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# How Heavy?

Alignments to Content Standards: 3.MD.A.2

Students will need various items, a balance scale, a large set of cubes such as unifix or snap cubes, and a recording sheet. They choose an item to measure. Using the balance scale, they put the item on one side of the balance scale. Then they put enough cubes on the other side of the scale to make it balance. They remove the cubes, count them, and record the result. For example, if a small book balances with 12 cubes, they should write, "The book has the same mass as 12 cubes." They continue same routine three more times with different items.

## IM Commentary

The purpose of this task is for students to measure the mass of objects with a balance scale. It would be worthwhile measuring the mass of the same object with a different units as well so that students understand that if you use a smaller unit, you will need more of them to equal the same mass. Students should follow up this measuring activity with finding the mass of an object with a scale that gives the mass as a numeric value, such as a kitchen scale. At some point, students should measure the mass of objects that are close to 1 gram or 1 kilogram so they can develop intuition for these units, and they should understand that if a book has a mass of approximately 1 kg, then it will take about 30 books to balance with a 30-kilogram child on an appropriately sized balance scale. Since the balance scale is not precise enough to measure objects that are close to 1 gram and might not fit objects that are approximately 1 kg in mass, other tools would be needed to do this.

It is common for teachers and curriculum writers to use a balance scale as a metaphor for an equation; for example, they might be asked to write an equation for a balance scale like this:

Note that without an experience of actually using a balance scale, this representation is almost as abstract as an equation like 32 = ? + 24 is. Moreover, the standard representations of a balance scale such as the one shown above disregard the concept of a moment-arm, which takes into account not just the mass of an object, but its distance from the balance point. Finally, the concept of mass itself is more abstract than many people realize, in part because it is measured indirectly through weight. In summary, using a balance scale to represent an equation should be used cautiously and only once students have a solid understanding of the concept of mass.

## Solution

The scissors have the same mass as 8 marbles.

The hacky sack has the same mass as 20 marbles.

The eraser has the same mass as 2 marbles.

The racquet ball has the same mass as 9 marbles.

#### redbaron says:

over 4 years

Not having taught in the elementary grades, I do not know how much of an issue this really is, but the 1st grade measurement progressions warn against working with non-standard units as a precursor to standard ones:

"Another important issue concerns the use of standard or nonstandard units of length. Many curricula or other instructional guides advise a sequence of instruction in which students compare lengths, measure with nonstandard units (e.g., paper clips), incorporate the use of manipulative standard units (e.g., inch cubes), and measure with a ruler. This approach is probably intended to help students see the need for standardization. However, the use of a variety of different length units, before students understand the concepts, procedures, and usefulness of measurement, may actually deter students’ development. Instead, students might learn to measure correctly with standard units, and even learn to use rulers, before they can successfully use nonstandard units and understand relationships between different units of measurement."

Using familiar objects as measuring units seems like an intuitive and worthwhile approach to me, but again I do not have the background to have experienced firsthand how legitimate these dangers spoken of in the progressions really are. From your vantage point, is this admonition viewed as an authorative guideline for shaping instruction and assessment, or is it posed merely as "something to consider"? Thanks

#### Kristin says:

over 4 years

Thanks for your insightful comments, redbaron. Note that the progression specifically speaks to the idea of using non-standard units for length. Length is students' first formal experience with measuring a quantity using an appropriate unit, and is work done in first grade. (Note that at some point students should understand the fact that one can choose an arbitrary unit to measure length because that is what is done every time one draws a number line. However, that does seem like an idea that should come after a student has a good grasp of the concept of length.) It is not at all clear that this warning applies to other kinds of measurement, especially in third grade where students have more experience measuring different attributes of objects. Having said that, I realized that the purpose of this task shouldn't be about standard vs. non-standard units, but about the proper use of a balance scale for measuring mass. So I completely rewrote the commentary. Please let me know what you think.

As for your question about how authoritative the admonishment to take care with using non-standard length units in instruction is, here is what I hope curriculum writers will do:

The K-6, Geometry Progression document seems to be referring to a research base around students' understanding of and proficiency with measuring length. Any curriculum writer designing a unit intended to support students' learning about length should be familiar with that literature and take it into consideration. On the other hand, it seems unlikely that the research in this area is absolutely definitive. So just as curriculum writers should take the existing literature into consideration when developing curriculum, so they should be reporting on the results of pilots with the materials they develop (and modifying what they have written accordingly), thus contributing to the professional and scientific knowledge of how students learn about length.

Note also that measuring in non-standard units, however it is employed, is not an end in and of itself in any case. So when developing assessments, it seems that the outcome of interest is whether students can measure using standard units, and assessments should focus on whether students can do that.

I'll see if I can find someone who contributed to writing the progression document to weigh in on this conversation (it might take awhile--folks are busy).