Mathematics level 3 - Patterns and algebra

Teaching context - Create number patterns using addition and subtraction

At this level students work with number patterns using addition and subtraction. For example, suppose you gave students the number sequence below and asked them to continue the pattern:

1, 3, 5, 7…

It is often difficult for students to instantly recognise how the pattern changes and in some instances, they may see the rule differently from others. A way to help students explain their thinking and build their experiences in finding patterns is to encourage them to use blocks, counters or drawings to represent their thinking before using mathematical symbols. 

The pattern below has been constructed with concrete materials such as square tiles. As students continue the patterns with materials or drawings, ensure they express what they see in words. In this case, the sequence can be explained in different ways. Below are examples of a student's findings:

• the numbers form a rectangle with one extra square
• these are odd numbers that cannot be equally divided into two groups (i.e. there will be a remainder)
• the numbers increase by two squares each time
• I double the term number then take one square tile each time e.g. if I double the second term I get four tiles.
  

From this, the student can generate a rule which is to add two to the previous number in the sequence. 1, 3, 5, 7, 9, 11, 13 etc.

In later years, they will learn to represent this information on a table and analyse the relationship between the terms and the number of blocks. For this level, students can use simpler rules such as adding two to the previous number (which is still recognisable on the table). 

When students begin exploring growing patterns, they can be challenged to find larger terms. For example, you could ask students to find the tenth term, which would require them to extend the table until they reach the desired number or generalise and check their rule.

Teaching idea - Create your own number sequence

In this activity students make a number sequence starting at a given number such as 83. Students create their patterns using multiple representations then explain how their pattern works. Then they swap these with a friend. They need to solve their friend's pattern by continuing the sequence using materials, drawings and symbols.

Extension ideas

Different number sequence starting points

Have students start their number sequence at a different point. For example, at a number in the hundreds or thousands, at a negative number, or at a number with one or two decimal places. This requires high-ability students to incorporate their understanding of place value, negative numbers and decimals into the task. They could be asked to explain the pattern used within their number sequence to like-ability peers.

Rules for number sequences

Give students some rules for their number sequence. For example, their number sequence must use at least two different operations, and end at the number 564. This adds complexity to the task for high-ability students as barriers have been placed in the way of a solution.

Locate existing number sequences and patterns

Have students locate number sequences and patterns that have already been developed, (for example, from nRich Maths). They can select questions of interest and work to find a solution. 

This task provides high-ability students with the scope to seek out tasks that are of interest to them, and are at, or just above, their point of need. It gives high-ability students a level of control over the tasks they choose to complete. This can be done individually or with like-ability peers, and students can share their solutions with others.