More Pizzas and Things
Solve problems that involve halves and thirds
Devise and use problem solving strategies (draw a picture, use equipment, guess and check, be systematic, think).
In this problem you need to know where you are going. In the first part, the thing that is common to both restaurants is the size of their tables. So this is going to have to be the basis of comparison. Relate everything to that.
In doing this problem the students can use their knowledge of quarters, can use division by four or can put 24 blocks into four groups. This makes this a good problem to use in a class with a range of abilities. However, be sure to allow each method of solution to be represented in the reporting back session.
The Pizza Place in my town has three tables of the same size. The Chicken N Chips restaurant has four of the same sort of table. The Chicken N Chips can just seat 24 people. How many people can The Pizza Place seat?
Now a third of the seats at The Chicken N Chips were empty and a half of the places at The Pizza Place were empty. If 18 more people wanted to eat out, how many would have to be turned away from the two restaurants?
- Introduce the problem by playing a guessing game:
I am a number which is half of 6 and 6, what am I?
- When the students give their answers ask them to explain their thinking. Ask if any one else used a different approach. For example some students will double 6 and then find half of 12. Others will see the "trick" in the problem use fact that doubling and halving are inverse operations.
- Read the problem with the class.
- Brainstorm for ways to solve the problem. Ask the students to explain their choice of operation. This encourages them to make connections with other problems that they have solved.
- As the students work on the problem ask questions that focus on their use of mental strategies and their ability to justify the reasonableness of their answers.
How are you solving the problem?
Is your answer reasonable? Why do you think that?
Can you think of a way to check your answer?
- Ask the students to record their solutions so that they can be displayed and shared with others.
- Look at and share solutions.
This problem might be about cars going to a netball game.
Extension to the problem
Now a third of the seats at The Chicken N Chips were empty and a half of the places at The Pizza Place were empty. If 18 more people wanted to eat out but they did not want to share a table with the people who were already in the restaurants, how many would have to be turned away from the two restaurants?
Now 24 people sit at four tables in The Chicken N Chips. So 4 by something equals 24. We have to undo this ‘fourness’ so we need an inverse operation (see Pizzas And Things). Dividing by four is the inverse operation. So dividing 24 by 4 gives 6. A table can seat 6 people.
In The Pizza Place there are 3 tables. As each can seat 6 people, The Pizza Place can seat 3 x 6 = 18 people.
For the second part of the question we need to know that a third of 24 is 8. So there are 8 seats vacant in The Chicken N Chips. We also need to know that a half of 18 is 9, so there are 9 seats vacant in the Pizza Place.
This means that there are 8 + 9 = 17 spare seats. As we have 18 people looking for a meal and 18 – 17 = 1, one person will have to go elsewhere (or perhaps have take-aways).
Solution to the Extension:
For the extension we have to think how the tables could be filled. Let’s start with the Chicken N Chips. Here there are 8 spare seats, so this means 24 – 8 = 16 seats filled. As each table seats 6 and there are four tables, the 16 could be seated in three seats or four. (But no less than three, why?) Under the conditions of the problem, none or 6 people could be seated.
At The Pizza Place, there are 9 people. They could be seated at two or three tables but not one. (Why?) Under the conditions of the problem, none or 6 people could be seated.
So there are three possible answers to this problem. First, all of the tables in both restaurants could be being used, so no further people could be seated. On the other hand, one table in one of the restaurants might be free. In this case 6 people could be seated. And finally, there might be a spare table in both restaurants. That would mean that 12 people could be seated.
The number turned away would then be 18, 12 or 6.