If you have found these intelligently varied sequences of questions and examples useful, then please consider submitting your own set, or offering an alternative version of an existing exercise that worked well with your class. This site can only continue to grow with your contributions.

Writing tips

If you are anything like me, you will find the process of creating a sequence of intelligently varied questions or examples quite tricky. I am by no means an expert, but the things I find useful to consider are:

• decide on a really precise area of maths (e.g. adding fractions)
• decide what concept or idea from that specific area you wish your students to understand (e.g. deciding if the fractions are in a form ready to be added)
• decide what are the critical features of that concept you wish to draw students’ attention to during the sequence. A useful way of doing this might be to consider where you have seen students make errors in the past – what are the common misconceptions, what are the gaps in their knowledge?
• decide which of the four types of activity (Practice, Rule, Pattern, Demonstration) is the best vehicle for developing an understanding of that concept
• start with some really simple questions or examples – this stops you making assumptions about what students can do, and also gives an important confidence boost for students starting the exercise
• when creating your sequence of questions or examples, try to only change a limited number of things from each question to the next. This invariance is key to draw students’ attention to the critical features you identified before you started writing the sequence. It may be elements of the question that contain this invariance, or it may be the answers.
• if in doubt, go for less variation over more variation
• do not be afraid to repeat the same type of variation in a few questions, especially if it is a concept/skill that you feel it is particularly important to draw students’ attention to
• don’t feel pressured to make the sequence of questions get harder
• try to include Boundary Examples – the kind of examples that may look a bit odd and catch students out if not dealt with early (e.g. including whole numbers, improper fractions, and things like 5/5)
• think how far you can push the concept (e.g. can you include an opportunity to generalise towards the end of the sequence?)
• maybe include high value areas of maths, such as negative numbers, decimals, fractions, the 4 operations – basically, the kind of skills that need constantly reinforcing
• THIS BIT IS SUPER IMPORTANT: have a look at your completed sequence – is it possible for students to form some sort of expectation about each answer before they work it out:
  • Will the answer increase/decrease/stay the same?
  • Will the answer look different?
  • Will the way you work out the answer change?
  • It is these expectations that set up the power of the explain part of Reflect, Expect, Check, Explain – now students have worked out the answer, are they surprised by it? Can they explain the answer to someone who is surprised?
• show your sequence to a colleague, and try it out with a class
• reflect – what worked, what didn’t
• tweak it where needed
• if possible, try it again
• then, when you are happy, download the relevant template from below and email it me using the address at the bottom of this page
• not only will you have my eternal gratitude, but you will be helping teachers and students all over the world 🙂

Templates

There are four different types of activities on this site, and you need to choose the one that best suits the concept you wish to develop. Each activity type has its own components. Here you can download an example of each. Please follow the structure of the templates so I don’t have to do too much editing at my end! And I know this sounds really fussy, but please could all text be written in Calibri, and all mathematical expressions by done using PowerPoint’s equation function. It just makes it look all nice and consistent. Thank you 🙂

Practice

Please read what is involved in a Practice activity here
Download Practice template
Purpose:  to enable students to gain vital practice in carrying out a mathematical procedure, whilst at the same time providing opportunities for students to make connections, predict, discuss, argue, reason, think harder, and not cruise through on autopilot.

Structure of this activity:

• Example-Problem Pair
• A sequence of intelligently varied practice questions
• Answers to those questions

Rule

Please read what is involved in a Rule activity here
Download Rule template
Purpose: To show students a carefully varied series of examples and non-examples so they can better discern for themselves what constitutes a rule or a definition 

Structure of this activity:

• A sequence of intelligently varied examples with an indication of whether they fit the rule or not
• Answers to these examples
• A selection of practice questions (ideally 5) to see if students have understood the rule
• Answers to those questions

Pattern

Please read what is involved in a Pattern activity here
Download Pattern template
Purpose: To present students with a logical pattern which helps them predict the answer to something they did not previously know because it forms a part of this pattern. 

Structure of this activity:

• A sequence of intelligently varied examples that follow a pattern, and at the end of this sequence a break (“the gap of understanding”) which students must cross to show they are not merely relying on the pattern
• Answers to these examples
• A selection of practice questions (ideally 5) to see if students have understood how to apply the concept within the pattern
• Answers to those questions

Demonstration

Please read what is involved in a Demonstration activity here
Download Demonstration template
Purpose: To demonstrate a concept to students using technology (e.g. Desmos or Geogebra) in a dynamic and intelligently varied way

Structure of this activity:

• A link to an interactive demonstration – ideally consisting of two examples, so students have something to compare to
• A selection of possible variation prompts

Ready to submit?

Please email your creations back to me at mrbartonmaths @ gmail.com (but without the space) and I will add them to the collection!

Thank you so much 🙂

My book

My book, Reflect, Expect, Check, Explain, contains many more tips for writing sequences of questions and examples, together with structures for writing sequences with colleagues.