# Truth Tiles 1

### Task 30 ... Years 2 - 8

#### Summary

Three equations have to be made true simultaneously using the digits 1 to 9 to fill the empty spaces.
• Can you find a solution?
• Can you find a different solution?
This cameo has a From The Classroom section which explains how the challenge was adapted to become a whole class investigation for one early years class.

#### Materials

• 9 numbered tiles

#### Content

• basic arithmetic skills
• problem solving strategies
• number patterns
• connection between addition and subtraction
• symmetry and combination theory properties
• fundamental laws of arithmetic - identity, commutative, inverse operations

#### Iceberg

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

Finding one solution is not too difficult, either by using a guess and check strategy, or by realising that the multiplication:

• can't be 1 x (something) because that would require a repeated digit
• must only be 2 x 3 or 2 x 4, otherwise the product would have two digits
Then comes a discussion about what makes a different solution. For example, if you have a solution using 2 x 3, is it different if everything else is the same but you make the multiplication 3 x 2? In effect, it doesn't matter, as long as you describe/define what you mean as different.

However, most agree that a different solution occurs when there is some change in the arrangement of digits in the other two equations. Two possible solutions are:

 5 + 4 = 9 8 - 7 = 1 2 x 3 = 6 7 + 1 = 8 9 - 5 = 4 2 x 3 = 6

The iceberg is driven by the mathematician's questions:

• How many solutions are there?
• How do I know when I have found them all?
Convincing someone else of the number of solutions requires application of problem solving strategies and fundamental laws of arithmetic.
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

#### 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.

To turn this task into a whole class investigation you could make a class set of number tiles. These would have many other uses. However, you could also ask students to rip up scrap paper so that they made nine pieces between each pair.

Start the problem in a central place using a large set of nine digits you prepared earlier. Gather the students in a central place (tables or floor space) and hand out the cards. Then place the +, -, x, and = cards you also prepared earlier on the space (as in the image on the card above) and explain the problem. Students with cards try to solve the problem, as others advise. As appropriate offer students the opportunity to find a solution for themselves using their number tiles.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 30, Truth Tiles, which is extended by companion software. Also, the Phillip-Steven-Harley Challenge in Calculating Changes was developed by these students during a whole class investigation of Truth Tiles 1.

#### 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 Truth Tiles task is an integral part of:

• MWA Number & Computation Years 3 & 4

The Truth Tiles lesson is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Number & Computation Years 7 & 8
This task is also included in the Task Centre Kit for Aboriginal Students and the Primary Library Kit. Solutions for tasks in the latter kit can be found here.

## From The Classroom

#### Lake Cargelligo Central School

Doug Williams
Consultant
I had the opportunity to work with a Stage 1 class (5/6 year olds) for a short while using this task in a whole class sense. Their teacher suggested the multiplication could be a problem. I took my lead from the companion Maths300 lesson and began a True/False game based on characteristics of myself, kids and things in the classroom. The children were sitting in a 'circle' around me on the mat. Then, we continued, using only one set of digits from 1 to 9...

 Okay, I can see you are good at the true/false game. Now let's try it with these signs. Firstly, tell me what they mean.

 True or false?

 True or false? How do you know it's true? Show me how you know. Can someone show me a different way? How many different ways can you convince me this is true?

 I introduced the subtraction sign and we tried a few more. True or false? How do you know it's true? Show me how you know. Can someone show me a different way? How many different ways can you convince me this is true?

 How about this? Something take away 1 equals something. Can you make this true? How many ways?

 Can you make this true? How many ways?

 Now I am getting really tricky. Can you make these both true at the same time using only the number cards 1 - 9?

 It's marvellous to see a smile of recognition when a student tries and gets to a position such as this. Uh oh! Nearly there, but you say it can't work. How do you know it can't work?

The next phase was to hand out strips of paper for pairs to quickly cut up and make digits 0-9 and +, - and = signs. The challenge became to find all the ways to make the two equations true at the same time.

Their teacher then realised that after a couple of days exploring that question and building a display of solutions, she could try introducing the multiplication part, along with exploring the x button on their calculator.