
SILO 6.1 (DRAFT) Year 6, Term 1: Cartesian geometry Focus: Coordinates Scope and sequence: Algebra
Learning intention: Students explore the Cartesian coordinate system linking geometry with algebra.
NSW Syllabus	Australian Curriculum (version 9.0)
"A student locates and describes points on a coordinate plane." (MA3-GM-01)	"Students learn to recognise situations, including financial contexts, that use integers; locate and represent integers on a number line and as coordinates on the Cartesian plane." (AC9M6N01)

Introduction to the topic

René Descartes

René Descartes (1596 – 1650) could be described as a 'triple threat' as he made significant contributions to science, philosophy and mathematics. This unit is about Cartesian geometry which is also known as as 'analytic geometry'. Analytic geometry is the study of geometry using a coordinate system.

(https://commons.wikimedia.org/wiki/File:Frans_Hals_-_Portret_van_Ren%C3%A9_Descartes.jpg)

Science

“It was René Descartes in 1637 who initially described the heart as a mechanical pump and the body as a machine” (Loxley, et al., 2018, p. 253).

Philosophy

Descartes is probably best known for his statement, “I think; therefore I am” which means we cannot doubt our existence while we doubt.

https://www.pinterest.com.au/pin/71353975316940566/

Using coordinates

The following 10 x 10 grid is labelled with letters from left to right and numbers from top to bottom. Using a letter and number specifies particular square on the grid. For example the square in the top-left corner is A 1. Coordinates are often written in brackets and separated with a comma so that would be (A,1). The square at the bottom-right would be (J,10).

Battleship game

This same grid can be used for the game Battleship. This game is commonly available as a board game with holes and pins or as a computer game where you play your opponent or the computer. It also works well on paper too. You can print an A4 page here which is double sided so you can play two games. Each player will need their own page.

Rules

Each player secretly draws all five ships onto the grid labelled 'My fleet'. The ships must be vertical or horizontal but they cannot be diagonal or overlap.
Decide who goes first.
The first play calls out a pair of coordinates such as (E,4).
If one of the other player's ships is located on (E,4) they must say, "Hit". If it is a miss then they say, "Miss". A miss is notated with an 'X' and a hit is notated with an 'O'.
Play continues back and forth in this manner.
When a ship has been destroyed by being hit in all squares the player must declare which ship has been sunk.
The winner is the first player to sink all of their opponent's ships.

The photo below shows how some clips and a piece of stiff board or cardboard can enable the paper pages to fit together as a two-player board game.

This introduction to coordinates will be extended into Cartesian geometry after looking at Euclidean geometry and algebra.

Euclidean geometry

Plane geometry

The following video (3:19) is a light-hearted look at the origins of Euclidean geometry.

Introduction to algebra

The following video (4:49) provides an introduction to algebra and equations using addition and subtraction.

Note about variables

Note that the video refers to x as a variable. They are using the word variable in a different way than we did in SILO 2.3 'Fair tests'. You will remember that there are many different types of variables such as independent, dependent and control. In the video they meant that x is a variable because it can change from one to question to another. However, in any given question, x simply means what is unknown but it is not really able to change. For example, in x + 3 = 9, x can only be 6 so it is not really a variable in the scientific sense.

Linking geometry with algebra

The following video (2:04) recounts a story about how Descartes devised his coordinate system while watching a fly on his ceiling. The story is considered to be a 'legend' as it might not have actually occurred in this way. Other versions of the legend mention square tiles on the ceiling which would help explain the layout of the grid.

Cartesian coordinates

(https://socratic.org/questions/where-is-the-x-axis-and-y-axis-located)

Using a table to help plot coordinates

The following image is from an A4 PDF worksheet which has table on the left and graph on the right. The double-sided worksheet can be downloaded by clicking here.