# Adding and Subtracting in Scientific Notation

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## Objective

SWBAT add and subtract numbers in scientific notation

#### Big Idea

Place value can help us quickly add and subtract numbers in scientific notation

## Start Up

15 minutes

The common approach to adding and subtracting numbers in scientific notation doesn't involve typical use of the laws of exponents, but it does tap into our place value system. I show students this possibility by sharing some quick examples in a string.

Write each  expression and its sum in decimal form and scientific notation:

5000 + 500

5000 + 50

5000 + 5

3 x 10^3 + 3 x 10^2

3 x 10^3 + 3 x 10^1

3 x 10^3 + 3 x 10^0

1,500,000 + 20,000

2,250,000 + 4 x 10^5

2,250,000 + 4 x 10^6

2,250,000 + 8 x 10^5

The string start off with three reference examples. Its a reminder of the the beauty of our place value system and a chance to tap into the way we use that system to think about adding. The next three examples all start with 3000 and then add hundreds, tens, and ones (like the first three reference examples). Then, there is an expression adding two quantities in decimal form (1,500,000 + 20,000). This example is meant to give students another reference with larger numbers (something that many students continually struggle with). The last three examples combine decimal form and scientific notation. The goal is to constantly reflect on the process of adding these numbers in an easy way. That might mean that students always rewrite in decimal form and then convert to scientific notation, but it also might mean that they recognize how the place values connect to solve a problem.

For example, when I add 3 x 10^3 + 3 x 10^2, I recognize that 3 x 10^3 represents 3 thousands and that 3 x 10^2 represents 3 hundreds. Together they make 3 thousands and 3 hundreds, which is 3.3 x 10^3. Students can always keep the higher power of 10 (in this case 10^3) and then add the first terms in their appropriate place. With 3 x 10^3, we have 3 thousands and 0 hundreds. So when we add the 3 hundreds to this number, we get 3 thousands and 3 hundreds. Students also need to recognize that when you have a number like 3.3 x 10^3, that the first three represents thousands and then the 3 after the decimal represents hundreds. We play more with this idea in the investigation.

## Investigation

25 minutes

Here I give students 4 strings to work with. Each string is not meant to be an endless series of repetitive exercises. Instead each expression is meant to help students make sense of a pattern in the process of adding and subtracting numbers in scientific notation.

As I circulate, I ask students to describe the patterns they see and remind them that even when patterns like this aren't listed, they could create simple expressions that help them solve tougher questions. That is the heart of the pedagogy behind strings. Its an opportunity to remind students that they can create simple problems to make sense of anything they have to solve.

Strings for Addition has all addition problems, but I ask them to turn some of them into subtraction for the summary.

## Summary

15 minutes

I could talk about strings for years and not get tired. This is because every string seems to hold some fun element of number sense that I planted and also some fun element that students and I discover together.

The first string is critical for its exposure of place value, but hard to share in an effective way:

1.23 x 106

1.23 x 105

1.23 x 104

1.23 x 103

1.23 x 102

1.23 x 101

1.23 x 100

1.23 x 10-1

The most ideal situation would be to share students work through the document camera. Some students will line up the numbers in a way that demonstrates the pattern in the string. If they set it up right, there is no need to write the names of place values over and over again. Instead, you can see the place value in the layout:

1.23 x 106 =  1230000

1.23 x 105 =    123000

1.23 x 104 =      12300

1.23 x 103 =        1230

1.23 x 102 =          123.

1.23 x 101 =             12.3

1.23 x 100 =               1.23

1.23 x 10-1 =                .123

The second string is meant to cover some basic elements of the scientific notation algorithm. Students understand that answers should be written in scientific notation, but I leave that instruction out (I want them to still break the numbers down to standard form if they want to).

32 + 4 x 102

32 + 4 x 103

32 + 4 x 104

32 + 4 x 105

32 + 4 x 106

3.2 x 102 + 4 x 102

3.2 x 103 + 4 x 102

3.2 x 103 + 4 x 103

3.14 x 104 + 6 x 105

This string starts by mixing standard and scientific notation, but keeps the number 32 constant throughout. This is meant to help students build confidence as they work. The final four problems add scientific notation only, but start by building off the same decimals in the number 32

The third string is meant to be a gentle introduction to decimal addition, something that is markedly more challenging for students.

4 x 101 + 6 x 101

4 x 101 + 6 x 100

4 x 101 + 6 x 10-1

4 x 100 + 6 x 10-1

4 x 100 + 6 x 10-2

4 x 100 + 6 x 10-3

4 x 10-1 + 6 x 10-3

4 x 10-2 + 6 x 10-3

4 x 10-3 + 6 x 10-3

The goal is to start with two small whole numbers and then work slowly towards all decimal addition.

The fourth string is "advanced" only because it deals with larger numbers and larger gaps in place value.

4 x 106 + 6 x 109

4 x 107 + 6 x 109

4 x 108 + 6 x 109

4 x 109 + 6 x 109

4.1 x 106 + 6.3 x 109

4 x 1060 + 6 x 1061

4 x 1061 + 6 x 1061

4 x 1064 + 6 x 1065

In the process of sharing these strings, I want students to recognize how to "line up" numbers by place value and recognize when this is effective and when it could be confusing (when there are large gaps in place value, it might always make more sense to transfer everything into standard form).

As for an algorithm, I am careful not to simply share my approach in lining up place value.

I picture the numbers stacked on top of each other. So if I am adding 3 x 10^3 and 2 x 10^2, I think:

3,000

+   200

3200 = 3.2 x 10^3

Students invariably think of similar strategies, but I will lose them with my strategy if I don't acknowledge their strategy first.