# Upper Lower Bounds

To provide a "hands-on" experience of the concept of upper and lower bounds, this activity from Chris Kacmar: Upper and Lower Measures  is a really nice, pair work measuring activity using 1 inch scale rulers (with marks only every inch, no subdivisions), 0.1 inch scale rules (no subdivisions), measuring masses and volumes (using graduated cylinders).

The below resource should help students "to do" upper and lower bound questions, and feel comfortable with using 0.5 instead of 0.49999..... as the upper limit (with the help of inequality symbols)!

The design of the slides and hand-out is influenced by cognitive load theory. The quantity of information has been minimised to focus only on the essential (with a visual image to back up the numerical solutions). It's the result of frustration, across a range of ability levels and ages, that despite conceptual approaches, measuring activities etc. where students seemed to show a good understanding of the imprecision of measurement, too many students continued to consistently get these questions incorrect in an exam situation. The below resources have been effective in improving student's outcomes.

## Teaching slides

Please find below some slides to help present and teach these ideas to students. You can click on each image to make it fill the screen and then click through them. You might also consider presentation mode when using these slides.

## Student worksheet

The below PDF is a worksheet that students can use to accompany, and take notes, as the class works through the teacher slides. It includes the Practice and Exam Questions that can, alternatively, BE completed directly on this webpage using the below self-checking quizzes.

## Practice questions

Upper and lower bounds Practice questions These questions offer practice at the concepts, techniques and ideas from this topic.

## IB Exam Style questions

Upper and Lower bounds Exam style questions Once you feel you have a good grasp of the techniques, try these exam style questions in preparation for the exams.

## Syllabus

SL1.6 Upper and lower limits/bounds of rounded numbers

## Description

• Students are not comfortable with having a "5" as the upper limit, preferring .4 or .41 and eventually understanding $$0.4\bar { 9 }$$, but it does not come intuitively to many, preferring 0.4 or 0.41 etc. Getting students comfortable with the less than '<' symbol as not including 0.5  is the first challenge (some students are then comfortable understanding the greater ease with which a non-recurring number, ending in "5", can be used in calculations).
• The term 'significant figures' is not to be found outside of science and mathematics classrooms (and professional papers/articles).
• The lack of familiarity in day-to-day life with the wide range of decimal numbers i.e. students don't tend to see numbers with more than 2 decimal places much outside of a mathematics/science classroom.
• The use of unfamiliar units e.g. cl, mg, micro and nano units etc.

It is the burden of this extra knowledge, and vocabulary (to which they have to associate meaning) that experience suggests makes this harder for students than the concise focus on the numerical understanding (that most students do have a fair to good notion of) that halfway between two place values in a base-10 numbers system will always end in a 5.

The above resource is designed to try and ensure students are not distracted by their lack of understanding of/familiarity with the above terminology and instead focus on the fact that in a base 10 system, 5 is always halfway.

The aim is to build confidence (by focusing on the "essential" ) that they have a means of seeing their way through to the upper and lower bound of a rounded measure. Ideally, understanding of the above two points will come also (but, in my limited experience, not always for all students).

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