COMBINING LIKE TERMS
A.1.1 and A.1.2
Algebra tiles are an important part of this course and will be used throughout. The tiles
provide students with the opportunity to “see” abstract algebraic expressions and
equations with two variables. Regular use of algebra tiles
will help students access abstract concepts through the use
y
x
of concrete physical representations.
1
x
In the figures at right, the dimensions of each tile are
shown along its sides, and the area is shown on the tile
itself. Algebra tiles are named by their areas. For
example, the x 2 -tile is in the upper left corner; it has an
area of x 2 .
x2
x
xy
1 1
1
x
x
1
y
y2
y y
y
In diagrams with combinations of tiles, as in Examples 1 to
3 that follow, the perimeter is the distance around the exterior of a figure. The area of a
figure is the sum of the areas of the individual tiles.
In algebraic expressions, as in Examples 4 to 6 below, combining terms that have the
same area to write a simpler expression is called combining like terms.
For additional information, refer to the Math Notes boxes in Lessons A.1.1, A.1.2, and
A.1.5.
Additional Math Notes boxes in Lesson A.1.3 and A.1.4 can help with evaluating
expressions. Students evaluate expressions in homework and Checkpoint 2.
A resource page for creating algebra tiles for home use is available at the CPM website,
www.cpm.org. Go to the student section, and download the Lesson A.1.1 Resource
Page.
Example 1
x
y
y
x
1
xy
x2
x x
x
1
y2
Perimeter = y + x + 1+ x + 1+ x + y + y + y + x = 4x + 4y + 2
(going clockwise from upper left)
Area = x 2 + xy + y 2 + x
y
y
136
Core Connections Algebra
Appendix A
Example 2
y
y
y
y2
y2
y
1 1 1 1
1 1
y y y 1 1
1
y–2
y
1 1 1
Example 3
Perimeter = y + y + 7 + (y ! 2) + 3 + y + y + y = 6y + 8
(going clockwise from upper left)
Area = y 2 + y 2 + y + y + y + 1+ 1 = 2y 2 + 3y + 2
Perimeter = 8x + 6
x2
x2
x2
x
x
x
x
x
x
x
1
1
Area = 3x 2 + 7x + 2
Example 4
The expression 3x 2 + 7x + 2 + x 2 + 5 + 2x can be rewritten as 3x 2 + x 2 + 7x + 2x + 2 + 5 and
simplified to 4x 2 + 9x + 7 by combining like terms.
Example 5
The expression x 2 + xy + y 2 + 3xy + y 2 + 7 can be rewritten as x 2 + xy + 3xy + y 2 + y 2 + 7 and
simplified to x 2 + 4xy + 2y 2 + 7 by combining like terms.
Example 6
3x 2 ! 4x + 3 + !x 2 + 3x ! 7
= 3x 2 ! x 2 ! 4x + 3x + 3 ! 7
= 2x 2 ! x ! 4
Parent Guide with Extra Practice
137
Problems
Determine the area and the perimeter of each figure.
1.
x
4.
7.
1 1 1
2.
x x
2
y
5.
y2
y
y
1
2
x
y
x2
x2
x x
x2
x2
x x
x
x
1 1
2
x
x
6.
x
x
9.
x2
x
x
1 1
1 1
y
y
1
1
y
8.
x2
3.
x
x2
xy
1
y2
Simplify each expression by combining like terms.
10. 2x 2 + x + 3 + 4x 2 + 3x + 5
11. y 2 + 2y + x 2 + 3y 2 + x 2
12. x 2 ! 3x + 2 + x 2 + 4x ! 7
13. y 2 + 2y ! 3 ! 4y 2 ! 2y + 3
14. 4xy + 3x + 2y ! 7 + 6xy + 2x + 7
15. x 2 ! y 2 + 2x + 3y + x 2 + y 2 + 3y
16. (4x 2 + 4x ! 1) + (x 2 ! x + 7)
17. (y 2 + 3xy + x 2 ) + (2y 2 + 4xy ! x 2 )
18. (7x 2 ! 6x ! 9) ! (9x 2 + 3x ! 4)
19. (3x 2 ! 8x ! 4) ! (5x 2 + x + 1)
Answers
1.
P = 4x + 4
A = x 2 + 2x
2.
P = 4y + 6
A = y2 + 3
3.
P = 4x + 8
A = 4x + 4
4.
P = 4y + 4
A = y 2 + 2y + 1
5.
P = 4y + 4
A = x 2 + 2y
6.
P = 2x + 2y + 6
A = 2x + 2y + 2
7.
P = 8x + 6
A = 4x 2 + 6x + 2
8.
P = 6x
A = x2 + x + 1
9.
P = 6x + 4y
A = 2x 2 + xy + y 2
10.
6x 2 + 4x + 8
11.
4y 2 + 2y + 2x 2
12.
2x 2 + x ! 5
13.
!3y 2
14.
10xy + 5x + 2y
15.
2x 2 + 2x + 6y
16.
5x 2 + 3x + 6
17.
3y 2 + 7xy
18.
!2x 2 ! 9x ! 5
19.
!2x 2 ! 9x ! 5
138
Core Connections Algebra