Big Numbers, Small Numbers

Background and Rationale

This next year I will be teaching eighth grade math at Brentano Math and Science Academy in Chicago’s Logan Square neighborhood. The school qualifies for title 1 funding and has an enrollment that is predominately Hispanic. After several years of teaching 8^th grade math, I have noticed that my students consistently have the same problem year after year with numbers that are out of their range of comfort - that is to say, any number beyond 100. Students rely on calculators to carry out basic operations and have great difficulty reasoning about larger numbers. These deficits in number sense present a significant obstacle whenever students are given a problem that involves large quantities. More acutely, their resistance to large numbers makes it difficult to teach the required unit on scientific notation and laws of exponents. As 8^th graders, students are required to master the following standards:

CCSS.MATH.CONTENT.8.EE.A.1   Know and apply the properties of integer exponents to generate equivalent numerical expressions.

CCSS.MATH.CONTENT.8.EE.A.3   Use numbers expressed in the form of a single digit times an integer power of 10   to estimate very large or very small quantities, and to express how many times as   much one is than the other. 

CCSS.MATH.CONTENT.8.EE.A.4   Perform operations with numbers expressed in scientific notation, including   problems where both decimal and scientific notation are used. Use scientific   notation and choose units of appropriate size for measurements of very large   or very small quantities.

In previous treatments of these topics I have explicitly taught the laws of exponents, how to represent numbers with powers of ten via scientific notation, and finally show students how to perform single step operations with scientific notation. My students have typically demonstrated a basic procedural mastery of the above standards. A typical problem from the 8^th grade curriculum is shown in Figure 1 below. In general, my 8^th graders would be able to address this problem by performing the required operation using laws of exponents, but would not have the ability to check if their answer is reasonable. Also, they would not know why we are using scientific notation in the first place. Most would show very little conceptual understanding of the given quantities and the procedure they were carrying out.

Figure 1

Here are the masses of the so-called inner planets of the solar system.

Mercury: 3.3022 x 10²³kg

Venus: 4.8685 x 10²⁴kg

Earth: 5.9722 x 10²⁴kg

Mars: 6.4185 x 10²³kg

What is the average mass of all four inner planets? Write your answer in scientific notation.

(Open-Up Resources, 2017)

Another concern I have with this topic is, that it has always felt isolated in the 8^th grade curriculum and does not receive much attention in the midst of linear algebra and functions, the major work of the grade. Once it is addressed, students are not tasked with applying the skills or concepts in any other unit. Instead of the unit giving students a set of tools to understand and operate with numbers of large and small magnitude in general, they are only able to use these skills when explicitly tasked to do so, as in the problem above. By not giving more weight and priority to these standards, I have missed opportunities for students to use laws of exponents and scientific notation when studying geometry, functions, and statistics.

My goal with this unit is to pay greater attention to this strand of standards, and to present this content in a way that leads to a conceptual and transferable mastery of skills for my students. My hope is that this unit helps students answer essential questions like: Why is it helpful to use scientific notation? What is actually happening when we increase by powers of 10? Why can we round numbers off to 2 or 3 digits when calculating big numbers and still have a reasonable answer? To get students to answer these important questions, I will design this unit to progressively teach the utility and rationale of each concept before focusing on procedural fluencies. This will require a careful sequencing of the key concepts and selecting prompts that give students opportunities to grapple with these essential concepts.