The Five-Representation Star
The five-representation star based on work of Preston and Garner, (2003  ) can be used to stimulate discussion. For example, at an appropriate time during the Summarize of a Problem, ask students to talk about which representations were useful in solving and communicating their thinking. The five-representation star can also be used to help students make connections between representations.
Algebraic relationships are common when considering these five representations. Below is an example of a problem relating the width of a rectangle to the area of a rectangle with five representations:
A rectangle has a perimeter of 12 meters with a width of x meters. What is the length? How does the area relate to the width, x?
Note that in the graph below, x values represent the width, and y values represent the area.
A = x(6-x) which was rewritten as y=x(6-x) for the graph above.
The perimeter of a rectangle is equal to twice the width plus twice the length. So, when the perimeter is 12 meters, the width plus the length must be 6 meters. This means that the maximum width is 6 meters, which would make the length 0 meters. The area would be equal to zero square meters.
These values correspond to the values in the table. Students may draw various rectangles and make a table of values for widths from 0 to 6 meters, noticing that this is the limit of the values for the width. They may also notice that the maximum area occurs when the width equals the length. In this case, the maximum area of9 square meters occurs when each dimension, the width and the length, is 3 meters.
Students may also explore rational numbers. For example, they may consider a width of 5.9 meters and a length of 0.1 meters, which results in an area of 0.59 square meters.
Students typically need some guidance to understand the difference between the domain and range of the problem situation, and the domain and range of the function used to model that problem situation. For instance, in this case, the values of the width must be between zero and 6 meters. The area must be between zero and 9 square meters. The domain of the function A = x (6 - x) is not limited, however, to positive values for x. Thus, x can be any real number. Similarly, the values for the range may be negative, with a maximum of 9.
Relationships between numeric equations may also be communicated using these five representations. Consider another example: the case of multiplying two mixed numbers, 51/2 x 43/4 = 261/8.
Jane is tiling her bathroom floor. The floor is 51/2 feet long by 43/4 feet wide. What is the area of the floor that Jane is tiling? If each tile is one square foot, how many tiles does Jane need to buy?
An area model fits this context nicely.
Students who understand multiplication as repeated addition may add 51/2 four times and then add 3/4 of 51/2, or 22 + 41/8 = 261/8
The problem can also be represented on a number line as repeated addition.
The equation 5½ x 4¾ = 26 1/8 can be illustrated in ways that correspond 248 to the solution methods student choose.
For instance, the area model corresponds with the distributive property:
So, Jane will need to purchase 27 or more tiles to tile her bathroom floor. More specifically, if Jane is accurate in cutting and fitting the tiles, she would likely need a minimum of 22 tiles plus 6 more tiles for the last row for a total of 28 tiles.