# Lesson 12

Applications of Arithmetic with Powers of 10

### Lesson Narrative

In this lesson, students first apply what they have learned about working with exponents (especially powers of ten) to solve rich problems in context. The style of questioning requires students to identify essential features of the problem and persevere to calculate and interpret the solutions in context (MP1, MP2, MP4). Students must attend to precision when considering appropriate units of measurement (MP6).

Next, the lesson formalizes what students have learned by introducing the definition of scientific notation. A number is said to be in scientific notation if it is written as a product of two factors: the first factor is a number greater than or equal to 1, but less than 10; and the second factor is an integer power of 10. This definition does not include negative numbers for simplicity. Students must attend to precision as they decide whether or not numbers are in scientific notation and convert to scientific notation (MP6).

### Learning Goals

Teacher Facing

• Determine what information is needed to answer a question about large numbers, and explain (orally) how that information would help solve the problem.
• Identify (in writing) numbers written in scientific notation, and describe (orally) the features of an expression in scientific notation.
• Use exponent rules and powers of 10 to solve problems in context, and explain (orally) the steps used to organize thinking.

### Student Facing

Let’s use powers of 10 to help us make calculations with large and small numbers.

### Required Preparation

The blackline master for the optional Activity Scientific Notation Matching has three sets of cards. Set A is for the teacher to demonstrate the process, so only one copy of set A is needed. Cut out one set of cards (either set B or set C) for every 2 students. If possible, copy each complete set on a different color of paper, so that a stray slip can quickly be put back.

### Student Facing

• I can apply what I learned about powers of 10 to answer questions about real-world situations.
• I can tell whether or not a number is written in scientific notation.

Building Towards

### Glossary Entries

• scientific notation

Scientific notation is a way to write very large or very small numbers. We write these numbers by multiplying a number between 1 and 10 by a power of 10.

For example, the number 425,000,000 in scientific notation is $$4.25 \times 10^8$$. The number 0.0000000000783 in scientific notation is $$7.83 \times 10^{\text-11}$$.