Using ratios to make sense of waste streams and density

This three-piece sequence features exploration in using ratios to make sense of waste and density. Reciprocally, we use waste and density to make sense of ratios and the related muddy mathematics. The density formula relies entirely on the relationship formed by a ratio.
A ratio is a comparison of two quantities or frequencies. A ratio earns a more specific name of “rate.” For example, when a ratio relates two quantities or frequencies that do not share a conversion formula, it is a rate. If we cannot “cancel out” the units (or get both units of measure to be the same), then the ratio is considered a rate.

A rate is a ratio that compares two units of measure quantifying aspects of the world that cannot be translated into the other. For instance, we cannot translate a measurement of any distance to be universally equivalent to a unit of time. This is where rates come in handy.

Rates are prevalent in our day-to-day experiences. Fire Marshalls have to determine the maximum occupancy of buildings or rooms. Occupancy per room is a rate that varies. Factors determining a maximum occupancy could be the number of exits and total area. But, the point is that we could not make a general conversion between rooms and people, so this ratio is a rate.
However, suppose we have a ratio comparing the area of tablecloth that we have to the area of the table we need to cover for a party. In that case, we can form a ratio that would not be considered a rate. This ratio compares two measures of area. First, a precise and generalized conversion may exist even if each value was measured in a different unit, for instance, square feet and square meters. A conversion helps us to move relatively freely between feet squared and meters squared.

I will generally round to the nearest one’s place in this lesson when measuring volume and mass. The scale I am using only measures mass to the one’s place, which is the place value for x 1, which lives directly to the left of the decimal point.
Density compares mass to volume. In building EcoBricks, we must be conscious of the density of our brick. Many components contribute to the likelihood that an EcoBrick will reside at a density represented by the variable d, in which 1/3 <= d <= 0.7.

In this video and worksheet series, we explore how ratios are a tool we use as humans to make sense of density.

We begin to explore both density and ratios. In this video series, we measure, use structure, attend to precision, construct viable arguments, wonder curiously, model visually and arithmetically, and make sense of 6th-grade mathematics.

To build an Eco-Brick, find the following materials:

1. Empty, dried, and clean plastic bottles
   *While any plastic bottle will due, to create interlocking modules for building, bottles should be the same shape and size.
2. A system to separate solid waste at home, clean, dry, and turn clean plastic into confetti
3. A drumstick or wooden handle that would have otherwise been thrown away
4. A kitchen scale that measures in grams
5. Scissors to trim plastic trash to the necessary size
6. Dedication to EcoBricking weekly because it is time-consuming and a bit of a workout

In the second video of the sequence, I completed a table of values generated from the density formula and some sample EcoBrick masses. The three examples show a variety of strategies to compute and calculate arithmetic related to ratios and furthermore division, equivalency, factors, and place value.

Since mathematics is organically intertwined, we will always skim many concepts in a single video sequence. While our focus is ratios, we visit some muddy mathematics involving:

1. Equivalent numeric expressions
2. Factors and powers of ten
3. Long division algorithm
4. Partial quotients strategy

Follow along with the video and take notes on your worksheet.

• How would you teach these concepts differently?
• What can you explain to someone else? ]
• What are you still wondering about?
• \What do you notice about the way Samm writes as she computes?
• How do you write as you calculate?

In the final video of this sequence, we visualize the order and magnitude of these various densities along a number line. Then, we can use the mathematical visuals and structures of inequalities to make sense of the range of densities that we can accept in our materials for constructing with EcoBricks.

Beyond using ratios to make sense of waste and density in a way that makes ratios more common to how we all see the world, I hope to build something with these EcoBricks. I hope to create moveable and comfortable furniture for the indoor classroom using these EcoBricks. (Although students working with me last school year were hoping to build a playground wall to use for games like Wall Ball.) Nonetheless, we want the EcoBricks to be strong enough for construction. They cannot exceed 0.70 g/mL because they would be too dense and likely cannot support the extra weight.

We can visualize the solution set of acceptable density values using a number line, filled in circles to plot the included minimum and included maximum, and a solid line to indicate the solution set between the minimum and maximum.

Here on Muddy Math, we take the world around us to make sense of muddy mathematics concepts. This time, we are using ratios to make sense of waste and density. Really, we are also using the mathematical tool of the ratio concept to make sense of waste and EcoBrick density. As we explore the details of EcoBricks and waste streams, we are bound to get muddied up more with ratios. So we will clear it up just in time to get muddy with mathematics all over again.
Until next time,
Samm

Update [9.28.23]

Coming soon… a two-dimensional interpretation of density.