## Core concept 4.1

## Sequences

4.1.1.1* Appreciate that a sequence is a succession of terms formed according to a rule

4.1.1.2 Understand that a sequence can be generated and described using term-to-term approaches

4.1.1.3 Understand that a sequence can be generated and described by a position-to-term rule

### 4.1.1.1 Appreciate that a sequence is a succession of terms formed according to a rule

●Understand the characteristics of sequences.

●Understand that some numeric sequences can be described by a non-mathematical rule.

Hide and reveal activity

Students work in pairs. Student 1 sits facing the board and student 2 has their back to the board. Reveal the image and ask student 1 to describe the image to student 2. Student 2 draws the image on squared paper.

Ask both students what the next term in the sequence will look like. What will the 10th term look like? What will the 100th term look like? Can you sketch the 100th term?

This activity draws out key language and encourages students to notice patterns and structure.

Ask students to make this sequence using Cuisenaire rods.

What will the 10th term look like? What will the 100th term look like? Can you sketch the 100th term?

Is 43 in this sequence?

Two student responses to "Is 43 in this sequence?"

Student A: "Yes because you can make 43 with 4 orange rods a 2 and a one."

Student B "Yes because you can have two rods of length 21 and a one."

Student B had noticed the structure of the sequence and went on to say, "Could you say it is 2x+1 because you have two lines and a one?"

(Student A and B have already worked on Base Blocks)

Compare this sequence to the one above.

Same mathematical structure, different rearrangements of the rods.

Show this sequence in a different way.

Students that had recognised the structure made sequences (below) using 4 rods of the same colour and 1 one.

Students that had not fully grasped the concept created sequences with different numbers of rods. When this was pointed out all students were able to make sequences with 4 rods and 1 one,

Moving on to the dynamic image below, students noticed the constants and the variables. Student B said "we can say this is 4x+1 because it is 4 lines and 1 one. "

Change the sliders below to create 4n+1 and compare with the sequence above. What do you notice?

This PowerPoint contains all the images for the activities above.

4.1.2.1 Understand the features of an arithmetic sequence and be able to recognise one

4.1.2.2* Understand that any term in an arithmetic sequence can be expressed in terms of its position in the sequence (nth term)

4.1.2.3 Understand that the nth term allows for the calculation of any term

4.1.2.4 Determine whether a number is a term of a given arithmetic sequence

### 4.1.2.2 Understand that any term in an arithmetic sequence can be expressed in terms of its position in the sequence (nth term)

●Appreciate that each term in an arithmetic sequence has the same structure and that the expression of that structure in terms of its position in the sequence is the nth term.

●Solve familiar and unfamiliar problems, including real-life applications.

●Solve problems where there is more than one answer and where there are elements of experimentation, investigation, checking, reasoning, proof, etc.

4.1.2.4 Determine whether a number is a term of a given arithmetic sequence

Is 31 a term in this sequence?

Allow students to move the bar to see that 31 is the 15th term.

Is 40 in this sequence?

Allow students to move the bar to see that 40 is not a term in this sequence.

Is 65 a term in this sequence?

Students could move on to use algebra tiles to reason why 65 is a term in this sequence. This could be linked to solving equations.

4.1.3.1 Understand the features of a geometric sequence and be able to recognise one

4.1.3.2 Understand the features of special number sequences, such as square, triangle and cube, and be able to recognise one

4.1.3.3 Appreciate that there are other number sequences