Proportional Problems 1

Prior Knowledge

• explain factors and multiples
• use arrays to model multiplicative problems
• use a wide range of adding, subtracting, multiplying, dividing strategies
• be developing decimal and fraction operation understanding.

Background:

The ideas of ratio, proportion and proportional thinking/reasoning are complex. We use fractions to represent these ideas but they do not always behave like fractions. Exercise 4 and 5 are designed to expose this misconception in particular.

The “unit” (one whole) to which we are referring can change during proportional problems and very often without this being obvious. This is a common feature of such problems and should always be kept in mind.

Relating to materials or representations can help – but sometimes this is also not obvious. For example, try drawing the ‘easy’ problem 3/4 divided by 2. Now try drawing the ‘hard’ problem 3/4 divided by 1/2. Most people cope with the first but find the second worthy of a good discussion.

A ratio compares compares ‘a part’ to ‘a part’ of a unit. A proportion compares “a part” to “a whole” or “unit”. Be aware that the statement a:b = a/b idea is not an obvious idea and needs thorough unpacking. a/b might look like a fraction but it rarely behaves like one in a proportional context. The definitions of ratio and proportion in Mathematics in the NZ Curriculum tell us very little which shows just how this topic has progressed since 1994.

Conceptual discourse is really important to engage with in these problems. This means allowing students to discuss their ideas at great length with high engagement and delving into questions, never ‘telling’ the answer. Listen and ask and listen again to see inside the mind of the student.

A fundamental concept in mathematics is that when comparing we must compare the same “size” unit. This is true with ratio, subtracting and dividing. Ratios can be operated on sensibly when the total parts involve the same “one”. Understanding this idea is a major step forward.

Further information to back up what they will meet:

• FIO Books – Check Proportion Books for extra activities.
• Digital Learning Objects
• National Archive of Virtual Manipulatives (via Google)

Comments on the Exercises

Exercise 1: The Gear Box

Asks students to solve problems involving gear ratios.

Teaching lessons preceding this exercise could include:

• activities with Lego gears for practical experience
• modelling ratio operations
• activities with bicycle gears and pedals to explore ratio

 The main misconception here is the two gearboxes are added but the ratio is multiplied. Billy is very wrong when he adds the ratios. The answer could be 5:2 if he added everything. Students who select addition as the first option to solve problems will see these as satisfactory answers. They should explore gearing in practical way with gears, pulleys, LEGO and bicycles to see that in practise adding is not an option.

Technical note. Gearing down means getting more torque or power to the driving wheel. It also means going slower, which is OK when driving steep or slippery, dangerous slopes.

Exercise 2: More 4WD Gear Boxes
Asks students to solve problems involving combining gear ratios.Do exercise 1 before attempting this one. Establish multiplying as the secret to find overall gear ratio. It may pay to ask the question “why is a gearbox called a gearbox?”

Exercise 3: Billy Finds Another Gear
Asks students to solve more complex gear ratio problems. Do exercise 1 and 2 before doing this one. Exercise 1 establishes gear ratios are multiplied and exercise 2 establishes combinations including reversing the gear box to make other ratios. This exercise is a more complicated version of exercise 2.

Students can be expected to develop their own table and investigate decide which ratios would be the best to produce, hence the answers are not shown on the answer sheet. The best ratios would be prime number ratios. Eg 1:2, 1:3, 1:5, 1:7 where to get 1:4 ratio two 1:2 boxes are needed. This problem is much like the ‘pasta maker’ problem.

Exercise 4: Proportions are not Fractions (Part 1)
Asks students to solve ratio problems with paint tins the same size.
Teaching lessons preceding this exercise could include:

• combining ratios using blocks
• exploring the meaning of ratio and proportion.

The relationship between ratios and fractions is a complex one and should not be taken for granted. The next 2 exercises explore some of the issues within that relationship that may be worth exploring with top students. In particular, they address the issue that while ratios can be converted to fractions, they don’t always act like fractions – at least not in obvious ways.

Question 7 exposes the error of representing proportions as fractions and expecting them to behave like fractions. This problem can be nicely represented using coloured blocks. For example,

+ =

 Here in the paint problem, 1/2  + 1/3 = 2/5. Two different “wholes” are being combined to make a new “whole”. Here the ratios can be expressed as fractions but they do not behave like fractions. They do not add like fractions either, for once they are combined, a “new whole” is created, rather than the retention of the “original whole” as happens when 2 fractions are added. They add like ratios, where no answer is bigger than one because a new unit is created.

Question 9 involves a major problem. What does halfway mean in this context? 1:1 is a proportion of 1/2 and 1:2 is a proportion of 1/3. On the number line, halfway between a half and a third is 5 twelfths. Fractions illustrate relationships that are fundamentally multiplicative in nature. Thus halfway is a multiplicative idea and not an additive idea so the number line shows the correct answer. One way to get this idea across to students is to get them to place the fractions 1/2, 1/3, and 1/4 on a number line. 1/3 is not halfway. Continuing with fractions up to 1/10 and drawing a graph is a nice way to reinforce the point. Half way between 1:1 and 1:2 is 1/12 or a ratio of 5:7. Many students may try to solve this problem additively, by looking for the number halfway between 1 and 2, giving an answer of either 1:1.5 or 2:3. Alternatively it is possible to look at halfway between 2:2 and 1:2, creating 1.5:2 or 3:4! The error here is to see the two parts of the ratio (or the fraction) as being independent numbers when in fact they are related.

Another example is what is halfway between 1:1 and 1:3? This is equivalent to asking what is half way between the proportions 1/2 and 1/4? The additive solution is 1/3 because 3 is halfway between 2 and 4, but the error in this thinking has been discussed above. Again, using the number line the fraction half way between 1/4 and 2/4 is 1.5/4 or 3/8, so the ratio 3:5 is the correct halfway ratio. This is not easy to understand.

Try finding halfway between 1:2 and 2:3. (the answer is 11:19)

Note that if we wish to keep the end volume the same and halve the strength of say a weedkiller mixture or an oil/petrol mixture the problem becomes even more complicated with decimal answers galore. What volume of oil to petrol do we mix to have exactly 5L mix wiith a 1:50 oil:petrol ratio? (answer is oil= 98mL petrol = 4902mL nearly!)

This sheet was meant to be quite a simple exercise but could become quite a lesson for students, teachers, and facilitators of numeracy as it did with maths advisors.

Exercise 5: Fractions are not Proportions (Part 2)
Asks students to problems involving ratios with paint tins of different sizes.
Teaching lessons preceding this exercise could include:

• combining ratios using blocks
• exploring the meaning of ratio and proportion.

Question 7 is a much harder problem than the equivalent question in exercise 4, due to having paint tins of the same size. The amounts are the same but the ratios are not, so when combined they do not simply add. For example, if trying to add the ratios, (as was done above) 1:1 and 1:2 gives 2:3 again. However common sense says that overall there is less red when combining 2L tins than when combining a 2L tin and a 3L tin, so the colours won’t be the same. Examining what is happening in the 2 tins gives us the clue. In a 1:1 2L tin, there is 1L of yellow and 1L of red. In the 1:2 2L tin, there is 2/3 of a Litre of yellow and 4/3 of a Litre of red. It seems that the numbers in the different ratios are not representing this “change in size of a slice.” Since ratios compare like to like, where “the size of the individual piece” is not the same, they do not apply in the same way. It seems the above exercise provides a simple case – one where we were comparing individual Litres. Those ratios could be combined as ratios ‘work’ with the same units.

 One way to introduce this discussion is to consider the 2 tins with the same unit of a Litre. 1:1 means 1 Litre of yellow and 1L of red. 1:2 means 1L of yellow and 2L of red, combine these and we get 2:3 or 2 Litres of yellow and 3 Litres of red. Oops! We only have a 4L tin so what have we done wrong?

Reverting to fractions shows that we have 1L + 2/3L of yellow or 5/3L yellow in the 4 Litre tin. We also have 1L + 4/3L or 7/3L of red, giving a ratio of 5:7 (which in terms of fractions is the fraction that is halfway between 1/2 and 1/3.

A very nice way to show this multiplicative relationship is to model 1:1 with 3 yellow and 3 red blocks and model 1:2 as 2 yellow and 4 red blocks, which allows comparison of the “same sized pieces” as there are the same number of “colour blocks” in each 2L tin. and when adding the yellow and red parts we see the relationship 5:7 quite easily. This is now the same as adding fractions which only ‘works’ when there are the same number of parts in a whole.

For question 9, Buddy has a problem. What does halfway mean? Buddy thinks by halving one of the numbers we halve the ratio. 1:1 is a proportion of 1/2 and 1:2 is a proportion of 1/3. His answer is 1:1.5 or by doubling 2:3 which is 2/5. On the number line halfway between a half and a third is 5 twelveths which is 1/60 bigger than the 2/5. Half way is 5/12 or a ratio of 5:7. This is an example of when fractions do represent proportions.

Exercise 6: Farmer Brown
Asks students to solve problems involving rates.
Teaching lessons preceding this exercise could include:

• speed trials
• use of the formula speed = distance/time
• graphs of distance time with constant speeds
• expanding brackets distributive rule

This problem has evoked much discussion and can be solved in a number of very different and creative ways. Neither distance nor time are given but both these variables are embedded in the problem by the rate speed. Comparing the resulting time difference completes the needed information.

 An easy way to solve the problem is to make a distance and see what happens. The lowest common multiple of 30 and 20 is 60 so we will go for 240 (4th multiple of 60). 240/30 = 8 and 240/20 = 12, here we have a 4 hour time difference. Halve everything leads to 120km and 5 hours and the average speed of 24km/hr.

Most additive students pick 25km/hr as their first guess as 25 is halfway between 20 and 30.

Other methods for question 6: An interesting sideline here is to average the angle each line makes with the x-axis. The answer in this case is only an approximation which gets better as the slope of the two lines get closer together. (This is an interesting investigation as well.)

Exercise 7: Different Farmer Brown Problems
Asks students to solve problems involving rates.
Teaching lessons preceding this exercise could include:

• speed trials
• use of the formula speed = distance/time
• graphs of distance time with constant speeds
• expanding brackets distributive rule

Problem #1
This adds an extra time difficulty but is much the same as Problem 4. Use any of the methods in Problem #4

Problem #2
This has a fractional answer for the speed. Use any of the methods in Problem #4

Problem #3
This problem introduces fractions of an hour. Use any of the methods in Problem #4

Exercise 8: Fun in the Bath Tub
Asks students to solve problems involving ratios in the contexts of capacity and temperature.

Teaching lessons preceding this exercise could include:

• rate problems with water filling containers
• modelling the problem with blocks

There are many ways to solve this poblem. The proportion FIO books have a good series along the same lines and different solution techniques.

Part 1
How long to fill whole bath with both taps? If we turn both taps on we get 3 bath fulls in one hour. Obviously this can not happen in practise but it can happen by modelling with blocks. The answer to this question is then 1/3 of an hour or 20 minutes. Half a bath is 10 minutes. Investigate other ways as well (FIO books).

Part 2
Introducing temperature of the water is diguising what we are really talking about and that is heat energy. The hot water at 60^oC warms the cold water at 15^oC. The rate at which they are combining is 1 hot to 2 parts cold since the cold comes in at twice the rate. The physicist would talk about specific heat, mass of water, absolute zero but all these factors vanish because we are dealing with one medium which in this case is water. The equation 60 x 1/3 + 15 x 2/3 = 30^oC hides all the complication. Final T = 30^oC.

Part 3
This can be quite a complicated exercise to get an exact answer! Making the hot on for twice as long gives equal cold and hot amounts. Using the idea in Part 2 the new equation would be 60 x 1/2 + 15 x 1/2 = 37.5 which shows that Pliney is a bit shy of the target 42^oC. A better solution might be three times as long for the hot. How much hot is this? If the cold was on for an hour and the hot on for three hours we would have 2 + 3 bathfuls of water or 3/5 from the hot and 2/5  from the cold. Our equation reveals 60 x 3/5 +15 x 2/5 = 36 + 6 which is 42^oC or exactly correct. This solution like all the others above excludes loss of heat from cooling. Doing these exercises practically is a good idea from a science point of view as well as mathematically.

Exercise 9: Some Right, Some Wrong, Some Easy.
Asks students to interpret if ratios and rate problems make sense. These are included for fun and discussion. There are many many others.

Last comment. The more time spent learning proportional reasoning the better. This document was supported by readings from Sue Lamon (Marquette University), Vince Wright (University of Waikato) and others.