Multilink Staircases

Up and Down

Let's start with the 'up and down' staircase.  The one on the right is 4 steps high and is made from 16 cubes.

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

• Can you draw a 3 high 'up and down' staircase?  What about a 2 high 'up and down' staircase?

• How many cubes are needed for a 3 high 'up and down' staircase? What about a 2 high 'up and down' staircase?

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: label the first row "height" and the second row "number of cubes".

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 number of cubes and the height?

• What is the name of the sequence of numbers in the 'number of cubes' row?

• How many cubes would be required to make an n high 'up and down' staircase?

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 cubes are used for a 6 high 'up and down' staircase. Do the answers for the term to term and position to term rules match?

• Draw a 6 high 'up and down' staircase and count the number of cubes.  Does the answer 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 cubes would be needed for a 20 high 'up and down' staircase.

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

Double Up

Next, let's look at the 'double up' staircase.  The one on the right is 4 steps high and is made from 20 cubes.

Step 1: predict what the next patterns may be.  

• Can you draw a 3 high 'double up' staircase?  What about a 2 high 'double up' staircase?

• How many cubes are needed for a 3 high 'double up' staircase? What about a 2 high 'double up' staircase?

Step 2: organise your findings in a systematic way.  

• Can you summarise your findings in a table? 

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?

• How many MORE cubes are needed for a 4 high 'double up' staircase from a 4 high 'up and down' staircase?

• How many cubes are needed ALTOGETHER for a 4 high 'double up' staircase from a 4 high 'up and down' staircase?

• Repeat the last two bullet points with the 3 high and 2 high staircases.  What patterns have you noticed?

• Position to term rule - can you find the relationship between the number of cubes and the height?

• How many cubes would be required to make an n high 'up and down' staircase?

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 cubes are used for a 6 high 'double up' staircase. Do the answers for the term to term and position to term rules match?

• Draw a 6 high 'double up' staircase and count the number of cubes.  Does the answer 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 cubes would be needed for a 20 high 'double up' staircase.

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

Up

Next, let's look at the 'up' staircase.  The one on the right is 4 steps high and is made from 10 cubes.

For the steps below, you won't be as heavily guided as in the two previous sections.  If in doubt, look at the previous sections for clues of what to do next.  As mathematicians, you need to be able to investigate these problems independently without extra guidance. 

Step 1: predict what the next patterns may be.   

Step 2: organise your findings in a systematic way. 

Step 3: come up with some general rules.  

• Hint: you may want to think about how to go from a 'double up' staircase to an 'up' staircase.

Step 4: verify (a.k.a. check) that your rules work.  

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

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