# Dynamic Resources

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

# Arrays

### Multiplication and Division

The counter model array (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**

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

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.

### 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

See Number to Algebra for a detailed explanation and further examples

# 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.

See Simplifying and manipulating algebra for further examples

### with Dynamic Algebra Tiles and Vectors

See Solving Equations for more examples

# 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. See Core concept 4.1: Sequences for examples.

See Algebra Tiles and Graphs for more models.

### with Dynamic Algebra Tiles

# Integers

See Integers for further examples.

# 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).*

# Double Number Line and Ratio Tables

See understanding multiplicative relationships for further examples