Additional Topics 1.2 How can I justify?
Algebraic Properties
This lesson is designed to follow Lesson 2.1.9 to give students practice with justifying their
statements with algebraic properties and reasoning.
Lesson Objective:
Students will identify and use algebraic properties (such as Commutative
Property of Addition) to decide if statements are true.
Length of Activity:
One day (approximately 50 minutes)
Core Problems:
Problems ACT-9 through ACT-13
Ways of Thinking:
Justifying
Materials:
Algebra tiles
Algebra Comparison Mats
Suggested Lesson
Activity:
This lesson relies on the familiarity of the algebraic properties introduced
in Math Notes boxes in Chapter 2. It is suggested that you start today’s
lesson with a review of where students can find these properties in their
textbooks. For your convenience, the page numbers for the student text
are printed below.
Property
Page #
Commutative Properties
45
Associative Properties
53
Identity Properties
53
Inverse Properties
72
Once students have reviewed the properties, ask a student to read the
lesson introduction. This ties today’s work with one of the Ways of
Thinking: justification. Stress that it is important when students make
mathematical claims or assertions that they be able to support them with
valid reasoning and properties.
Distribute algebra tiles and expression comparison mats to each team.
Then give teams roughly 10 minutes to work on problem ACT-9, which
requires them to test the Associative Property of Addition by building
and comparing two expressions. Students then test whether the
Associative Property holds when terms are not added.
In part (d), students are asked to formulate another question and test for
the Associative Property. Since this is left open, some students may just
alter the original expressions with new terms. Although this can be
helpful (for example, what happens when some of the terms are added
986
Algebra Connections
and others are subtracted? Does it matter which is added?), emphasize
that students should ask a meaningful math question that could not be
answered by the previous tests. For example, students could ask and
design a test to determine if the Associative Property holds over division.
Note that some questions may require numerical testing, as opposed to
using algebra tiles.
Pull the class together and select some teams to share their responses to
part (d) of ACT-9. Then move teams on to problems ACT-10 through
ACT-13.
Closure:
(10 minutes)
Close today’s lesson by leading a whole-class discussion. Use the
questions below as prompts to help stimulate the discussion:
What is a property? What makes something a property?
Why is it important to justify math statements?
What other questions can we ask about algebraic properties?
Homework:
Problems ACT-14 through ACT-19
Additional Topics 1.2
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Algebraic Properties
This lesson is designed to follow Lesson 2.1.9.
One of the Ways of Thinking used in this course is Justifying. Up until now, you have justified
your conclusions using explanations and reasoning to convince others. Another way to
demonstrate valid conclusions is to use algebraic properties (sometimes referred to as “laws”).
During this lesson, you will use the Commutative, Associative, Identity, and Inverse Properties
of addition and multiplication to justify conclusions and determine if other conclusions are valid.
ACT-9.
Use algebra tiles and an expression comparison mat to show that the expressions
below are equivalent.
3 + (2x 2 + 3x) and (3 + 2x 2 ) + 3x
a.
State the algebraic property that justifies that they are equivalent. [ Associative
Property of Addition ]
b.
Use the same property to rewrite the expression (4 + y) + 2y . Then simplify the
expression as much as possible. [ 4 + (y + 2 y) = 4 + 3y ]
Problem continues on next page →
Appendix A: Investigating the Growth of Patterns
987
ACT-9.
Problem continued from previous page.
c.
What if the terms are not being added? Does the Associative Property still
hold? Use algebra tiles and an expression comparison mat to compare
2x 2 ! (3x ! 8) with (2x 2 ! 3x) ! 8 . Are these expressions equivalent?
Explain. [ No, they are not equivalent. The Associative Property only holds
over addition and multiplication. ]
d.
Your work for part (c) answered the question “What if the terms are not being
added?” What is another question you could ask about the Associative
Property? Create another test for the Associative Property that will help you
better understand the conditions under which it holds true. Be creative. When
you are finished, be prepared to share your results with your class. [ Answers
vary. See Suggested Lesson Activity notes for sample questions. ]
ACT-10. When finding the perimeter of the shape at right for part (b) of problem
2-13, Kiet and Corvell saw the shape differently. Their resulting
expressions are shown below.
Kiet:
988
2 + 2x + 2
Corvell:
x
2 + x + 3+ x !1
a.
Examine each expression. Is each expression valid for the perimeter of the
shape? Justify your conclusion. [ yes ]
b.
Show that the expressions are equivalent. What algebraic properties can be
used to demonstrate that Kiet’s expression is equivalent to Corvell’s
expression? [ The Commutative Property of Addition and Identity
Property of Addition can be used to simplify and show that both
expressions are equivalent to 2x + 4 . ]
Algebra Connections
ACT -11. Examine the work below of a
Note: This stoplight icon will
student simplifying the
appear periodically throughout
expression 9 ! (3 ! 5y) . Is the
the text. Problems with this icon
work valid? If so, justify your
display common errors that can
conclusion by naming the
be made. Be sure not to make the
algebraic properties. If not,
same mistakes yourself!
explain which statement (or
statements) is incorrect and
provide a valid strategy to simplify the expression. [ This work is invalid. Step (b)
incorrectly applied the Associative Property of Addition to terms that are not
added. Alternative strategies vary. ]
a.
b.
c.
d.
e.
9 ! (3 ! 5y)
(9 ! 3) ! 5y
(6 + 3 ! 3) ! 5y
(6 + 0) ! 5y
6 ! 5y
ACT-12. Using algebra tiles, Brad and Donis each used different steps to simplify the
expression !5 + 2x + 8 . Their work is show below. For each student’s work, justify
each step by naming the property that was used or other valid reasoning. Then
compare their strategies. [ a: Commutative Prop. of Add.; b: 8 = 3 + 5 ; c: Inverse
Prop. of Add.; d: Identity Prop. of Add.; e: 8 = 3 + 5 ; f: Commutative Prop. of
Add.; g: Inverse Prop. of Add.; h: Identity Prop. of Addition ]
Donis’s work
Brad’s work
Statement
Reason
Statement
Reason
1. !5 + 2x + 8
Given
1. !5 + 2x + 8
Given
2. 2x + 8 + (!5)
a.
2. !5 + 2x + 5 + 3
e.
3. 2x + 3 + 5 + (!5)
b.
3. !5 + 5 + 2x + 3
f.
4. 2x + 3 + 0
c.
4. 0 + 2x + 3
g.
5. 2x + 3
d.
5. 2x + 3
h.
ACT-13. Determine if the following statements are always true, sometimes true, or never true.
Justify your conclusion. [ a: always true, b: sometimes true (only true when
x = 3 ), c: always true, d: never true, e: sometimes true ]
a.
3x 2 + 7 = 7 + 3x 2
b.
2x ! 1 = 5
c.
!(!18) = 18
d.
!2 + 8 = !10
e.
A rectangle with perimeter of 10 units has an area of 6 square units.
Appendix A: Investigating the Growth of Patterns
989
ACT-14. Using the pattern shown at right, copy and complete the following
Diamond Problems. Then answer parts (e) and (f) below. [ Note:
The answer to the Diamond Problems are contained in the
Diamonds below. One possible answer is provided for part (b). ]
a.
b.
–64
8
–8
c.
–25
5
0
–5
0
8
0
y
x+y
d.
–64
–8
xy
x
1
2
! 16
! 13
1
6
e.
What properties guarantee that (a) and (c) have the same solutions? Justify your
answer. [ Commutative Properties of Addition and Multiplication ]
f.
Use the same property to find another solution to (b) above. [ The 5 and –5
can be switched. ]
ACT -15. Examine the work below that shows one way to simplify the
expression !8 + 0 + (1·5x + 13) . Is the work valid? If so, justify
your conclusion by naming the algebraic property for each step. If
not, explain which statement (or statements) is incorrect and provide
a valid strategy to simplify the expression. [ This work is valid.
Justification: a: Given; b: Identity Property of Addition;
c: Identity Property of Multiplication; d: Commutative
Property of Addition; e: Associative Property of Addition;
f: 8 + 5 = 13 ; g: Additive Inverse Property ]
a.
b.
c.
d.
e.
f.
g.
990
!8 + 0 + (1·5x + 13)
!8 + (1·5x + 13)
!8 + (5x + 13)
!8 + (13 + 5x)
(!8 + 13) + 5x
(!8 + 8 + 5) + 5x
5 + 5x
Algebra Connections
ACT-16. Review the algebraic properties found in the Math Notes boxes for Chapter 2. Then
identify which property is being used below. [ a: Identity property of mult.;
b: Commutative prop. of add.; c: Inverse prop. of add.; d: Associative prop. of
add.; e: Identity prop of add.; f: Inverse prop. of multiplication ]
a.
1!·!(3y) = 3y
b.
16 + 2x = 2x + 16
c.
!4 + 4 = 0
d.
3 + (6 + 2x) = (3 + 6) + 2x
e.
18x 2 + 0 = 18x 2
f.
3 4
4 !·! 3
=1
ACT-17. Decide if the following statements are true or false. Justify your conclusion.
[ a: True, because both sides of the equation equal 9; b: True, because if c = d ,
then c + g = d + g ; c: False, because 10 ! 2!·!5 = 0 ; d: True, because
2(6) ! 5 = 7 ]
a.
1
2
(18) = 18 ÷ 2
b.
If a + 2 = b , then a + 3 = b + 1
c.
10 ! 2 · 5 = 40
d.
If!2x ! 5 = 7, !then!x = 6
ACT-18. The set of integers can be represented with the notation below.
{..., !3, !2, !1, 0, 1, 2, 3, ...}
Note that the “…” symbol indicates that the numbers continue to follow the pattern
without end. Use set notation to represent the set of numbers described below.
[ a: {…,–4, –2, 0, 2, 4, …}; b: {0, 1, 2, 3, …} ]
a.
The set of even numbers
b.
The set of natural numbers (i.e., integers greater than or equal to zero)
ACT-19. Decide if the following statements are always, sometimes, or never true. Explain
your answer. [ a: Sometimes true (not true when the number is 0); b: Always
true; c: Sometimes true, such as when b = 1 ; d: Never true (if x is an even
integer, then x + 2 must also be even). ]
a.
When a number is squared, the result is always positive.
b.
The product of two negative numbers is positive.
c.
If a and b are integers, then
d.
If the number x is an even integer, then x + 2 must be odd.
Appendix A: Investigating the Growth of Patterns
a
b
is an integer.
991
Additional Topics 1.2
•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Algebraic Properties
This lesson is designed to follow Lesson 2.1.9.
One of the Ways of Thinking used in this course is Justifying. Up until now, you have justified
your conclusions using explanations and reasoning to convince others. Another way to
demonstrate valid conclusions is to use algebraic properties (sometimes referred to as “laws”).
During this lesson, you will use the Commutative, Associative, Identity, and Inverse Properties
of addition and multiplication to justify conclusions and determine if other conclusions are
valid.
ACT-9.
Use algebra tiles and an expression comparison mat to show that the expressions
below are equivalent.
3 + (2x 2 + 3x) and (3 + 2x 2 ) + 3x
a.
State the algebraic property that justifies that they are equivalent.
b.
Use the same property to rewrite the expression (4 + y) + 2y . Then simplify
the expression as much as possible.
c.
What if the terms are not being added? Does the Associative Property still
hold? Use algebra tiles and an expression comparison mat to compare
2x 2 ! (3x ! 8) with (2x 2 ! 3x) ! 8 . Are these expressions equivalent?
Explain.
d.
Your work for part (c) answered the question “What if the terms are not being
added?” What is another question you could ask about the Associative
Property? Create another test for the Associative Property that will help you
better understand the conditions under which it holds true. Be creative. When
you are finished, be prepared to share your results with your class.
Additional Topics
527
ACT-10. When finding the perimeter of the shape at right for part (b) of
problem 2-13, Kiet and Corvell saw the shape differently. Their
resulting expressions are shown below.
Kiet:
2 + 2x + 2
Corvell:
x
2 + x + 3+ x !1
a.
Examine each expression. Is each expression valid for the perimeter of the
shape? Justify your conclusion.
b.
Show that the expressions are equivalent. What algebraic properties can be
used to demonstrate that Kiet’s expression is equivalent to Corvell’s
expression?
ACT -11. Examine the work below of a
student simplifying the
expression 9 ! (3 ! 5y) . Is the
work valid? If so, justify your
conclusion by naming the
algebraic properties. If not,
explain which statement (or
statements) is incorrect and
provide a valid strategy to simplify the expression.
a.
b.
c.
d.
e.
Note: This stoplight icon will
appear periodically throughout
the text. Problems with this icon
display common errors that can
be made. Be sure not to make the
same mistakes yourself!
9 ! (3 ! 5y)
(9 ! 3) ! 5y
(6 + 3 ! 3) ! 5y
(6 + 0) ! 5y
6 ! 5y
ACT-12. Using algebra tiles, Brad and Donis each used different steps to simplify the
expression !5 + 2x + 8 . Their work is show below. For each student’s work, justify
each step by naming the property that was used or other valid reasoning. Then
compare their strategies.
Donis’s work
528
Brad’s work
Statement
Reason
Statement
Reason
1. !5 + 2x + 8
Given
1. !5 + 2x + 8
Given
2. 2x + 8 + (!5)
a.
2. !5 + 2x + 5 + 3
e.
3. 2x + 3 + 5 + (!5)
b.
3. !5 + 5 + 2x + 3
f.
4. 2x + 3 + 0
c.
4. 0 + 2x + 3
g.
5. 2x + 3
d.
5. 2x + 3
h.
Algebra Connections
ACT-13. Determine if the following statements are always true, sometimes true, or never true.
Justify your conclusion.
a.
3x 2 + 7 = 7 + 3x 2
b.
2x ! 1 = 5
c.
!(!18) = 18
d.
!2 + 8 = !10
e.
A rectangle with perimeter of 10 units has an area of 6 square units.
ACT-14. Using the pattern shown at right, copy and complete the following
Diamond Problems. Then answer parts (e) and (f) below.
xy
x
y
x+y
a.
b.
8
–25
–8
c.
d.
–8
8
0
1
2
! 13
e.
What property guarantees that (a) and (c) have the same solutions? Justify
your answer.
f.
Use the same property to find another solution to (b) above.
ACT -15. Examine the work below that shows one way to simplify the
expression !8 + 0 + (1·5x + 13) . Is the work valid? If so, justify
your conclusion by naming the algebraic property for each step.
If not, explain which statement (or statements) is incorrect and
provide a valid strategy to simplify the expression.
a.
b.
c.
d.
e.
f.
g.
Additional Topics
!8 + 0 + (1·5x + 13)
!8 + (1·5x + 13)
!8 + (5x + 13)
!8 + (13 + 5x)
(!8 + 13) + 5x
(!8 + 8 + 5) + 5x
5 + 5x
529
ACT-16. Review the algebraic properties found in the Math Notes boxes for Chapter 2. Then
identify which property is being used below.
a.
1!·!(3y) = 3y
b.
16 + 2x = 2x + 16
c.
!4 + 4 = 0
d.
3 + (6 + 2x) = (3 + 6) + 2x
e.
18x 2 + 0 = 18x 2
f.
3 4
4 !·! 3
=1
ACT-17. Decide if the following statements are true or false. Justify your conclusion.
a.
1
2
(18) = 18 ÷ 2
b.
If a + 2 = b , then a + 3 = b + 1
c.
10 ! 2 · 5 = 40
d.
If!2x ! 5 = 7, !then!x = 6
ACT-18. The set of integers can be represented with the notation below.
{..., !3, !2, !1, 0, 1, 2, 3, ...}
Note that the “…” symbol indicates that the numbers continue to follow the pattern
without end. Use set notation to represent the set of numbers described below.
a.
The set of even numbers
b.
The set of natural numbers (i.e., integers greater than or equal to zero)
ACT-19. Decide if the following statements are always, sometimes, or never true. Explain
your answer.
530
a.
When a number is squared, the result is always positive.
b.
The product of two negative numbers is positive.
c.
If a and b are integers, then
d.
If the number x is an even integer, then x + 2 must be odd.
a
b
is an integer.
Algebra Connections