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Reteaching Name 31 Math Course 2, Lesson 31 • Reading and Writing Decimal Numbers • To read the decimal number 123.123: Say “and” for the decimal point. 123 . 123: “One hundred twenty-three and one hundred twenty-three thousandths” Say “thousandths” to conclude naming the number. 3 Examples: 0.3 = ___ Both are read “three tenths.” 10 21 0.21 = ____ Both are read “twenty-one hundredths.” 100 Practice: Write each number as a decimal number. 1. Forty-nine and thirteen hundredths 2. Seven hundred eight and sixty-four thousandths Use words to write each decimal number. 3. 38.459 4. 500.05 5. In the number 74.5604, which digit is in the thousandths place? 34 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 2 Reteaching Name 32 Math Course 2, Lesson 32 • Metric System Units of Length Examples of Metric Prefixes Unit 10 millimeters (mm) = 1 centimeter (cm) kilometer (km) 1000 meters hectometer (hm) 100 meters dekameter (dkm) 10 meters 1000 millimeters (mm) = 1 meter (m) 100 centimeters (cm) = 1 meter (m) 1000 meters (m) = 1 kilometer (km) Relationship meter (m) Units of Liquid Measure decimeter (dm) 1 __ meter 1000 milliliters (mL) = 1 liter (L) centimeter (cm) 1 ___ meter millimeter (mm) 1 ____ meter 10 100 1000 Units of Mass Celsius/Fahrenheit Scales 1000 grams (g) = 1 kilogram (kg) 1000 milligrams (mg) An increase of 100°C on the Celsius scale is equivalent to an increase of 180°F on the Fahrenheit scale. = 1 gram Practice: Write the length of each segment in centimeters and in millimeters. " ! $ # ___ ___ ___ 1. AB 2. AD 3. BD 4. The bag of grapes has a mass of 500 grams. a. How many kilograms is that? b. How many milligrams is that? Saxon Math Course 2 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 35 Reteaching Name 33 Math Course 2, Lesson 33 • Comparing Decimals • Rounding Decimals • To round a number: 1. Circle the place value you are rounding to. 2. Underline the digit to its right. 3. Ask “Is the underlined number 5 or more?” Yes Add 1 to the circled number. No Circled number stays the same. 4. Replace the underlined number (and any numbers after it) with zero. 67 ○ 329 ○ Examples: 70 300 • Terminal zeros are zeros at the end of a number to the right of the decimal point. They have no value. Example: 1.3 = 1.30 = 1.300 = 1.3000 • To compare decimal numbers, it helps to insert or delete terminal zeros so that both numbers have the same number of digits after the decimal point. Then compare digits in each place from left to right. Examples: 0.12 ○ 0.012 0.4 ○ 0.400 0.120 ○ > 0.012 0.4 ○ = 0.4 • After rounding decimal numbers, remove all terminal zeros after the decimal point. Example: Round 3.14159 to the nearest hundredth. 3.1 ○ 4 159 3.14000 3.14 Practice: Compare 1–3. 1. 4.73 ○ 4.09 2. 0.27 ○ 1.0 3. 5.618 ○ 5.861 4. Estimate the sum of 7.08, 4.901, and 3.521 by rounding each number to the nearest whole number before adding. 5. Round 1154.07085 to the nearest thousandth. 6. Round 1154.07085 to the nearest thousand. 36 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 2 Reteaching Name 34 Math Course 2, Lesson 34 • Decimal Numbers on the Number Line • The metric system uses decimals, not fractions. Example: Find the length of this segment: cm 1 2 3 a. in millimeters. 23 mm b. in centimeters. 2.3 cm Example: Find the number on the number line indicated by each arrow. A 4.05 4.0 B 4.38 4.1 4.2 4.3 C 4.73 4.4 4.5 4.6 4.7 4.8 4.9 5.0 Sometimes adding a zero to the end of each number on the decimal number line helps you to locate decimal numbers. Example: 4.38 is between 4.30 and 4.40. The number halfway between two numbers is the average (mean) of the two numbers. Practice: 1. How many millimeters equal 40.7 cm? 2. What decimal number is halfway between 20 and 21? 3. What decimal number is halfway between 2.7 and 2.8? 4. What decimal number names each point on this number line? A 70 a. A Saxon Math Course 2 B C 70.1 b. B © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. c. C 37 Reteaching Name 35 Math Course 2, Lesson 35 • Adding, Subtracting, Multiplying, and Dividing Decimal Numbers Decimals Chart +/− × ÷ by whole ÷ by decimal A. Line up the B. Multiply. Then count C. Decimal D. Decimal point decimal points. decimal places. point is up. is over, over, up. E. 1. Place a decimal point to the right of a whole number. F. 2. Fill empty places to the right of the decimal point with zeros. Examples: A, E, and F 3.6 0.36 + 36. 39.96 5.00 – 4.32 _______ 0.68 0.23 × 0.4 ______ 0.092 A and E B C and F 2 places 0.0018 _______ 8)0.0144 1 places 3 places Practice: Simplify 1–4. 1. (6.3)(2.4)(1.2) 2. 1.55 ÷ 5 3. 29.71 – 3.087 4. 2.2 + 0.54 + 12 5. If the product of seven tenths and two tenths is subtracted from the sum of eight tenths and five tenths, what is the difference? 6. What is the area and perimeter of a rectangle that is 1.3 meters wide and 0.9 meter long? a. area: b. perimeter: 38 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 2 Reteaching Name 36 Math Course 2, Lesson 36 • Ratio • Sample Space • Ratio is a way to describe a relationship between two numbers. Example: In a class of 28 students, there are 12 boys. What is the boy-girl ratio? What is the girl-boy ratio? 1. Find the number of girls in the class. 12 boys 12 boys + ? girls 28 total + 16 girls 28 total 2. Write the ratios as reduced fractions. boys 3 12 = __ _____ = ___ 16 4 girls girls 16 4 _____ ___ __ boys = 12 = 3 number of favorable outcomes • Probability = ____________________________ number of possible outcomes • A sample space is the list of all possible outcomes for an event. Example: One coin toss has two possible outcomes. Sample space = {heads, tails} Coin 1 • A tree diagram shows all the possible outcomes of two events that occur at the same time, such as tossing two coins. Coin 2 Outcome H H H T H T H T H T T T H T • The Fundamental Counting Principal says: If there are m ways for A to occur, and n ways for B to occur, then there are m × n ways for A and B to occur together. Example: There are 2 × 6, or 12 outcomes for tossing a coin and a number cube. Practice: 1. In a box of pens, the ratio of red pens to blue pens is 3 to 2. What fraction of the pens are blue? 2. What is the probability of tossing heads with one coin toss? 3. A coin is tossed and a number cube is rolled. One possible outcome is H3 (heads, 3). What is the sample space for the experiment? Saxon Math Course 2 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 39 Reteaching Name 37 Math Course 2, Lesson 37 • Area of a Triangle • Rectangular Area, Part 2 A triangle has three sides, and any side can be the base. A triangle may have three base-height orientations, as shown by rotating these triangles. One Right Triangle Rotated to Three Positions height height height base base base One Obtuse Triangle Rotated to Three Positions height height height base base base bh 1 bh or ___ Area of a triangle = __ 2 2 • To find the area of a complex shape: Example: 10 cm 1. Divide the shape into rectangular parts. 2. Find the area of each part. 3. Add the areas to find the total area. Sometimes subtracting a “ghost” area (the area that is missing) from a larger rectangle that includes the entire figure is easier. 10 7 cm 23 cm There are three ways to find the area of this shape. 10 C 50 (13) A 120 12 12 cm B (65) (5) (5) 12 12 B 91 (13) 10 D 161 7 A 276 7 7 23 23 23 120 91 211 cm2 50 161 211 cm2 276 65 211 cm2 Practice: Find the area of each figure. 1. 3 in. 6 in. 5 in. 2. 13 cm 5 cm 3. 8.6 ft 8.6 ft 7 ft 6 cm 6 cm 7 ft 10 ft 40 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 2 Reteaching Name 38 Math Course 2, Lesson 38 • Interpreting Graphs Bar graph Number of Aluminum Cans Collected by Each Homeroom Number of Aluminum Cans Pictograph Doughnut Sales Jan. Feb. Mar. Represents 10,000 doughnuts 10,000 8,000 6,000 4,000 2,000 0 12 14 16 18 Score Room Number Line graph Circle graph Paul’s Game Scores Where Dina Spends Her Day 100 90 80 70 60 50 40 30 20 10 0 Elsewhere 4 hr At home 12 hr 1 2 3 4 5 At school 8 hr 6 Game Practice: Use information from the graphs above to answer each question. 1. How many more doughnuts were sold in March than in January? 2. Which homeroom collected twice as many cans as homeroom 18? 3. To the nearest whole number, what was Paul’s average score? 4. True or false? Dina spends 60% of her time at home. Saxon Math Course 2 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 41 Reteaching Name 39 Math Course 2, Lesson 39 • Proportions • A proportion is a statement that two ratios are equal. Example: 5 · 16 = 80 16 = ___ 20 4 __ 20 · 4 = 80 5 Solve a proportion by finding the missing term. 1. Find the cross products. 2. Divide the known product by the known factor. Example: 3 = __ 6 __ 5 W 3 ∙ W = 5 ∙ 6 3W = 30 30 W = ___ 3 W = 10 Practice: Solve 1–6. 0.4 16 = ___ 1. ___ r 4 a 7 = ___ 2. ___ 40 20 21 24 = ___ 3. ___ t 32 b 1.5 = ___ 4. ___ 45 2.5 c 0.45 = ___ 5. _____ 2.7 8.1 28 w = ___ 6. ___ 70 50 42 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Course 2 Reteaching Name 40 Math Course 2, Lesson 40 • Sum of the Angle Measures of a Triangle • Angle Pairs 90° • A square has four angles and each one measures 90°. 90° 90° By drawing a diagonal segment from one corner to the opposite corner, the square divides into two congruent triangles. 90° • A triangle has three angles and the sum of those angles is 180°. 45° Adjacent angles share a common side. 90° 45° Examples: ∠1 and ∠4, ∠x and ∠y 50° Supplementary angles are two angles whose sum is 180°. y x Examples: ∠y and ∠z, ∠3 and ∠4 z A Complementary angles are two angles whose sum is 90°. Example: ∠A and ∠B C B 2 3 1 Vertical angles are a pair of non-adjacent angles formed by two intersecting lines and have the same measure. Examples: ∠1 and ∠3, ∠2 and ∠4, ∠x and ∠z 4 Practice: 1. If one angle of a right triangle measures 60˚, then what are the measures of the other two angles? Find each angle measure in this figure. d c o b 60 55o a 2. m∠ a = 3. m∠b = 4. m∠ c = 5. m∠d = Saxon Math Course 2 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 43