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Using Dynamic Representations
Using Dynamic Representations
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Number Arrays and Area Models
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.
Division (bus stop)
Division (bus stop)
This array model partitions when the divisor or the quotient are greater than 10. It can be used to show the connection between the array, multiplication and division. It represents the structure behind the bus stop!
Quotative division (grouping)
Quotative division (grouping)
Partitive division (sharing)
Partitive division (sharing)
Area model for multiplication linked to grid model
Area model for multiplication linked to grid model
Common Factors
Common Factors
In this model the height is the common factor. Change the slider to change the height. The two numbers have common factors when both bases are integers.
Area model and 2D Dienes blocks for multiplying decimals between 0 and 1
Area model and 2D Dienes blocks for multiplying decimals between 0 and 1
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
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.
(x + a)(x + b)(x + c)
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
x2+bx + c = (x+b/2)2 -b/22 + c
b/22 and -b/22 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 for Fractions, Decimals and Percentages
Area and Bar Models for Fractions, Decimals and Percentages
Fractions, Decimals and Percentages
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
Percentage increase and decrease
Bar model, double number line and ratio table. Instruction video link.
Base Blocks (3D Algebra Tiles)
Base Blocks (3D Algebra Tiles)
Base Blocks from Base 1 to 10
Base Blocks from Base 1 to 10
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.
Base x Interactive Blocks
Base x Interactive Blocks
Base 2 Interactive Blocks
Base 2 Interactive Blocks
Base 3 Interactive Blocks
Base 3 Interactive Blocks
Dynamic Algebra Tiles
Dynamic Algebra Tiles
Linear Expressions with 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.
variable variable with expressions cropped.mp4Equations and Inequalities
Equations and Inequalities
Sequences and Graphs
Sequences and Graphs
Sequences using Cuisenaire Rods
Sequences using Cuisenaire Rods
Sequences using Dynamic Algebra Tiles
Sequences using Dynamic Algebra Tiles
sequences with table cropped.mp4These 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.
y = mx + c with Dynamic Algebra Tile
y = mx + c with Dynamic Algebra Tile
Change m, x and c using the sliders.
x + 3 and 3x
x + 3 and 3x
Which is bigger?
Quadratic Graphs with Dynamic Algebra Tiles
Quadratic Graphs with Dynamic Algebra Tiles
Integers
Integers
Adding Integers as Vectors
Adding Integers as Vectors
Counters and Zero Pairs
Counters and Zero Pairs
Counters, Vectors and Bar Model
Counters, Vectors and Bar Model
Subtracting Integers - Additive Inverse
Subtracting Integers - Additive Inverse
Subtracting integers - additive inverse
Subtracting integers - additive inverse
Multiplying integers using vectors
Multiplying integers using vectors
Multiplying Integers using a number line
Multiplying Integers using a number line
multiply integers on a number line.mp4Trigonometry
Trigonometry
The Unit Circle
The Unit Circle
See Trigonometry with unit circle for examples
See Trigonometry for further examples
Area
Area
Area of a Circle
Area of a Circle
Number Lines
Number Lines
Variable number lines
Variable number lines
Arithmetic and Algebra in Early Mathematics Education Adapted from D.W. Carraher et al. (2006).
Multiplicative reasoning
Multiplicative reasoning
Speed 1
Speed 1
Speed 2
Speed 2
Multiplication as Scaling
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
Double Number Line with multipliers
Double Number Line with multipliers
Statistics
Statistics
Counters, Bar Chart, Bar Model and Pie Chart
Counters, Bar Chart, Bar Model and Pie Chart
Cumulative frequency curve and stacked bar chart
Cumulative frequency curve and stacked bar chart
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