Teaching mathematics
Teaching mathematics

Start this free course now. Just create an account and sign in. Enrol and complete the course for a free statement of participation or digital badge if available.

Free course

Teaching mathematics

Changing shapes

Learners will come up with several different ways of looking at the way these ‘houses’ grow. As you saw in Week 4, having multiple solution methods can give rise to fruitful algebraic reasoning if there is time planned in the lesson for learners to compare their methods, correct errors and explain why they are the same.

One very common method of describing the changing shapes is to focus on the 3 horizontal rows of matches. Another method is to note that each house has the same beginning and the same end. The middle section grows each time by adding 3 matches (1 each to the roof, the floor and the middle). We will now follow this second method to show its details.

The method is described in pictures, and then words and symbols. This sequence can be used for any methods.

Pictures (see Figure 8)

Figure 8 Growing houses


The number of matches for the 10th shape will be 5 (for the beginning) plus 2 (for the end) plus 3 for each of term added on after the first. You add one less group of 3 than the shape number.

Number symbols

It can be helpful for learners to see the calculations written as a sequence.

Shape 1: 5 + 2 or 7

Shape 2: 5 + 3 + 2 or 7 + 3

Shape 3: 5 + 3 + 3 + 2 or 7 + 3 + 3

Shape 4: 5 + 3 + 3 + 3 + 2 or 7 + 3 + 3 + 3

Now that we have got past 3 × 3, it is worth moving to multiplication instead of repeated addition.

Shape 5: 5 + 4 × 3 + 2 or 7 + 4 x 3

Words and algebraic symbols

Looking at the calculations shows a pattern. A method for finding the nth term is (starting with the part that changes) to calculate the number that is one less than n, multiply that by 3 and add it to 7.

In symbols this is (n-1) × 3 + 7. We can simplify this by expanding the brackets:

3n – 3 + 7

3n + 4.

So, the nth term of the sequence is 3n + 4.


Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to University-level study, we offer two introductory routes to our qualifications. You could either choose to start with an Access module, or a module which allows you to count your previous learning towards an Open University qualification. Read our guide on Where to take your learning next for more information.

Not ready for formal University study? Then browse over 1000 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus371