Purpose

This unit supports students to understand ratio problems such as: It takes 4 balls of wool to make 7 beanies. How many balls of wool does it take to make 19 beanies? It provides an entry into solving ratio problems using one simple algebraic equation.

Specific Learning Outcomes
  • investigate situations involving ratios;
  • understand that there are many ways to solve ratio problems;
  • solve simple equations of the form ax = b;
  • see the relevance of algebra to ratio problems.
Description of Mathematics

Ratio problems appear to be difficult ones. Relatively few people in the community seem to be able to solve the Beanie Problem easily. It is therefore not expected that all students will be able to handle equations or simple ratio problems confidently after this unit. These things are difficult for even good students. These topics are ones that you need to keep coming back to and working on in a variety of ways.

We are trying here to lead students into solving ratio problems while at the same time giving them an introduction to algebra. This unit is not a comprehensive attack on algebra but it is a way of getting starting. At the end of this unit your students should see that you can use algebra to represent simple problems and even to solve these problems. But they may still feel much more comfortable solving the problems by an alternative method. The idea of the later sessions of this unit is to help them understand how ratios work and how to apply algebra to solve these ratio problems. However, it is better for the students to know and apply a method they understand than to use a rote method without understanding.

The Beanies’ Problem has been used in the Staff seminar: Four Problems and a Funeral While it is not necessary to have read that seminar, there are some questions in the seminar section that might be worth considering before you teach this unit.

Because most of the calculations in this unit are fairly straightforward we think it is in the students’ best interest to not use a calculator. However, things get a little sticky in session 5 so they might be encouraged there.

Required Resource Materials
A calculator for the harder calculations (preferably only in session 5)

Copymaster 6

Copymaster 5

Copymaster 4

Copymaster 3

Copymaster 2

Copymaster 1

Key Vocabulary

 ratios, linear equations, multiples, inverse operations

Activity

Session 1

The prime aim of this session is to reinforce the fact that multiplying and dividing are inverse operations, that is, that one can ‘undo’ the other.

Teachers’ Notes

What happens if you multiplying something by 5 and then divide the answer by 5?

What happens if you divide something by 5 and then multiply the answer by 5?

Clearly you end up with the ‘something’ that you started with. Surprisingly this seemingly simple step doesn’t come easily to even good students. It takes some practice before it is accepted as fact.

This is an important concept however, because it is fundamental in solving equations. After all, the reason that we can solve 5y = 17 by dividing both sides of the equation by 5, is simply because multiplying y by 5 and then dividing 5y by 5 gives you y back again. So

                                                   5y = 17
leads to                                    5y/5 = 17/5,
and so                                           y = 17/5 = 3.4

From this it might seem a relatively simple step to be able to solve simple equations but even good students don’t find this easy.

You might get several possible general statements from your class. Here are a few possibilities. Encourage them to word things the way they understand best.

If you multiply something by 5 and then divide the answer by 5 you get the original something as the answer.

If you multiply something by any number and then divide by the answer by the same ‘any number’, you get the something you started with as the answer.

5 by x divided by 5 is just x, where x is any number that you can think of except zero.

5x/x = 5.

a by x divided by a is x, where a is any number except zero and x is any number too.

ax/x = a.

If you divide something by 5 and then multiply the answer by 5 you get the original something as the answer.

If you divide something by any number except zero and then multiply the answer by the same ‘any number’ you get the something as the answer

x/5 multiplied by 5 is just x, where x is any number that you can think of.

x/5 x 5 = x.

a by x divided by a is x, where a is any number except zero and x is any number too.

x/a x a = x.

At this point we note that there is the same inverse relation between addition and subtraction. We will leave exploration of that pair of operations for another unit but note that students can’t master the more difficult equations of the type ay + b = c until they are confident about these two pairs of inverse relations.

Teaching Sequence

 

  1. Start with a whole class activity.
    Write down a number on a piece of paper and turn the paper over.
    Multiple your number by 5.
    Divide the result by 5.
    Josh, what is your answer?
    Anyone, what number did Josh start with?
  2. Repeat this with a few more students.

  3. Repeat this with numbers other than 5.

  4. Try to give most students a turn one way or the other.

  5. Depending on the class, and what calculations you want to reinforce, get them to think of a number that is a small integer (less than 20), a large integer, a decimal, or a fraction. If necessary let them use a calculator.

  6. Discuss what is going on here.
    What’s the effect of multiplying a number by 4 and then dividing by 4?
    Does this always happen no matter what the number?

  7. Repeat with a number other than 4.
    Does this happen for all numbers? (Almost – zero is a special case, why?)

  8. Write a general statement composed by the class on an A2 sheet and put it on display. (Try to lead them away from guessing and checking.)

  9. Get them to go work in pairs to explore what happens when you divide by a number and then multiply by the same number. (You might want to give them the instructions from copymaster 1.)

  10. Discuss what is going on here.
    What’s the effect of dividing a number by 4 and then multiplying by 4?
    Does this always happen no matter what the number?

  11. Repeat with a number other than 4.
    Does this happen for all numbers? (Almost – zero is a special case, why?)

  12. Write a general statement composed by the class underneath the statement in 2.  Put the A2 sheet on display. (Encourage but do not insist on, an algebraic formulation. Help them to realise that each way of saying it is the same.)

  13. Now try this game with the whole class.
    Twice a number is 14. What is the number?
    Petra what is your answer?
    Twice a number is 30. What is the number?
    Vijay what is your answer?

  14. Repeat with various doubling questions.
    What is happening here?
    How is this related to what we did earlier in this session?
    How can we find the answer?

  15. Point out that ‘twice’ is multiplying by 2 and to ‘undo’ multiplying by 2 we need to divide by 2.

  16. Some students might say that we have to halve the 14 or the 30. Discuss why halving is the same as dividing by 2.

  17. Then ask different members of the class to suggest a problem along the lines “So many times a number is such and such. What is the number?”
    What is the general rule here?

  18. Discuss and write a student composed statement for display. (Try to lead them away from guessing and checking. Encourage but do not insist on, an algebraic formulation.)

  19. In their pairs let them take turns to construct and solve problems of the form “A third of a number is 8. What is the number?

  20. Allow the use of calculators.

  21. Discuss the way that they went about solving these problems.
    What is the general rule here?

  22. Discuss and write a student composed statement for display. (Try to lead them away from guessing and checking. Encourage but do not insist on, an algebraic formulation.)

  23. If you think that the students need more practice on these concepts make up word problems along the lines of :
    My salary just doubled to $50,000. What was my salary before that?
    Tomatoes have tripled in cost this week to $2:40 a kilo. What price were they last week?

    The Uruguayan pesito halved in value this week to go down to $NZ 2.30 . What was the pesito’s value last week?

Session 2

In this session we look at problems that lead to equations of the form ax = b, where a and b can be fractions.

Teachers’ Notes

The aim here is to be able to tackle simple problems that arise in word problem situations and that can be written in the form ax = b. Here’s a simple example:

Problem 1: If Joan had five times her bus money she’d have $15. How much is her bus money?

There are lots of ways of doing this problem.

Method 1: You might draw up a table of multiples of 5 and stop when you get to 15.

Method 2: You might guess that Joan needs $4 for the bus. 5 x 4 = 20 and that’s bigger than 15. So she needs less than $4. Try $2. 5 x 2 = 10. That’s too small. Now try $3.

Method 3: Let b be the bus money. Then 5b = 15. How can we solve this? If we divide 5b by 5 we get b. Since 15 and 5b are the same thing, if we are going to divide 5b by 5, then we have to divide 15 by 5 too.

                                            5b = 15.
Divide both sides by 5 gives equation..
Tidying up gives                    b    = 3.

Method 4: What other methods can you or your class think of?

Praise any correct method that works but keep leading them back to the algebraic method.

Problem 2: If Joan divided the money she needs for the bus by two, she’d have $2. How much does she need for the bus?

This problem can be solved using any one of the methods above.

Teaching Sequence

  1. Introduce a simple problem like Problem 1. (Personalise it to include the name of someone in the class or the principal!) Make sure everyone understands what the problem is asking.

  2. Ask:
    How would you solve this problem?
    Then discuss the methods they think up.
    Is Wing’s answer correct? Is his method correct?
    What other ways are there of doing that problem?

  3. List the methods on the board. Be sure to cover at least the three methods above.
    Do they all give the same answer?
    Are the methods all correct?
    Which methods do you understand?
    Which is the best method for you?
    Which is the best method of all?
    How would you show your working for your favourite method? (Try to cover all the methods by asking different members of the class.)

  4. Now give them some of the problems from copymaster 2 to work on in pairs. Insist that they show how they got their answers. If any group finishes a lot earlier than the others, ask them to make up and solve some questions of their own.

  5. Bring the class together to discuss what they have done.
    Which was the easiest problem?
    Which was the hardest problem?
    Which was the easiest method?
    Which was the hardest method?
    Peta, can you show us how you did one of the problems?

    What other method can you use to solve this? (Make sure that they come up with an algebraic one.)

  6. Now repeat the sequence 1 to 4 above. This time problems for the pairs can be found in copymaster 3.

  7. At the end, ask them
    Was it harder to do the problems of Copymaster 2 or 3?
    Why do you think that was the case?

  8. You may want them to write something in their books as a record of the lesson.

Session 3

Here we look at simple problems like the Beanie Problem, where only multiplying by a number or dividing by a number is necessary.

Teachers' Notes

The basis for the next three sessions is the problem below. In this session we’ll concentrate on beanie-like problems that can be done in a number of ways but only require algebraic equations of the form ax = b, where a and b are whole numbers. In the next session we’ll deal with problems where the same equation will have a and b as fractions, and in session 5 the numbers a and b will be decimals.

The Beanies’ Problem: It takes 4 balls of wool to make 7 beanies. How many balls of wool does it take to make 19 beanies?

It is not obvious how much wool is needed  to make one beanie. The point is to explore this situation and the ratios that are a part of it, and to find out as many ways as we can to solve the problem. So let’s start off to explore the situation with some slightly easier numbers than those in the Beanies’ Problem.

Problem 3: It takes 8 balls of wool to make 2 beanies. How many balls of wool does it take to make 9 beanies?

Method 1: Perhaps the simplest method is to note that you can make 1 beanie from 4 balls of wool. So you’ll need 9 times as much wool for 9 beanies. So 9 x 4 = 36 balls of wool should make 9 beanies.

Method 2: We can use a table. Here we have used a guess and check approach. We can see that 32 balls give us 8 beanies so we need more than 32 balls. But how much more? 40 would give us too many beanies. But we the number of beanies is halfway between 8 and 10 so the number of balls must be halfway between 32 and 40. So 36 is the answer.

Number of balls of wool

Number of beanies

 8

 2

16

  4

32

  8

40

10

36

  9

Method 3: Let’s aim for the 9 more directly. We can easily calculate the number of balls of wool needed for an even number of beanies, so note that 9 = 8 + 1, 8 = 4 x 2 and so 8 beanies need 4 x 8 = 32 balls of wool. But 1 beanie only needs 4 balls. So since 9 = 8 + 1 we require 32 + 4 = 36 balls of wool.

Method 4: But this is really a ratio problem so let’s look at it from that perspective. If we are going to get 9 beanies, then we have increased the number of beanies in the ratio 9:2. (What do we have to do to 2 to get 9? Multiply it by 9/2.) So we have to increase the number of balls of wool in the same ratio. In other words

new number of balls:8 = 9: 2.

To calculate this in practice amounts to multiplying the original number of balls (8) by 9/2. This is because if we take the ratio equation above we can turn it into the fractional equation 

below.

equation..

And so

equation.


 

Method 5: But the ratio idea works the other way too.

equation.


 

And that leads to the answer we are getting used to. What’s more it is a slightly more efficient method here than Method 4 because 8/2 = 4 and the final calculation turns out to be nicer.

Method 6: But we can formalise the approaches of the last two ratios by replacing the ‘new number of balls’ by a variable, b say. Then the first of these equations becomes

equation.,

while the second becomes

equation.


 

Either equation can then be solved algebra. (Use the idea of the first two sessions to see that to isolate the b in the last of these two equations, for instance, you need to multiply the left side by 9. So you have to multiply the right side by 9 too. The rest is simple arithmetic.) If you simplify the left hand side of the last equation before doing anything else, The calculation is a little easier.

So there appear to be at least six ways to do this kind of problem. You can

 

  1. find how many balls of wool are needed for 1 beanie;
  2. use a table of values and your favourite strategy for working with number patterns;
  3. break up the number of balls into simple multiples of the 8;
  4. use the ratio of the new number of balls of wool to 8 and multiply 9/2 by that ratio;
  5. take the ratio of 8 to 2 and multiply the new number of balls of wool divided by 9 by that fraction;
  6. use algebra to formalise the approaches of 4 and 5.

But while we’re on this kind of problem, we should see that there is another type of problem that can be asked here. Instead of wanting the number of balls of wool given a number of beanies, we can ask how many beanies we could get to from a given number of balls. All the methods we have mentioned so far can be used to solve the inverse problem of the one we tackled above. However, the ratio approach shows a much deeper understanding of what is happening we just use the ratio around the other way. The fact is that

equation.


 

So the beanies to balls ratio is always 2: 8 (or 1: 4) and we can put whatever number of balls we like into the equation above to find the number of beanies we can make.

And, of course, we can do this using algebra too. We are now in a position to tackle Problem 1 and any variation of that problem.

Note: Since the numbers here are relatively straightforward we do not encourage the use of calculators in this work.

Teaching Sequence

  1. Give Problem 3 to your class in their groups. Let them tackle the problem but their aim is not just to solve it but to see how many different ways they can find to solve it. Be generous with your interpretation of ‘different’ here.

  2. In a whole class setting, discuss Problem 3 and the various ways there are to solve the problem. If not all of the methods you want discussed are mentioned by the students you may need to add in one or two of your own. Ask
    How did your group solve the problem?
    Why does your method work?
    If you get an algebraic equation, how do you solve it? (Remind them of the work of the last two sessions.)
    Which of the ways that people have suggested do you find the easiest to use?
    Which do you find the hardest?

    Can you tell me why?

  3. Now send them back to their groups to tackle the problems of copymaster 4. Again let them know that you are looking for as many ways to solve this problem as they can find.

  4. With the class together, discuss the problems that they have just tackled and the methods they used. Ask
    Which of these ways do you find the easiest to use?
    Which do you find the hardest?
    Can you tell me why?
    Encourage them to use the algebraic approach.

  5. When they have finished those problems let them make up three problems that they should arrange in order of difficulty. They should be able to solve each of these problems in at least two ways.

  6. Get groups to swap their problems with another group and solve the other group’s problems.

  7. In a class discussion determine what methods the students prefer to use and why. Also ask them
    Did you agree with the order of difficulty that the problems were listed them in?
    What makes a problem like this difficult?

Session 4

Here we look at problems like the Beanie Problem, where fractions are involved.

Teachers’ Notes

This session is based on the same ideas as the last one except that we are not going to use such nice numbers. Here we won’t get such nice cancelling as we did when we use the 8:2 ratio of the last session.

Problem 4: The Beanies’ Problem: It takes 4 balls of wool to make 7 beanies. How many balls of wool does it take to make 19 beanies?

This problem can be done using any of the methods from the last session. The difficulty level is increased though, by the fact that the numbers don’t simplify as easily.

There is one further tease with this kind of problem. Do you accept non-whole number answers? For instance, in Problem 4, the answer is 76/7. Now if you were going to buy the wool you’d have to buy 11 balls of wool rather than 10.86. Your better students at least should understand the subtlety here.

Note: Since the numbers here are relatively straightforward we do not encourage the use of calculators in this work.

Teaching Sequence

  1. Give Problem 4 to your class in their groups. Let them tackle the problem but their aim is not just to solve it but to see how many different ways they can find to solve it. Be generous with your interpretation of ‘different’ here.

  2. In a whole class setting, discuss Problem 4 and the various ways there are to solve the problem. If not all of the methods you want discussed are mentioned by the students you may need to add in one or two of your own. Ask
    How did your group solve the problem?
    Why does your method work?

    If you get an algebraic equation, how do you solve it? (Remind them of the work of the first two sessions.)
    Which of the ways that people have suggested do you find the easiest to use?

    Which do you find the hardest?
    Can you tell me why?

  3. Now send them back to their groups to tackle the problems of copymaster 5. Again let them know that you are looking for as many ways to solve this problem as they can find.

  4. With the class together, discuss the problems that they have just tackled and the methods they used. Ask
    Which of these ways do you find the easiest to use?
    Which do you find the hardest?
    Can you tell me why?

    Encourage them to use the algebraic approach.

  5. When they have finished those problems let them make up three problems that they should arrange in order of difficulty. They should be able to solve each of these problems in at least two ways.

  6. Get groups to swap their problems with another group and solve the other group’s problems.

  7. In a class discussion determine what methods the students prefer to use and why. Also ask them
    Did you agree with the order of difficulty that the problems were listed them in?
    What makes a problem like this difficult?

Session 5

Here we consider and solve, bald algebraic equations of the form ax = b, where a and b are decimals.

Teachers’ Notes

We are going over the same ground here but this time we’re using decimals. The same methods apply. A typical problem in this session looks like:

Problem 5: It takes 4 balls of wool to make 1.2 beanies. How many balls of wool does it take to make 3 beanies?

The use of calculators should be encouraged in this session.

Teaching Sequence

  1. Give Problem 5 to your class in their groups. Let them tackle the problem but their aim is not just to solve it but to see how many different ways they can find to solve it. Be generous with your interpretation of ‘different’ here.

  2. In a whole class setting, discuss Problem 5 and the various ways there are to solve the problem. If not all of the methods you want discussed are mentioned by the students you may need to add in one or two of your own. Ask
    How did your group solve the problem?
    Why does your method work?
    If you get an algebraic equation, how do you solve it? (remind them of the work of the first two sessions)
    Which of the ways that people have suggested do you find the easiest to use?
    Which do you find the hardest?

    Can you tell me why?

  3. Now send them back to their groups to tackle the problems of copymaster 6. Again let them know that you are looking for as many ways to solve this problem as they can find.

  4. With the class together, discuss the problems that they have just tackled and the methods they used. Ask
    Which of these ways do you find the easiest to use?
    Which do you find the hardest?
    Can you tell me why?
    Encourage them to use the algebraic approach.

  5. When they have finished those problems let them make up three problems that they should arrange in order of difficulty. They should be able to solve each of these problems in at least two ways.

  6. Get groups to swap their problems with another group and solve the other group’s problems.

  7. In a class discussion determine what methods the students prefer to use and why. Also ask them
    Did you agree with the order of difficulty that the problems were listed them in?
    What makes a problem like this difficult?

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