# Using Dynamic 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 (<<)

# Arrays

### Multiplication and Division

The array model (above) uses double sided counters. Yellow positive and red negative. See integers for more details.

This array model partitions when the divisor or the quotient are greater than 10. This model can be used to show the connection between the array, multiplication and division. It represents the structure behind the bus stop!  # Area and Bar Models

### Multiple Representations

• Multiple representations allow students to gain a deeper understanding of the mathematical structure.

• 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.

### 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.mp4

### Factors

Finding common factors using area models. In this model the height is the common factor. Change the slider to change the height. They have common factors when both bases are integers.

### Multiply Decimals

Multiply decimals between 0 and 1 using the area model.

Click and drag the coloured Dienes blocks over the shaded area to count the number of 0.1's, 0.01's, 0.001's and 0.0001's.

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. ### Division with Dienes and Area Model

These representations for base 10 use the same colours as algebra tiles for ones, x and x squared including red tiles for negatives.

### (ax + b)(cx + d)

Multiplication of two linear factors is modelled as an area using dynamic algebra tiles.

x is variable in all models. See the model below to link the area model with the graphical representation.

Negatives are represented by red tiles and make zero pairs. In the three models above and left, you can vary the size of a, b, c, d and x.

### Difference of Two Squares

Change a and b to change the size of the squares. The a+b and a-b labels are draggable to allow students to place them by the correct side.

Completing the square model to the left allows students to see the structure of the general form of x^2+bx + c = (x+b/2)^2 - (b/2)^2 + c.

(b/2)^2 and -(b/2)^2 make a zero pair.

# Base Blocks (3D Algebra Tiles)

### Base Blocks with Negative Indices

Base blocks allow students to make the link between Dienes blocks (base 10) and algebra tiles (base x) by exposing the mathematical structure of the place value of the base system.

# Dynamic Algebra Tiles

### Linear Expressions with Dynamic Algebra Tiles

Compare expressions with algebra tile representations using the dynamic variable sliders to change x.

This representation models the x and 1 algebra tiles. The green tile x is a variable and can be changed with the slider under the x axis.

# Sequences and Graphs

### Sequences using Dynamic Algebra Tiles sequences with table cropped.mp4

These dynamic Cuisenaire rods show the first 5 terms and the nth of linear sequences of the form mn+c. Connections can be make to graphical representations and y=mx+c.

# Trigonometry

### The Unit Circle Trig with unit circle cropped.mp4

# Area

See Perimeter, Area and Volume for further examples

# Number Lines

### Variable number lines

Arithmetic and Algebra in Early Mathematics Education Adapted from D.W. Carraher et al. (2006).