Tiles

Introduction

In this task, we will practising the mathematical thinking outlined in the unit overview.  In particular, we will follow the following steps when investigating patterns.

Step 1: predict what the next patterns may be.  One good strategy is to draw out the next few patterns.

• Can you draw out the next few patterns?

Step 2: organise your findings in a systematic way.  One good strategy is to put your observations in a table. 

• Can you summarise your findings in a table? 

• Hint: you may wish to have four rows: label the first row "pattern number", the second row "drawing", the third row "number of yellow tiles" and the fourth row "number of blue tiles".

Step 3: use your table to come up with some general rules (term to term rule and position to term rule).

• Term to term rule - looking at your table, can you find a relationship going from one term to another?

• Position to term rule - can you find the relationship between the pattern number and the number of yellow tiles? The pattern number and the number of blue tiles? The number of yellow tiles and the number of blue tiles?

Step 4: verify (a.k.a. check) that your rules work.  This can be achieved by comparing your drawings with the answers found using your rules.

• Use your rules to find how many yellow and blue tiles are used in pattern number 6. Do the answers for the term to term and position to term rules match?

• Draw pattern number 6.  Do the number of yellow and blue tiles match the ones found using your rules?

Step 5: justify (a.k.a explain) why the rules work.

• Can you explain why your rules work?  You may wish to illustrate this by using diagrams.

• Use your rule to predict how many yellow and blue tiles would be needed for pattern number 20.

• If there are 35 yellow tiles how many blue tiles are there?

• If there are 93 blue tiles how many yellow tiles are there?

• If there are 238 blue tiles how many yellow tiles are there?

• If there are 104 blue tiles how many yellow tiles are there?  Explain your answer.

• Are Sushma and Tracy's explanations easy to follow?  Which one do you prefer?

• Did you find your rules these ways?  If not, how else can you explain why your rules work?  You may wish to use diagrams.

Take a look here for suggested answers for steps 1 to 5.

• Use your rule to predict how many red and white tiles would be needed for pattern number 20.

• If there are 35 red tiles how many white tiles are there?

• If there are 93 white tiles how many red tiles are there?

• If there are 237 white tiles how many red tiles are there?

• If there are 104 white tiles how many red tiles are there?  Explain your answer.

Practice

Now repeat the steps above with the following questions:

For more guided practice click on the worksheet here with the textbook questions here and answers are here.

Extension

For more interesting visual patterns, take a look at this website.

Try this Shifting Times Tables task on nRich.  If you need further support, use this to help you.

Try this Linear Sequence Puzzle on mathspad.  If you need further support, use this to help you.

For something not really related to linear sequences but still about tiles, try this Paving Path task on nRich.