Using Dynamic Representations

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All highlighted points and arrows are interactive (top right).

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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.

This array model partitions when the divisor or the quotient are greater than 10.  This model can be used to show the connection between the array, multiplication and division.  It represents the structure behind the bus stop!

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

FDP.mp4

Percentage increase and decrease

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

Common Factors

Finding common factors using area models.  In this model the height is the common factor.   Change the slider to change the height.  They have common factors when both bases are integers.

Multiply Decimals

Multiply decimals between 0 and 1 using the area model.

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.

Multiply whole numbers and decimals

Multiply two decimals up to 1 decimal place using the area model.  Click and drag the coloured Dienes blocks over the shaded area to count the number of 1's, 0.1's and 0.01's.

Area and Grid Models

Compare with algebra tiles

Division with Dienes and Area Model

Comparing the part whole model, bus stop method and the area model.

Quotative division (grouping)

Groups are represented by columns.

Quotative division (grouping)

Partitive division (sharing)

These representations for base 10 use the same colours as algebra tiles for ones, x and x squared including red tiles for negatives.

(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 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

variable variable with expressions cropped.mp4

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.

Sequences and Graphs

Sequences using Dynamic Algebra Tiles

sequences with table cropped.mp4

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.

y = mx + c  with Dynamic Algebra Tile

Change m, x and c using the sliders.

Which is bigger?

Integers

Multiplying Integers using Vectors

multiply integers on a number line.mp4

Trigonometry

The Unit Circle

See Trigonometry for further examples

Trig with unit circle cropped.mp4

Area

Area of a Circle

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).