Multiple Representations

Multiple representations allow students to gain a deeper understanding.

Comparing multiple methods allows students make connections and use the most appropriate method with fluency.

Comparing different representations uncovers the structure of the mathematics and allows students to make generalisations.

Using Multiple Representations

Hide and reveal each representation by clicking the circle in the left of the control panel (left).

All highlighted points and arrows are interactive (top right).

Click points to reveal hidden text (bottom left)

Click the image to open the interactive representation in Desmos.  You can edit this version and save it to your own account.   

DO NOT WORRY ABOUT BREAKING IT!  If you accidentally delete something or rescale it and cannot get it back simply come back to this page and click the image again.

Some representations have been embedded into this site.  To open in full screen on Desmos, click edit graph on Desmos in the bottom right.  

Minimise the control panel in Desmos for the best view (<<)


Fractions, Decimals and Percentages


Fractions, decimals and percentages are represented by a 100 grid, decimal number line, percentage number line and bar model.  Instruction video link

FDP mult rep template.docx

blank printable templates 

Percentage Increase and Decrease

Bar model, double number line and ratio table. Instruction video link.

Students often find it hard to visualise beyond 100% using bar models.  A double number line allows students to compare % and decimals and links to ratio tables - see Multiplicative Reasoning.

Percentage increase and decrease.docx

blank printable template

Multiply two digits by two digits

The area model uses 2D representations of Dienes blocks (below).  Compare this to the grid model (centre).    

Multiply Decimals

The area model for decimals is represented alongside Dienes blocks.  Drag the blocks over the area to clear up any misconceptions.

For example 0.31 x 0.2 = 0.62

The area above is covered by 6x0.01s and 2x0.001s not 6x0.1s and 2x0.01s.

0.31 x 0.2 = 0.062

multiplying decimals 2.docx

blank printable template

multiply decimals.docx

blank printable template

Multiply whole numbers and decimals

Multiply two decimals up to 1 decimal place using the area model.  Click and drag the coloured Dienes blocks over the shaded area to count the number of 1's, 0.1's and 0.01's.

Sequences and Graphs