 Unit 1: Number Sense

The first unit of this course involves extending your prior knowledge of fractions and decimals. You will use models to show how the division equation works. This will help you make sense of finding quotients involving fractions. During this unit, you will also work with negative integers along a number line. This might involve negative temperatures (cold!), having credit or debt with money, or changing elevation as you hike up a mountain or scuba dive into the ocean. During this number-sense unit, make your best effort to do just as the unit title implies - make sense of numbers. Don’t try to memorize equations; rather, understand the numbers and operations, so you can continue to make sense of the math in future situations.

Completing this unit should take you approximately 13 hours.

☐ Subunit 1.1: 5 hours and 15 minutes

☐ Subunit 1.2: 30 minutes

☐ Subunit 1.3: 2 hours and 45 minutes

☐ Subunit 1.4: 1 hour and 45 minutes

☐ Subunit 1.5: 2 hours and 45 minutes

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to:

• Compute quotients of fractions.

• Solve word problems involving quotients of fractions.

• Use visual models and equations to represent division of fractions.

• Fluently compute multi-digit division of numbers.

• Fluently add, subtract, multiply, and divide multi-digit decimals.

• Solve for the greatest common factor of two numbers.

• Solve for the least common multiple of two numbers.

• Use the distributive property to change expressions.

• Define the opposite of a number.

• Extend a number line to represent negative rational points.

• Order rational numbers using lists and inequalities.

• Define absolute value and interpret how it is used in real world situations.

1.1 Fractions

Fractions are used in our daily lives in a variety of ways. The U.S. customary measurement system uses fractions as a measuring tool (example: 3/8 inch). Fractions are also used in cooking and baking (example: 3/4 cup), and fractions can even be used to describe partial situations (example: I completed 1/4 of the project). These are just a few general examples of how fractions are used. In this subunit you will review addition and subtraction of fractions, as well as work with multiplication and division. These concepts might already be familiar to you. However, adding models and fluency will be a key concept that you should focus on while working.

1.1.1 Operations with Fractions   1.1.1.1 Adding and Subtracting Fractions

You and a friend have decided to bake cookies. You both want to make chocolate chip cookies, but you each have a different recipe. The two of you decide to have a “bake-off” in order to do a taste test and figure out which recipe is the best. You plan to go grocery shopping together, buy everything that you need, and come back to your house to bake the cookies. Since both recipes call for the same ingredients, just different amounts of each, you decide to add your amounts together.

Adding fractions, as well as subtracting them, is an important concept to understand when dealing with recipes because most ingredients are listed as parts of a whole cup.

Instructions: Work through the problems with the instructor in the videos. Pay attention to strategies that help you find a common denominator.

Watching this video and taking notes should take you approximately 15 minutes.

Instructions: Work through the problems with the instructor in the video. Pay attention to the two different strategies of subtracting fractions.

Watching this video and taking notes should take you approximately 15 minutes.

• Did I Get This? Activity: Howard County Public School System’s “Cookie Calculations”

 Ingredients Calculations Amount to Buy 1 2/3 cups of butter         1 2/4  cups of butter 1 2/3 = 1 8/12  1 2/4 = 1 6/12                1 8/12 + 1 6/12 = 2 14/12 = 3 2/12 3 1/6  cups of butter

Completing this activity should take you approximately 30 minutes.

1.1.1.2 Multiplying Fractions

In the previous subunit you combined ingredients before you went to the store. What happens if you want to double or triple a recipe to make lots of cookies? This would involve multiplying each ingredient by two or three. In this subunit you will see that multiplying fractions can be modeled using a fraction bar. As you work through this subunit, pay attention to how the fraction models work so you are able to use models when you multiply fractions.

• Explanation: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 4, Section 2: Multiplying Fractions”

Link: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 4, Section 2: Multiplying Fractions” (PDF)

Instructions: In the table of contents, click on “4.2: Multiplying Fractions.” Read through the explanation on page 249 about how to draw a visual model of multiplying fractions. Copy the sample model in your notebook. Do “Example 1” using a visual model. Make sure to recognize how the word of is used in math.

Read about the multiplication rule on pages 250 - 251 to understand the formula for multiplying fractions. Try “Example 2” using this method.

Reading this section, working the examples, and taking notes should take approximately 30 minutes.

Instructions: While watching this video, pay attention to how the instructor changed a whole number into a fraction and how you can simplify the fraction after multiplying or before multiplying. Sometimes it’s easier to simplify before multiplying because then you don’t have to deal with digits that are very large.

The last 1:20 on the video shows how this problem can also be solved using a model. Make sure you recognize how to model multiplication of fractions.

Write a paragraph explaining how the model of a fraction multiplication problem works. Your audience is a younger student who has never multiplied fractions using a model. Make sure to use math vocabulary. Drawing an example model and using arrows/labels might be helpful.

Watching this video, taking notes, and writing a paragraph should take you approximately 30 minutes.

Instructions: While watching this video, pay attention to how to change a mixed number into an improper fraction.

Watching this video and taking notes should take you approximately 15 minutes.

• Checkpoint: Wade Ellis, Jr. and Denny Burzynski’s “Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions”

Link: Wade Ellis, Jr. and Denny Burzynski’s Fundamentals of Mathematics: “Introduction to Fractions and Multiplication and Division of Fractions: Multiplication of Fractions” (HTML)

Instructions: Scroll through the models and samples (feel free to refer to these as additional resources, if needed) until you get to “Practice Set A.” In your notebook, complete all problems in the practice set. Check the solutions as you go. Make sure to always simplify your fraction.

Complete the problems in “Practice Set B.” Use the “simplify before multiplying” strategy. Check the solutions as you go.

Complete the problem in “Practice Set C.” Remember when you have a mixed number you have to convert it to an improper fraction before you can use the multiplication formula. Check the solutions as you go.

Skip all parts of “Sample Set D” and “Practice Set D.”

Continue practicing the problems beginning with “Exercise 28” and working through “Exercise 89.” Do all the even problems. Use the odd numbered problems as extra practice, as needed. The types of questions vary as you work. Check the solutions as you go.

With all of the above problems, if you are not able to solve for the correct product, figure out your mistake. Read back through the sample items in each section or re-watch the videos provided.

Completing these practice sets should take you approximately 45 minutes.

1.1.1.3 Dividing Fractions

How do you feel about sharing? Sometimes we have to share, or split up, parts of a whole. Other times, we have to share something that isn’t whole -  like 2/3 of a leftover pizza shared among four family members. How much pizza will you get? In this unit you will work to model division of fractions. It is important to know and use the formula that works for always dividing fractions. However, make sure you are able to follow the models given to you in this subunit. Modeling division of fractions is an important skill to have in this course.

• Explanation: CK-12: “Division of Fractions by Whole Numbers”

Link: CK-12: “Division of Fractions by Whole Numbers” (HTML)

Instructions: Read about Julia trying to figure out how many strips of paper she can cut from her poster board. Continue reading under the “Guidance” section about how you can model division of fractions. Do examples A - C. Write down the vocabulary words (Inverse Operation and Reciprocal) and their definitions in your notebook, as these concepts will be used often in this subunit.

Watch the first video, “Khan Academy Dividing Fraction Example.” Work the problems while the video instructor completes them. Pay attention to the model he shows you at the end.

Watch the third video, “James Sousa Example of Dividing Fractions.” Work through the problems while the video instructor completes them. Pay attention to the different strategies of how to simplify the fractions before solving.

Reading this explanation, watching the videos, working the problems, and taking notes should take you approximately 30 minutes.

Instructions: Watch this video about dividing mixed numbers. Similar to multiplication of fractions, you will want to pay attention to how the instructor first makes the mixed numbers into improper fractions.

Watching this video and taking notes should take you approximately 15 minutes.

1.1.2 Solving Word Problems with Fractions

Understanding “how to” operate with fractions is important; however, more importantly you need to know how to put your fraction skills into real-life situations. When you encounter fractions - whether it’s in baking, measurement, traveling distances, or just working with parts of a whole - it is essential that you can define the necessary operation and put your math skills into action. This subunit will help you break down word problems and solve them successfully.

Instructions: Begin watching this video. However, click the pause button after you have a chance to read the problem provided. Try solving the problem before you watch the video. Check your process and your work once you have completed the problem on your own.

Watching this video and working the problem on your own should take you approximately 15 minutes.

Instructions: Begin watching the video. However, click the pause button after you have a chance to read the problem provided. Try solving the problem before you watch the video. Check your process and your work once you have completed the problem on your own.

Watching this video and working the problem on your own should take you approximately 15 minutes.

Instructions: Begin watching the video. However, click the pause button after you have a chance to read the problem provided. Try solving the problem before you watch the video. Check your process and your work once you have completed the problem on your own.

Watching this video and working the problem on your own should take you approximately 15 minutes.

Instructions: Begin watching the video. However, click the pause button after you have a chance to read the problem provided. Try solving the problem before you watch the video. Check your process and your work once you have completed the problem on your own.

Watching this video and working the problem on your own should you approximately 15 minutes.

• Checkpoint: Khan Academy’s “Multiplying Fractions Word Problems”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice the word problem questions until you feel confident that you understand how to successful multiply fractions within word problems (recommended a minimum of 10 minutes of practice).

Completing these practice problems should take you approximately 15 minutes.

• Checkpoint: Khan Academy’s “Dividing Fractions Word Problems”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice the word problems questions until you feel confident that you understand how to successful divide fractions within word problems (recommended a minimum of 10 minutes of practice).

Completing these practice problems should take you approximately 15 minutes.

1.2 Long Division

Fluently using division is valuable to your daily life. This subunit will cover the formula for using long division as well as the different options for what to do when you have a remainder.

Instructions: Depending on your current level of understanding with division - and specifically long division - watch the first problem as an example, but stop it when you are given the problem for the second example. Try to solve it yourself before watching the solution.

Watching this video and working the problem on your own should take you approximately 15 minutes.

• Explanation: YouTube: James Olsen’s “Long Division with Remainders - Three Ways to Handle Them”

Instructions: The purpose of this video is to make you aware of the different ways mathematicians deal with remainders. As you continue in your math career, you want to make sure you are confident with writing a remainder as a fraction and a decimal. Also, it is important in word problems to understand what the remainder might mean for that particular word problem.

Watching this video and taking notes should take you approximately 15 minutes.

1.3 Decimals

Decimals are used all around us. Most commonly, we use decimals when we deal with money. It is important to know how to find an exact answer while using decimals, and to use estimation skills. In this subunit you will work with addition, subtraction, multiplication, and division of decimals. You should have some prior knowledge about decimal placement, place value with decimals, and saying a decimal number using proper math grammar (example: 2.35 = two and thirty-five hundredths). If you don’t feel confident with place value of decimals, do a little research on your own at this time. An Internet search for “decimal place value” should give you a good place to start.

``````Link: YouTube: Mathispower4u: James Sousa’s [“Adding and Subtracting

Instructions: While watching this video, pay close attention to the
decimal placement throughout the addition and subtraction problems.
Otherwise, you’ll notice that the addition and subtraction
strategies remain the same.

Watching this video and taking notes should take you approximately
15 minutes.

-   [CCSS.Math.Content.6.NS.B.3](http://www.corestandards.org/Math/Content/6/NS/B/3)

attributed to James Sousa, and the original version can be found
``````
• Did I Get This? Activity: Wade Ellis, Jr. and Denny Burzynski’s “Decimals: Addition and Subtraction of Decimals”

Instructions: Click on the link above and scroll down to the “Exercises” section. Do exercises 12 - 36 and check your solutions as you go.

Completing these exercises should take you approximately 30 minutes.

1.3.2 Multiplying Decimals   - Explanation: CK-12: “Decimal Multiplication”

``````Link: CK-12: [“Decimal
Multiplication”](http://www.ck12.org/arithmetic/Decimal-Multiplication/lesson/Decimal-Multiplication/) (HTML)

Notice that decimal multiplication is not much different from
regular multiplication. Ignore the decimal until you have the final
product, and then place the decimal in the proper location. Both
strategies of “counting decimal places” and “estimation” work for
knowing where to place the decimal in the product.

Do the example problems and check your solutions as you read through
the text.

Of the four videos accompanying the text, watch the first video
“James Sousa Multiplying Decimals.” Pause the video at each example,
solve it on your own, and then check your solution and steps with
the instructor. (You are welcome to watch the other videos for

Do the practice problems beneath the videos, and check your solution
on a calculator.

problems should take you approximately 45 minutes.

-   [CCSS.Math.Content.6.NS.B.3](http://www.corestandards.org/Math/Content/6/NS/B/3)

attributed to CK-12, and the original version can be found
[here](http://www.ck12.org/arithmetic/Decimal-Multiplication/lesson/Decimal-Multiplication/).
``````

1.3.3 Dividing Decimals   - Explanation: CK-12: “Decimal Division”

``````Link: CK-12: [“Decimal
Division”](http://www.ck12.org/arithmetic/Decimal-Division/lesson/Decimal-Division/) (HTML)

that decimal division is not much different from regular long
division. Pay attention to how and why mathematicians (you!) move
the decimal place in both the divisor and dividend.

through the text.

While watching the Khan Academy video, pause the video at each
example, solve it on your own, and then check your solution and
steps along with the instructor.

Do the practice problems beneath the video, and check your solution
on a calculator.

problems should take you approximately 45 minutes.

-   [CCSS.Math.Content.6.NS.B.3](http://www.corestandards.org/Math/Content/6/NS/B/3)

attributed to CK-12, and the original version can be found
[here](http://www.ck12.org/arithmetic/Decimal-Multiplication/lesson/Decimal-Multiplication/).
``````
• Did I Get This? Activity: Howard County Public School System’s “School Store Task”

Instructions: On the webpage above, scroll down until you find the activity titled “School Store.” (You will have to search carefully to find this.) Download the “6.NS.3 School Store Task Resource Sheet.” Follow the instructions on the document to solve each of the problems about buying school supplies. Think of different strategies that could be used to solve the problems.

Use this answer key (PDF) to check your answers. This document has extension questions that you can use to challenge yourself. The solutions are at the bottom.

Completing these problems should take you approximately 30 minutes.

1.4 Factors and Multiples

Factors and multiples are used when you are finding patterns with numbers. Developing fluency in factors and multiples will help you when working with expanding and reducing fractions. As you will also see in this unit, factors can be used to help with the distributive property.

1.4.1 Greatest Common Factors   - Explanation: YouTube: Mathispower4u: James Sousa’s “Factors”

``````Link: YouTube: Mathispower4u: James Sousa’s

Instructions: While watching the video, find the factors of each
number on your own as the instructor does it. Fluently knowing your
multiplication facts will benefit you greatly when finding factors.

Watching the video and taking notes should take you approximately 15
minutes.

-   [CCSS.Math.Content.6.NS.B.4](http://www.corestandards.org/Math/Content/6/NS/B/4)

attributed to James Sousa, and the original version can be found
``````
• Explanation: YouTube: Mathispower4u: James Sousa’s “Example: Determining the Greatest Common Factor”

Instructions: Find the prime factorization of each number on your own as the instructor does it in the video. Pause the video if you need more time.

Watching this video and taking notes should take you approximately 15 minutes.

1.4.2 Least Common Multiples   - Explanation: YouTube: Mathispower4u: James Sousa’s “Example: Determining the Least Common Multiple Using Prime Factorization”

``````Link: YouTube: Mathispower4u: James Sousa’s [“Example: Determining
the Least Common Multiple Using Prime

<span>Instructions: While watching the video, notice how the
instructor shows a different method for determining the LCM. Either
method works; however, this method using prime factorization will be
a lot less time consuming if you need to find an LCM of larger
numbers.</span>

<span>Watching this video and taking notes should take you
approximately 15 minutes.</span>

-   [CCSS.Math.Content.6.NS.B.4](http://www.corestandards.org/Math/Content/6/NS/B/4)

a </span>[Creative Commons Attribution 3.0 Unported
attributed to James Sousa, and the original version can be found
``````
• Explanation: YouTube: Mathispower4u: James Sousa’s “Example: Determining the Least Common Multiple Using a List of Multiples”

Link: YouTube: Mathispower4u: James Sousa’s “Example: Determining the Least Common Multiple Using a List of Multiples” (YouTube)

Instructions: While watching the video, notice how the instructor makes a list of multiples for each number before looking for the least common multiple (LCM).

Watching this video and taking notes should take you approximately 15 minutes.

1.4.3 Distributive Property   - Explanation: YouTube: Mathispower4u: James Sousa’s “Using the Distributive Property to Multiply Quickly”

``````Link: YouTube: Mathispower4u: James Sousa’s [“Using the Distributive
Property to Multiply

Instructions: While watching this video, think about how this
strategy could be beneficial to you when multiplying mentally. You
will continue to use the distributive property in many ways

Watching this video and taking notes should take you approximately
15 minutes.

-   [CCSS.Math.Content.6.NS.B.4](http://www.corestandards.org/Math/Content/6/NS/B/4)

attributed to James Sousa, and the original version can be
``````
• Did I Get This? Activity: College of the Sequoias: Ross Rueger’s Math 360 - PreAlgebra: “Chapter 1, Section 5: Greatest Common Factor and Least Common Multiple”

Link: College of the Sequoias: Ross Rueger’s Math 360 - PreAlgebra: “Chapter 1, Section 5: Greatest Common Factor and Least Common Multiple” (PDF)

Instructions: Do “Exercise Set 1.5,” problems 1 - 27 (ODD) and 61 - 64 (ALL). Answers can be found on pages 61 - 62.

Completing these problems and checking your answers should take you approximately 30 minutes.

1.5 Rational Numbers

Using negative numbers in life usually means that it’s very cold outside (temperatures below zero), you lost points in a game you are playing, or possibly you are in debt to someone and owe them money. Knowing and understanding the number line as it extends into negative numbers is important as you relate the entire rational number system to your real life.

• Explanation: Cheryl Wilcox’s Free PreAlgebra: “Chapter 4, Lesson 19: Negative Numbers”

Link: Cheryl Wilcox’s Free PreAlgebra: “Chapter 4, Lesson 19: Negative Numbers” (PDF)

Instructions: This textbook introduces concepts covered in this entire subunit. This is a chance for you to begin to get an understanding of rational numbers. Separate your notebook paper into three sections. Label the sections “Definitions,” “Real-Life Situations,” and “Number Lines.” As you read pages 1 - 5, fill in the columns with information from text. These are the vocabulary words that you should look for to put in the definitions column: negative numbers, positive numbers, opposites, and absolute value. In the text, many real-life situations are mentioned, explained, and even illustrated with images. The more of these you can learn, the more beneficial this text will be to your understanding of rational numbers. Under the “Number Lines” column, you should use the type 2 models to draw a few number lines and plot points to help compare greater than and less than.

Do the entire worksheet on page 6 as practice. You could also do practice problems 8 and 9 on the two homework pages, page 8 and page 12.

Reading this chapter, taking notes, and working the practice problems should take you approximately 30 minutes.

• Explanation: YouTube: Mathispower4u: James Sousa’s “Integers”

Instructions: While watching this video, add notes to your notebook paper that has three columns from the previous activity. You should add the word integers to the definitions column. Also, pay attention when the video gives examples of how negatives numbers are used in real-life situations.

Watching this video and taking notes should take you approximately 15 minutes.

1.5.1 Ordering Rational Numbers   - Did I Get This? Activity: Khan Academy’s “Number Line 2”

``````<span style="font-size: 12px;">Link: Khan Academy’s </span>[“Number
Line

you can answer and check online. Each question has a solution worked
out step-by-step if you need hints along the way. Practice these
exercises to make sure you feel confident with number lines.

Practicing these problems will take you approximately 15 minutes.

-   [CCSS.Math.Content.6.NS.C.6c](http://www.corestandards.org/Math/Content/6/NS/C/6/c)
-   [CCSS.Math.Content.6.NS.C.7a](http://www.corestandards.org/Math/Content/6/NS/C/7/a)

``````
• Did I Get This? Activity: Khan Academy’s “Ordering Negative Numbers”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice these exercises to make sure you feel confident with ordering negative numbers.

Practicing these problems will take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Number Line 3”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice these exercises to make sure you feel confident with number lines.

Practicing these problems will take you approximately 15 minutes.

• Checkpoint: The Mathematics Common Core Toolbox: “Cake Weighing, Part A” and “Part B”

Link: The Mathematics Common Core Toolbox: “Cake Weighing, Part A” and “Part B” (HTML)

Completing these two questions and checking your work should take you approximately 15 minutes.

1.5.2 Absolute Value   - Did I Get This? Activity: Khan Academy’s “Comparing Absolute Value”

``````Link: Khan Academy’s [“Comparing Absolute

you can answer and check online. Each question has a solution worked
out step-by-step if you need hints along the way. Practice the
absolute value questions until you feel confident that you
understand how to compare integers and absolute value (recommended a
minimum of 10 minutes of practice).

Practicing these problems will take you approximately 15 minutes.

-   [CCSS.Math.Content.6.NS.C.6c](http://www.corestandards.org/Math/Content/6/NS/C/6/c)
-   [CCSS.Math.Content.6.NS.C.7c](http://www.corestandards.org/Math/Content/6/NS/C/7/c)

``````
• Checkpoint: The Mathematics Common Core Toolbox: “Cake Weighing, Part C” and “Part D”

Link: The Mathematics Common Core Toolbox: “Cake Weighing, Part C” and “Part D” (HTML)

Completing these two questions and checking your work should take you approximately 15 minutes.

• Checkpoint: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 2, Section 1: An Introduction to the Integers”

Link: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 2, Section 1: An Introduction to the Integers” (PDF)

Instructions: If needed for extra guidance, read through the sections under “2.1: An Introduction to the Integers.” As a checkpoint to see how well you know these concepts, click on “Exercises” under section 2.1 or scroll to page 106. Do all the ODD problems 1 - 69. Check your answers starting on page 110.

Completing this checkpoint should take you approximately 30 minutes.

`````` Instructions: Test your knowledge by completing this checkpoint.