Helping a Struggling Maths Student: Investigating Quadrilaterals

Goals of this week’s maths fun:

• To discover the shapes and names of all the quadrilaterals
• To be able to recognise each of them as plain mathematical shapes and as shapes in everyday life.
• To sort, classify and display their findings relating to a bag of quadrilaterals
• To discover and fully understand the terms 2D and 3D and introduce the girls to the terms cube, cuboid and square  based pyramid
• To discover the basic properties of a cube net.  To draw their own net and make a cube from it

All of these things they have already covered in their maths in years gone by.  I think for the girls, seeing diagrams and being told facts about the diagrams doesn’t help them to know and understand what they are, and ultimately what can be learnt from them.  My girls, it seems, need to touch and experience maths in far greater ways than a simple piece of paper.

2D QUADRILATERALS

Last week I had the girls begin their new curriculum free maths with an investigation of shapes.  First up was the classification of different types of two-dimensional shapes.  I gathered together all sorts of shapes of different sizes and form.  I let the girls take a good look before asking them to begin to classify them.  What would be the easiest way to classify?  What criteria should they use?  The purpose of this was for them to really see the shapes and what it was that made up the shapes.  They had done lots of classification in their science so I knew they would understand the concept.  Here are the groups they came up with, including the criteria they used (I also used polygons which were on another tray.  I thought I had taken a photo but it wasn’t on my camera so who knows?):

I asked them to name each group based on the criteria they had chosen.  The purpose of this was for them to see for themselves that mathematic terms are often very logical, using Latin or Greek roots to explain what they mean.  From this we looked at the roots of the words ‘triangles’, ‘rectangles’ and so forth.

I pulled all of the quadrilaterals together and asked collectively what they would name them, if they had the chance.  They chose cuatrangles.  Cuatr- from their knowledge of Spanish numbers (Cuotro) and angles from the fact that there are four angles.  I then had them name the shapes I had.  I’d been quite impressed by their first attempt with cuatrangles, but they seemed to have a bit more fun naming these.  Not quite such sensible names:

I then wrote on the board what they were called in real life.  I had written out cards of the names of the ones I wanted them to know about and we discussed what each of the shapes were and they matched them up to the card.

They repeated this each day to help them remember the terms.  In addition I told a mathematical story each day, giving them tooth picks and marshmallows to build the shape in the story.  If they could name it they could eat it.  May I make an observation that the brain works much better if there is a bribe reward at the end:

I also had them use elastic bands and pin board to make up variations of the quadrilaterals:

I gave them a bag of shapes, all different types of quadrilaterals and asked them to classify, count and illustrate their findings by making up a pictograph on the table:

We also made play dough squares and rectangles and I let them discover for themselves how the various quadrilaterals could be made from these two shapes.  That a rhombus was very similar to a squashed square and a trapezium is a square with a squashed top or bottom, a parallelogram was a squashed rectangle and a deltoid was a square or rectangle where the two sides had been manipulated.  Not strictly mathematical but hands on learning always appeals to my girls, so I was hoping this would also aid their memory:

Lastly, just to illustrate how far-reaching mathematical names can reach, I had them find out where the Trapezium, Rhomboid and Deltoid muscles were and make a hypothesis using mathematical language why they were called by those names.  The term ‘mathematical language’ is coming up a lot in our household these days.  I want the children to be using the terms they know in their every day language so they become comfortable with them.  I believe many questions in exams are forfeited not on a lack of mathematical understanding but a lack of knowledge pertaining to mathematical vocabulary.  I am now a self-proclaimed mathematical bore.

3D SHAPES with a QUADRILATERAL BASE

I had bought a pack of 3D shapes for A5’s use and we looked at all of them, separating the ones with a quadrilateral as a base.  I had them describe in their own words what the difference was between 2D and 3D.  I was looking for 2D having length and width and 3D having those plus depth:

I then had them make up some 2D play dough models and label them and some 3D play dough models and asked them to discuss what their names might be.  I brought out the Magnetix and had them build to their heart’s content.  First they made 2D then concentrated their efforts on 3D:

I gave them a sheet of squared paper and asked them if they could make me a 3D model of a cuboid.  I thought they would find this really hard……I encouraged them to unravel some of the models they had made previously so they could really start to investigate how the cube was put together.  They made all the nets they could think of for a cube from Magnetix.  This introduced the idea of variations:

I asked them to investigate the combinations they found of squares which make up a cube.  They came up with the following rules:

1. No more than four squares in a row
2. There must be a length of four but there doesn’t necessarily need to be four in a row ( see top left)
3. If you start with a row of four squares the two squares which make up the required six faces needed to be placed on the side of the row of four.  However they need to be placed one on each side and can never be on the same side
4. The extra two squares can be placed in any position along a row of four squares so long as there is one on each side.

They found six.  There are eleven, but I want the girls to discover these for themselves.  I had them make up rules only for those they found on this occasion.  I have not told them that there are four more to find.  However, next week they will be given time to find more.  They will also be encouraged to review their rules and see if they still apply.  This replicates mathematical investigations done by adults who work in a mathematic sphere.  They find out, hypothesise, find out some more and re-hypothesise.  I think this is a great way to learn maths because right now, for the girls, there is no right or wrong, simply lots of fun things to find out.  They don’t need to fear failure.  Each week they will unveil more of maths to themselves and hopefully start to experience its beauty.

The next day I had them transfer their discoveries onto paper and build me a cube:

In their final lesson I had them make rice crispy, marshmallow and toffee tray bakes.  Once the mixture had cooled and hardened somewhat I asked the girls to cut out each quadrilateral they had learnt about, name each and describe the properties of it.  You could eat what you made, named and described!  Never have two children been so excited about the possibility of more maths!!

During the cutting out and describing process, the girls used the term parallel lines a lot.  The deltoid doesn’t have any parallel lines and the girls were asking what those types of lines would be called.  I thought it would be a useful thing to investigate so after they had finished with the tray bakes I asked them to draw a deltoid on the white board and continue on their lines to show that they intersected each other.  Whilst they could not use the terms intersecting to describe the lines which make up a deltoid, it was useful to illustrate that two straight lines drawn in a 2D plane, if continued on from both ends, will eventually meet, unless they are parallel.  So two straight lines when drawn in a 2D plane will either be intersecting or parallel.  I asked the girls how they could prove two straight lines were parallel.  Between them, they came up with the following:

I didn’t confuse them by talking about skewed lines in a 3D plane.  They’ll be time for that at a later date!

These lessons were designed to help the girls become familiar with quadrilaterals in general.  Throughout the week I used mathematical language rather than every day language during our maths lesson.  The girls did over 1 1/2 hours of maths each day and I didn’t hear one complaint!

We did not go into specific properties at all, but it is some of those properties that will make up the bulk of this weeks lesson, along with extending their knowledge of nets.