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© 2010 College Board. All rights reserved. Unit 1 • Patterns and Equations 19 My Notes ACTIVITY 1.3 Properties of Real Numbers Properties by the Pound SUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall e local girls’ track team is strength training by liﬅing weights. One of the runners puts weights on each end of the weightliﬅing bar while she is getting ready to bench press. 1. On one end she places a ten-pound plate and then a ﬁve-pound plate. On the other end, she places ﬁrst a ten-pound plate and then a ﬁve-pound plate. Does the bar have the same amount of weight at each end? Explain using a diagram in the My Notes section. In symbols, this situation described above is written as a + b = b + a, where both a and b are real numbers. is is a statement of the Commutative Property. 2. What does the commutative property state? 3. Demonstrate the commutative property of addition by writing an example using: a. negative integers b. rational numbers represented as fractions Another runner needs to increase the amount of weight on the bar for another liﬅ. On each side she places three diﬀerent plates: one ﬁve pound plate, one ten pound plate, and one twenty pound plate as shown in the diagram below. 20 20 10 10 5 5 To ensure that the weight on both sides is the same, she wants to ﬁnd the total weight on both sides. For the leﬅ side of the bar, she used the expression (5 + 10) + 20 and on the right side of the bar, she used the expression 5 + (10 + 20). 4. Use the order of operations to evaluate each expression and compare the results. Order of Operations: 1 st : Evaluate expressions within grouping symbols. 2 nd : Evaluate powers and radicals. 3 rd : Do all multiplication and division from left to right. 4 th : Do all addition and subtraction from left to right.

# Properties of Real Numbers ACTIVITY Properties by … of Real Numbers Properties by the Pound ... c. Explain whether the identity properties apply to division and subtraction

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Unit 1 Patterns and Equations 19

My Notes

ACTIVITY

1.3Properties of Real NumbersProperties by the PoundSUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall

Th e local girls track team is strength training by lift ing weights. One of the runners puts weights on each end of the weightlift ing bar while she is getting ready to bench press.

1. On one end she places a ten-pound plate and then a fi ve-pound plate. On the other end, she places fi rst a ten-pound plate and then a fi ve-pound plate. Does the bar have the same amount of weight at each end? Explain using a diagram in the My Notes section.

In symbols, this situation described above is written as a + b = b + a, where both a and b are real numbers. Th is is a statement of the Commutative Property.

2. What does the commutative property state?

3. Demonstrate the commutative property of addition by writing an example using:

a. negative integers

b. rational numbers represented as fractions

Another runner needs to increase the amount of weight on the bar for another lift . On each side she places three diff erent plates: one fi ve pound plate, one ten pound plate, and one twenty pound plate as shown in the diagram below.

20 2010 10

5 5

To ensure that the weight on both sides is the same, she wants to fi nd the total weight on both sides. For the left side of the bar, she used the expression (5 + 10) + 20 and on the right side of the bar, she used the expression 5 + (10 + 20).

4. Use the order of operations to evaluate each expression and compare the results.

Order of Operations:

1st: Evaluate expressions within grouping symbols.

3rd: Do all multiplication and division from left to right.

4th: Do all addition and subtraction from left to right.

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20 SpringBoard Mathematics with Meaning Algebra 1

My Notes

Properties of Real Numbers ACTIVITY 1.3continued Properties by the Pound Properties by the Pound

Th e expression (5 + 10) + 20 = 5 + (10 + 20) can be written in symbols as (a + b) + c = a + (b + c), where a, b, and c are real numbers. Th is 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 integers

b. rational numbers represented as decimals

7. Th e properties you have been using were described as properties of addition. Determine if these properties also apply for the operations that follow. If the property applies, then write an example that demonstrates it. If not, write a counterexample.

Commutative Property Associative PropertySubtraction

Multiplication

Division

Th e identity properties apply to both addition and multiplication.

8. Use what you know about identity properties to complete these statements. Th en state the identity properties in words.

a. a + = a, and + a = a.

b. x = x, and x = x.

Identity Property of Multiplication:

c. Explain whether the identity properties apply to division and subtraction. Include an example or counterexample with each explanation.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Interactive Word Wall, Group Presentation

A counterexample is an example or case that proves a conjecture or theory wrong.

MATH TERMS

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Unit 1 Patterns and Equations 21

My Notes

ACTIVITY 1.3continued

Properties of Real Numbers Properties by the Pound Properties by the Pound

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 additive inverse of a is a: a + (a) = 0, and a + a = 0

9. Write an equality showing a number and its additive inverse.

Th e Multiplicative Inverse Property states that a number multiplied by its multiplicative inverse yields a product of one. In symbols, themultiplicative inverse of a is 1 __ a : a (

1 __ a ) = 1 and 1 __ a a = 1

10. Write an equality showing a number and its multiplicative inverse.

11. Which real number does not have a multiplicative inverse? Explain.

TRY THESE A

Complete the following number sentences by fi nding each:

5 + = 0 -b + = 0 4 __ 5 + = 0

b. Multiplicative inverse

6 = 1 2 __ t = 1 0.23 = 1

12. Th e distributive property can be written in symbols as a(b + c) = a(b) + a(c). Th e factor a is said to be distributed to

a. Describe the order of operations used on the left side of the equation 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 of the equation a(b + c) = a(b) + a(c) in terms of a, b, and c.

c. What is true about the two sides of the equation?

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation, Activating Prior Knowledge, Quickwrite, Think/Pair/Share

distributive property

The distributive property can also be written as (b + c)a = ba + ca.

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22 SpringBoard Mathematics with Meaning Algebra 1

My NotesMy Notes

Properties of Real Numbers ACTIVITY 1.3continued Properties by the Pound Properties by the Pound

TRY THESE B

Use the distributive property to rewrite and evaluate each of the following expressions.

a. 5 (-7 + 12) b. 1 __ 2 ( 7 __ 3 +

5 __ 3 ) c. -3(-24 + 16)

d. (2 + 9) 5 e. 4 (x + 2y)

13. Does the distributive property apply to operations other than multiplication and addition? Write an example or a counterexample 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 is performed on members of that set, the result also is included in the set.

14. Th e set of whole numbers is closed under addition because the sum of any two whole numbers is also a whole number. Write an equality that demonstrates the closure of the whole numbers under addition.

15. Not all number sets are closed under an operation. Write an equality that demonstrates that natural numbers are not closed under subtraction.

16. Determine for which operations the sets of numbers listed are closed. Use whole numbers in the fi rst table and integers in the second table.

Operation Example or CounterexampleClosed or

Not Closed OperationExample or

CounterexampleClosed or

Subtraction SubtractionMultiplication Multiplication

Division Division

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Activating Prior Knowledge, Group Presentation

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Unit 1 Patterns and Equations 23

My Notes

ACTIVITY 1.3continued

Properties of Real Numbers Properties by the Pound Properties by the Pound

17. Th ink of some examples of rational and irrational numbers.

a. Is the sum of a rational number and an irrational number rational or irrational? Explain.

b. Is the product of a rational number and irrational number rational or irrational? Explain.

18. Th ere are properties of equality that can make it easier to solve equations and to work with proofs in geometry. Th e weights that the track team is lift ing can be used to illustrate these.

a. Th e refl exive property states that a = a.20 20=

Tell what the refl exive property means in your own words using at least one example.

b. Th e symmetric property states if a = b, then b = a.

Tell what the symmetric property means in your own words using at least one example.

c. Th e transitive property states if a = b and b = c, then a = c.

What does the transitive property mean? Explain in your own words using at least one example.

20 201010 1010

20 201010 1010

205 5 5 5 5 5 5 51010 1010

20205 5 5 5 5 5 5 51010 1010

20

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation

CONNECT TO FITNESSFITNESS

The illustrations of different kinds of plates on a barbell in this activity are meant to demonstrate the properties of equality. If you were actually placing plates on a barbell at a gym, however, you would use exactly the same plates, placed in the same order from each end.

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24 SpringBoard Mathematics with Meaning Algebra 1

My Notes

Properties of Real Numbers ACTIVITY 1.3continued Properties by the Pound Properties by the Pound

19. If 3x + 2 = y and y = 8 what property allows you to conclude that 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 a numerical example or counterexample to explain.

a. refl exive

b. symmetric

c. transitive:

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation

1. Identify the property illustrated in each of the following equations.a. 5(2 + 3) = (2 + 3)5b. 25 (0.04) = 1c. 4 + (9 + 3) = (9 + 3) + 4d. (x + 3)4 = 4x + 12

2. Write an equality that illustrates the associative property of multiplication.

3. Write an equality that illustrates the additive inverse of 3.

4. Th e expressions below are the result of distributing multiplication over addition. Rewrite each to show the expression prior to distributing.a. 3(x) + 3(8)b. 10y + xyc. 6w + 9z

5. Using the table below, determine for which operations the set of natural numbers is closed or not closed.

Operation Example or CounterexampleClosed or

SubtractionMultiplication

Division

6. Explain why the sum of __

2 and 3 is irrational.

Identify the property illustrated in each of the following equations.

7. y = y 8. 10 5 = 5 10 9. If y = 11 and y = 3x + 2, then 11 = 3x + 2. 10. If 7 + x = 12, then 12 = 7 + x 11. MATHEMATICAL

R E F L E C T I O N Which of the properties of operations of real

numbers is most useful to you? Explain using more than one example.

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