Properties Of Real Numbers ACTIVITY Properties By The Pound 1
Properties of Real NumbersProperties by the PoundACTIVITY1.3SUGGESTED LEARNING STRATEGIES: Create Representations,Activating Prior Knowledge, Think/Pair/Share, Interactive Word WallMy NotesThe local girls’ track team is strength training by lifting weights.One of the runners puts weights on each end of the weightliftingbar while she is getting ready to bench press.1. On one end she places a ten-pound plate and then a five-poundplate. On the other end, she places first a ten-pound plate and thena five-pound plate. Does the bar have the same amount of weightat each end? Explain using a diagram in the My Notes section.In symbols, this situation described above is written asa b b a, where both a and b are real numbers. This is astatement of the Commutative Property.2. What does the commutative property state?3. Demonstrate the commutative property of addition by writingan example using:a. negative integers 2010 College Board. All rights reserved.b. rational numbers represented as fractionsAnother runner needs to increase the amount of weight on the barfor another lift. On each side she places three different plates: onefive pound plate, one ten pound plate, and one twenty pound plateas shown in the diagram below.5102020105Order of Operations:1st: Evaluate expressions withingrouping symbols.2nd: Evaluate powers and radicals.To ensure that the weight on both sides is the same, she wants tofind the total weight on both sides. For the left side of the bar, sheused the expression (5 10) 20 and on the right side of the bar,she used the expression 5 (10 20).3rd: Do all multiplication anddivision from left to right.4th: Do all addition andsubtraction from left to right.4. Use the order of operations to evaluate each expression andcompare the results.Unit 1 Patterns and Equations19
ACTIVITY 1.3Properties of Real NumberscontinuedProperties by the PoundMy NotesSUGGESTED LEARNING STRATEGIES: Activating PriorKnowledge, Create Representations, Interactive WordWall, Group PresentationThe expression (5 10) 20 5 (10 20) can be written insymbols as (a b) c a (b c), where a, b, and c are realnumbers. This is an example of the associative property.5. What does the associative property state?6. Demonstrate this property of addition by writing an example using:a. negative integersb. rational numbers represented as decimalsMATH TERMSA counterexample is anexample or case that proves aconjecture or theory wrong.7. The properties you have been using were described asproperties of addition. Determine if these properties also applyfor the operations that follow. If the property applies, then writean example that demonstrates it. If not, write a counterexample.Commutative PropertyAssociative PropertySubtractionMultiplicationThe identity properties apply to both addition and multiplication.8. Use what you know about identity properties to complete thesestatements. Then state the identity properties in words.a. a a, and a a.Identity Property of Addition:b. x · x, and· x x.Identity Property of Multiplication:c. Explain whether the identity properties apply to divisionand subtraction. Include an example or counterexample witheach explanation.20SpringBoard Mathematics with Meaning Algebra 1 2010 College Board. All rights reserved.Division
Properties of Real NumbersACTIVITY 1.3continuedProperties by the PoundSUGGESTED LEARNING STRATEGIES: Think/Pair/Share,Group Presentation, Activating Prior Knowledge,Quickwrite, Think/Pair/ShareMy NotesEach real number has an associated number called its additiveinverse. All but one real number has an associated number calledits multiplicative inverse or reciprocal.The Additive Inverse Property states that a number added to itsadditive inverse gives a sum of zero. In symbols, the additive inverseof a is –a: a (–a) 0, and –a a 09. Write an equality showing a number and its additive inverse.The Multiplicative Inverse Property states that a number multipliedby its multiplicative inverse yields a product of one. In symbols, the111multiplicative inverse of a isa : a · ( a ) 1 and a · a 110. Write an equality showing a number and its multiplicativeinverse.11. Which real number does not have a multiplicative inverse?Explain.TRY THESE AComplete the following number sentences by finding each: 2010 College Board. All rights reserved.a. Additive inverse5 0-b b. Multiplicative inverse2·6· 1t 0 14 50.23 · 0 112. The distributive property can be written in symbols asa(b c) a(b) a(c). The factor a is said to be “distributed” toboth addends in the parentheses.ACADEMIC VOCABULARYdistributive propertya. Describe the order of operations used on the left side of theequation a(b c) a(b) a(c) in terms of a, b, and c.b. Describe the order of operations used on the right side ofthe equation a(b c) a(b) a(c) in terms of a, b, and c.The distributive property can alsobe written as (b c)a ba ca.c. What is true about the two sides of the equation?Unit 1 Patterns and Equations21
ACTIVITY 1.3Properties of Real NumberscontinuedProperties by the PoundSUGGESTED LEARNING STRATEGIES: Think/Pair/Share,Activating Prior Knowledge, Group PresentationMy NotesTRY THESE BUse the distributive property to rewrite and evaluate each of thefollowing expressions.7 51a. 5 (-7 12)b.c. -3(-24 16)2 3 3d. (2 9) 5e. 4 (x 2y)()13. Does the distributive property apply to operations other thanmultiplication and addition? Write an example or acounterexample for each of the following.a. Does multiplication “distribute” over subtraction?b. Does division “distribute” over addition?A set has closure under an operation if, when an operation isperformed on members of that set, the result also is included in the set.15. Not all number sets are closed under an operation. Write anequality that demonstrates that natural numbers are not closedunder subtraction.16. Determine for which operations the sets of numbers listedare closed. Use whole numbers in the first table and integers inthe second vision22Example orCounterexample1 4 5Closed orNot ClosedClosedSpringBoard Mathematics with Meaning Algebra nExample orCounterexampleClosed orNot Closed 2010 College Board. All rights reserved.14. The set of whole numbers is closed under addition becausethe sum of any two whole numbers is also a whole number.Write an equality that demonstrates the closure of the wholenumbers under addition.
Properties of Real NumbersACTIVITY 1.3continuedProperties by the PoundSUGGESTED LEARNING STRATEGIES: Think/Pair/Share,Group PresentationMy Notes17. Think of some examples of rational and irrational numbers.a. Is the sum of a rational number and an irrational numberrational or irrational? Explain.b. Is the product of a rational number and irrational numberrational or irrational? Explain.18. There are properties of equality that can make it easier to solveequations and to work with proofs in geometry. The weights thatthe track team is lifting can be used to illustrate these.a. The reflexive property states that a a.2020 Tell what the reflexive property means in your own wordsusing at least one example.b. The symmetric property states if a b, then b a. 2010 College Board. All rights reserved.2010 102010 10CONNECT TO FITNESSTell what the symmetric property means in your own wordsusing at least one example.c. The transitive property states if a b and b c, then a c.20101010105 5 5 5205 5 5 5The illustrations of different kindsof plates on a barbell in this activityare meant to demonstrate theproperties of equality. If you wereactually placing plates on a barbellat a gym, however, you would useexactly the same plates, placed inthe same order from each end.What does the transitive property mean? Explain in yourown words using at least one example.Unit 1 Patterns and Equations23
ACTIVITY 1.3Properties of Real NumberscontinuedProperties by the PoundSUGGESTED LEARNING STRATEGIES: Think/Pair/Share,Group PresentationMy Notes19. If 3x 2 y and y 8 what property allows you to concludethat 3x 2 8?20. If 10 y, which property allows you to conclude that y 10?21. Do each of these properties apply to inequalities? Use anumerical example or counterexample to explain.a. reflexiveb. symmetricc. transitive:CHECK YOUR UNDERSTANDING1. Identify the property illustrated in each ofthe following equations.a. 5(2 3) (2 3)5b. 25 · (0.04) 1c. 4 (9 3) (9 3) 4d. (x 3)4 4x 122. Write an equality that illustrates theassociative property of multiplication.3. Write an equality that illustrates theadditive inverse of 3.4. The expressions below are the result ofdistributing multiplication over addition.Rewrite each to show the expression priorto distributing.a. 3(x) 3(8)b. 10y xyc. 6w 9z24SpringBoard Mathematics with Meaning Algebra 15. Using the table below, determine forwhich operations the set of naturalnumbers is closed or not closed.OperationExample orClosed orCounterexample Not ClosedAdditionSubtractionMultiplicationDivision6. Explain why the sum of 2 and 3 is irrational.Identify the property illustrated in each of thefollowing equations.7. y y8. 10 · 5 5 · 109. If y 11 and y 3x 2, then 11 3x 2.10. If 7 x 12, then 12 7 x11. MATHEMATICAL Which of the propertiesR E F L E C T I O N of operations of realnumbers is most useful to you? Explainusing more than one example. 2010 College Board. All rights reserved.Write your answers on notebook paper. Showyour work.
Each real number has an associated number called its additive inverse. All but one real number has an associated number called its multiplicative inverse or reciprocal. Th e Additive Inverse Property states that a number added to its additive inverse gives a sum of zero. In symbols, the addit
system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers. Complex Numbers Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Natural Numbers The imaginary unit i is defi
Introduction to Algebra 8.1 The Real Number Line and Absolute Value 8.2 Addition with Real Numbers 8.3 Subtraction with Real Numbers 8.4 Multiplication and Division with Real Numbers 8.5 Order of Operations with Real Numbers 8.6 Properties of Real Numbers 8.7 Simplifying and Evaluating Algebraic Expressions 8.8 Translating English Phrases
(ex. 53.66563146 .). All square roots of non-square numbers are also irrational (ex. 8 and 33). Real Numbers are all rational and irrational numbers. Check all the sets of numbers to which each number belongs. Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers 1. 0 2. 0.15 3.
Substituting Variable Values in Program Understand how real numbers and integers are treated by the CNC control Real numbers are any number rational or irrational Real numbers include integers 1.25, -.3765, 5, -10 are all real numbers Integers are whole numbers 2500, 3, 20 Numbers with decimals are not integers i.e. 1.25 Some NC words only allow integers
To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. a bi, bi b a bi. a i2 1. i 1 1. x x2 1 0 Appendix E Complex Numbers E1 E Complex Numbers Definition of a Complex Number For real numbers and the number is a complex number.If then is called an imaginary number.
Chapter 1 Tools of Algebra . 1.1 Using Variables 1.2 Exponents & Order of Operations 1.3 Exploring Real Numbers 1.4 Adding real numbers 1.5 Subtracting real numbers 1.6 Multiplying & Dividing real numbers 1.7 The Distributive Property 1.8 Properties
Apr 9 Numbers 27 Proverbs 19 Philippians 3-4 Apr 10 Numbers 28 Proverbs 20 John 1 Apr 11 Numbers 29-30 Proverbs 21 John 2-3 Apr 12 Numbers 31 Proverbs 22 John 4 Apr 13 Numbers 32 Proverbs 23 John 5 Apr 14 Numbers 33 Proverbs 24 John 6 Apr 15 Numbers 34 Proverbs 25 John 7 Apr 16 Numbers 35 Proverbs 26 John 8
One Gorilla: A Counting Book, by Atsuko Morozumi. Dear Family. 1 1. Trace the Numbers To help your child learn to write numbers correctly, have . Activity 7 Let’s Write Numbers 1–50 Look at page 11. Have your child trace the numbers shown. Then have your child fill in the missing numbers. 10. 11. 13 My Number Chart. 11379.1 WRITES NUMBERS .
