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Chapter 1 Decimals and Integers MATH 7 Using Estimation Strategies LESSON 1-1 Definition Numbers that are easy to compute mentally Facts/Characteristics Also called “friendly numbers” as mental calculations are easy Vocabulary Word 3.67 x 42.5 4 x 40 = 160 3.67 x 42.5 ≈ 160 Examples Compatible Numbers 10.93 + 3.25 11 + 3 10.93 + 3.25 ≈ 14 Non-Examples Definition Add the “front-end digits,” estimate the sum of the remaining Facts/Characteristics Gives a higher estimate as involves the cents, less likely to be short of money Vocabulary Word $3.98 6.49 then look 9.08 at the cents +3.47 adjust the estimate 21 Front-End Estimation $3.98 + $6.4 4 + 7 = 11 $3.98 + $6.49 ≈ $11.00 21 + about $2 = $23 Examples Non-Examples Adding and Subtracting Decimals LESSON 1-2 COMMUTATIVE PROPERTY OF ADDITION Changing the order of the addends does not change the sum. 5.78 + 9.3 = 9.3 + 5.78 a+b=b+a ASSOCIATIVE PROPERTY OF ADDITION Changing the grouping of the addends does not change the sum (3.2 + 8) + 4 = 3.2 +( 8 + 4) (a + b) + c = a + (b + c) IDENTITY PROPERTY OF ADDITION The sum of 0 and any number is that number 4.5 + 0 = 0 + 4.5 = 4.5 a + 0 = 0 + a =a FINDING DECIMAL SUMS Line up the decimal points!!! Add: 3.842 2.450 write zeros so that all +1.300 decimals have the same 7.592 number of digits to the right of the decimal point FINDING DECIMAL DIFFERENCES Line up the decimal points!! 7 10 Subtract: 68.0 - 51.8 16.2 rename as 7 and 10 tenths Estimate answers first then check for reasonableness! Multiplying and Dividing Decimals LESSON 1-3 COMMUTATIVE PROPERTY OF MULTIPLICATION Changing the order of the factors does not change the product. 5.78 x 9.3 = 9.3 x 5.78 a∙b=b∙a ASSOCIATIVE PROPERTY OF MULTIPLICATION Changing the grouping of the factors does not change the product (3.2 x 8) x 4 = 3.2 x ( 8 x 4) (a ∙ b) ∙ c = a ∙ (b ∙ c) IDENTITY PROPERTY OF MULTIPLICATION The product of 1 and any number is that number 4.5 x 1 = 1 x 4.5 = 4.5 a∙1=1∙a ZERO PROPERTY OF MULTIPLICATION The product of 0 and any number is 0 6x0=0x6=0 a∙0=0∙a=0 DECIMAL X DECIMAL 0.25 x .015 125 + 025 .00375 two decimal places three decimal places move the decimal point five places to ………………………………………………..the left Add the number of decimal places in the factors to find the number of decimal places in the product. WHOLE NUMBER X DECIMAL 0.25 x 5 1.25 two decimal places 0 decimal places Add the number of decimal places in the factors to find the number of decimal places in the product. DECIMAL ÷ DECIMAL Multiply both the dividend and the divisor by the same number, (multiple of 10), so the divisor becomes a whole number. ANNEXING ZEROS TO DIVIDE Measuring in Metrics LESSON 1-4 USING METRIC UNITS Type Unit Reference Example Length millimeter (mm) centimeter (cm) meter (m) kilometer (km) about the thickness of a dime about the width of your little finger about the distance from the doorknob to the floor about the length of 11 football fields Capacity milliliter (mL) liter (L) a small spoon holds about 5 mL a little more than 1 quart Mass milligram (mg) gram (g) kilogram (kg) about the mass of a mosquito about the mass of a paper clip about the mass of a bunch of bananas MASS VS. WEIGHT Mass is a measurement of how much matter is in an object; weight is a measurement of how hard gravity is pulling on that object. Your mass is the same wherever you are--on Earth, on the moon, floating in space--because the amount of stuff you're made of doesn't change. But your weight depends on how much gravity is acting on you at the moment; you'd weigh less on the moon than on Earth, and in interstellar space you'd weigh almost nothing at all. CONVERTING METRIC UNITS Kids Have Dropped Over Dead Converting Metrics HOW DOES THIS WORK? K H D O 5 liters D C M 5 L = ____________ mL Start at Over and move to the right as mL are smaller than L. (Think of O as original unit) We moved 3 places. 5 L = 5000 mL HOW DOES THIS WORK? K H D O D 5 liters C M 5 L = _____?____ kL Start at Over and move to the left as kL are larger than L. (Think of O as original unit) We moved 3 places. 5 L = .005 kL CAUTION: Remember all whole numbers have a decimal point Comparing and Ordering Integers LESSON 1-6 Definition Facts/Characteristics Two numbers that are the same distance from 0 on a number line, but in opposite directions Another name for opposites is additive inverse as a number + its additive inverse = 0 Vocabulary Word Opposites 7 + (-7) = 0 -0.3 + 0.3 = 0 Ordinal numbers do not have opposites. There is no negative 3rd. Examples Non-Examples Definition The set of positive whole numbers, their opposites, and zero. Facts/Characteristics Fractions are not integers Decimals are not integers Vocabulary Word Integers Examples Non-Examples Definition Facts/Characteristics The distance the number is from 0 Always use bars to show you mean absolute value |a| Vocabulary Word Absolute Value |3| = 3 |-3| = 3 Both are 3 places from 0 Examples (3) [3} these notations do not mean absolute value Non-Examples ORDERING INTEGERS Positive numbers are always greater than negative numbers Negative numbers are always less than positive numbers When using a number line, numbers increase as you move to the right When using a number line, numbers decrease as you move to the left INTEGERS ON A NUMBER LINE Adding and Subtracting Integers LESSON 1-7 ADDING INTEGERS WITH SAME SIGN The sum of two positive numbers is positive. 3+5=8 The sum of two negative numbers is negative. -3 + (-5) = -8 ADDING INTEGERS WITH DIFFERENT SIGNS Find the absolute value of each. Subtract the lesser absolute value from the greater. The sum has the sign of the integer with the greater absolute value -3 + 5 = 2 3 + (-5) = -2 (that is, subtract the lower number from the higher and keep the higher sign) SUBTRACTING INTEGERS To subtract integers, add its opposite Examples: 3- 5 = 3 + (-5) = -2 -7 – (-3) = -7 + 3 = -4 4 – 5 = 4 + (-5) = -1 -6 – 1 = -6 + (-1) = -7 Multiplying and Dividing Integers LESSON 1-8 MULTIPLYING INTEGERS The product of two integers with the same sign is positive 3(4) = 12 -4(-6) = 24 The product of two integers with different signs is negative 3(-4) = -12 -4(6) = -24 DIVIDING INTEGERS The quotient of two integers with the same sign is positive 14 ÷ 7 = 2 -14 ÷ (-7) = 2 The quotient of two integers with different signs is negative -14 ÷ 7 = -2 14 ÷ (-7) = -2 HOW ABOUT THIS - -35 -7 = 5 25 -5 = -5 - Since a fraction is a division problem, the rules apply Order of Operations and the Distributive Property LESSON 1-9 PLEASE EXCUSE MY DEAR AUNT SALLY Do all operations within parentheses first Work the exponents Multiply and divide in order from left to right Add and subtract in order from left to right USING THE DISTRIBUTIVE PROPERTY The distributive property lets you multiply a sum by multiplying each addend separately and then add the products. 4 ∙ 25 = 100 or 4(20 + 5) 4(20) + 4(5) 80 + 20 = 100 CONTINUED……. other examples 6(7 – 5) =12 or 6(7 – 5) 6(7) – 6 (5) 42 – 30 = 12 AND WITH DECIMALS 5(6.8) = 34 or 5(6.8) 5(7 - .2) 5(7) – 5(.2) 35 – 1 = 34 Why do this? The Distributive Property makes numbers easier to work with and mental calculations become easier. Mean, Median, Mode and Range LESSON 1-10 Definition The mean of a set of data is the average: sum of the set divided by the number of items Facts/Characteristics You can use the mean to find your grade average Vocabulary Word Grades: 90, 95, 88 Mean: 273 ÷ 3 = 91 Examples Mean Not some one who is selfish or unkind Non-Examples Definition The median is the middle value when the data are in numerical order Facts/Characteristics The median always separates the data into two groups of equal size Vocabulary Word Median 23, 36, 45, 46, 89 The median is 45 Examples 46, 23, 89, 45, 36 The median is not 89. Arrange data in numerical order FIRST Non-Examples Definition Facts/Characteristics The mode of the data set is the item that occurs with the greatest frequency There can be more than one mode in a data set Vocabulary Word Mode 1,2,2,3,4,4,5,6,7,7 the mode is 2, 4, 7 Examples There is no mode when all data items occur the same number of times Non-Examples Definition Facts/Characteristics The range of a data set is the difference between the greatest and least values Order the data from least to greatest before finding the range Vocabulary Word Range 14, -12, 7, 0, -5, -8, 17, order least to greatest -12, -8, -5, 0, 7, 14, 17 Range is 17- (-12) = 17 + 12 = 29 Examples Non-Examples