### 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.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Number Sense and Operations

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### 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 × 10
^{3}) + (3.4 × 10^{5}) as 4.7 × 10^{8}.

- For example, students may miscalculate (1.3 × 10
- Some students may incorrectly apply multiplication across a problem.
- For example, students may miscalculate (2 × 10
^{4})(3 × 10^{5}) as 6 × 10^{20}. - Some students may incorrectly represent their final answer not in scientific notation.
- For example, students may write (2.0 × 10
^{4})(6.0 × 10^{5}) as 12.0 × 10^{9}instead of 1.2× 10^{10}.

- For example, students may write (2.0 × 10

### 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 × 10
^{4}, they should be multiplying 5 × 2, and not multiply by the exponent.

- For example, if the student is asked what is five times larger than 2 × 10
- 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 × 10
^{2}) and (4 × 10^{4}), students can rearrange the expression as (3 × 4)(10^{2}× 10^{4}) to determine 12 × 10^{6}which is equivalent to 1.2 × 10^{7}.

- For example, when multiplying (3 × 10
- 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 × 10
^{18}.

- 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 × 10
- 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?

### Instructional Tasks

*Instructional Task 1*

**(MTR.6.1)**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 × 10

^{6}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 × 10

^{7}gallons of water into the Atlantic Ocean every second. There are about 3.2 × 10

^{9}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

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Problem-Solving Tasks

## STEM Lessons - Model Eliciting Activity

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.

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. Click here to learn more about MEAs and how they can transform your classroom.

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

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

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

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

## Student Resources

## Problem-Solving Task

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.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task