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Why do this problem?
The geometric representation of sequences in this problem gives students something concrete to work with as they begin working with quadratic sequences. As well as introducing or consolidating sequences expressed term-to-term or position-to-term, this problem offers a chance to gain fluency in creating and manipulating algebraic expressions and recognising equivalence.
Squared paper and mini-whiteboards could be useful.
Show students the first sequence of patterns. Check that they can draw the next pattern in the sequence. How did they know what it would look like?
Next, ask students to count the number of squares in each shape. Is there a rule? Is the number pattern (+8 each time – they are likely to spot the term-to-term rule, rather than a rule for the $n$th term) related to the geometric rule (add 2 more squares to each side of the shape)? Allow some time for students to work alone or in small groups, and then to share their work with the class.
Then ask students how many squares there would be in the 20th pattern, or the 50th pattern. Let them work alone or in small groups so that you end up with a variety of methods and ideas. You could encourage students to focus on the geometric representation by suggesting they sketch the 20th pattern – how long will the sides be? You could suggest that they make a table showing $n$ and the number of squares in the $n$th pattern to encourage them to find a position-to-term rule. Students can then extend their reasoning to the 50th pattern, and so on, and generalise to the $n$th pattern.
Several different representations are likely to arise, and so this is a good time to share students’ work. Give students time to confirm that their different methods lead to the same results (or to find out why they don’t) and to understand the connections between the representations. You could leave examples of the different methods on the board so that students can refer to them as they move on to the next part of the problem. You might want to agree on some notation at this point.
For the next part, students are likely to feel ready to begin investigating using the representations from the first part. The sequence is a quadratic sequence, and so may be more challenging for students to generalise. As well as the strategies used previously, they may notice that each pattern is a square with one corner missing – or that the number of squares used is always one less than a square number. Some may recognise the sequence as the sum of the patterns in the first sequence.
Once again, several different methods and representations are likely to emerge as students work in small groups. You might want to bring the students together for a whole class discussion to share their work as the end of the lesson, or before moving all students on to the third part of the problem. Alternatively, you could move some students onto the third part once you feel they are ready, and share their findings at the end of the lesson.
How are we going to record our findings?
What is the difference between each shape in the sequence?
Can you find a pattern between the position in the sequence and the number of squares in the shape?
How do you know your pattern will continue?
Encourage students to draw out the patterns and describe them verbally. Then formalise the description mathematically and introduce a notation in a structured way, possibly by using a table for e.g.
How many different methods can you find to solve this problem?
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