# Doctor Dart

### Task 89 ... Years 4 - 10

#### Summary

Embedded in a story shell that is almost like a video game scenario, the challenge involves considerable arithmetic practice as the students search to fulfil the Evil Professor's rules and gain the required score to open the door. How many possible scores are there? How many of these give the one that opens the door.

#### Materials

• Blocks numbered:
8, 8, 8, 9, 9, 10, 10, 11, 11

#### Content

• basic arithmetic practice as mental calculation
• odd & even numbers
• tree diagrams
• problem solving

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The language on the card makes more sense if the blocks stand on top of each other, rather than lying flat. Then they look more like a door with a number pad. The first section of the card (above the double line) can be answered by guess and check, but it is worth chatting with the students about some of the possibilities and restrictions.

• The final total of 60 is even.
• The second number scored must contribute an even amount to this final total because it is twice the number zapped.
• Therefore, if the first number zapped is even, the third number zapped must also be even, so that the total is even, since three times an even number is even, but three times an odd number is odd.
• However, if the first number zapped is odd, the third number zapped must also be odd, so that the total will be even.
This type of thinking raises the investigation from playing with numbers to playing with types of numbers. Applying these thoughts to the choices and thinking ahead can lead to solving Doctor Dart's problem.

The next set of questions is designed to lead students towards exploring every possibility. Perhaps they will use a table, or perhaps they will reach into their skill toolbox and use a Tree Diagram...

Doctor Dart has only 3 choices at the start, each of these has three consequential choices and each of these has another 3 consequential choices. That's a total of 27 possible choices.

Screen capture from Maths300 software that supports this investigation.

In answering these questions a possible reason for choosing the total 60 appears, and it also becomes clear that another total could have been chosen which would have been equally difficult to find.

The iceberg of the task begins to appear when students notice connections between the numbers in the original problem.

• They are consecutive.
• One is used three times (which one in the sequence?) and the others are used only twice.
• There is a pattern in the way they are arranged.
So what happens if...? We choose 4 other consecutive numbers, eg: 1, 2, 3, 4 and arrange them in the same way:

 2 4 1 4 1 3 1 3 2
• Is there a door opening number now?
• What happens if the Evil Professor changes the calculating rules?
• What happens if the numbers are arranged a different way?
• What happens if it isn't the lowest number that occurs three times?
• What happens if the key pad is 4 x 4? What numbers would you choose? How would you arrange them? What rules would you use? Is there a door opening total that only appears once?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

One way to begin this investigation with the whole class is to use large digit cards and act the problem out. Make three rows of students - one sitting on a three chairs, one standing behind them and one standing on chairs behind them. Students hold the digit cards in front of their chests and become the number pad. As a number is 'pressed' the student turns their card to the blank side and the next number down is seen to be on top.

Set up the problem in this way, then have student pairs tear up a piece of paper into 9 parts, number them and arrange them as in the array. Continue investigating guided by the questions above.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 61, Doctor Dart, which is extended by companion software.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Doctor Dart task is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 9 & 10

The Doctor Dart lesson is an integral part of:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 9 & 10