Using Dynamic Representations

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Number Arrays and Area Models

Multiplication and Division

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

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!

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 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

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)

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

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 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.

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.mp4

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 a number line

multiply integers on a number line.mp4

Trigonometry

The Unit Circle

See Trigonometry for further examples

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