# Delta Area - Subunity

- Author:
- Walter M. Stroup

- Topic:
- Area, Calculus, Definite Integral, Fractions, Functions, Function Graph, Numbers, Ratios

## Movement and the area model of multiplication

INTRODUCTION: A rectangle with a width of 2 units and a height of 3 units has an area of 6 square units.
This can be represented with multiplication as 2 units x 3 units = (2 x 3 ) x (units x units) = 6 units squared.
The area model of division starts with the area and one of the sides and produces the 'other' side. 6 square units ÷ 2 unit side (the base) results in 3 units, which is the 'other' side (the height in this case).
The AREA MODEL continues to work EVEN WHEN THE BLOCKS OF AREA NO LONGER REPRESENT A SURFACE, like a desktop, but can represent a unit of change in the position (delta block) of an elevator.
6 blocks of area can now represent the four floors an elevator travels.
If the width of the 6 block area is 2 seconds, then the height can be found by using the ÷ model.
6 floors ÷ 2 seconds
or 6 floors per 2 seconds
or 6 fl / 2 seconds
or 3 floors / 1 second
or 3 fls/second ("three floors per second").
The AREA MODEL of multiplication, unlike the repeated addition model of multiplication, can support understanding products (the result) when the factors (the numbers multiplied) are NO LONGER INTEGERS. Using repeated addition, 3 x 2 can mean something like 3 groups of 2 objects or 6 objects. Using the area model the factors are the sides of a rectangle and, as is discussed above, 6.
The repeated addition model does not help us makes sense of 0.3 x 0.2. It is not clear what 0.3 groups of 0.2 objects would mean, or how it would be represented with addition. And it doesn't really help us understand whether the result is 0.6 or 0.06 (where, it turns out, 0.06 is the product or 0.3 x 0.2, not 0.6).
The following sequence illustrates how the AREA MODEL can help us make sense of non-integer products.
THE SEQUENCE; (1) Start with an integer on one side (3) and an integer on the other side (1) and explore what the product -- the area -- means for the motion of the elevator. (2) Start with an integer on one side (3) and a non-integers on the other side (0.5 or 1/2) and explore what the product -- the area -- means for the motion of the elevator. (3) Similar to (2) but now with both sides the non-integer 0.5 (1/2). For (1)-(3) the meaning of the area stays the same for the motion of the elevator.
NOTE: As a unit of reference for all the graphs, a unit block of area is shaded green.
(1) Starting with the first graph below. try running the simulations. Then try moving points A and B down (or up) in ways that keeps the line segment between them horizontal. Predict what will happen with the motion of the elevator. How is this motion related to the area of the rectangle?
SOME HOW TO's:
The resent, start and stop buttons work as expected (especially if you've done previous delta blocks activities). You can move the point A or B by clicking and dragging. But it is hard to get "exact" locations this way.
EXACT movements of points:
>> To move by ONE unit - CLICK on a point (e.g., A), then PRESS the ARROW KEYS.
>> To move by tenths of a unit - CLICK on a point (e.g., A), HOLD DOWN THE SHIFT KEY, then PRESS the ARROW KEYS.

## [2] Starting with Product of an Integer (3) and a Non-Integer (0.5):

Beginning with the first graph below. try running the simulation.
Then try moving points A and B down (or up) in ways that keeps the line segment between them horizontal. Predict what will happen with the motion of the elevator. How is this motion related to the area of the rectangle?

## [3] Starting with the Product of Two Non-Integers

Try running the simulation.
Then try moving points A and B around. up or down in ways that keeps the line segment between them horizontal. Predict what will happen with the motion of the elevator. How is this motion related to the area of the rectangle? What fraction of the green unit square do the tiny squares represent. (Might want to 0.3 x 0.2 and see if what result makes sense in terms of the area and the motion of the elevator.)
HOW TO:
>> If you want to move B to the side, you'll first have to move C to the side to keep the graph from disappearing (no longer a function, so it disappears). If you want to go past the location of D, you'll have to first move D, then C then B.