# Highest Number 2

### Task 128 ... Years 2 - 10

#### Summary

This task is a variation on Task 127, Highest Number 1. By using digits 1 to 9 and a ten-sided dice, it extends the place value component and challenges students to reconsider the higher order thinking involved when faced with the question of choosing a strategy for placing cards.

#### Materials

• One ten-sided dice
• Two sets of cards A to 9
• Two playing boards

#### Content

• basic place value skills
• basic arithmetic skills, especially addition
• probability, including calculating chances of simple events, long run expectation & probability distribution patterns
• statistics, including data collection in stem and leaf form and potential to introduce or revise, range, mode, median & mean in the context of first hand data

#### Iceberg

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

The first part of the task is a game, so has not specific answers. It is Question 2 that focuses attention on strategy for placing the cards. The following example indicates the type of reasoning that the task is encouraging students to employ.

5 is rolled first so place it in the centre position because it is in the middle of 1-9.

 5

The next roll is a 6. After that there are 7 numbers left for the third roll. If 6 is placed in the units, four of these (4, 3, 2, 1) would be a poor result, and only three (7, 8, 9) are good.

But if the 6 is placed in the hundreds:

 6 5

The score will 650+ whatever the next roll.

Possible scores the 1st way are:
156, 256, 356, 456, 756, 856, 956 (Total = 3792)
Possible scores the 2nd way are:
651, 652, 653, 654, 657, 658, 659 (Total = 4584)
Which decision is better in the long run?

Having analysed several (if not all) situations like this, students can be challenged to put their reasoning to work. Ask them to play a game composed of five rounds each. The winner is the person with the higher total score at the end of the five rounds, rather than the number of rounds an individual wins.

This can be taken further by suggesting that there is the opportunity for the game to be used at the school fair. Players would pay \$1 to play a five round game. If they achieved above a certain total the would win \$5. The challenge is to decided what that certain total would be.

To come to this decision students would need to play many more 5-round games and record their data (perhaps as a stem-and-leaf plot). The investigation can be taken as deeply as you wish, for example by asking what happens if the amount to play and/or the prize are changed. See the Investigation Guides in Highest Number 1 for ideas.

Another aspect of the task is its place value component. Especially in younger classes teachers might use the task primarily for this purpose. If so, the task can be varied by:

• Including a zero card.
• Adding an extra column to make Thousands.
• Adding extra columns and a decimal point.
The task could be included in a unit of work on Place Value that also used these activities which are from the Free Tour section of Calculating Changes: and the more advanced Calculator Slido.

#### 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.

• an invitation to two students to work like a mathematician
• a deeper investigation led by an Investigation Guide
• a whole class investigation to model how to work like a mathematician
The whole class lesson for this task could be based around the ideas above or on variations in the whole class investigation plan provided for Task 127, Highest Number 1. This has been extracted from the Chance & Data Replacement Unit and a variation of this plan, which includes photos from classrooms, is available in Maths300 Lesson 26, Highest Number. This Maths300 lesson includes software relating to strategy placement, but it only uses a six-sided dice.

#### 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 Highest Number 2 task is an integral part of:

• MWA Chance & Measurement Years 5 & 6

This task is also included in the Task Centre Kit for Aboriginal Students and the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.