Before we launch into the world of unbelievable spice, we give students a few moments to make some basic comparisons in scientific notation. I like to set them up in a string (a list of expressions that can all be individually solved but also relate to each other in some way). Here students need to write out the string and include the appropriate inequality.
1 or 9
1 x 10 or 9
1 x 10 or 9 x 10
1 x 10^2 or 9 x 10^2
1 x 10^2 or 9 x 10^3
1 x 10^4 or 9 x 10^3
1 x 10^10 or 9 x 10^9
1 x 10^10 or 10 x 10^9
I also place this string next to the original, because I don't want students thinking that the way in which we compare scientific notation is as simple as "a greater exponent means a greater number." The list is basically the same with all values reversed. I usually add in one thats different, I bolded it in the list below (the last comparison is as simple as + versus -, where all + have a greater value than - numbers). The last expression is a reminder that our number sense always applies (and its never a good idea to simply memorize algorithms).
-1 or -9
-1 x 10 or -9
-1 x 10 or -9 x 10
-1 x 10^2 or -9 x 10^2
-1 x 10^2 or -9 x 10^3
-1 x 10^4 or -9 x 10^3
-1 x 10^10 or -9 x 10^9
-1 x 10^10 or -10 x 10^9
1 or -9 x 10^9
This investigation is drawn around the Scoville Heat scale and the obsession of many people with extremely hot food. I like to show the "Man versus Food" episode where he eats the hottest curry in the USA. We are fortunate that it is right around the corner from our school (there are hotter curry's now, but I like to stick local). I show the video and cut out the part we drinks beer at the end.
The video is also on the restaurant's New York City page at the bottom.
I then give a brief introduction about the Scoville Heat Scale and a review of Capsaicin.
I like to print and laminate color images of the peppers and write the names of the item on the back and the Scoville units in scientific notation. For each group of four, I place two bins on each table and label one as "hot" and the other as "not." Students are asked to start picking peppers and write the measurements in scientific notation. The instructions are simple (I sometimes switch up instructions for fun):
1. Pick 1 pepper from the "hot" bin and then roll a number cube and pick that many peppers from the "not" bin.
2. Write the Scoville Units for the "hot" pepper and the total Scoville Units for the "not" pepper in Scientific Notation
3. Which group has more Capsaicin (a higher total heat on the Scoville Scale)?
I circulate and make sure students are comparing and explaining how they know which group of pepper's are "hotter"? I ask questions like, how did you know that those peppers were hotter? Did you need to write out all the numbers in standard form? How could you use the scientific notation to your advantage?
At the end I update the investigation by asking to order all of the groups from least to greatest. The discussion around that process is meant to tap into the idea that the "exponents" always matter more than the first factor in scientific notation.
I like to review their findings by sharing their "pepper recipes" and comparing them with the class. The goal is to agree on some standard algorithms for comparing numbers in scientific notation. Using student inferences, we work our way towards the following take aways:
I would never ask students to write all this down, but we would work together to create concise statements that work for them. I think the point is simple, you can write down these rules as a reference. But its pointless to memorize them. Instead, you can quickly infer these rules by using simple strings or comparisons. Students can figure this out by playing with some simple examples of positive and negative values with different combinations of exponents. Knowing that they can tinker with the math to understand the content is a wonderful moment for students.