357. It’s not square!

I do love a little challenge for A-level Further Maths students. They are often confident and very capable mathematicians, but occasionally overlook the small details. This challenge looks into which strategies students use when working with 3D vectors, lines and angles.

The most annoying thing? There is no single correct answer.

What is the investigation?

Students start with two points, create a line, construct two perpendicular lines and then join up the lines – did they create a square? How do you know? Justify it?

Download the instructions here: It’s not square (docx), It’s not square (PDF)

Skills required

• Distance between two points
• Equation of a line in three dimensions
• Scalar (dot) product

Solution/Discussion point

• Students need to use the same direction vector for both perpendicular lines too create a square
• The two new corners need to be n the same direction away from the original line (not one above and one below)
• It’s interesting to discuss what non-squares they made. Technology could be used to plot them in 3D.

211. Hidden Rectangle problem

Cool vectors can be exciting! They can describe the motion of a particle, they can represent the acceleration of a rocket, they can tell you about the angle an impact takes place at!

Uncool vectors describe lines, they can intersect, they could be perpendicular, they could even describe skew lines in three-dimensions. Not quite as exciting. It isn’t difficult to see that revising standard C4 vectors can be a tad dull. How about an investigation? An investigation without an obvious answer. A question so simple that the answer is a single number. It’s the steps in between that make things interesting…

• I asked my A-Level class to find the area of a rectangle … simple so far, how is this worthy of C4?
• The rectangle is bounded by four vector equations … ok, points of intersection, line segment length, bit of Pythagoras there
• The vector equations are 3D … ooh, that makes it a bit harder
• There are eight equations to choose from … that’s mean, that means finding the angle between lines, checking for skewness, identifying parallel vectors
• There are plenty of ‘red herrings’ … now that is just unfair (great!)

The solution to the problem is a simple surd. If you do ‘Crack the Code’ or ‘Locked Box’ problems you could use the digits under the square root sign as your padlock code.

You can download the worksheet and teachers notes here: C4 Vectors Hidden rectangle (pdf)

Depending on the engagement/ability of the students this could take between 20 and 40 minutes. It would also make an easy to assess homework.

143. Jumping the gap

The transition from GCSE to A-level Maths is as smooth as can be for some students. Others need the London Underground sign:

This time last year the biggest issue (amongst many others) was the lack of logic and rigour in their algebraic solutions and graphs. I tried giving model answers (‘Thank you, Miss’, then file it in the recycling …. Grrr!). I tried explaining why it was important (you could almost hear the shutters slide down in most of their heads). I tried sharing the best student’s work on the board using the visualiser (type of document camera), but all to no avail. The majority of students thought they knew best and ignored all advice.

Now rather than go all Professor Umbridge on them*, I switched things around. They critiqued each other’s work.

Activity

1. You will need an exam (style) question, paper and post-it notes.

2. Ask students to complete the question on a sheet of paper – do not write names on it.

3. Put all the solutions out at the front or stick them to the board.

4. Give each student three post-its. They should write something good and something to improve and stick it on the work. Do this three times.

5. Each student reclaims their work and reads the notes. They then discuss the feedback and draw up a list of keypoints for improvement.

That could be the end of it, but I wanted to remind them of the task so:

6. Collect in the work and notes and mount them on half a noticeboard.

7. In the middle of the board put the question, the model solution and their list of key points for improvement.

8. For the next week or so keep referring to the wall display in lesson.

9. Set another question and repeat steps 2-5. Discuss how their work has (hopefully) improved.

10. Fill the remainder of the wall display with the work and comments.

This could be a useful activity to do at the start and end of a topic. It would also be a good BLP (Building Learning Power) activity.

* Professor Umbridge had a particularly sadistic detention task in Harry Potter, where whatever you wrote on the paper was etched into the detainee’s skin. Vile woman, odd ideas on education.

125. View from … Pensthorpe

The Sandpit is currently in Norfolk and on a daytrip to Pensthorpe. In the ‘Wild Rootz’ adventure playground there is a bridge – except it’s not.

It’s a damped ‘see-saw’, which you can run up and down. This is great fun – people run from one end to the other and try to reach the end before it goes down, with a bump.

However, you can also use the principle of moments to try and balance the bridge. It took a bit of running up and down and fine tuning (bouncing), but we got the bridge balanced with five members of the family, age range: 84, weight range: wouldn’t be polite to say.

If you know somewhere near you, with a similar piece of equipment, this would be a brilliant way to demonstrate moments with a Sixth Form class. Obviously the calculations would be slightly out, due to the effect of the dampers, but it’s still fun – and a lot safer than piling a whole class on a see-saw.

68. Another GCSE revision idea

You will be surprised to hear that this activity doesn’t involve cutting up a GCSE paper! See Foundation GCSE student review and How to make GCSE past papers fun.

Equipment
Digital version of a GCSE (or A-Level or Functional Skills) paper
Individual whiteboards & pens
Digital projector

Set up
Split the class into groups of 4-6. They will need a whiteboard each. Allocate a team number/letter or name. Project the first question on the screen.

Activity
Use the GCSE paper to set (part of) a question for the class.
They all answer on their whiteboards and hold up the answer when you say.

The beauty of this method is you can adapt the questions to the understanding of the class and focus on specific skills, as opposed to issuing a paper version and going through every question.

Scoring
It’s quite common for a few bright/strong characters to take over team games, unless you can find a way to avoid this. The scoring is quite easy.

2 points if every member of the group gets the answer right.

1 point for each team, if more than one team is 100% correct.

You may think that this will encourage copying, but there is a third score: