# Discovering Math: Number Concepts Teacher s Guide

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1 Teacher s Guide Grade Level: 3 5 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Number Concepts From prime and composite numbers to fractions and decimals to number models, introduce elementary students to more advanced properties and concepts of numbers. Lesson Plan Student Objectives Describe and apply basic number theory concepts (recognize and use prime and composite numbers, factors, multiples, numbers, and divisibility). Demonstrate the ability to convert basic fractions to percents and decimals, to the nearest 10 and 10 th. Identify odd and even numbers. Explain the basic meaning of place value. Explain the relative magnitude and relationships among whole numbers, fractions, decimals, and mixed numbers. Use models, such as a number line, to identify, order, and compare numbers. Materials Discovering Math: Number Concepts video Number Grid (see below) Number Grid Direction Sheet (see below) Yellow highlighter Red pencil or crayon Prime, composite, factors, and multiples chart Picture Cards (see below) 10 playing cards

2 Teacher s Guide 2 Playing Cards Chart (see below) Number fact sheet (see below) Number Chart (see below) Strip of paper (6 x 2 ) for number line Procedures Prime and Composite Numbers, Factors and Multiples, and Even and Odd Numbers 1. Display the terms prime number and composite number. Discuss these number concepts with the class. Prime numbers have only two factors, 1 and the number (e.g., 7). Composite numbers have more than two factors (e.g., 12). Ask students to give examples or give them numbers to classify as prime or composite. 2. Display the words factor and multiple. Discuss these number concepts with the class. Factors are the numbers multiplied together to get a given number (e.g., 3 and 5 are factors of 15). Multiples are products of a given number and any other number (e.g., a multiple of 3 is 9). Ask students to give examples or give them numbers to identify the factors or multiples. 3. Students can work in pairs or independently, depending on ability level. Give them a copy of the Number Grid and Number Grid Direction Sheet. Review the directions. Identify prime numbers with highlighter. Identify odd numbers by circling in red. Choose ten composite numbers and identify four factors of each. Choose five prime numbers and identify two multiples of each. 4. When students have completed the assignment ask them study the chart to see if they can identify any patterns (e.g., look for odd and even numbers in connection with prime and composite). Have students share their thoughts and ideas with the class. Fractions, Decimals, and Percents 1. Display the Picture Cards or distribute a copy to each student. Ask students what percent of the cards are letters and display the fraction ( 4 10 ). Model how to convert a fraction to a decimal and ask students how to write this fraction as a decimal ( 4 10 = 0.4). Model how to write a decimal as a percent and ask students how to write this decimal as a percent (40%). Continue modeling these conversions until the students understand the process. Possible questions: What fraction of the cards are *? What fraction of the cards are M? What fraction of the cards are and?

3 Teacher s Guide 3 2. Have students work in pairs. Distribute sets of ten playing cards (each set should contain five red cards, numbers 2, 4, 7, 9, and a jack and five black cards, numbers 2, 5, 7, 9, and 10) and a Playing Cards Chart to each pair. Review the directions with students (read the question in the question column and identify the fraction, decimal, and percent that answers the question) and have them complete the chart. If any teams finish before the others, have them create their own questions to challenge each other. When teams complete the activity, have them share their work and explain how they converted fractions, decimals, and percents. Extension Have students work with a set of 20 cards and identify fractions of the whole set and convert to decimals and percents. Place Value, Fractions, Decimals, Whole Numbers, and Mixed Numbers 1. Display the following numbers: 2, Ask students to classify them as whole numbers, fractions, decimals, or mixed numbers. Discuss the classification of each and have students share their ideas with the class. Ask them to give examples of other whole numbers, fractions, decimals, and mixed numbers. Model how to determine the smallest number by using place value, size of fraction, or decimal and ask students to order the numbers from least to greatest. Have them share their ideas and elicit responses. The numbers should be ordered in the following way: 1 4, 3.45, 7 1 2, 2, Distribute a Number Fact Sheet and Number Chart to each student. Tell them to identify each number as a whole number, fraction, decimal, or mixed number and record it in the correct place on the chart. Have students compare their work with a partner to check if their answers are correct. 3. Divide students into pairs and distribute a strip of paper to each. Tell students they will be creating a number line, ordering the numbers from the Number Fact Sheet from greatest to least. Depending upon ability levels, students may need a template or model of a number line as a guide. Have each pair share their number line and explain how they decided the position of each number.

4 Teacher s Guide 4 Assessment Use the following three-point rubric to evaluate students work during this lesson. 3 points: Students produced complete charts and number line, including all the requested information; clearly demonstrated the ability identify prime, composite, even, and odd numbers; clearly demonstrated the ability to convert between fractions, decimals, and percents to the nearest 5 and 20 th ; and clearly demonstrated the ability to identify whole numbers, fractions, decimals, and mixed numbers and order the numbers on a number line. 2 points: Students produced adequate charts and number line, including most of the requested information; satisfactorily demonstrated the ability identify prime, composite, even, and odd numbers; satisfactorily demonstrated the ability to convert between fractions, decimals, and percents to the nearest 10 and 10 th ; and satisfactorily demonstrated the ability to identify whole numbers, fractions, decimals, and mixed numbers and order the numbers on a number line. 1 point: Students produced incomplete charts and number line with little or none of the requested information; did not demonstrate the ability identify prime, composite, even, and odd numbers; did not demonstrate the ability to convert between fractions, decimals, and percents; and did not demonstrate the ability to identify whole numbers, fractions, decimals, and mixed numbers or order the numbers on a number line. Vocabulary composite number Definition: any number with more than two factors Context: 12 is a composite number because it has more than two factors. The factors of 12 are 1, 2, 3, 4, 6, and 12. decimal Definition: a number with one or more digits to the right of the decimal point Context: The student wrote the decimal 4.8. even number Definition: any number that can be divided into two equal groups and ends with the digits 0, 2, 4, 6, 8, or 0 Context: The numbers 12, 20, 34, 56, and 68 are even numbers because they can be divided into two equal groups and they end in the digits 0, 2, 4, 6, or 8. factor Definition: numbers that are multiplied together to get a given product Context: Two factors or 24 are 4 and 6 because 4 x 6 = 24.

5 Teacher s Guide 5 fraction Definition: a number that names a part of a whole or part of a group Context: 3 10 of the people were girls because there were 10 people and 3 of them were girls. mixed number Definition: an amount given as a whole number and a fraction Context: The recipe called for cups of flour is a mixed number because it has a whole number, 2, and a fraction, 1 2. multiple Definition: the product of a given whole number and another whole number Context: Two multiples of 4 are 12 and 20 because 4 x5 = 20 and 4 x 3 = 12. odd number Definition: any number that cannot be divided equally into two groups and ends in the digit 1, 3, 5, 7, or 9 Context: The numbers 15, 31, 43, 57, and 89 are odd numbers because they cannot be divided equally into two groups and they end in the digits 1, 3, 5, 7, or 9. percent Definition: the ratio or comparison of a number to 100 Context: There were 100 apples in the basket and 34 were eaten, so 34 percent of the apples were eaten. place value Definition: determines the value of a digit in a number, based on the location of the digit Context: The students used the place value of the numbers to determine the value of each number. They were able to order the numbers from least to greatest once the place values were determined. prime number Definition: a number that has only two factors, 1 and itself Context: The number 23 is a prime number because it has only two factors, 1 and 23. whole number Definition: a number that represents a whole amount Context: The number 12, 14, 65, and 84 are whole numbers because they refer to whole amounts.

6 Teacher s Guide 6 Academic Standards Mid-continent Research for Education and Learning (McREL) McREL s Content Knowledge: A Compendium of Standards and Benchmarks for K 12 Education addresses 14 content areas. To view the standards and benchmarks, visit This lesson plan addresses the following benchmarks: Understands basic number theory concepts. Understands equivalent forms of basic percents, fractions, and decimals, and when one form of a number might be more useful than another. Understands the basic difference between odd and even numbers. Understands the basic meaning of place value. Uses models to identify, order, and compare numbers. National Council of Teachers of Mathematics (NCTM) The National Council of Teachers of Mathematics (NCTM) has developed national standards to provide guidelines for teaching mathematics. To view the standards online, go to This lesson plan addresses the following standards: Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals. Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers. Explore numbers less than 0 by extending the number line and through familiar applications. Describe classes of numbers according to characteristics such as the nature of their factors. Use representations to model and interpret physical, social, and mathematical phenomena. Recognize and use connections among mathematical ideas.

7 Teacher s Guide 7 Support Materials Develop custom worksheets, educational puzzles, online quizzes, and more with the free teaching tools offered on the Discoveryschool.com Web site. Create and print support materials, or save them to a Custom Classroom account for future use. To learn more, visit DVD Content This program is available in an interactive DVD format. The following information and activities are specific to the DVD version. How to Use the DVD The DVD starting screen has the following options: Play Video This plays the video from start to finish. There are no programmed stops, except by using a remote control. With a computer, depending on the particular software player, a pause button is included with the other video controls. Video Index Here the video is divided into chapters indicated by title. Each chapter is then divided into four sections indicated by video thumbnail icons; brief descriptions are noted for each section. To play a particular segment, press Enter on the remote for TV playback; on a computer, click once to highlight a thumbnail and read the accompanying text description and click again to start the video. Quiz Each chapter has four interactive quiz questions correlated to each of the chapter s four sections. Standards Link Selecting this option displays a single screen that lists the national academic standards the video addresses. Teacher Resources This screen gives the technical support number and Web site address. Video Index I. Prime and Composite Numbers (7 min.) Prime and Composite Numbers: Introduction Prime and composite numbers are defined and modeled. Prime numbers have only two factors, and composite numbers have more than two factors. Example 1: Composite Numbers A composite number can be divided evenly by numbers other than itself and one. A soccer team is used to define and model composite numbers.

8 Teacher s Guide 8 Example 2: Prime Numbers A prime number cannot be divided evenly by any numbers other than itself and one. An air show is used to define and model prime numbers. Example 3: Prime and Composite Numbers Rows and columns of squares are called a rectangular array. Any number that can be formed by more than one rectangular array is a composite number. II. III. Factors and Multiples (8 min.) Factors and Multiples: Introduction Baseball is used to introduce factors numbers multiplied to get a given number and multiples the product of a number and any whole number greater than zero. Example 1: Multiples of Whole Numbers A multiple of a number is the product of that number and any other whole number other than zero. Boxes of caps are used to model multiples. Example 2: Factors and Composite Numbers A composite number is any number with more than two factors. Rows of seats in a baseball stadium are used to model factors and composite numbers. Example 3: Odd and Even Numbers Even numbers can be divided evenly by 2 and have a 0, 2, 4, 6, or 8 in the ones place. Odd numbers cannot be divided evenly by 2 and have a 1, 3, 5, 7, or 9 in the ones place. Baseball teams are used to model even and odd numbers. Fractions and Decimals (8 min.) Fractions and Decimals: Introduction Fractions and decimals are numbers less than one that describe part of a whole or parts of a group. Fractions have numerators and denominators. The first place after the decimal point is the tens place. Example 1: Quarters and Halves Groups of flamingos are used to model fractions. The numerator represents the part of the whole, and the denominator represents the total number in the whole group. Example 2: Fractions and Decimals Writing fractions and decimals is modeled with bowling scores. Fractions with a denominator of 10 can be rewritten as a decimal. The first place to right of the decimal is the tens place. Example 3: Fractions, Decimals, and Percents Fractions and decimals show the value of money. The two places after the decimal point are the tens and hundreds places. Rewrite a fraction with a denominator of 100 as a percent by writing the numerator with a percent sign.

9 Teacher s Guide 9 IV. Odd and Even Numbers (6 min.) Odd and Even Numbers: Introduction There are examples of even and odd numbers in nature. An even number can be divided evenly by two. An odd number cannot be divided evenly by two. Example 1: Even Numbers Dancers are used to model even numbers. Even numbers can be divided into pairs. Even numbers have a 0, 2, 4, 6, or 8 in the ones place. Example 2: Odd Numbers Animals are used to model odd numbers. Odd numbers cannot be divided evenly into two groups. Odd numbers have a 1, 3, 5, 7, or 9 in the ones place. Example 3: Even and Odd Numbers The numbers in a bowling alley are used to model the sums of number sentences with even and odd numbers. (odd + odd = even; even + even = even; odd + even = odd) V. Place Value (6 min.) Place Value: Introduction Flowers are used to explore place value. The ones, tens, and hundreds places are explained and modeled. Example 1: Place Value to the Thousands Stadium seats are used to model ones, tens, hundreds, and thousands. There are 10 tens in 100 and 10 one hundreds in 1,000. Example 2: Place Value Greater Than Thousands Numbers in the hundreds, thousands, ten thousands, hundred thousands, and millions are used to record populations. The more place values a number has, the larger the number. Example 3: Decimal Place Value Decimals are used when writing money amounts. One dime is 0.1 of a dollar and one penny is 0.01 of a dollar. VI. Relationships Among Numbers (8 min.) Relationships Among Numbers: Introduction Different types of numbers are used every day. Numbers are infinite. We use whole numbers, fractions, and decimals. Example 1: Whole Numbers, Fractions, and Mixed Numbers Snow is measured in inches, and the depth is usually recorded as a mixed number, which has a whole number and a fraction. Comparing whole numbers, fractions, and mixed numbers is explained and modeled. Example 2: Decimals Prices in a grocery store are used to model and compare decimals. Numbers to the left of the decimal represent whole numbers. Numbers to the right of the decimal represent less than one whole.

10 Teacher s Guide 10 Example 3: Relationships Among Numbers Expressed Differently A football field is used to show how to label, compare, and convert between fractions and decimals. Fractions and decimals are numbers between whole numbers. VII. Number Models (9 min.) Number Models: Introduction Fishermen s crab pots are used to explore number models, or ways to represent numbers and their relationships. Example 1: A Model for Time Scientists look at ice cores to study the past. An ice core is used as a number line, a type of number model, to compare periods of time, temperature, and rain or snowfall. Example 2: Modeling Units and Multiples Sets of bowls are used as number models to represent numbers by units and multiples of the units. Example 3: Graphing Numbers Researchers use graphs to record and analyze data on carbon dioxide levels in the atmosphere. A graph shows the relationships between numbers and allows people to analyze data visually. Quiz I. Prime and Composite Numbers 1. Identify the prime number. A. 6 B. 11 C. 21 D. 24 Answer: B 2. Identify the composite number. A. 7 B. 11 C. 18 D. 23 Answer: C

11 Teacher s Guide A prime number can be divided evenly by itself and. A. 0 B. 1 C. 2 D. 10 Answer: B 4. Which number has more than one rectangular array? A. 3 B. 11 C. 17 D. 20 Answer: D II. Factors and Multiples 1. Identify two factors of the number 15. A. 1 and 7 B. 3 and 5 C. 2 and 15 D. 10 and 5 Answer: B 2. Han has four boxes of hats. Each box holds eight hats. How many hats does Han have in all? A. 4 B. 12 C. 32 D. 48 Answer: C 3. How many groups of 6 are in 18? A. 3 B. 4 C. 5 D. 6 Answer: A

12 Teacher s Guide Joan needs an even number of players for her new game. Which number of players could she choose? A. 11 B. 16 C. 21 D. 29 Answer: B III. Fractions and Decimals 1. Mark has seven marbles. Four of the marbles are red. What fraction of Mark s marbles are red? A. 3 4 B. 7 4 C. 4 7 D. 3 7 Answer: C 2. Sue cuts her cookie into four equal pieces. She eats one piece of the cookie. What fraction of Sue s cookie is left? A. 4 2 B. 3 4 C. 2 6 D. 1 4 Answer: B 3. Collin knocked down 4 out of the 10 bowling pins. What decimal represents the number of pins Collin knocked down? A. 4.0 B C. 0.4 D. 4.4 Answer: C

13 Teacher s Guide What percent of a dollar is two dimes? A. 2% B. 20% 2 C. 100 D. 200% Answer: B IV. Odd and Even Numbers 1. Which object represents an odd number? A. a 4 leaf clover B. an ant with 6 legs C. a tree with 13 branches D. a butterfly with 2 symmetrical wings Answer: C 2. Identify the even number. A. 332 B. 413 C. 447 D. 523 Answer: A 3. What is the remainder when 7 is divided into two groups? A. 0 B. 1 C. 2 D. 3 Answer: B 4. Which number sentence will always equal an odd sum? A. odd + odd B. even + odd C. even + even D. even + even +even Answer: B

14 Teacher s Guide 14 V. Place Value 1. What digit is in the tens place? 563 A. 0 B. 3 C. 5 D. 6 Answer: D 2. In what place value is the digit 9 in the number 9,354? A. ones B. tens C. hundreds D. thousands Answer: D 3. What digit is in the ten-thousands place? 287,951 A. 2 B. 5 C. 8 D. 9 Answer: C 4. Nick has one dollar bill, three dimes, and two pennies. How much money does Nick have? A. \$1.20 B. \$1.23 C. \$1.30 D. \$1.32 Answer: D

15 Teacher s Guide 15 VI. Relationships Among Numbers 1. Four friends measured how deep the snow was at their houses and recorded the data in a chart. Who had the deepest snow? A. Rob B. Tom C. Lara D. Margaret Name Lara Snow inches Answer: C Rob inches Tom inches Margaret inches 2. Joe is shopping for lunch. He sees the following prices at the deli. Turkey \$5.99 Ham \$6.99 Salami.\$3.99 Cheese.\$5.50 Joe wants to buy the least expensive item. What should Joe buy? A. ham B. cheese C. salami D. turkey Answer: C A. < B. = C. > D. + Answer: C

16 Teacher s Guide 16 VII. Number Models 1. There are 12 crab pots in one layer. There are 4 layers of crab pots on the boat. How many crab pots are there in all? A. 8 B. 16 C. 24 D. 48 Answer: D 2. What type of number model can a scientist use to compare periods of time, temperature, rainfall, or snowfall? A. ice B. number line C. number chart D. number layers Answer: B 3. The waiter is stacking bowls. He puts five bowls in each row. There are three rows of bowls in one layer. Then he puts another layer of bowls on top of the first layer. He has two bowls left over, so he puts them on top of the two layers. How many bowls has the waiter stacked in all? A. 15 B. 17 C. 30 D. 32 Answer: D 4. John collected data on how many books he read throughout the year. He recorded his data in a graph. During what season did John read the most books? A. fall Number of Books Read B. spring C. winter 25 D. summer Answer: D Number of Books Spring Summer Fall Winter Season

17 Teacher s Guide 17 Number Grid

18 Teacher s Guide 18 Number Grid Direction Sheet 1. Highlight every prime number on the grid with your highlighter. 2. Circle the odd numbers on the grid with your red crayon or pencil. 3. Choose ten composite numbers from the grid and write them in the chart. Identify four factors of each composite number and write them in the chart. Composite Number Four Factors 4. Choose five prime numbers from the grid and write them in the chart. Identify two multiples of each prime number and write them in the chart. Prime Number Two Multiples

19 Teacher s Guide 19 Picture Cards * T M M * M

20 Teacher s Guide 20 Playing Cards Chart What fraction of the cards are red? Question Fraction Decimal Percent What fraction of the cards are black? What fraction of the cards have even numbers? What fraction of the cards have odd numbers? What fraction of the cards have a 2? What fraction of the cards have a 5? What fraction of the cards have a 4 or 9? What fraction of the cards have a 2, 5, or 9? What fraction of the cards have a 9 or 10? What fraction of the cards have numbers? What fraction of the cards are prime numbers? What fraction of the cards are composite numbers? What fraction of the cards have a picture? What fraction of the cards are red or black?

21 Teacher s Guide 21 Number Fact Sheet Directions Identify the numbers in the following sentences. Classify them as whole numbers, fractions, decimals, or mixed numbers and record them on the Number Chart. Then order the numbers from least to greatest and create a number line, placing them in the correct order. There are 24 students in Miss Smith s class. Add cups of flour to the recipe. 6.4 of the money will go to charity. 3 4 of the pizza was eaten. Sam has 45 books on the bookshelf. Jill built 1 2 of the model car on Saturday. Harry is working 8.4 hours on Monday. The girls decided to buy 5 lollipops for a treat. The teacher let the students stay at recess for 10 extra minutes. There are 540 students in the 4 th grade. Justin made jars of homemade jelly. There are 60 minutes in an hour. There are 15 minutes in 1 4 of an hour. There are 52 weeks in 1 year. Frank drew a line 14.7 inches long. The perimeter around the yard is feet. Sally bought pounds of hamburgers for her party. The little girl weighed 22.5 pounds.

22 Teacher s Guide 22 Number Chart Whole Numbers Fractions Decimals Mixed Numbers

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