Estimation by approximation applets
For questions with numerical answers, I have always implored students to sense check their answers. While many do, there always seems to be a core that simply never check whether an answer looks reasonable. For example, they might be working out 53% of 247 and somehow get an answer that is less than 120. A quick check ought to suggest the answer is wrong, since they should have obtained something greater than 50% of 240.
Having pondered why some students never seem to check for reasonableness, I’ve identified a reason why they don’t, and a possible way to address the problem. For some students, as soon as they’ve got an answer, they feel their work is done, and they don’t feel motivated to put the effort into checking. Often these students don’t realise how quick it is to use approximation to make estimates. To address this, rather than getting students to sense check their answers, I’ve had some success with getting students to roughly estimate the answer before performing a calculation. I’ve encouraged this by using question generators that generate multiple choice questions where all the options are approximations, so there is little point performing an exact computation. With such questions, it quickly becomes obvious that the fastest way to answer these is to use approximations and a bit of logic:
The above screenshot shows a multiple choice percentages question. The idea here is that students should spot that the answer must be a little greater than 20, and that of the available options, 21 is clearly be the closest. Using this question generator, we can give students as much practice as needed to get them comfortable with approximation and estimation. Once they see how quick it is, I’ve found it easier to encourage them to make estimations before exact computations when doing typical exercises.
Desmos classroom activities
We have added some more Desmos classroom activities to our resources, including a couple of activities on rounding, available in the Teacher Resources boxes for both N15a Parts 1 and 2. These activities contain randomised questions which means that for any given tasks, students in general will get similar, but not identical, questions to each other. As a teacher, you can quickly see the type of questions that students are seeing, but you can also dig deeper to see the specific question that each student sees. The video below shows the mobile phone screen of a student working through a task (on the right) and the teacher’s dashboard view (on the left). Whilst teachers can monitor students’ progress on a question-by-question basis during a live lesson, you don’t have to do this in real time. You can simply check how students fared at a later time, if you wish. This makes these activities suitable for use as homework tasks.
Where available, you will find links to these Desmos classroom activities in the expandable Teacher resources boxes under a topic’s set of slides.
We have added more Target X sheets, containing questions from a mix of topics, are designed to be used with students targeting GCSE grades between 4 and 9. Target 4, Target 5, and Target 6 sheets contain four questions, whilst Target 7, Target 8, and Target 9 sheets each contain two longer questions. The sheets are designed to take students around 10 minutes. Each sheet is available in three “fixed” versions A, B, and C; these are PDFs containing fixed questions. Additionally, each sheet is also available as a “dynamic” version, which you can use to generate and unlimited number of questions similar to those in the fixed sheets. This video shows dynamically regenerating version of Target 8 Sheet 03:
We have continued to add Delta exercises to a few topics, to follow on from the Alpha, Beta and Gamma exercises. Here are a couple of exercises from A22a, on linear inequalities.