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This printable worksheet may be useful: Growing Surprises.
The worksheet is also available in French: Suites Surprises
We are grateful to Clara Huizink for translating the original.
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 termtoterm or positiontoterm, this problem offers a chance to gain fluency in creating and manipulating algebraic expressions and recognising equivalence.
Possible approach
Squared paper and miniwhiteboards 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 termtoterm 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 20^{th} pattern, or the 50^{th} 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 20^{th} 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 positiontoterm rule. Students can then extend their reasoning to the 50^{th} 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.
Key questions
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?
Possible support
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.
Position in the sequence 
1 
2 
3 
Squares in the shape 



Link between position and number of squares 



Possible extension
How many different methods can you find to solve this problem?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?
How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?