# Decimal Notations for Fractions Number and Operations Fractions /4.NF

Size: px
Start display at page:

Transcription

1 Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions. 4.NF.C.5 Express a fraction with denominator as an equivalent fraction with denominator, and use this technique to add two fractions with respective denominators and. For example, express 3/ as 30/, and add 3/ + 4/ = 34/. 4.NF.C.6 Use decimal notation for fractions with denominators or. For example, rewrite 0.62 as 62/; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. The standards covered in this section fall under the cluster for understanding decimal notation for fractions and comparing decimal fractions. This cluster connects to the second critical area of focus for fourth grade which requires students to develop an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions. More specifically within this cluster, we ll be dealing with understanding fraction equivalence between fractions with the denominator and (and adding such fractions), fraction equivalence with decimals, and comparing two decimals to the hundredths. The standards within this cluster should be covered after the standards in the cluster for building fractions from unit fractions. The knowledge gleaned from understanding and generating equivalent fractions will be invaluable as students embark on working with fractions with the denominator as well as building an understanding of decimals and decimal notation. Moreover, the skills learned within this cluster will serve as a foundation in fifth grade as students perform operations with decimals to the hundredths. Before we dig in to each of the standards presented within this section, be sure to review the helpful literature and resource books that can complement your lessons and provide additional support for students.

2 Literature Books to Support Teaching the Standard(s) in this Section Fractions, Decimals, and Percents by David A. Adler Do You Know Dewey? by Brian P. Cleary Bob the Alien Discovers the Dewey Decimal System by Sandy Donovan Parting is Such Sweet Sorrow by Linda Powley Great Videos to Support What You re Teaching There are many instructional videos that you can use to supplement your lessons. Save the following links to videos for future viewing during class time, within your math center, as part of a lesson, or for students to use at home for review or supplemental learning. Benjamin Light 4.NF.5 Converting Decimals to Fractions Math Antics Converting Base Fractions Expanded Notation Decimals and Decimal Fractions Writing Tenths and Hundredths with Decimals Decimals in Standard and Word Form Decimals on a Number Line Decimals in the Tenths and Hundredths Tenths and Hundredths on a Number Line Comparing and Arranging Decimals Great Virtual Games and Resources to Support What You re Teaching Fractions to Decimals Fruit Shoot: Shoot the fruit with the decimal equivalent of the fraction.

3 The Decimal Detectives: Students are quizzed on their knowledge of a decimal s place value on a number line to help catch a crooked decimal! Matching Math Tenths: Match the decimals to the tenths with its model. Matching Math Hundredths: Match the decimals to the hundredths with its model. Comparing Decimals Fruit Shoot: Compare the two decimals and shoot the fruit with the appropriate symbol (<, >, or =). Scooter Quest Decimals: Students use their knowledge of decimal place value to deliver the newspaper to the correct house. 4.NF.C.5 / Express a fraction with denominator as an equivalent fraction with denominator, and use this technique to add two fractions with respective denominators and. For example, express 3/ as 30/, and add 3/ + 4/ = 34/. Thus far, students have been working mostly with like denominators, specifically the denominators 2, 3, 4, 5, 6, 8,, and 12. As part of this domain, students must also be familiar with fractions with the denominator. Fractions with the denominator come into play in this standard as students compare their values with fractions with the denominator. Students will use their knowledge of equivalent fractions to generate fractions with the denominator into equivalent fractions with the denominator. Additionally, they will be adding fractions with the denominators and. First and foremost, students must be taught that a fraction with the denominator and a fraction with the denominator both represent the same whole. Previously, students learned that fractions with the same denominator represent the same whole. Now, they are beginning to learn that numbers with different denominators can also represent the same whole. This, of course, varies depending on the denominators. Working with the denominators and is a great starting point to help students learn this concept. We know that fractions can be represented by a number line. Let s take, for example, a fraction with the denominator. The denominator tells us that we must divide the number line into ten equal parts; each part represents 1 /. Notice that / is the same as 1 whole because the denominator and the numerator are the same.

4 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / Now, what if we wanted to create a number line with a fraction with the denominator? Well, we d use the same process except this time, we ll divide the number line into equal parts. Each part now represents 1 / / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / Do you notice some similarities between the two number lines? Both number lines are the same length and have the endpoints 0 and 1. Notice that / is equal to 1 just like / is equal to 1. That must mean that both denominators refer to the same whole! Still not convinced? Let s compare / and / to 1 whole using models. 1 whole ( 1 / 1 ) / / Above we have three models, 1 whole, shaded parts out of equal parts, and shaded parts out of equal parts. You can see that / and / both have the same whole. In other words, they are equal to 1 whole. Keeping this idea in mind, take a look at the problem below. Is 5 / equivalent to 50 /?

5 How can we determine if the two fractions are equivalent? Start by looking at the denominators. We have a denominator of and a denominator of. We now know that the denominators and refer to the same whole. That means we can create identical models to represent each fraction. That is, the models can be of the same shape and size. For this example, we chose a square. 5 / 50 / In the first model representing 5 /, the denominator tells us to divide the model into ten equal parts. The numerator tells us to shade in five parts. In the second model representing 50 /, the denominator tells us to break the model into equal parts and the numerator 50 tells us to shade in 50 parts. Compare the two models. Do they look the same? Yes! We already know they have the same whole. We can now see from the shaded portions that they are equivalent. 5 / and 50 / are equivalent fractions. Now, try the problem below. Find a fraction with a denominator that is equivalent to 40 /.

6 In the problem above, we have the fraction 40 /. Write down the fraction. We want to find a fraction that is equivalent, so we ll write an equal sign. Our equivalent fraction must have the denominator. Write a fraction with a denominator. What we don t know is the numerator. So, write a question mark to represent the unknown numerator. 40 =? Next, we ll draw two identical models because we know the denominators and refer to the same whole. To represent the denominator, we ll divide the first model into equal pieces. The second model is divided into ten equal parts to represent the denominator. We know the numerator for the first fraction is 40, so we ll shade in 40 parts in the hundreds model. To create an equivalent fraction in the second model, we ll shade in the same amount. 40 /? / The models now represent equivalent fractions. Now, we must simply determine what the denominator is for the second model to find the answer. We know that the numerator

7 represents the number of parts shaded. So, let s count the number of shaded parts. We have four shaded parts so we know the numerator is 4! The answer is 4 /. 4 / is equivalent to 40 /. 40 = 4 To check to make sure the answer is correct, we can use number lines. We ll create two identical number lines for the two fractions / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / We can see that both number lines have the same points and the same shaded portions which tells us that the fractions are equivalent. The answer is correct! Once students have fully grasped the idea that a fraction with a denominator has the same whole as a fraction with the denominator, it s time to get them started on adding fractions with the denominators and. We ll continue to use models as tools for finding the answer. Take a look at the problem below

9 the numerator? This is quite a tedious process, but once the shaded parts are counted, we know that we have a numerator of 80. The answer is 80 /! Is there an easier way to write 80 /? How about 8 /? How do we know for sure that 80 / is equivalent to 8 /? Well, suppose we did the reverse and instead of adding the shaded portion of the first model into the second model, we combined the shaded portion of the second model into the first model. We would have equal parts and 8 shaded parts, giving us a fraction of 8 /. = 8 / 80 / Using modeling is an excellent way to help students make the connection between equivalent fractions with the denominators ten and one hundred. Models allow students to visually see that although two fractions may appear different, they both represent the same quantity. The goal of this standard is for students to understand fractions beyond the digits they consist of, and acquire a deeper understanding of fractions as parts of a whole. 4.NF.C.6 / Use decimal notation for fractions with denominators or. For example, rewrite 0.62 as 62/; describe a length as 0.62 meters; locate 0.62 on a number line diagram. This standard introduces students to the concept of decimals and is directly connected to the aforementioned standard on fractions with the denominators and. In order for students to master this standard, they need a deep understanding of place value. Previously, students learned to work with place value charts with numbers up to one million. It is important that students fully understand the concept of place value in whole numbers as that concept will now be extended to fractions and their decimal equivalents. In this section, we teach students how to convert decimals to fractions and fractions to decimals using place value charts, number lines, and division.

10 Before we begin, let s do a brief review. As you recall, fractions can be represented in a number of ways. Parts of a group Parts of a whole Number line 0 ¼ 1 Today, we ll introduce yet another way to represent fractions: as decimals! Reviewing a number s place value is very important as well. We know that a digit s location in a number is significant. Why? Because the location tells us the value of the digit. For example, the value of 5 is different than the value of 50 which is also different from 500. We worked with a place value chart such as the one below to help us understand the values of digits in a number. Periods Hundred Millions Millions Thousands Ones Ten Millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones Sections At the top, we have the periods and below we have the sections. Remember that when writing a number in numerical form, we separated the periods with a comma. Now, we ll be adding another period to the place value chart. This time, the period will represent numbers that are less than one. In other words, they are fractions or decimals. The period is called thousandths.

11 Mil. Millions Thousands Ones Thousandths Mil. Mil. Thous. Thous. Thous. s s,,. 1s Tenths Hundredths Thousandths Within the thousandths period, we have the sections representing the tenths place, hundredths place, and thousandths place. Notice that although there is a comma separating the whole number periods, there is a period separating the ones period from the thousandths period. This is because fractions (with a numerator that is smaller than the denominator) are less than one whole, which means they are not whole numbers; they are parts of a whole. Therefore, the fractions represented in decimal form must go after a period. To reiterate, whole numbers go to the left of the period and decimal fractions go to the right of the period. Let s try an example to help bring this concept home. Convert 0.4 to a fraction. Above we have the decimal 0.4. The zero in front of the period tells us there are no wholes in the number. The 4 after the period tells us that this number is a decimal. To convert the decimal to a fraction, we can use a place value chart. Since we are working with decimals, we ll simplify the chart to include only the periods ones and thousandths. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths 0 4. With consideration to the location of the period in the decimal, we entered the numbers into the place value chart. We can now see more clearly that there are no whole numbers and 4 tenths. 4 tenths sounds a lot like a fraction, doesn t it? That s because it IS a fraction! Since 4 is in the tenths place, we know that the fraction would have 4 as the numerator and (TENths) as the denominator. 0.4 = 4

12 There we have it, 4 / is the fraction form of 0.4! To gain a better understanding of the value of 0.4, let s represent the decimal fraction as a model and plot 4 / on a number line. The denominator tells us to divide the model into ten equal parts and the numerator tells us to shade in four. Take a look at how 0.4 looks as a model. 0.4 ( 4 / ) Next, we ll represent 0.4 on a number line. We know that a decimal is less than one whole so we ll create a number line between 0 and 1. The denominator tells us to divide the number line into ten equal parts and the numerator 4 tells us to plot a point on the fourth dash and shade up to that point / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / Now, let s try another problem. Put 0.27 into fraction form. Once again, we are given a decimal and asked to turn it into a fraction. As we ve done previously, we ll use a place value chart to help us better understand the value of 0.27 and convert it to a fraction.

13 Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths Following the location of the period, we have zero ones (or wholes) in front of the period. After the period, we can see that we have two tenths and seven hundredths. Or, we can simply say that we have twenty-seven hundredths. To create a fraction, we can use 27 as the numerator and (HUNDREDths) as the denominator is equivalent to 27 /! 0.27 = 27 What if we wanted to do the reverse and turn a fraction into a decimal? Well, that s a little trickier but previous practice with expanded form and division can help us master this skill! Rewrite 52 / as a decimal. As you can see, we are now given a fraction and asked to convert the fraction to decimal form. One strategy we can use is to put the fraction into expanded form then input it into a place value chart. Looking at the numerator, we know that the expanded form of 52 is The denominator stays the same so we have 50 / + 2 /. We know that 50 / is the same as 5 /, so the expanded form of 52 / is 5 / + 2 /. 52 (50 + 2)

14 Now that we have the fraction in expanded form, we can easily input the numbers into a place value chart. Since the 5 has a denominator of, we know that 5 goes into the tenths place. 2 has a denominator of which puts it in the hundredths place. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths A period goes before the thousandths period and a zero goes before the period to represent no whole numbers. Together, we have a decimal of Be sure to say out loud, point-fifty-two and fifty-two tenths to students so they make the connection between decimals and fractions. 52 = 0.52 Another strategy students can use to convert fractions to decimals is to use division. You may have noticed from a previous section that although we ve taught students to divide multi-digit numbers, we have not yet introduced long division. The CCSS does not officially introduce dividing multi-digit numbers in the standard algorithm until the sixth grade. In the interim, students have learned to divide by using their knowledge about multiplication which, in many cases, is the more efficient method for finding multi-digit quotients. We bring this up because the strategy that we introduce below will utilize a bit of long division method so it may be too advanced for some students. However, the long division is quite basic and may be useful to many students when converting fractions to decimals. Using the next method, it is important that students see fractions as a division problem. For example, instead of seeing 52 / as 52 parts out of, they should recognize that 52 is being divided by. 52

15 52 To solve for this division problem, we ll put the dividend (52) inside the house and the divisor () on the outside. To start, we ll ask ourselves how many times the dividend can go into the divisor. That is, how many times can go into 52? Zero! Because is larger than 52. So, we ll write a zero above 52. Then, we ll multiply zero by. We know anything multiplied by zero is zero so we ll write a zero below 52. Subtract the zero from the divisor. 52 minus 0 is still is the number leftover after we divided so, essentially, we can treat 52 as the remainder (part of a whole). Previously, we wrote the remainder as r 52. However, that doesn t work in this case because we have no whole number. Also, the goal is to create a decimal. Instead, we ll add a period to the answer and bring 52 up as a decimal: NF.C.7 / Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. In this standard, students must compare two decimals to the hundredths. To properly compare two decimals, students must understand the relationship between decimals and fractions with the denominators and. They must also grasp the concept of place value for decimals. Students have already worked with comparing multi-digit whole

16 numbers and fractions with different numerators and denominators so the process of comparing quantities should be very familiar to them. The tricky part now is that they are dealing with decimals and there are some extra steps involved in finding the answer. Nonetheless, we continue to use modeling and number lines as familiar tools that students can use to compare decimals To compare these two decimals, we begin by converting the decimals to fractions. We can use place value charts to do the conversions. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths Both numbers are in the tenths place which tells us that the denominator is. Therefore, we have 7 / and 4 / as our fractions. 7 4 Now, we ll draw models to represent each fraction. Since the fractions have the same denominator, we know we are referring to the same whole. Consequently, the models should be identical and divided into ten equal parts. In the first model, we shade in 7 parts; in the second model, we shade in 4 parts. 7 /

17 4 / We can see from the models that 7 / is greater than 4 /. In other words, 0.7 is greater than 0.4. We have the answer! > Now, we ll try a problem that s a bit trickier! This problem is trickier because upon first glance, students may think that 0.16 is greater than 0.9 because 16 is greater than 9. But is that really the case? Again, we ll use a place value chart to convert the decimals into fractions. Ones Thousandths Hundreds Tens Ones Tenths Hundredths Thousandths

18 Looking at the chart, we can see that we have 16 hundredths which tells us the first fraction has a denominator of : 16/. The second decimal is 9 tenths which means we have a denominator of : 9/ Again, we ll draw models to represent each fraction. Since the fractions have the denominators and, we know we are referring to the same whole. Consequently, the models should be identical. The first model represents 16 / which means it must be divided into equal parts with 16 parts shaded. In the second model representing 9 /, we divide the model into ten equal parts and shade in 9 parts. 16 / 9 / The models clearly show that 16 / is less than 9 /. Thus, 0.16 is less than 0.9!

19 < To make sure the answer is correct, we can use a number line to compare the decimals as well. Since the decimals represent fractions that are less than one whole, we ll draw a number line of 0 to is equivalent to 16 / which tells us to divide the first number line into equal parts and shade to the 16 th dash. 0.9 is equivalent to 9 / so we create a second number line with ten equal parts and shade up to the ninth dash / 20 / 30 / 40 / 50 / 60 / 70 / 80 / 90 / / / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / / The number lines confirm that 16 / is indeed less than 9 /. Perfect! The standards covered within this section serve as great milestones for students as they journey through the CCSSM. The standards require students to use skills acquired from other standards and grade levels on the place value system and make connections to build on their prior knowledge to develop new skills. Moreover, these standards also prepare students for their work in fifth grade as they will be required to perform operations with decimals to the hundredths. In addition to the strategies provided above, allowing students to physically work with base ten blocks is always a good idea when dealing with fractions. You should also encourage students to pronounce decimals and fractions out loud. For example, 0.32 as point thirty-two, thirty-two hundredths, or even the sum of three tenths and two hundredths. As students master the skills presented within these standards, the method of connecting fractions and decimals to their word form will help students develop a greater understanding of decimal value without having to refer to the place value chart.

20 For practice with the skills taught in this section of the course, provide students with the Denominators & Decimals worksheet in the Course Guides folder.

### Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### Sample Fraction Addition and Subtraction Concepts Activities 1 3

Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations

### Dr Brian Beaudrie pg. 1

Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, 3 0.14 means add 0.14 three times, regroup, and simplify,

### What Is Singapore Math?

What Is Singapore Math? You may be wondering what Singapore Math is all about, and with good reason. This is a totally new kind of math for you and your child. What you may not know is that Singapore has

### + = has become. has become. Maths in School. Fraction Calculations in School. by Kate Robinson

+ has become 0 Maths in School has become 0 Fraction Calculations in School by Kate Robinson Fractions Calculations in School Contents Introduction p. Simplifying fractions (cancelling down) p. Adding

### Grade 5 Math Content 1

Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - DECIMALS AND WHOLE NUMBERS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### Fourth Grade Math Standards and "I Can Statements"

Fourth Grade Math Standards and "I Can Statements" Standard - CC.4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and

### BPS Math Year at a Glance (Adapted from A Story Of Units Curriculum Maps in Mathematics K-5) 1

Grade 4 Key Areas of Focus for Grades 3-5: Multiplication and division of whole numbers and fractions-concepts, skills and problem solving Expected Fluency: Add and subtract within 1,000,000 Module M1:

### 5 th Grade Common Core State Standards. Flip Book

5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### Sunny Hills Math Club Decimal Numbers Lesson 4

Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions into improper fractions, just to multiply and convert them back? Are you tired of reducing fractions

### 3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### The Crescent Primary School Calculation Policy

The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has

### Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

### Section 1.5 Exponents, Square Roots, and the Order of Operations

Section 1.5 Exponents, Square Roots, and the Order of Operations Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Identify perfect squares.

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### The Euclidean Algorithm

The Euclidean Algorithm A METHOD FOR FINDING THE GREATEST COMMON DIVISOR FOR TWO LARGE NUMBERS To be successful using this method you have got to know how to divide. If this is something that you have

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS

COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS Compiled by Dewey Gottlieb, Hawaii Department of Education June 2010 Operations and Algebraic Thinking Represent and solve problems involving

### Chapter 1: Order of Operations, Fractions & Percents

HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

### NF5-12 Flexibility with Equivalent Fractions and Pages 110 112

NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### Five daily lessons. Page 23. Page 25. Page 29. Pages 31

Unit 4 Fractions and decimals Five daily lessons Year 5 Spring term Unit Objectives Year 5 Order a set of fractions, such as 2, 2¾, 1¾, 1½, and position them on a number line. Relate fractions to division

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### Chapter 4 -- Decimals

Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789

### WSMA Decimal Numbers Lesson 4

Thousands Hundreds Tens Ones Decimal Tenths Hundredths Thousandths WSMA Decimal Numbers Lesson 4 Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions

### 1 BPS Math Year at a Glance (Adapted from A Story of Units Curriculum Maps in Mathematics P-5)

Grade 5 Key Areas of Focus for Grades 3-5: Multiplication and division of whole numbers and fractions-concepts, skills and problem solving Expected Fluency: Multi-digit multiplication Module M1: Whole

Grade 6 Math Oak Meadow Coursebook Oak Meadow, Inc. Post Office Box 1346 Brattleboro, Vermont 05302-1346 oakmeadow.com Item #b064010 Grade 6 Contents Introduction... ix Lessons... Lesson 1... 1 Multiplication

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### This lesson introduces students to decimals.

NATIONAL MATH + SCIENCE INITIATIVE Elementary Math Introduction to Decimals LEVEL Grade Five OBJECTIVES Students will compare fractions to decimals. explore and build decimal models. MATERIALS AND RESOURCES

### Solution Guide Chapter 14 Mixing Fractions, Decimals, and Percents Together

Solution Guide Chapter 4 Mixing Fractions, Decimals, and Percents Together Doing the Math from p. 80 2. 0.72 9 =? 0.08 To change it to decimal, we can tip it over and divide: 9 0.72 To make 0.72 into a

1 Decimals Adding and Subtracting Decimals are a group of digits, which express numbers or measurements in units, tens, and multiples of 10. The digits for units and multiples of 10 are followed by a decimal

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### Fractions as Numbers INTENSIVE INTERVENTION. National Center on. at American Institutes for Research

National Center on INTENSIVE INTERVENTION at American Institutes for Research Fractions as Numbers 000 Thomas Jefferson Street, NW Washington, DC 0007 E-mail: NCII@air.org While permission to reprint this

### Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations

Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)

### Grade 5 Common Core State Standard

2.1.5.B.1 Apply place value concepts to show an understanding of operations and rounding as they pertain to whole numbers and decimals. M05.A-T.1.1.1 Demonstrate an understanding that 5.NBT.1 Recognize

### Converting from Fractions to Decimals

.6 Converting from Fractions to Decimals.6 OBJECTIVES. Convert a common fraction to a decimal 2. Convert a common fraction to a repeating decimal. Convert a mixed number to a decimal Because a common fraction

### Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots

### DIVISION OF DECIMALS. 1503 9. We then we multiply by the

Tallahassee Community College 0 DIVISION OF DECIMALS To divide 9, we write these fractions: reciprocal of the divisor 0 9. We then we multiply by the 0 67 67 = = 9 67 67 The decimal equivalent of is. 67.

### Introduction to Decimals

Introduction to Decimals Reading and Writing Decimals: Note: There is a relationship between fractions and numbers written in decimal notation. Three-tenths 10 0. 1 zero 1 decimal place Three- 0. 0 100

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### CHAPTER 4 DIMENSIONAL ANALYSIS

CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.

### 1. The Fly In The Ointment

Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### NS6-50 Dividing Whole Numbers by Unit Fractions Pages 16 17

NS6-0 Dividing Whole Numbers by Unit Fractions Pages 6 STANDARDS 6.NS.A. Goals Students will divide whole numbers by unit fractions. Vocabulary division fraction unit fraction whole number PRIOR KNOWLEDGE

### Unit 11 Fractions and decimals

Unit 11 Fractions and decimals Five daily lessons Year 4 Spring term (Key objectives in bold) Unit Objectives Year 4 Use fraction notation. Recognise simple fractions that are Page several parts of a whole;

### Domain of a Composition

Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such

### Lesson 1: Fractions, Decimals and Percents

Lesson 1: Fractions, Decimals and Percents Selected Content Standards Benchmarks Addressed: N-2-H Demonstrating that a number can be expressed in many forms, and selecting an appropriate form for a given

### FRACTIONS MODULE Part I

FRACTIONS MODULE Part I I. Basics of Fractions II. Rewriting Fractions in the Lowest Terms III. Change an Improper Fraction into a Mixed Number IV. Change a Mixed Number into an Improper Fraction BMR.Fractions

### Calculation Policy Fractions

Calculation Policy Fractions This policy is to be used in conjunction with the calculation policy to enable children to become fluent in fractions and ready to calculate them by Year 5. It has been devised

### Numerator Denominator

Fractions A fraction is any part of a group, number or whole. Fractions are always written as Numerator Denominator A unitary fraction is one where the numerator is always 1 e.g 1 1 1 1 1...etc... 2 3

### Math 0306 Final Exam Review

Math 006 Final Exam Review Problem Section Answers Whole Numbers 1. According to the 1990 census, the population of Nebraska is 1,8,8, the population of Nevada is 1,01,8, the population of New Hampshire

### Georgia Standards of Excellence Grade Level Curriculum Overview. Mathematics. GSE Fifth Grade

Georgia Standards of Excellence Grade Level Curriculum Overview Mathematics GSE Fifth Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

### DECIMAL COMPETENCY PACKET

DECIMAL COMPETENCY PACKET Developed by: Nancy Tufo Revised: Sharyn Sweeney 2004 Student Support Center North Shore Community College 2 In this booklet arithmetic operations involving decimal numbers are

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern.

INTEGERS Integers are positive and negative whole numbers, that is they are; {... 3, 2, 1,0,1,2,3...}. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

### Using Proportions to Solve Percent Problems I

RP7-1 Using Proportions to Solve Percent Problems I Pages 46 48 Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving

### Introduction to Fractions, Equivalent and Simplifying (1-2 days)

Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use

### Graphic Organizers SAMPLES

This document is designed to assist North Carolina educators in effective instruction of the new Common Core State and/or North Carolina Essential Standards (Standard Course of Study) in order to increase

### 1004.6 one thousand, four AND six tenths 3.042 three AND forty-two thousandths 0.0063 sixty-three ten-thousands Two hundred AND two hundreds 200.

Section 4 Decimal Notation Place Value Chart 00 0 0 00 000 0000 00000 0. 0.0 0.00 0.000 0.0000 hundred ten one tenth hundredth thousandth Ten thousandth Hundred thousandth Identify the place value for

### CCSS Mathematics Implementation Guide Grade 5 2012 2013. First Nine Weeks

First Nine Weeks s The value of a digit is based on its place value. What changes the value of a digit? 5.NBT.1 RECOGNIZE that in a multi-digit number, a digit in one place represents 10 times as much

### GRADE 5 SKILL VOCABULARY MATHEMATICAL PRACTICES Evaluate numerical expressions with parentheses, brackets, and/or braces.

Common Core Math Curriculum Grade 5 ESSENTIAL DOMAINS AND QUESTIONS CLUSTERS Operations and Algebraic Thinking 5.0A What can affect the relationship between numbers? round decimals? compare decimals? What

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### 5 Mathematics Curriculum

New York State Common Core 5 Mathematics Curriculum G R A D E GRADE 5 MODULE 1 Topic B Decimal Fractions and Place Value Patterns 5.NBT.3 Focus Standard: 5.NBT.3 Read, write, and compare decimals to thousandths.

### Previously, you learned the names of the parts of a multiplication problem. 1. a. 6 2 = 12 6 and 2 are the. b. 12 is the

Tallahassee Community College 13 PRIME NUMBERS AND FACTORING (Use your math book with this lab) I. Divisors and Factors of a Number Previously, you learned the names of the parts of a multiplication problem.

### FRACTIONS OPERATIONS

FRACTIONS OPERATIONS Summary 1. Elements of a fraction... 1. Equivalent fractions... 1. Simplification of a fraction... 4. Rules for adding and subtracting fractions... 5. Multiplication rule for two fractions...

### A Prime Investigation with 7, 11, and 13

. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain six-digit numbers A c t i v i t y 3 Materials TI-73 calculator A Prime Investigation

### CCSS-M Critical Areas: Kindergarten

CCSS-M Critical Areas: Kindergarten Critical Area 1: Represent and compare whole numbers Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such

### Warm-Up ( 454 3) 2 ( 454 + 2) 3

Warm-Up ) 27 4 ST/HSEE: 4 th Grade ST Review: 4 th Grade ST t school, there are 704 desks to place into classrooms. If the same number of desks is placed in each classroom, how many desks will be in each

### 5.1 Introduction to Decimals, Place Value, and Rounding

5.1 Introduction to Decimals, Place Value, and Rounding 5.1 OBJECTIVES 1. Identify place value in a decimal fraction 2. Write a decimal in words 3. Write a decimal as a fraction or mixed number 4. Compare

### Mathematics Instructional Cycle Guide

Mathematics Instructional Cycle Guide Fractions on the number line 3NF2a Created by Kelly Palaia, 2014 Connecticut Dream Team teacher 1 CT CORE STANDARDS This Instructional Cycle Guide relates to the following

### An Introduction to Number Theory Prime Numbers and Their Applications.

East Tennessee State University Digital Commons @ East Tennessee State University Electronic Theses and Dissertations 8-2006 An Introduction to Number Theory Prime Numbers and Their Applications. Crystal

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

Arithmetic 1 Progress Ladder Maths Makes Sense Foundation End-of-year objectives page 2 Maths Makes Sense 1 2 End-of-block objectives page 3 Maths Makes Sense 3 4 End-of-block objectives page 4 Maths Makes

### QM0113 BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION)

SUBCOURSE QM0113 EDITION A BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION) BASIC MATHEMATICS I (ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION) Subcourse Number QM 0113 EDITION

### Mathematical goals. Starting points. Materials required. Time needed

Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;

### Overview. Essential Questions. Grade 4 Mathematics, Quarter 4, Unit 4.1 Dividing Whole Numbers With Remainders

Dividing Whole Numbers With Remainders Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Solve for whole-number quotients with remainders of up to four-digit dividends

### Training Manual. Pre-Employment Math. Version 1.1

Training Manual Pre-Employment Math Version 1.1 Created April 2012 1 Table of Contents Item # Training Topic Page # 1. Operations with Whole Numbers... 3 2. Operations with Decimal Numbers... 4 3. Operations

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Introduction to Fractions

Section 0.6 Contents: Vocabulary of Fractions A Fraction as division Undefined Values First Rules of Fractions Equivalent Fractions Building Up Fractions VOCABULARY OF FRACTIONS Simplifying Fractions Multiplying

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Sequential Skills. Strands and Major Topics

Sequential Skills This set of charts lists, by strand, the skills that are assessed, taught, and practiced in the Skills Tutorial program. Each Strand ends with a Mastery Test. You can enter correlating