sense & Algebraic Thinking, Book 1 L2-3
Level Three Unpacking & Attacking • Model/suggest double number lines, ratio tables • Order of problem helps guide towards finding solution • Need to search for common factors and equivalent fractions • . Possible Challenges • Reading abilities • Knowledge of time as a measure • Requires some knowledge of inverse relationships • Understanding of equivalent fractions • Knowledge of multiplication factors, e.g. looking for common factor of both 30 and 45 (easier to work with 3 rather than 5min intervals to simplify fraction work) With thanks Figure It Out, Proportional Reasoning, Book 2 L3-4
Level Four • The results: • Ratio: 5:3 • Fractions: 5/8 and 3/8 • Rachel: $500 ÷8= $62.50, $62.5x5=$312.50 • Hemi: $500÷8=$62.50, $62.50x3=$187.50 • Check: $187.50 +$312.50 =$500 ✓ Paul Mounsey Year 10 Proportions and Ratios -HGHS
NZ Maths on Ratios and Proportions Big Ideas: Ratios allow us to compare the relative sizes of two quantities. Background points for teaching An understanding of ratios involves understanding the following: Ratios can compare a part to the whole. An example of a part to a whole ratio is the number of females in a class to the number of students in the class. If there are 8 females in the class of 20 students the ratio of girls to students can be expressed 8:20 (females to students). Because this ratio is relating a part to a whole it can also be expressed as a fraction (8/20) or as a percentage (40%). Ratios can compare parts to parts. An example of a part to a part ratio is where the number of females in a class is compared to the number of males. If there are 8 females in a class of 20 the ratio of females to males is 8:12. It is important when using ratios to clearly state what the comparison is made in relation to. One of the most common uses of part-to-part ratios are odds. The odds of an event happening is a ratio of the number of ways an event can happen to the number of ways it cannot happen. Ratios can also be a rate. Part-to-whole and part-to-part ratios compare two quantities of the same thing. Rates on the other hand are examples of ratios where a comparison is made between quantities of different things. In rates the measuring units are different for the quantities being compared and the rate is expressed as one quantity per the other quantity. For example the value of food can be expressed as price per kilogram, fuel efficiency can be expressed as litres per 100 km. A proportion expresses the relationship between two ratios. A proportion is a statement of the equality between two rations. For example if it takes 10 balls of wool to make 15 beanies, 6 beanies will take 4 balls of wool. In this example the ratio of 2:3 (balls to beanies) can be applied to each situation. Solving proportional problems involves applying a known ratio to situations that are proportionally related and finding one of the measures when the other is given. For example, in the beanie situation the ratio of 2:3 (balls to beanies) can be applied to the problem where you want to find out how many balls of wool are needed to make 33 beanies. 2:3 = ?:33, ? = 22
Where to from here? Where to go ...when we don’t know. • Book 7 The Numeracy Professional Development Projects documents • www.nzmaths.co.nz • http://www.ncwilkinsons.com/wilkieway/pages/educational_resources.php • http://www.primaryresources.co.uk/maths/mathsB7.htm • http://www.oxfordowl.co.uk/maths/any-questions/