Even More Pizzas And Things
Solve problems involving fractions
Devise and use problem solving strategies (draw a picture, use equipment, guess and check, be systematic, think)
Interpret information and results in context.
This problem is a multi-step one that involves fractions and the four arithmetic operations. It also requires a careful analysis if what is known, what is unknown and what can be obtained from what with what sequence of operations.
Like a number of similar problems that have a lot of inter-related information, this problem can be handled by algebra. So algebra is a natural follow on from this type of problem. However, like a number of so-called algebra problems, this one can be sorted out by careful reasoning.
The students should be asked – what do you know? and what can you find out from this?
The pizza place has three tables. The biggest one seats three times as many people as the smallest one. The middle sized table seats twice as many people as the smallest one.
On Tuesday night three-quarters of the seats were taken. Then twelve more people arrived. Unfortunately there were only enough seats for half of them.
How many people can sit at the smallest table?
- Introduce the problem by posing fraction questions to be solved mentally. Ask that the students explain the mental strategies that they used.
I am ¾ of 16, what number am I?
I am 2/3 of 30, what number am I?
I am 4/6, what other names could I be called?
- Pose the problem to the class. Check that they understand what is required by asking volunteers to retell the problem.
- As the students solve the problem ask questions that focus on their understanding of fractions.
How are you solving the problem?
Why are you using those numbers?
Are you convinced that your answer is correct? Why?
Can you convince me that you have the correct answer?
How would you describe what a fraction was to a friend who had forgotten what they were?
- Remind the students to record their work so that it can be shared with others.
- Display and discuss solutions.
Letting the tables be containers and the people, sand or water can turn the problem into one relating to measurement.
Extension to the problem
The number of tables or the number of people that can be seated at the smaller table can be increased.
What do we know? There are three tables and they are of different sizes. We even know the number that the two bigger tables will seat with respect to the first one. But at the moment we have no information to be able to work out the size of any table.
We know that The Pizza Place is three-quarters full. But again this is no use to us at the moment.
We know that 12 people arrive and half of them are turned away. So we know how many are turned away. A half of 12 is 6. So that means too that 6 people can be seated.
How can we tie that in with the fact that the Pizza Place is three-quarters full? If it is three-quarters full, then it is one-quarter empty. So 6 people is a quarter of a full house. So the full house must be 4 times 6 = 24.
Now we can go back to the tables. At this stage we could guess that the smallest table seats 3. In that case the next table seats 6 and the biggest table seats 9. Since 3 + 6 + 9 = 18, our guess is a bit low. So guess 5. Then we get 10 and 15 for the other tables. Now 5 + 10 + 15 = 30 and that’s too high. So the smallest table must seat 4. That can be easily checked.
Another way to tackle the last part is to suppose that the smallest table seats some. Then the next table seats two lots of some. Altogether, this is three lots of some. The biggest table seats three lots of some. So now we have six lots of some in total. Now six lots of some is 24. So some is 4 and that’s the number of people that can suit at the smaller table.
(If you think that this problem is too hard for your class, then tell them that the smaller table holds 4 people and ask them how many people The Pizza Place has to turn away when 12 new people arrive. Remember that by reversing what is known we can make a problem harder or easier. In this case it can be made easier. Try doing this with some of the other problems that we have on the site.)