Using Dynamic Representations
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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 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
Area Models for Multiplication
Area and Grid Models
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
The Unit Circle
Variable number lines
Arithmetic and Algebra in Early Mathematics Education Adapted from D.W. Carraher et al. (2006).