Number Line: Comparing and Ordering Integers (page 6)


 Verity Gardner
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1 LESSON Name 1 Number Line: Comparing and Ordering Integers (page 6) A number line shows numbers in order from least to greatest. The number line has zero at the center. Numbers to the right of zero are positive. Numbers to the left of zero are negative. A set is a collection of numbers. Sets are shown inside braces { }. Ellipsis points( ) show that a set is infinite. Counting numbers are the numbers we use to count: {1, 2, 3, 4, 5, } Whole numbers are the counting numbers and zero: {0, 1, 2, 3, 4, 5, } Integers are the whole numbers and their opposites: {, 2, 1, 0, 1, 2, } Teacher Notes: Students who have difficulty with subtraction, multiplication, or division will benefit from working Targeted Practices 1A, 1B, and 1C before Lesson 1. Introduce Hint #8, Positive and Negative Numbers, and Hint #9, Comparing Numbers. Refer students to Number Line on page 9 and Number Families and Definitions on page 10 in the Student Reference Guide. Post reference chart, Number Families. A numberline manipulative is available in the Adaptations Manipulative Kit Opposites are two numbers that are the same distance from zero but in opposite directions. 3 and 3 are opposites. The absolute value of a number is the distance of that number from zero. 5 = 5 5 = 5 The absolute value The absolute value of 5 is 5. of 5 is 5. The absolute value of a number is always positive. We use symbols to compare the values of numbers. 5 < = 5 0 > 2 5 is less than 4. 3 plus 2 equals 5. Zero is greater than 2. To graph a number, draw a point to correspond to that number on the number line. Practice Set (page 9) a. Arrange these integers in order from least to greatest: 4, 3, 2, 1, 0, 1,,, b. Which number 4, 1, 0, 2, 3 is an even number but not a whole number? Cross out the odd numbers. c. Compare: 2 4 The larger the negative digit, the smaller the number is. Saxon Math Course 3 L11 Adaptations Lesson 1
2 Practice Set (continued) (page 9) d. Graph the numbers in this sequence on a number line: 4, 2, 0, 2, 4, Simplify. e. 3 = f. 3 = g. Write two numbers that are ten units from zero., h. Write an example of a whole number that is not a counting number. Written Practice (page 9) least to greatest Use a number line. 5, 3, 2, ,,, 3. whole numbers 4. even numbers 5. Which number is an even number? 5, 3, 2, 1 {, 1, 2,, 4,...} {..., 2, 0,, 4,,...} 6. Which whole number is not a counting number? See page 10 in the Student Reference Guide. 7. Write the graphed numbers {...,,,,,... } 8. Name the type of numbers in problem See page 10 in the Student Reference Guide.. z is a whole number but not a c number. e integers Saxon Math Course 3 L12 Adaptations Lesson 1
3 Written Practice (continued) (page 10) 10. See page 10 in the Student Reference Guide.. All c numbers are w numbers Integers include c numbers, their o, and zero. 12. What is the absolute value of 21? If n 5, then n can be which two numbers? 20. Write two numbers that are five units from zero n,, Saxon Math Course 3 L13 Adaptations Lesson 1
4 Written Practice (continued) (page 10 ) Write 2 numbers that are 3 units from , What number is the opposite of 10? What number is the opposite of 2? Choose all correct answers. 7 A counting numbers C integers B whole numbers D none of these 29. Choose all correct answers 30 A counting numbers C integers B whole numbers D none of these 30. Choose all correct answers A counting numbers C integers 1 3 B whole numbers D none of these,, Saxon Math Course 3 L14 Adaptations Lesson 1
5 LESSON Name 2 Operations of Arithmetic (page 12) The fundamental operations of arithmetic are addition, subtraction, multiplication, and division. More than one symbol can show multiplication and division: Symbols for Multiplication and Division three times five 3 5, 3 5, 3(5), (3)(5) six divided by two 6 2, 2 6, 6 2 A property of addition or multiplication is something that is always true no matter what numbers are used. Properties are written with letters instead of numbers. These letters are called variables. The letters could represent any number. Teacher Notes: Introduce Hint #10, Fact Families, and Hint #11, Properties of Operations. Refer students to Properties of Operations on page 20 in the Student Reference Guide. Review Division on page 5 in the Student Reference Guide. Triangle fact cards are available in the Adaptations Manipulative Kit. Some Properties of Addition and Multiplication Name of Property Representation Example Commutative Property of Addition a + b = b + a = Commutative Property of Multiplication a b = b a 3 4 = 4 3 Associative Property of Addition (a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5) Associative Property of Multiplication (a b) c = a (b c) (3 4) 5 = 3 (4 5) Identity Property of Addition a + 0 = a = 3 Identity Property of Multiplication a 1 = a 3 1 = 3 Zero Property of Multiplication a 0 = = 0 The Commutative and Associative Properties do not work with subtraction or division. Practice Set (page 16) Name each property illustrated in a d. a. 4 1 = 4 I Property of b = C Property of c. (8 + 6) + 4 = 8 + (6 + 4) A Property of Saxon Math Course 3 L25 Adaptations Lesson 2
6 Practice Set (page 16) d Property of M e. What s is the difference when the sum of 5 and 7 is subtracted from the product of 5 and 7? (5 7) (5 7) f. See Example 4 on pg Answer: g. What properties did Lee use to simplify his calculations? What are the properties that cover multiplication? 5 (7 8) Given 5 (8 7) Property of (5 8) 7 Property of h. Explain how to check this subtraction problem. What is the opposite of subtraction? i. Explain how to check this division problem. What is the opposite of division? For j and k, find the unknown. j. 12 m m k. 12n 48 Saxon Math Course 3 L 26 Adaptations Lesson 2
7 Written Practice (page 16) 1. ( ) ( ) product sum 2. ( ) ( ) product sum 3. fact family fact family a. What makes 100? ( ) 6. set of counting numbers b. The product is. c. C Property of A Property of {1,,,...} 7. set of whole numbers 8. set of integers {, 1, 2,...} 9. See page 10 in the Student Reference Guide.. Zero is a w number but not a c number. {...,,,,...} 10.. All numbers are. Saxon Math Course 3 L27 Adaptations Lesson 2
8 Written Practice (continued) (page 17) 11. least to greatest Use a number line. 12. a. 12 b. 11 0, 1, 2, 3, 4,,,, 13. See page 20 in the Student Reference Guide a 0 a Property of Multiplication Property of Addition 15. (5) (0) (10 15) (5 10) 15 Zero of Property of Addition a. The four operations of arithmetic are (), (), (), and (). b. The Commutative and Associative Properties apply to and. Property of Multiplication Saxon Math Course 3 L28 Adaptations Lesson 2
9 Written Practice (continued) (page 17) 19. n 10 n a. 0 1 b. 2 3 c. 2 3 a. b., c opposite of Which integer is neither positive nor negative? 24. Choose all correct answers. 100 A counting numbers B whole numbers C integers D none of these A counting numbers C integers B whole numbers D none of these A counting numbers C integers B whole numbers D none of these Saxon Math Course 3 L29 Adaptations Lesson 2
10 Written Practice (continued) (page xx) rearrange long division _ 5075 Saxon Math Course 3 L210 Adaptations Lesson 2
11 LESSON 3 The plot in a word problem tells us what equation to write to solve the problem. A formula is an equation written with letters, also called variables. Each kind of word problem has a formula. Stories about combining have an addition pattern: some more total s m t Stories about separating have a subtraction pattern: starting amount some went away what is left s a l Stories about comparing also have subtraction patterns: To solve a word problem: greater lesser difference g l d later earlier difference l e d Name 1. Look for keywords that will help you find the plot of the story: combining, separating, or comparing. Use the key words chart on page 35 in the Student Reference Guide. 2. Write an equation for the problem using the formula and numbers from the story. Use a variable for the missing number 3. Find the missing number. Use the missing numbers chart on page 4 in the Student Reference Guide. 4. Check to see that your answer makes sense. Practice Set (page 22) Addition and Subtraction Word Problems (page 19) a. What are the three kinds of word problems described in this lesson? Teacher Notes: Introduce Hint #12, Word Problem Cues, Hint #13, Finding Missing Numbers, and Hint #14, Abbreviations and Symbols. Refer students to Equivalence Table for Units on page 1, Time on page 2, Missing Numbers on page 4, and Word Problem Keywords on page 35 in the Student Reference Guide. Post reference chart, Word Problem Keywords. c s c b. In example 2 on pg. 21, we solved a word problem to find how much money Alberto spent on milk and bread. Using the same information, write a word problem that asks how much money Alberto gave to the clerk. At the store, Alberto bought milk and bread that cost $. The clerk gave Alberto $ in change. How much money did give to? When you see, refer to your Saxon Math Course 3 textbook. Saxon Math Course 3 L311 Adaptations Lesson 3
12 Practice Set (continued) (page 22) c. Write a story problem for this equation. $20.00 a $8.45 Abby went to the store with $. She bought a. The clerk gave her $ in change. money did Abby spend? For problems d f, identify the plot, write an equation, and solve the problem. d. From 1990 to 2000 the population of Garland increased from 180,635 to 215,768. How many more people lived in Garland in 2000 than in 1990? plot: equation: d answer: people e. Binh went to the theater with $20.00 and left the theater with $ How much money did Binh spend at the theater? Explain why your answer is reasonable. plot: equation: a answer: ; The answer is reasonable because half of $20 is $. The money left, $10.50, is a little m than half, so the money spent should be a little l than half. f. In the three 8thgrade classrooms at Washington school, there are 29 students, 28 students, and 31 students. What is the total number of students in the three classrooms? plot: equation: t answer: students g. Circle the equation that shows how to find how much change a customer should receive from $10.00 for a $6.29 purchase. A $10.00 $6.29 c B $10.00 $6.29 c Written Practice (page 23) 1. plot: min sec min sec d 2. plot: $ $ $ t d t Saxon Math Course 3 L312 Adaptations Lesson 3
13 Written Practice (continued) (page 23) 3. plot: 1 ft in. in. in. d 4. plot: $ m $ d m 5. plot: 6. plot: d 1 dozen d d d 7. Sam earned $ mowing yards. He spent $4.05 on. 8. n 3 n 3 How much?, 9. least to greatest Use a number line. 6, 5, 4, 3, a. 5 1 b. 1 2 a.,,,, b. Saxon Math Course 3 L313 Adaptations Lesson 3
14 Written Practice (continued) (page 23) 11. a b (the distance from 0 to 5 on a number line.) Use a number line. a. b. 13. See page See page 13. a. a. b. b. 15. a. What makes 100? ( ) b. C Property of A Property of c. ( ) = 17. Commutative Property of Addition Associative Property of Addition ( ) ( ) Saxon Math Course 3 L314 Adaptations Lesson 3
15 Written Practice (continued) (page 24) rearrange fact family 22. fact family Which whole number is not a counting number? 24. opposite of t 5 Property of u 5 Property of t u Saxon Math Course 3 L315 Adaptations Lesson 3
16 Written Practice (continued) (page 24) 27. 4x 0 Property of 28. $ $90.90 x long division $ $72.18 Saxon Math Course 3 L316 Adaptations Lesson 3
17 LESSON 4 Some word problems have an equal groups plot. Stories about equal groups have a multiplication pattern: number of groups number in group total n g t The keywords for equalgroups problems are in each. If the missing number is a product, multiply. Name Example: There were 24 rows of chairs with 15 chairs in each row. How many chairs were there in all? n g t 24 groups 15 in a group t There were 360 chairs in all. To solve this problem, we multiplied. If the missing number is a factor, divide. Multiplication and Division Word Problems (page 27) Example: There were 360 chairs arranged in rows with 15 chairs in each row. How many rows were there in all? n g t n 15 in a group 360 total There were 24 rows. To solve this problem, we divided. Some equalgroups problems have a remainder. Example: Cory sorted 375 quarters into groups of 40 so that he could put them in rolls. How many rolls can Cory fill with the quarters? n g t n 40 in a group 375 total 9 r Teacher Note: Review Missing Numbers on page 4 and Word Problem Keywords on page 35 in the Student Reference Guide. Cory can make 9 full rolls of quarters and have 15 quarters left over. The problem asks how many rolls Cory can fill. The answer is 9 rolls. Saxon Math Course 3 L417 Adaptations Lesson 4
18 Practice Set (page 28) a. Carver gazed at the ceiling tiles. He saw 30 rows of tiles with 32 tiles in each row. How many ceiling tiles did Carver see? tiles t offset b. Four student tickets to the amusement park cost $ The cost of each ticket can be found by solving which of these equations? (Circle one.) A 4 $ t B t $ C 4 x $95 t D 4t $95.00 c. Amanda has 632 dimes. How many rolls of 50 dimes can she fill? Explain why your answer is reasonable. n long division 632 rolls; 632 by 50 is with dimes left over. Written Practice (page 28) 1. g 2. 1 dozen ) 98 t g t 3. n 5000 Cancel matching zeros divided by 800 is with left over. 4. In which equation is the total cost missing? A c $ C $1.98 c 3 B c 3 $1.98 D 3 $1.98 c Only markers be used. n Saxon Math Course 3 L418 Adaptations Lesson 4
19 Written Practice (continued) (page 29) 5. n Cancel matching zeros divided by 60 is with over. be used. left buses will 7. $ Jake had $5. He spent all but $. 10. Tajuana paid $ for concert tickets. did Jake spend? Each ticket cost $. How many did Tajuana buy? Saxon Math Course 3 L419 Adaptations Lesson 4
20 Written Practice (continued) (page 29) 11. least to greatest Use a number line. 1, 2, 3, 4, a. 7 8 b. 5 6 a.,,,, b. 13. a b a. b. 15. n odd numbers n 1, {,, 1,,,, } Saxon Math Course 3 L420 Adaptations Lesson 4
21 Written Practice (continued) (page 29) 17. Graph the set of odd numbers. 18. See page 4 in the Student Reference Guide a. b. 19. See page 4 in the Student Reference Guide. 20. a. ( ) b. Property of Property of a. b fact family 23. fact family 24. opposite of 10 Saxon Math Course 3 L421 Adaptations Lesson 4
22 Written Practice (continued) (page 30) groups of in. to in whole numbers tickets for {,,,,... } 29. See page 10 in the Student Reference Guide. 30. n 1 2 n 1 2, Saxon Math Course 3 L422 Adaptations Lesson 4
23 LESSON 5 Fractional Parts (page 31) Name We can use fractions to describe part of a group. Fractions have two parts: a numerator (top number) and a denominator (bottom number). numerator denominator To find a fraction of a group: 1. Divide by the denominator. 2. Multiply by the numerator. number of parts described number of equal parts Example: Two fifths of the 30 questions on the test were multiplechoice. How many questions were multiplechoice? 1. Divide by the denominator: Multiply by the numerator: of the questions were multiple choice. One way to compare fractions is to compare each fraction to 1_ 2. Take half of the denominator and compare it to the numerator. If the numerator is greater than half the denominator, the fraction is greater than 1_ 2. If the numerator is less than half the denominator, the fraction is less than 1_ 2. If the numerator is equal to half the denominator, the fraction is equal to 1_ 2. Example: Arrange these fractions from least to greatest. 3 6, 3 5, 3 8 Compare each fraction to 1_ 2. 3_ 6 3_ 5 3_ 8 : half of 6 is 3. The numerator is equal to 3, so 3_ is equal to 1_ 6 2. : half of 5 is 21_. The numerator is more than 21_, so 3_ is more than 1_ 2. : half of 8 is 4. The numerator is less than 4, so 3_ is less than 1_ < 3 6 < Teacher Notes: Introduce Hint #15, Naming Fractions/Identifying Fractional Parts, Hint #16, Fraction of a Group, and Hint #18, Comparing Fractions. Students who are not ready for the abstract nature of fractions will benefit from fraction manipulatives. Fraction tower manipulatives are available in the Adaptations Manipulative Kit. Hint #17, Fraction Manipulatives, describes how to make your own fraction tower manipulatives. Refer students to Fraction Terms on page 12 in the Student Reference Guide. Practice Set (page 33) a. How many minutes is 1_ of an hour? 6 1 hr min Divide by the denominator. Multiply by the numerator. Saxon Math Course 3 L523 Adaptations Lesson 5
24 Practice Set (continued) (page 33) b. Three fifths of the 30 questions on the test were multiplechoice. How many multiplechoice questions were there? Explain why your answer is reasonable. Divide by the denominator. questions; 3_ 5 Multiply by the numerator. is than 1_, and is greater than 1_ of c. Greta drove 288 miles and used 8 gallons of fuel. Greta s car traveled an average of how many miles per gallon of fuel? 288 d. Arrange these fractions from least to greatest.,, Compare each to , 5 6, 5 12 least,, greatest Written Practice (page 33) 1. Divide by the denominator. Multiply by the numerator. 1_ of a mile 4 2. least to greatest Compare to mi yd,, 3. 2_ of Divide by the denominator. Multiply by the numerator. 4. short division Saxon Math Course 3 L524 Adaptations Lesson 5
25 Written Practice (continued) (page 33) 5. t 6. 1 hr min 2 _ 1 hr min t 7. long division $ of Divide by the denominator. Multiply by the numerator. 10. total A c 19 $2.98 B 19 $2.98 = c C $2.98 c = 19 D $ = c 11. offset x 12. $10 Saxon Math Course 3 L525 Adaptations Lesson 5
26 Written Practice (continued) (page 34) 13. Cyndie is microwaving frozen. 14. Erika had pounds of compost. Each takes 20 seconds. How She used all but pounds in her long should she microwave? garden. How many did she use? of compost least to greatest Use a number line. 5, 7, 4, 3, 0,,,,, 17. Compare to n 6 n 6, ,, Saxon Math Course 3 L526 Adaptations Lesson 5
27 Written Practice (continued) (page 34) 21. Use parentheses. 22. = What two numbers are 50 units from zero? 24. fact family , 25. fact family 26. opposite of 5 5 Saxon Math Course 3 L527 Adaptations Lesson 5
28 Written Practice (continued) (page 35) List some integers and counting numbers. The 1, 2, 3,... ; are also numbers. 29. List some integers and fractions (17 50) Given ; An example of a f that is not an. is 6 ( ) ( ) 17 Commutative Property of Multiplication Property of Multiplication 300 Multiplied Multiplied and and Saxon Math Course 3 L528 Adaptations Lesson 5
29 LESSON 6 Converting Measures (page 36) Name We measure weight, length, amount of liquids (capacity), and temperature. There are two systems of measurement: The metric system is used throughout the world. The United States uses both the U.S. Customary System and the metric system. The U.S. Customary System uses fractions. The metric system uses decimal numbers. To convert from one unit to another, use five steps. Example: The 5000meter run is an Olympic event. How many kilometers is 5000 meters? 1. Name the two units in the problem and write them in a column: km m 2. Fill in what you know from the Equivalence Table. Write the amounts next to the units: km m 3. Write what you are looking for. Put a question mark in the unknown spot: km 4. Draw a loop around the two diagonal numbers. The loop should never include the question mark. km m ? 5000? Multiply the numbers in the loop. Divide by the number outside the loop There are 5 km in 5000 m. Teacher Notes: Introduce Hint #19, Converting Measures and Rate and Hint # 20, Measuring Liquids and Capacities of Containers. Refer students to Liquids on page 1 and Proportion (Rate) Problems on page 19 in the Student Reference Guide. Review Equivalence Table for Units on page 1 in the Student Reference Guide. Post reference chart, Liquids. Measure U.S. Customary Metric Length 12 in. = 1 ft 1000 mm = 1 m 3 ft = 1 yd 100 cm = 1 m 5280 ft = 1 mi 1000 m = 1 km Capacity Equivalent Measures 1 in. = 2.54 cm 1 mi 1.6 km 16 oz = 1 pt 2 pt = 1 qt 1000 ml = 1 L 4 qt = 1 gal 1 qt 0.95 Liters Weight/ 16 oz = 1 lb 1000 mg = 1 g Mass 2000 lb = 1 ton 1000 g = 1 kg 1000 kg = 1 tonne 2.2 lb 1 kg 1.1 ton 1 metric tonne Saxon Math Course 3 L629 Adaptations Lesson 6
30 Practice Set (p. 38) a. A room is 15 feet long and 12 feet wide. What are the length and width of the room in yards? ft 1 yd Multiply the loop. Divide by the outside number. length width length: width: ft yd ? b. Nathan is 6 ft 2 in. tall. How many inches tall is Nathan? First, convert 6 feet to inches. Then add the 2 inches. 1 ft in. ft in 1 12? c. Seven kilometers is how many meters? 1 km m Multiply the loop. Divide by the outside number. km m 1? Written Practice (page 38) 1. 1 hr min Multiply the loop. Divide by the outside number. hr 1? min pt oz Find half of that kg g Double that of Divide by the denominator. Multiply by the numerator. Saxon Math Course 3 L630 Adaptations Lesson 6
31 Written Practice (continued) (page 38) Shade two of 18 the parts. What fraction is that? 8. ) Drawing a picture may help week days 1 day hr Multiply the loop day hr 1? of 12 oz. 4 Divide by the denominator. Multiply by the numerator. 14. least to greatest Use a number line. Compare 5 7 to , 5 7, 1, 0, 1 2,,,, Saxon Math Course 3 L631 Adaptations Lesson 6
32 Written Practice (continued) (page 39) 18. Commutative Property of Addition 19. even counting numbers Draw the braces ,,, fact family fact family opposite of See p. 10 in the Student Reference Guide. 26. See p. 10 in the Student Reference Guide. 27. ; 3 is a number and it is an 28. ; Every number is greater than zero, and every negative number is than zero. 29. ; 2 is equal to x 7 2 x 7. x, Saxon Math Course 3 L632 Adaptations Lesson 6
33 LESSON Name 7 Rates and Average Measures of Central Tendency (page 41) A rate is a relationship between two measures. Rates use the word per to mean in one. 65 miles per hour (65 mph) means 65 miles in one hour. 32 feet per second (32 ft/sec) means 32 feet in one second. To solve rate problems, use the loop method from Lesson 6. The average (or mean) describes what number is in the center of a group of numbers. 1. Add the numbers. 2. Divide by the number of items. The average must be between the smallest and largest numbers. Example: Find the average of 5, 1, 3, 5, 4, 8, and Add the numbers Divide by the number of items The average is 4. Four is between the smallest number (1) and the largest number (8). The median is the middle number when the numbers are put in order. Example: Find the median of 5, 1, 3, 5, 4, 8, and Write the numbers in order. 1, 2, 3, 4, 5, 5, 8 2. Count the numbers. There are 7 numbers. Counting from the first number or the last number, 4 is the middle number. The median is 4. In this group of numbers, the average and the median are the same. This is not always true. The mode is the number that occurs most often. Example: Find the mode of 5, 1, 3, 5, 4, 8, and 2. The mode is 5. Five is the only number that occurs more than once in these numbers. The range is the difference between the largest and smallest numbers in a group. Example: Find the range of 5, 1, 3, 5, 4, 8, and 2. The range is 7. The largest number is 8 and the smallest number is Teacher Notes: Introduce Hint #21, Average. Refer students to Average on page 7 and Statistics on page 23 in the Student Reference Guide. Review Hint #19, Converting Measures and Rate. A line plot is a way to show a group of numbers. Each number is shown by an X above a number line. Example: Display this group of numbers on a line plot. {5, 1, 3, 5, 4, 8, 2} x x x x x x x Saxon Math Course 3 L733 Adaptations Lesson 7
34 Practice Set (page 44) a. Alba ran 21 miles in three hours. What was her average speed in miles per hour? mi hr 21 3? 1 per Multiply the loop. Divide by the outside number. b. How far can Freddy drive in 8 hours at an average speed of 50 miles per hour? mi hr 1? 8 c. If a commuter train averages 62 miles per hour between stops that are 18 miles apart, about how many minutes does it take the train to travel the distance between the two stops? 62 miles per hour is about 60 miles per hour and 60 miles per hour is 1 mile per minute. mi min 1 1? 8 d. If the average number of students in three classrooms is 26, and one of the classrooms has 23 students, then which of the following must be true? (Circle one.) The average must be between the smallest and largest numbers. A At least one classroom has fewer than 23 students. B At least one classroom has more than 23 students and less than 26 students. C At least one classroom has exactly 26 students. D At least one classroom has more than 26 students. e. What is the mean of 84, 92, 92, and 96? f. The heights of five basketball players are 184 cm, 190 cm, 196 cm, 198 cm, and 202 cm. What is the average height of the five players? The price per pound of apples sold at different sold at different grocery stores is reorted below. Use this information to answer problems g i. $0.99 $1.99 $1.49 $1.99 $1.49 $0.99 $2.49 $1.49 g. Display the data in a line plot. $1.00 $1.50 $2.00 $2.50 Saxon Math Course 3 L734 Adaptations Lesson 7
35 Practice Set (continued) (page 45) h. Compute the mean, median, mode, and range of the data. i. Rudy computed the average price and predicted that he would usually have to pay $1.62 per pound of apples. Why is Rudy s prediction incorrect? The mode is the most common price for apples. The mode is $. Written Practice (page 45) 1. long division of 35 5 Divide by the denominator. Multiply by the numerator. 3. Convert pounds to ounces. Then add 7 ounces. 4. Multiply the loop. Divide by the outside number. 1 lb. oz. lb. oz. 1 7? km hr? 1 5. mi hr 1? 5 6. See p. 1 in the Student Reference Guide. 7. average plot: equation: t t Saxon Math Course 3 L735 Adaptations Lesson 7
36 Written Practice (page 45) 9. plot: 10. plot: equation: m equation: l m l terms yr 12. Each bag of weighed pounds. plot: If there were bags, how much did all equation: e the bags weigh? e 13. Ginger gets an allowance of $ each week. At the end of one week, she had $ left. How much money did Ginger spend? 14. least to greatest Use a number line. Compare each fraction to , 1, 2 3, 1, a b c,,,, addend d e f factor Use parentheses. Saxon Math Course 3 L736 Adaptations Lesson 7
37 Written Practice (continued) (page 46) Rearrange fact family fact family ; is a number that is not a number. 25. opposite of 6 26 a. x Property of Multiplication b. y Property of Addition 27. Rearrange ,101 $ Saxon Math Course 3 L737 Adaptations Lesson 7
38 Written Practice (continued) (page 46) 29. long division 12 $ (12 75) Given 4 ( ) Property ( ) Property Multiplied and Multiplied and Saxon Math Course 3 L738 Adaptations Lesson 7
39 LESSON 8 Perimeter and Area (page 47) Name A rectangle is a foursided shape with two dimensions, length ( l ) and width (w). 5 ft. The perimeter of a rectangle is the distance around the rectangle. Perimeter is measured in units of length such as ft, in., cm, and m. To find a perimeter, add all sides. The perimeter of the rectangle above is 16 ft. l w l w perimeter ft. The area of a rectangle is the amount of surface. Area is measured in square units such as ft 2, in. 2, cm 2, and m 2. 3 ft. Area length width The area of the rectangle above is 15 ft 2. l w Area 5 ft 3 ft 15 ft 2 To find the perimeter and area of some shapes, we divide the shape into more than one area and find each unknown side length. 1. Subtract to find each unknown side length. h 12 cm 5 cm 7 cm v 10 cm 6 cm 4 cm 2. Perimeter: Add all sides. Perimeter cm 3. Area: Find the area of each small rectangle. Then add the two areas. Area of A 4 cm 5 cm 20 cm 2 Area of B 6 cm 12 cm 72 cm 2 Total area 20 cm 2 72 cm 2 92 cm 2 Teacher Notes: Introduce Hint #22, Perimeter and Area Vocabulary and Hint #23, Perimeter and Area of Complex Shapes. Refer students to Perimeter, Area, Volume on page 16 and Length and Width on page 17 in the Student Reference Guide. Color tiles for demonstrating perimeter and area are available in the Adaptations Manipulative Kit. 6 cm h v B 12 cm 5 cm A 10 cm Saxon Math Course 3 L839 Adaptations Lesson 8
40 Practice Set (page 51) a. 12 ft How many feet of baseboard does Jared need? 8 ft Perimeter How many tiles will he need? Area floor tiles Find the perimeter and area of each rectangle. b. P c. P 3 cm 9 cm 4 cm A A 12 cm d. The length and width of the rectangle in c are three times the length and width of the rectangle in b. The perimeter of c is how many times the perimeter of b? The area of c is how many times the area of b? e. Pete made a rectangle using 12 tiles side by side. This is a 12 1 rectangle. Name two other rectangles Pete can make with 12 tiles. and Use square tiles for help. f. Find the perimeter and area of a room with these dimensions. x y P A 13 ft 11 ft ft ft 3 ft 14 ft Saxon Math Course 3 L840 Adaptations Lesson 8
41 Written Practice (page 51) 1. Perimeter: Add all sides. Area length width 2. Label the sides in the figure. Find its perimeter and area 15 yd 20 yd 3. The perimeter of 2 is how many times the perimeter of 1? 4. Use the color tiles for help. The area of 2 is how many times the area of 1? 5. t 6. Multiply the loop. lb oz 1? 7. 2 of $45 3 Divide by the denominator. Multiply by the numerator. 8. a. Put in order.,,,, mean: mode:,,,,, median: range: b. Which describes the difference from greatest to least? ) Saxon Math Course 3 L841 Adaptations Lesson 8
42 Written Practice (continued) (page 52) 9. t 10. ( ) ( ) sum difference 11. See p. 1 in the Student Reference Guide. 12. Multiply the loop. Divide by the outside number. mi hr? Murat drives miles every day to. 14. A carpenter has a board that measures inches. After cutting, the board How many will Murat measures inches. How many drive in 4 days? did the carpenter cut off? 15. least to greatest. Use a number line. Compare each fraction to , 1, 5 7, 2, 2 6, ,,,, Saxon Math Course 3 L842 Adaptations Lesson 8
43 Written Practice (continued) (page 52) Commutative Property fact family opposite of n 5 n 5, Saxon Math Course 3 L843 Adaptations Lesson 8
44 Written Practice (continued) (page 53) 25. ; is a number but is not an 26. ; The contain the numbers ; The sum of two positive integers is a p 28. x 15 x 15 i., Three out of four did not order the special. 37 What fraction did? Saxon Math Course 3 L844 Adaptations Lesson 8
45 LESSON 9 Prime Numbers (page 54) Name A factor is a counting number that divides evenly into another number. Think of factors in pairs that multiply to make the number: 6 has two factor pairs: and The factor pairs can be shown with rectangles: 6 The factors of 6 are 1, 2, 3, and 6. A prime number is a counting number greater than 1 that has exactly two factors (one factor pair). 7 has one factor pair: Teacher Notes: Introduce Hint #24, Factors of Whole Numbers, Hint #25, Tests for Divisibility, Hint #26, Prime Factorization Using the Factor Tree, and Hint #27, Prime Factorization Using Division by Primes. Refer students to Factors and Tests for Divisibility on page 5 and Prime Numbers on page 9 in the Student Reference Guide. Post reference chart, Primes and Composites. The factors of 7 are 1 and 7. Therefore, 7 is a prime number. The factors of a prime number are always 1 and the number itself. Some prime numbers: 2, 3, 5, 7, 11, 13, A counting number that has more than two factors is a composite number. Any composite number can be written as the product of factors that are prime numbers. This is called prime factorization. We can use a factor tree to write the prime factorization of a number: 1. List two factors of the given number under branches of the tree. (If you have trouble, start with 2, 3, or 5.) 2. Check if either factor is prime. If the factor is prime, circle it. If it is not, continue to draw branches and factor until each number is prime. 3. Write the prime factors in order. You may have to write some numbers more than once Saxon Math Course 3 L945 Adaptations Lesson 9
46 Another way to do prime factorization is division by primes: 1. Write the number in a division box. 2. Divide by the smallest prime number that is a factor. (Try 2, 3, or 5.) 3. Divide that answer by the smallest prime number that is a factor. 4. Repeat until the quotient is The divisors are the prime factors of the number. Division by Primes 1 7 ) 7 5 ) = Tests for divisibility will help you find factors of numbers or tell if a number is prime: Tests for Divisibility A number is able to be divided by... 2 if the last digit is even. 4 if the last two digits can be divided by 4. 8 if the last three digits can be divided by 8. 5 if the last digit is 0 or if the last digit is 0. 3 if the sum of the digits can be divided by 3. 6 if the number can be divided by 2 and by 3. 9 if the sum of the digits can be divided by 9. Practice Set (page 57) a. Is 9 a prime or composite number? Use 9 color tiles to make a square. b. Write the first 10 prime numbers. 2,,,, 11, 13,,, 23, See p. 9 in the Student Reference Guide. c. Complete this factor tree for d. Find the prime factors of 60 using division by primes. 60 Write the prime factorization of each number in e g. e Saxon Math Course 3 L946 Adaptations Lesson 9
47 Practice Set (continued) (page 57) f g h. A multidigit number might be prime if its last digit is. A 4 B 5 C 6 D 7 See the tests for divisibility. i. Show that 12 is composite by drawing three different rectangles using 12 squares. Use color tiles for help. Written Practice (page 57) of Divide by the denominator. Multiply by the numerator. 2. Multiply the loop. m cm 1 2? 3. Multiply the loop hours 3 hours 4 hours? 4. a. plot: equation: b. average 150 m a. 2 ) b. Saxon Math Course 3 L947 Adaptations Lesson 9
48 Written Practice (continued) (page 58) 5. a. median: Put the numbers in order. b. mean: ,,,,,, 7 ) 6. Multiply the loop. ft yd ? 7. first eight prime numbers,,,,,,, c. The is closer to most of the numbers cm 68 cm 9. $ Cancel matching zeros ) 11. short division 3 $ Perimeter: Add all sides. Area length width 13. Perimeter: Add all sides. 44 in. 32 in. 12 m 15 m 19 m Saxon Math Course 3 L948 Adaptations Lesson 9
49 Written Practice (continued) (page 58) 14. Perimeter: Add all sides. area length width 4 in ft y 15 ft 3 in. x 6 in. a. Perimeter: Add all sides. b. Area = length x width 9 in. a. b a b c 20 Addition Property of Property of Multiplication Property of Multiplication a b c d 0 Property of Multiplication 20. a. 9 b a. d b. 22. long division Choose all correct answers. 27 A whole number B counting number C integer 24. Choose all correct answers. 0 A whole number B counting number C integer Saxon Math Course 3 L949 Adaptations Lesson 9
50 Written Practice (continued) (page 59) 25. Choose all correct answers. 2 A whole number B counting number 26. Choose all correct answers. Use tests for divisibility A 2 B 3 C 4 D 5 C integer 27. See p. 9 in the Student Reference Guide ,,,, (23 50) Given 40 ( ) Property ( ) Property Multiplied and Multiplied and Saxon Math Course 3 L950 Adaptations Lesson 9
51 LESSON 10 Rational Numbers Equivalent Fractions (page 60) Rational numbers are numbers that can be expressed as a ratio (division) of two integers. All integers, whole numbers, and counting numbers are rational numbers. All fractions are rational numbers. Equivalent fractions are different names for the same number. 4_ 8 3_ 6 4 8, 3 6, 2, and 1 are equivalent fractions , 3 and 2 all reduce to _ 4 1_ 2 Name Teacher Notes: Introduce Hint #28, Finding the Greatest Common Factor, and Hint #29, Improper Fractions. Refer students to Mixed Numbers and Improper Fractions and Fraction Families Equivalent Fractions on page 12 in the Student Reference Guide. Review Hint #17, Fraction Manipulatives, and Hint #19, Comparing Fractions. Review Number Families on page 10, Factors on page 5, and Fraction Terms on page 12 in the Student Reference Guide. To reduce a fraction: 1. Write the prime factorization of the numerator and denominator. 2. Cancel all the pairs of factors To form equivalent fractions, multiply by a fraction equal to 1 (same numerator and denominator) = = = 4 8 As a shortcut, use the loop method. Example: Write a fraction equivalent to 1_ that has a denominator of Multiply the loop. Divide by the outside number. 1 2 = is the equivalent to Fractions that have the same denominator have common denominators. 5 and and 7 10 Common Denominators Not Common Denominators Saxon Math Course 3 L1051 Adaptations Lesson 10
52 To compare fractions that do not have common denominators, crossmultiply: Example: Compare: , so An improper (top heavy) fraction is a fraction that is greater than 1 or equal to 1. A mixed number is a whole number and a fraction. To write an improper fraction as a mixed number or integer: 1. Divide the denominator into the numerator. 2. Write the quotient as the whole number. 3. Write the remainder as the numerator of the fraction. 4. Keep the same denominator. Example: Express 3 as a mixed number. 10 3R1 3 ) The quotient is 3, so that is the whole number. The remainder is 1, so that is the numerator. The denominator stays the same: 3. Practice Set (page 65) Each number in a c is a member of one or more of the following sets of numbers. Write all the letters that apply. A Whole numbers B Integers C Rational numbers a. 5 b. 2 c. 2 5 Use prime factorization to reduce each fraction in d f ) 3 ) ) 9 d e f ) 10 2 ) 20 ) ) ) ) 36 ) ) ) 75 2 ) 18 2 ) Saxon Math Course 3 L1052 Adaptations Lesson 10
53 Practice Set (continued) (page 65) Complete each equivalent fraction in g i. Multiply the loop. Divide by the outside number. g h i j. Compare: Crossmultiply k. Draw and label points on the number line for these numbers: 1, 3 4, 0, 3 2, l. Convert the improper (topheavy) fraction to a mixed number. Shade the circles to show that the numbers are equal ) R 9 m. Equivalent fractions can be formed by multiplying by a fraction form of 1 (same numerator and denominator). What property of multiplication states that any number multiplied by 1 is equal to the original number? Property of Multiplication n. Write a subtraction equation using whole numbers and a difference that is an integer. Saxon Math Course 3 L1053 Adaptations Lesson 10
54 Written Practice (page 66) Use Tests for Divisibility ) ) Multiply the loop. Divide by the outside number Perimeter: Add all sides. Area length width 13 ft. 10 ft. Saxon Math Course 3 L1054 Adaptations Lesson 10
55 Written Practice (continued) (page 66) 11. Put the numbers in order: 12. Which measure is the halfway point in order?,,,,,, median: mode: range: mean: ) hr min 2 hr min 1 hr min of 80 4 Divide by the denominator. Multiply by the numerator. 15. c Dwayne had pounds of apples. After he made a pie, he had pounds left. How many of apples did Dwayne use? 18. Each cost $2. If the total price was $, how many were bought? Saxon Math Course 3 L1055 Adaptations Lesson 10
56 Written Practice (continued) (page 66) 19. least to greatest , 3, 4, 1, Convert 4 to a 3 mixed number.,,,, n 7 n 7, 25. sometimes, always, or never? 26. ; A mixed number has a f part, so ; Every number is an integer. it cannot be a number. 27. ; The set of numbers contains the. 28. long division $ $ ( ) ( ) 63 product sum $17 6 $17 6 Saxon Math Course 3 L1056 Adaptations Lesson 10
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