**A learning and teaching resource to demonstrate the concept of equivalence**

This resource was produced by purchasing two 60cm X 90cm pieces of advertising board from Bunnings Warehouse. Using a utility knife I cut out four 10cm X 35.5cm sections from one of the pieces of board. Using Adobe software I created the images of the rooms of the house, and the windows and front door decal which has been glued onto the board with a hot glue gun. The second piece of board acts as a backing to the house, so that the windows can sit snugly in the spaces that have been cut out. The windows have been divided into sections with a utility knife which will represent the fractions that will be worked with using this resource. The resource represents a house that has it’s windows missing, and they need to be put back into the house.

I have chosen to create this resource as it is large enough to use with the whole class, is visually engaging with the images and colours, and the concept is practical i.e.: the house is missing it’s windows and they need to be replaced. The 10cm X 35.5cm sections I cut from the board are all the same size, and therefore equivalent in size. I have then cut each of these sections into smaller pieces to represent different fractions (1/2, 1/4, 1/3, 1/6). The sections are still equivalent to each other but the pieces are different.

The way this resource will be used is in front of the whole class, perhaps as an introductory exercise to a full blown lesson on fractions. As a teacher I will introduce the concept of the house without windows to the class, I will have the window pieces removed from the resource and placed in a box. I will invite a volunteer to come to the front of the class and pick one piece of window from the box and push it into the space provided. The student could pick up a 1/2, 1/4, 1/3 or 1/6. I will then ask another student to come up and pick a second piece that is a different size to the first one and put that into the window section. This process will continue with the students until all windows are in tact. Once this is completed as a classroom we can identify the each window section is the same size. Each window section was broken up into different sized pieces. Each window section represents an equivalent whole section, but each one is broken up into different sized groups, and therefore represents a different fraction of the whole.

**The purpose of the equivalent fractions resource**

The resource can be used to demonstrate the concept of equivalence. The concept of equivalence ‘focuses on different representations of the same amount’ (Reys, Et Al 2012) . In this case different representations of a whole window broken into different groups which represent different fractions. The resource would be used to demonstrate that each window is of equivalent size, and by separating each window into different sized groups it will be clear to children that even though the windows are the same size or equivalent, the windows can be broken down into different sized groups or fractions.

The appropriate ACARA (2010) reference comes from the year 3 curriculum which states ‘*Understanding* includes connecting number representations with number sequences, partitioning and combining numbers flexibly, representing unit fractions, using appropriate language to communicate times, and identifying environmental symmetry.’ This directly relates to the sub strand ‘Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058).’ I have chosen this ACARA reference as it directly mentions the fractions represented on my resource, and talks about modeling and representing these fractions as part of a complete whole. My resource has four representations of equivalent whole areas and the fractions make up these complete wholes.

This concept will be introduced to the children at the third step of the language model, or the mathematics language stage of the model. The reason for this is that the concept of equivalence is a mathematical concept and therefore is not categorized as a materials language or student language word. Once the children have been introduced to the concept of equivalence it would make sense to introduce a resource such as this one because it is a graphic representation of a familiar object. The graphic representation and hands on nature of the resource combine to present the concept of equivalence and that these equivalent objects can be broken into equal groups, to children in year three according to the Australian Curriculum (ACARA, 2010).

**Using the equivalent fractions resource to facilitate learning**

In a lesson environment I will use this resource as a teaching aid in the front of the classroom, perhaps on an easel or sitting against the whiteboard, so long as all children can see it. I will have all pieces of the window sections in a box, so the house will be window-less as the picture demonstrates.

I will then begin by asking a child to come to the front of the room and pick out a piece of a window from the box, and place it into the window section of the house. This is demonstrated below.

This process will be repeated until each row is full, as can be demonstrated in the picture below.

Once this has been completed, I will then get the class to tell me how many pieces of window is in each window section, and therefore what fraction or group this window represents for example, the top window section has two pieces. It is a group of two. The whole window has been broken into a group of two.

So far the resource has been used to demonstrate the concept of equivalence in relation to the whole window section i.e.: I will ask the children if they can tell me what is the same about each window, and highlight that each window section is the same, or equivalent in size, but it has been broken into or divided into different sized groups or fractions.

I will then use this resource to demonstrate the concept of equivalence in terms of pieces that each window has been broken into, or how half a window is the same size as two quarters of a window, and how one third of a window is the same size as two sixths of a window.

I will conclude the use of this resource by asking the children to write down a sentence explaining what we have gone though today demonstrated by the house resource. I will write these down on the board and ask the children to copy the sentences down.

I will write 1/2 of 2 = 1, 2/4 of 4 =2, 1/3 of 3 = 1, 2/6 of 6 = 2. I will then ask the children to write down that 1/2 and 2/4 of window pieces cover the same space as does 1/3 = 2/6, to confirm and highlight the concept of equivalence in writing.

**References**

ACARA (2012), *The Australian Curriculum, *retrieved 24th May 2014 from http://www.australiancurriculum.edu.au/Year3

Reys, E., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., Falle, J., … Bennett, S. (2012), *Helping children learn mathematics*