## Dancing Vectors

‘Explore and practise the concept of position vectors through dancing!’

Moves you never knew you had! Dancing Vectors introduces the idea of vectors as units of movement using the analogy of a dance routine! Each move from the routine is defined by a displacement vector (embellished a little for fun). Practise and perform a dance routine to music based on the instructions given as vectors that tell you how far to move and in which direction. Then use the analogy to help solve vector problems!

Resources

You will need these Routine Cards to help. There is a printable Dancing vectors quick guide for teachers, a follow up worksheet and Dancing Vectors teacher notes for this activity. You will also need a coopy of ‘Hot Stuff’ by Donna Summer to listen to! Hot Stuff on Youtube.

Activity in Action

Here is a video of a class doing the dancing vectors routine. Notice that some of the students are holding the vector map as their guide!

The Moves

Four dance moves are defined and demonstrated, with the use of volunteers.

Vector a – the jump
Vector b – the slide
Vector c – the diagonal reach
Vector d – the diagonal twist

Here is a short video demonstrating the four different vectors involved in this routine to help get the class going!

These moves are practised without music and then with. Volunteers/students are given Laminated ‘Routine cards’ with dance routines shown as Vector combination diagrams.

Discussion – vectors c and d can be expressed as combinations of a and b.

Preparation

This activity can be a little heavy on preparation but it really, really is worth it! Not only is it a lot of fun but it is really powerful way of understanding the concept. There are a few things needed to make this activity really work;

A print out of the vector maps, ‘Routine Cards’ and ‘Five went Dancing’ (link included above)
A space large enough for the class to move freely in
A copy of ‘Hot Stuff’ by donna summer (available also through youtube here)
Some speakers!
Some preparation time
A video camera if possible!

Combinations

The combinations are ‘Introduction’, ‘Verse’ and ‘Chorus’ and these are practised individually before being combined. This is then done to music ‘Hot Stuff’ Donna Summer.

The introduction uses only a and b, the verse introduces a different combination with some negatives and the chorus uses c and d.

5 Went Dancing!

For the second ‘verse’, there is an alternative routine, where 5 dancers each have their own routine to follow involving – in one case – half vectors. This is performed between the intro and the chorus for the 2nd verse.

The whole routine can be put together for a run through the first 2 verse and chorus cycles!

Routine Cards

The following is just a screen shot of the follow up worksheet to give an idea of what it contains. The worksheet builds on the ideas from the dance.

Description

Here follows an outline of the task;

• Teacher introduces and defines the concept of a displacement vector and uses the four vectors a, b, c and d as defined by the activity to help them.
• Students practise the four different displacement vectors and some simple combinations.
• Teacher and students begin the put the routine together by doing an ‘introduction’ as defined by the routine first without and then perhaps with the music.
• Then the next part of the routine without then with music.
• Try to run through the whole routine a few times working towards a final performance (this should be recorded on video!)
• There should follow a discussion on the collective effect of the different combinations of vectors.
• Students complete the associated worksheet.

‘Programming with equations to create reflective, artistic masterpieces!’

The big step up from primary school to secondary school reflections is in understanding that where you choose to place the mirror line is essential in determining the position of the reflected image – as in the position of the lake in front of the Taj Mahal! Watch the overview (with technical help hints & tips) of the activity below for how to use the embedded geogebra files under “Resources” to remind yourself how equations define perfectly the relationship between ‘x’ and ‘y’ coordinates on a line. Once you have understood how to position a line using an equation, put your knowledge to the test in Reflection Activity 1 and 2 below. You will then be ready to programme, using equations, a reflective, artistic masterpiece. In geogebra you can then drag these reflection lines to extend your fixed picture into a dynamic animation!

Resources

Use the three embedded geogebra files under “Geogebra” below (no software required) to explore how equations define perfectly the relationships between the ‘x’ and ‘y’ coordinate points that lie on each line. You can use the Equation Reflections activity sheet to record your experiments and findings.

Geogebra

Now try the Reflection activity 1 and Reflection Activity 2 task below. You can double click anywhere on these embedded geogebra files, at any time, to open them in the geogebra software, provided the software is already downloaded on your computer (FREE download here). See the “help” video under “Geogbra” below for an overview of the task.

Once you have completed Reflection Activity 1 and 2 above, or when your teacher instructs you to, create your own shapes and programme in equations, to produce a reflective, artistic masterpiece. In geogebra you can then drag these reflection lines to extend your fixed picture into a dynamic animation (see the Geogebra “animation” video below for an overview of this task).

Autograph

If you are using Autograph, scroll to the bottom of this page for Reflection Activity 1 and 2 and associated “help” video.

Geogebra Files

You can double click at any time on these embedded geogebra files and they will open in Geogebra (provided Geogebra is installed on your computer). For “Geogebra Reflection Activity 1” and “Reflection Activity 2” it is better to double click on the files and open these in Geogebra, provided the software is downloaded onto your computers (FREE download here) so that you can see the whole page.

Geogebra Reflection Activity 1

Geogebra Reflection Activity 2

Geogebra Animated Art

Geogebra Help Video

Autograph

If you are using Autograph open these autograph files: Reflection activity 1 and Reflection Activity 2. Find each of the equations necessary to complete the activity – see the “help” video below for an overview of the task.

Autograph Help Video

Description

The embedded geogebra files at the start of the lesson allow students to trace and create a table of coordinate points that lie on each line. Students can then relate the pattern they see between the X and Y coordinate to the equation of the lines.
Once students have understood how to define the position of the mirror lines using equations, they can attempt the reflection activity 1 and reflection activity 2. The “Help” videos under the “Resources” section provide the necessary technical support for using Autograph or Geogebra. Students should have these help videos open and ready to fast forward, rewind and pause as required.
The use of geogebra and autograph forces students to have to enter equations to perform the necessary reflections. Geogebra is helpful for students that require a little more assistance as they can drag their mirror lines to see how this changes the equation of the line. This can be a helpful, additional step to understanding the use of equations to define the mirror line.
In the last 10-20 minutes of the lesson, with their newly acquired skills, using a maximum of four shapes and as many different reflections as they wish, students should make their own artistic designs (as in the Images gallery above). For students that need an extension, get them to animate their reflections (see “Geogebra Animated Art” above).

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