# MA.8.NSO.1.6

Solve real-world problems involving operations with numbers expressed in scientific notation.

### Clarifications

Clarification 1: Instruction includes recognizing the importance of significant digits when physical measurements are involved.

Clarification 2: Within this benchmark, for addition and subtraction with numbers expressed in scientific notation, exponents are limited to within 2 of each other.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Scientific Notation
• Significant Digits

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students developed an understanding of Laws of Exponents with numerical expressions. They focused on generating equivalent numerical expressions with whole-number exponents and rational number bases. In grade 8, students use the knowledge of Laws of Exponents to work with scientific notation. In Geometry, students will solve problems involving density in terms of area and volume which can be represented using scientific notation when the numbers are large. Additionally, students can apply their scientific notation knowledge in science courses.
• Instruction includes opportunities to engage in virtual or physical situations to understand the importance of significant digits.
• Instruction includes student understanding of the following aspects:
1. zeros at the beginning of a number are never significant,
2. zeros at the end of a number are only significant if there is a decimal point and
3. zeros in the middle of a number are always significant.
• Students should develop fluency with and without the use of a calculator when performing operations with numbers expressed in scientific notation.
• For mastery of this benchmark, students are expected to express the product or quotient with the appropriate number of significant digits. In general, the number of significant digits in the result will be the least number of digits in the operands.
• For example, when multiplying two numbers together, one that has 4 significant digits and the other that has 2 significant digits, then only two significant digits should be retained for the product.

### Common Misconceptions or Errors

• Students may incorrectly identify zeros as significant digits.
• Some students may incorrectly apply addition and subtraction across a problem.
• For example, students may miscalculate (1.3 × 103) + (3.4 × 105) as 4.7 × 108.
• Some students may incorrectly apply multiplication across a problem.
• For example, students may miscalculate (2 × 104)(3 × 105) as 6 × 1020.
• Some students may incorrectly represent their final answer not in scientific notation.
• For example, students may write (2.0 × 104)(6.0 × 105) as 12.0 × 109 instead of 1.2× 1010.

### Strategies to Support Tiered Instruction

• Instruction includes making connections of a number written in standard form to the same number written in scientific notation by noticing patterns. Key connections include recognizing the similarities in the first two digits of both numbers and the connections between the place value of the number in standard form and the exponent of the power.
• Teacher provides opportunities for students to utilize appropriate calculators and provides instruction on the various calculator notations for scientific notation.
• Instruction includes rewriting whole numbers in scientific notation when finding products or quotients with scientific notation to demonstrate correct use of operations and laws of exponents.
• For example, if the student is asked what is five times larger than 2 × 104, they should be multiplying 5 × 2, and not multiply by the exponent.
• Instruction includes making connections to the use of place values when adding and subtracting numbers written in standard form to place values with scientific notation. Teacher should demonstrate how rewriting numbers in scientific notation utilizing the same power of 10 represents numbers with the same place value.
• Instruction includes modeling the correct use of operations and laws of exponents when finding the products and quotients of numbers represented in scientific notation, paying close attention to the solution to ensure it is in scientific notation.
• For example, when multiplying (3 × 102) and (4 × 104), students can rearrange the expression as (3 × 4)(102 × 104) to determine 12 × 106 which is equivalent to 1.2 × 107.
• Teacher provides opportunities for students to check their work by rewriting numbers in standard form and applying any necessary operations before comparing their solution to the solution found with the use of a calculator.
• Instruction includes the use of manipulatives such as Base Ten blocks to make connections to the purpose of utilizing scientific notation.
• For example, the teacher could pose the question: “What would be the best way for us to represent 2430 using Base Ten Blocks? We could use 2430 individual Base Ten Unit blocks, or we could 2 Base Ten Cubes, 4 Base Ten Flats, and 3 Base Ten Rods. Student can then see that it would be easier to represent 2430 using the Cubes, Flats, and Rods as opposed to the large amount of individual Unit blocks. When students see how it would be easier to use the larger blocks to represent the number, Teachers can explain how it is similar to using scientific notation to write out very large or very small numbers. Instead of writing 2873000000000000000, they can write 2.873 × 1018.
• Teacher provides opportunities for students to complete problems using scientific notation and standard form in order to check for the reasonableness of their solutions and build on connections between the two.
• Instruction includes the use a three-read strategy. Students read the problem three different times, each with a different purpose (laminating these questions on a printed card for students to utilize as a resource in and out of the classroom would be helpful).
• First, read the problem with the purpose of answering the question: What is the problem, context, or story about?
• Second, read the problem with the purpose of answering the question: What are we trying to find out?
• Third, read the problem with the purpose of answering the question: What information is important in the problem?

Measures of population density indicate how crowded a place is by giving the approximate number of people per square unit of area. In 2009, the population of Puerto Rico was approximately 3.98 × 106 people.
• Part A. How many significant digits are there in the population of Puerto Rico?
• Part B. If the population density was about 1000 people per square mile, what is the approximate area of Puerto Rico in square miles?
• Part C. Does the number of significant digits change when finding the population density? Why or why not?

### Instructional Items

Instructional Item 1
The Amazon River releases 5.5 × 107 gallons of water into the Atlantic Ocean every second. There are about 3.2 × 109 seconds in a year. How many gallons are released into the ocean in one year? Express your answer with the appropriate number of significant digits.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.NSO.1.AP.6: Given a real-world problem, perform operations with numbers expressed in scientific notation using a calculator and interpret the answer in context.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Sums and Differences in Scientific Notation:

Students are asked to add and subtract numbers given in scientific notation in real-world contexts.

Type: Formative Assessment

Scientific Multiplication and Division:

Students are asked to multiply and divide numbers given in scientific notation in real-world contexts.

Type: Formative Assessment

Mixed Form Operations:

Students are given word problems with numbers in both standard and scientific notation and asked to solve problems using various operations.

Type: Formative Assessment

## Lesson Plans

Make My Number:

Students will apply their knowledge of mathematical operations to provide inputs in a Scratch program to try and make a specific given output, in this lesson plan.

Type: Lesson Plan

How Many Smoots Does It Take to Reach the Moon?:

In this discovery oriented lesson, students will explore the use of non-standard units of measurement. They will convert linear measurements within the metric system and convert measurements given in astronomical units (AU) into more familiar units, specifically meters and kilometers. The unit conversions will be completed with measurements that are expressed in scientific notation. Students will recall their prior knowledge of how to add and subtract numbers given in scientific notation. They will also use their knowledge of exponent rules to determine an efficient method for multiplying and dividing numbers expressed in scientific notation.

Type: Lesson Plan

3-D Printers: Cost Effective for the Quality?:

Students will compare the cost of pre-made solar car kits to cars made from a 3-D printer. In the second part of the activity, students will research other available 3-D printers and determine what attributes are important to consider. There is also an optional solar panel car race for day 3.

Type: Lesson Plan

Quest For Life: Space Exploration:

Students must decide the destination of a multi-billion dollar space flight to an unexplored world. The location must be selected based on its potential for valuable research opportunities. Some locations may have life, while others could hold the answers to global warming or our energy crisis. Students must choose the destination that they feel will be most helpful to human-kind.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

Type: Lesson Plan

Ants versus humans:

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Giantburgers:

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

## STEM Lessons - Model Eliciting Activity

3-D Printers: Cost Effective for the Quality?:

Students will compare the cost of pre-made solar car kits to cars made from a 3-D printer. In the second part of the activity, students will research other available 3-D printers and determine what attributes are important to consider. There is also an optional solar panel car race for day 3.

Quest For Life: Space Exploration:

Students must decide the destination of a multi-billion dollar space flight to an unexplored world. The location must be selected based on its potential for valuable research opportunities. Some locations may have life, while others could hold the answers to global warming or our energy crisis. Students must choose the destination that they feel will be most helpful to human-kind.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx

## MFAS Formative Assessments

Mixed Form Operations:

Students are given word problems with numbers in both standard and scientific notation and asked to solve problems using various operations.

Scientific Multiplication and Division:

Students are asked to multiply and divide numbers given in scientific notation in real-world contexts.

Sums and Differences in Scientific Notation:

Students are asked to add and subtract numbers given in scientific notation in real-world contexts.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Giantburgers:

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.