We just discussed a math practice solution for soup ladles in kitchen math.
We focused on the relationship between the diameter of a soup ladle bowl and the standard sizes used in recipes and food service plans.
Now, we will look at how mathematics goes into 3D modeling of soup ladles and similar objects. TINKERCAD is the main tool we will use to visualize some important math concepts.
This activity looks closely at the geometry related to soup ladle bowls.
Positive and negative volumes
TINKERCAD provides a wonderful platform for applying math knowledge from the classroom. For example, the solid objects represent “positive” volumes (or volume values with positive signs). In contrast, the hole objects represent “negative” volumes (or volume values with negative signs).
During this activity, let us also keep track of the total amount of volume on the workplane. Right now, there are no objects on the workplane. The total volume is zero.
V_TOTAL = 0
Let’s dive right into Part 1.
Okay, there is a lot here.
Before we began 3D modeling in Part 1, we chose a system of units. This project used inches.
The hemisphere was placed on the “work plane” with the flat side down. We wanted the flat side up, so we rotated the hemisphere 180 degrees about a single axis.
CCSS.MATH.CONTENT.HSG.GMD.A.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
The shape that was added from the library was a hemisphere (half-sphere). The length and the width (the diameters) of the hemisphere were equal, but the height was half the diameters.
These rules were applied when the length and width were changed. To maintain the regular hemisphere shape, the height scaled (“proportionally”) at the same rate as the length and width.
Also, when we duplicated the solid hemisphere object, we doubled the total volume of the objects sitting on the “work plane”..
Let us continue with Part 2.
In Part 2, we reduced the length and the width of the second object to 1.8 inches. The height was proportionally reduced to be half that measurement, which was 0.9 inches.
Also, something significant happened.
There was a sign change. The positive volume of the second object was converted into a negative volume.
Now, the total volume on the “work plane” comprises the positive volume of the first object (represented by the solid) and the negative volume of the second object (represented by the hole).
V_TOTAL = (+)V_SOLID + (-)V_HOLE
V_TOTAL = V_SOLID — V_HOLE
The next step is to combine both volumes. Perhaps adding a negative volume to a positive volume will result in less volume. Let us find out in Part 3.
As we saw in Part 3, when both objects occupy the same space, we can apply additive properties. The (negative) hole object removed material from the (solid) positive object.
The result is a new shape that represents the positive (solid) object minus the negative (hole) object.
This is the same as digging a hole. Whenever we dig a hole, we are adding a negative volume to dirt.
Bigger holes, thinner walls
Let us suppose we want to decrease the wall thickness of the soup bowl ladle. One solution would be to increase the size of the (negative) hole object. Let us see what happens when the (negative) hole object’s volume is increased while the (positive) solid object’s volume remains unchanged.
The hole is bigger.
The size of the length and width were increased to 1.95 inches — and the diameter (0.975 inches) was changed to proportionally match the new value.
Now, we must add the volumes again to form the new shape.
This activity required a special type of computational thinking. This is visuospatial computational thinking, which is important in manufacturing, logistics, fashion, and the performing arts.
There are also some important takeaways. Volume and density are related to weight (and mass).
Weight = Density * Volume
If density stays constant, then a decrease in wall thickness (an increase in the hole size) would cause the soup ladle bowl weight to decrease. Less material is less expensive to produce and transport. However, minimal material may affect product quality and user experience.
These are design issues to think about in cooperative and collaborative projects. Math is no longer about the “answer in the back of the book” in the workplace. Instead, math is now about creating new solutions and using critical thinking to continuously improve products and create new opportunities.