The unit investigates links between recursive rules and patterns on the hundreds board and other types of arrangements for whole numbers. Students are encouraged to find rules for relationships as shown through patterns on number grids and calculator outputs.
- Show number patterns using the hundreds board and other grid arrangements for whole numbers.
- Find the rule for a pattern of numbers shown on a hundreds board or for input/output pairs from a calculator.
- Relate sequential spatial patterns to how they appear as a number sequence on a hundreds board.
This unit looks at number patterns and their links with various shapes. At the end of the day, the students should be able to find rules for number patterns and see how to generate these numbers on a regular calculator. This aspect of the unit is a very useful preparation for later algebra. It is only a small step from the calculator instructions used here to a formal algebraic expression. The unknown in that expression is just the number of presses of the calculator = button.
Links to Numeracy
This unit provides an opportunity to focus on the strategies students are using to solve number problems, in particular strategies involving multiplication and division.
In session two students are involved in shading hundreds boards, 40-boards, 70-boards and 99-boards. It will become evident that shaded columns will emerge if the patterns used involve numbers that are multiples of the row length of the board. This provides a useful context to investigate multiplication and division.
Questions to develop strategic thinking.
Will the pattern of adding 4s fit on the 40-board in columns? Why / Why not?
Would a pattern of adding 6s make columns in a board with 12 squares in each row? How did you work that out?
What numbers would form patterns in a board with 8 squares in each row?
How do you know?
Encourage students to explain their reasoning. The patterns can be seen as repeated additions across each row or used to investigate the relationships between multiplication facts e.g. 4 x 4 = 2 x 8 becomes clear in the following table.
- Copymaster four
- Copymaster three
- Copymaster two
- Copymaster one
- Dice (1-6)
- Square tiles (optional)
- Grid paper
This session develops pattern seeking by looking at patterns on the hundreds board using the constant function of the calculator as well as students’ mental calculation.
- Introduce the students to this activity. Each student will begin by starting on the number 1. They show this by shading 1 on their copy of the hundreds board. To find the next shaded number they roll a standard dice (1-6) and add on that number of spaces, eg. if they throw 4 then 5 will be shaded on the board. They continue rolling and shading until they reach 100 or pass it. The shaded numbers form the students’ patterns.
- Put the students together in groups of four or five to share their patterns. Tell them that they are to find things that are the same about all five patterns. As the patterns were created by random number selection using a dice they will naturally vary considerably. The students may conclude that there is no consistent pattern. (Though some students may only have odd numbers, there might be some similarities between patterns though they are unlikely to be exactly the same.)
- How could we find out if any shaded patterns in this class are the same?
Ask the students for their ideas about how this might be checked.
One systematic way to do this would be to put the students into groups according to which square was shaded on their first throw. Within these smaller groups they could look for an identical pattern. In a class of thirty students this would be highly improbable.
- Next tell the students to use another colour and shade all the answers to their five times multiplication tables (multiples of 5): 5, 10, 15, 20, 25, …, etc. Show them that this pattern of numbers could be generated using the constant function on the calculator by keying in + 5 = = = = …etc.
- Have we got any patterns that are the same this time?
The students will report that they are all the same because each person followed the same rule and that the numbers lie in parallel vertical columns.
Why do you think this happens?
Some students may offer that because 5 divides evenly into 10 and each row of the array has ten numbers, the shaded numbers will be in the same relative position on each row.
- Now use the 99-board. What will happen when you shade the numbers that are multiples of 5 on this board?
Encourage ideas like: "They won’t be in columns like before because 5 doesn’t divide evenly into 9. So I think they’ll give an angled pattern."
What happens if we use the 40-board or the 70-board?
(The 40-, 70-, and 99-boards are given on copymasters.)
Get the students to shade in the multiples of 5 on each grid to check their predictions.
In this session we get the students to extend their observations from Session 1 by considering what patterns sets of multiples will form on the different number grids.
- Tell the students that you are going to get them to shade the multiples of 3 on both the hundreds and 99-boards. Get their ideas about what patterns they expect the shaded squares to make.
- After students have shaded each grid ask why they think the patterns occurred. Expect ideas related to divisibility again like, "9 can be divided evenly by 3 so we got a column pattern. When you divide 10 by 3 you get a remainder of 1 so the pattern slips 1 to the left in each row."
- If we had a grid with the consecutive numbers 12 to a row, what multiples would give column patterns?
(The answer is 2, 3, 4, 6 as they divide 12 exactly).
Students may wish to check this for themselves.
What pattern would the multiples of 5 make on the 12 grid?
12 divided by 5 leaves a remainder of 2 so the pattern will slip 2 places to the left on each new row.
- Invite the students to investigate what patterns the multiples of 2, 4, 6, 7, and 8 will make on each of the grids of copymaster one. In each case encourage prediction based on knowledge of divisibility. For example, on the seven grid the multiples of eight create a pattern that shifts 1 space to the right with each new row as it takes 1 full row and 1 more square to make each additional set of 8 squares.
In this session students try to work out the calculator rule that was used to create a given pattern of shaded squares on the hundreds board or other number grids.
- Show students copymaster two where some squares are shaded. Remind them that to create the multiples of 5 they entered + 5 === ... on their calculator. Ask them what calculator rule they think was entered to create the shaded patterns. Their predictions can be checked by keying them into a calculator to see if they produce numbers that match the shaded squares.
- The first pattern is the multiples of 3 and can be generated with the calculator input of + 3 ==== etc. The second rule is less obvious. Students may note that there is an increase of 4 between squares which suggests that + 4 == was used. But 4 is not the first number shaded. Here the rule was 3 + 4 === etc.
- Set students the problem of entering a rule using the constant function of their calculator, colouring in the numbers which come up in the calculator window when the rule is used, and giving the completed hundreds board to another student. That student then attempts to figure out what rule they used.
In this session the patterns created by certain types of repeating rules on the hundreds board are linked to predicting how many square tiles will be needed to make the fifteenth member of a spatial pattern.
- Show the students the sequential spatial pattern at the top of copymaster three. Ask them to draw what they think the fifth member of the pattern will look like using the squares in their exercise book. Tell them to work out how many squares make up the fifth member.
- Using a hundreds board highlight the numbers which give the total number of squares that the first, second, third, and fourth members have (5, 9, 13, 17, ...).
What calculator rule would give us this pattern of numbers? (1 + 4 = = =…)
Tell the students to investigate why the pattern numbers are increasing by 4 (1 is added to each arm of the figure each time.)
- Challenge the students to predict the number of squares that would be in the fifteenth member of the pattern using whatever strategy they choose. Students may elect to use the calculator, shade the hundreds board or draw the pattern. Discuss which method is most efficient and why.
(Using rules that allow manipulation of the numbers is considerably quicker than building the figure.)
- Some students may develop a functional rule for the relationship between the ordinal number of the member of the pattern (or term) and how many square tiles it takes to build it. In the case of the first pattern on copymaster three the functional rule is (n x 4) + 1 where n is the ordinal number of the term and the calculation yields the number of square tiles. Students may find other forms of the same type of rule such as (n - 1 x 4) + 5 or as words, "take one off the term number, multiply by four, and add 5".
- Get students to look at the second patterns on copymaster three. Tell them to work out how many square tiles will be needed to make the fifteenth term of the pattern. Share their methods in a whole class setting with particular focus on using rules to make a laborious task easy.
- The second pattern increases by two each time. The function rule for the number of tiles is (n x 2) + 2 where n is the term number. The fifteenth member will take (15 x 2) + 2 = 32 squares.
- Tell the students to make up their own growing pattern of square tiles using grid paper and use their pattern as a problem for others to solve.
Students explore the pattern created by triangular and square numbers on different grids.
- Give the students the following pattern of numbers.. 1, 3, 6, 10, 15, 21, 28, 36, … Ask them to predict the next few numbers in the pattern. Some may notice that the difference between members of the pattern increases by 1 each time, eg. 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, etc. and 4 is 1 more than 3 is 1 more than 2. This will allow them to predict the next few numbers 36 + 9 = 45, 45 + 10 = 55, 55 + 11 = 66.
- Get them to shade these numbers on the hundreds grid, and the two different staircase grids shown in copymaster four. Ask them to look for patterns. On the hundreds grid these numbers make no clear pattern but on the first triangular staircase they occupy the right hand squares. Ask the students why they think this occurs. Some may notice that the number of squares in each layer of the pattern increases by 1 as does the number pattern.
- Explain that these numbers are called triangular numbers because you can make larger and larger triangles with them. We show you how in the diagram below.
Ask the students to draw the next 2 triangles.
(Note: You need to use your imagination when making the 3 square triangle. Also note that these numbers make no clear pattern on the other staircase grid.)
- Provide the students with square tiles and tell them that you want them to make larger and larger squares with the tiles noting down how many tiles are used in total each time. (You may want to use the triangles above as a model.) This should give a pattern of 1, 4, 9, 16, 25, 36, 49, … Students may note that the pattern is increasing by odd numbers each time, eg. 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, etc. Challenge them to explain why this occurs. (To build the next square 2 lots of the existing side length are added plus 1 for the corner.)
- Get the students to mark the square numbers on the 2 staircase grids and to look for patterns. No clear pattern emerges on the triangular staircase but on the pyramidal staircase the square numbers occupy the right hand squares. Ask students why they think this occurs. (The layers added to the staircase are also the odd numbers). The pyramid might also lead to discovering that any square number is the sum of 2 consecutive triangular numbers. For example, 16 is the sum of 10 and 6, 25 is the sum of 15 and 10.
Can you see how to put together the triangular drawings for 6 and 10 together to make the square with 16 tiles?
- What about the 10 and 15 triangles? Can they be combined to make a 5 x 5 square?
- Students may enjoy investigating the patterns made by multiples of 2, and 3 on each staircase grid.