In this unit students look for and describe the patterns they see in different types of staircases.
- Continue a sequential pattern.
- Develop bar charts to show relationships.
The students will have experienced and explored simple non-numerical patterns in lessons such as Snakes and Scarves. In this unit we look at geometrical patterns and the number patterns that are associated with them.
There are two aspects of these number patterns that are worth thinking about. If possible it is nice to be able to find the actual numbers involved. For instance in the up-and-down staircases, the number of blocks is a square number. So if we have a 7-step up-and-down staircase, we need 72 = 49 blocks, while for a 100-step up-and-down staircase, we need 1002 = 10000 blocks. Here there is a simple functional relationship between the number of steps and the number of blocks. (See Algebra Information.)
But for some number patterns it is not so easy to find a functional relationship. Consider the straight up staircases of Problem 1. Here it is much easier to see how the number of blocks changes from one stair to the next. In this problem going from a 4-step to a 5-step staircase we add on 5 blocks, and going from a 53-step to a 54-step staircase we add on 54 blocks. This type of link is called a recurrence relationship. (See Algebra Information.)
In general we would really like to have a functional relationship. It turns out that sometimes we can use a recurrence relationship to get a functional relationship. But even if we can’t we can always get the exact number of blocks for a given situation by starting from the smallest case and adding on the appropriate numbers until we get to the case that we want. For instance, to get the number of blocks in a 6-step staircase in Problem 1 we start with 1 block for the 1-step staircase. Then we add 2 to get the 2-step; 3 to get the 3-step; 4 to get the 4-step; 5 to get the 5-step; and 6 to get the 6-step. Altogether then we need 1 + 2 + 3 + 4 + 5 + 6 = 21 blocks.
- Multi-link cubes
- Staircase problems
- Graph paper
Today we explore up-and-down staircases to find the pattern in the number of blocks they are made from.
- Begin the session by telling the students about up-and-down staircases:
One block is needed to make a 1-step up-and-down staircase. It takes one step to get up and one step to get down.
This is a called a 2-step staircase as it takes two steps to go up and two steps to go down.
- Together count the steps so that the students understand why it is called a 2-step staircase.
How many blocks are in the staircase?
How many blocks do you think would be in a 3-step up-and-down staircase?
How could you work it out?
- Give the students time to work out the number of blocks. Share the ways that they used to work it out.
- Ask the students to guess how many cubes they think would be needed to make a 5-step staircase. Get them to build a 5-step staircase. Check their guesses.
- Get the students to build more staircases. As they do, ask them about any patterns they see.
Some may recognise the horizontal layers as being the sequence of odd numbers. Some may see the vertical stacks.
Some may see that square numbers of blocks are always involved. This can be checked by rearranging the blocks to make square numbers (see the diagram below). This is an important discovery. Let them make it. This may require some careful scaffolding on your part.
- Some students may wish to continue to find numbers that make larger up-and-down staircases. To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. However, if the students prefer to use cubes, then they should record their results in a table.
Over the next 2-3 days the students work in pairs or individually to solve the following problems. Show the students how to use grid paper to continue the patterns. As they complete the problems ask them about any patterns they see and encourage the students to record these observations with the patterns on the graph paper.
Problem 1: Straight up the stairs
How many blocks are in this 4-step-up staircase?
How many blocks would there be in a 5-step-up staircase?
How many blocks would there be in a 6-step-up staircase?
How many blocks in a 10-step-up staircase?
How many more blocks will an 11-step-up staircase need?
What is the largest up staircase that you can tell us about?
(Note: the numbers of blocks in this pattern are the triangular numbers, see Algebra Information.)
Problem 2: Climbing ladders
How many pieces of wood have we used in this 1-rung ladder?
How many pieces of wood have we used in this 2-rung ladder?
How many pieces of wood would there be in a 4-rung ladder?
How many pieces of wood would there be in a 6-rung ladder?
What is the largest ladder that you can tell us about?
How many pieces of wood will you need to add to a 7-rung ladder to get an 8-rung ladder?
(Note: the number of pieces of wood is three times the number of rungs.)
Problem 3: Small steps
Watch out! You need to take small steps to walk up and down these stairs.
How many blocks are in the 4-step staircase?
How many blocks are in the 6-step staircase?
What is the largest staircase that you could tell us about?
Does this remind you of something you have done before?
(Note: the count here is the same as that in Problem 1.)
Problem 4: Star patterns
|This is a 1-star||This is a 2-star||This is a 3-star|
How many blocks are in a 4-star?
How many blocks are in a 5-star?
What do you notice about the stars?
How many blocks do you need to add to a 7-star to make an 8-star?
What is the largest star that you could tell us about?
(Note: the pattern here is 1, 5, 9, 13, … At each stage you add on 4 blocks. To make a 100-star you need to have 99 lots of 4 plus one block for the centre.)
Problem 5: L-shapes
|This is a 1-L||This is a 2-L||This is a 3-L|
How many blocks are in a 4-L?
How many blocks are in a 5-L?
What do you notice about the pattern in the L’s?
What is the largest L that you could tell us about?
(Note: to make a 100-L you need 100 + 100 – 1 = 199 blocks.)
In this session we share our solutions to the problems of the previous days. We listen carefully as the patterns are explained. We then make some block patterns of our own which we give to our classmates to continue.
- Begin the session by asking the students to attach their solutions to the problems to a display wall. Give the students time to look at the solutions of other students. Ask for volunteers to share their solutions.
- Give pairs of students a supply of blocks and grid paper and ask them to invent their own block pattern. Tell them to record the first three elements in the pattern on a piece of grid paper.
- Ask the students to swap patterns with another pair. Work together to discover the pattern and then continue it.
- Repeat with another pair’s pattern.
- Leave the patterns on a table for the students to solve in their own time.