**Dynamic Resources**

# Using Dynamic Representations

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

All highlighted points and arrows are interactive

Click points to reveal hidden text

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.

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# Number Arrays and Area Models

**NCETM **** KS3 Representations, arrays and area models**** **

**NCETM**

**KS3 Representations, arrays and area models**

**Multiplication and Division**

**Multiplication and Division**

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

**Area model and 2D Dienes blocks for multiplying decimals greater than 1**

**Area model and 2D Dienes blocks for multiplying decimals greater than 1**

**Area model and 2D Dienes blocks for multiplying two digit numbers linked to grid model**

**Area model and 2D Dienes blocks for multiplying two digit numbers linked to grid model**

Compare with algebra tiles (or see below)

All representations for 2D Dienes blocks (base 10) use the same colours as algebra tiles for ones, x and x squared including red tiles for negatives.

# Algebra Arrays and Area Models

**Dynamic Algebra Tiles**

**Dynamic Algebra Tiles**

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.

**Completing the square**

**Completing the square**

**Difference of Two Squares**

**Difference of Two Squares**

Completing the square model 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.}

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.

# Area and Bar Models

### 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.

# 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

### Linear Expressions with Dynamic Algebra Tiles

Compare the expressions with the dynamic algebra tile representations using the 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

See Trigonometry with unit circle for examples

See Trigonometry for further examples

# Area

# Number Lines

### Variable number lines

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

# Multiplicative reasoning

### Multiplication as Scaling

This representation models stretching a springs and compares this with a table and graph of two points on the spring. See https://www.ncetm.org.uk/media/mqfp3xb3/ncetm_ks3_cc_3_1.pdf page 9