# Dynamic Resources

# Using Dynamic Representations

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# Arrays

### Multiplication and Division

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

### Multiplication as Scaling

# 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

### Percentage increase and decrease

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

### Factors

### Area Models for Multiplication

### Area and Grid Models

Compare with algebra tiles

### 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.

### Quadratic Expressions with Algebra Tiles

### (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 from Base -10 to 10

### 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

### Dynamic Variable x

### 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.

### Equations and Inequalities

# Sequences and Graphs

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.

### Quadratic Graphs with Dynamic Algebra Tiles

# Integers

# Trigonometry

### The Unit Circle

# 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).