Expressions About Length

1
Expressions About Length
Lesson 2.3.1
2
Lesson 2.3.1
Expressions About Length
California Standards Algebra and Functions
1.2 Write and evaluate an algebraic expression
for a given situation, using up to three
variables. Algebra and Functions 1.3 Apply
algebraic order of operations and the
commutative, associative, and distributive
properties to evaluate expressions and justify
each step in the process. Algebra and Functions
3.1 Use variables in expressions describing
geometric quantities (e.g., P  2w  2l, A  ½bh,
C  pd the formulas for the perimeter of a
rectangle, the area of a triangle, and the
circumference of a circle, respectively). Algebra
and Functions 3.2 Express in symbolic form simple
relationships arising from geometry.
What it means for you Youll practice using
expressions to represent the length and perimeter
of objects.

• Key words
• length
• perimeter
• expression
• commutative
• associative

3
Lesson 2.3.1
Expressions About Length
Now you need to apply what you know to other
situations.
4
Lesson 2.3.1
Expressions About Length
Expressions Can Describe Length
If you dont know the length of something, use a
variable to represent it.
5
Lesson 2.3.1
Expressions About Length
Example 1
Write an expression for the total height of the
window on the right.
? in.
35 in.
Solution
Choose a letter to represent the unknown height
w, say.
Then the total height of the window is (35 w)
inches.
Solution follows
6
Lesson 2.3.1
Expressions About Length
Pictures can help you understand questions. So if
theres no picture in a question, you should
draw one yourself.
7
Lesson 2.3.1
Expressions About Length
Example 2
Imeldas brother is 56 inches tall. The top of
his head comes to Imeldas shoulders. Write an
expression for the height of Imelda.
Solution
Draw a picture a simple one will do fine.
The unknown length is the height of Imeldas
head.
So choose a letter to represent it x, say.
Then Imeldas height is (56 x) inches.
Solution follows
8
Lesson 2.3.1
Expressions About Length
Guided Practice
1. Write an expression for the length of these
boxes.
(10 r) yards, where r is the length of the box
on the right
2. Susans backyard has a swimming pool at the
end. She measures the distance from the house
to the swimming pool as 50 meters. By choosing
a letter to represent the length of the swimming
pool, write an expression for the length of the
entire yard.
(l 50) meters, where l is the length of the
swimming pool.
Solution follows
9
Lesson 2.3.1
Expressions About Length
Perimeter Means the Distance Around a Shape
The perimeter is the distance around the outside
of a shape.
You find a shapes perimeter by adding together
the lengths of its sides.
If you dont know the length of a side, use a
variable.
10
Lesson 2.3.1
Expressions About Length
Example 3
Write an expression for the perimeter of the
shape below.The two unknown sides have the same
length, y inches.
Solution
The perimeter is found by adding the lengths of
all the sides together.
So, the perimeter of the shape is 11 14 y
y.
This can be simplified to (25 2y) inches.
Solution follows
11
Lesson 2.3.1
Expressions About Length
When you write an expression for the perimeter of
a rectangle, there could be two unknown lengths
the length and the width.
l
length
w
w
width
width
length
l
So you might need to use two variables. For
example, l and w.
12
Lesson 2.3.1
Expressions About Length
Example 4
l
Write an expression for the perimeter of the
rectangle on the right.
w
w
l
Solution
The perimeter of the shape is w l w l 2w
2l.
Or you could write this expression as 2l 2w.It
means exactly the same.
Solution follows
13
Lesson 2.3.1
Expressions About Length
Add Numbers Together in Any Order
When you add the lengths together to find the
perimeter of a shape, you can add numbers
together in any order.
This is because addition is commutative and
associative.
14
Lesson 2.3.1
Expressions About Length
The Commutative Property of Addition
10 3 133 10 13
The Associative Property of Addition
(1 3) 5 4 5 91 (3 5) 1 8 9
15
Lesson 2.3.1
Expressions About Length
Guided Practice
3. Find the perimeter of a rectangle with length
8 inches and width 5 inches.
26 inches
4. Write and simplify an expression for the
perimeter of this shape.
42 n inches, where n is the length of the
unknown side
5. A triangle has one side of length 3 inches,
one side of length 4 inches, and a third side of
length q inches. Write and simplify an
expression for the perimeter of this triangle.
3 4 q 7 q inches
Solution follows
16
Lesson 2.3.1
Expressions About Length
Solve Geometry Equations by Manipulating Them
If youre given enough information, you can set
up an equation to find a missing length or a
perimeter.
17
Lesson 2.3.1
Expressions About Length
Example 5
Write an expression for the perimeter of the
shape below. If the full perimeter is 100 inches,
find the length of the unknown side.
Solution follows
18
Lesson 2.3.1
Expressions About Length
Example 5
Solution
The perimeter is the distance all the way around
the outside.
Using the picture, this is (15 19 26 16
x) inches, where x is the length of the unknown
side.
This can be simplified to 76 x inches.
Solution continues
19
Lesson 2.3.1
Expressions About Length
Example 5
Solution (continued)
Now you can write an equation using the other
information in the question that the full
perimeter is 100 inches.
So 76 x 100.
You can solve this equation by subtracting 76
from both sides
x 76 100
76 x is the same as x 76
x 100 76
Subtract 76 from both sides, and simplify.
x 24
So the length of the unknown side is 24 inches.
20
Lesson 2.3.1
Expressions About Length
Guided Practice
6. By first writing an expression, find the
unknown length on the shape shown to the right,
which has a perimeter of 60 m.
Perimeter 38 n So, 38 n 60, n 60 38,
n 22 m
7. Liz knows that the perimeter of her kitchen is
50 yards. She knows from measuring that three of
its four walls have a combined length of 32
yards. What is the length of the remaining wall?
Perimeter 32 n So, 32 n 50, n 50 32,
n 18 yards
Solution follows
21
Lesson 2.3.1
Expressions About Length
Independent Practice
1. Write an expression for the length of the
piece of wood shown below.
20 x, where x is the unknown length
20 in.
? in.
2. Use guess and check to find the unknown
length in Exercise 1 if the total length of the
wood is 32 inches.
12 inches
Solution follows
22
Lesson 2.3.1
Expressions About Length
Independent Practice
Write an expression for the perimeter of each
shape given in Exercises 34.
3.
4.
61 i, where i is the unknown length
23 q, where q is the unknown length
Solution follows
23
Lesson 2.3.1
Expressions About Length
Independent Practice
In Exercises 56, form and solve an equation to
find the unknown length, given that the perimeter
of each shape is 100 inches.
5.
6.
66 x 10066 x 66 100 66 x 34 inches
78 x 10078 x 78 100 78x 22 inches
Solution follows
24
Lesson 2.3.1
Expressions About Length
Round Up
Dealing with expressions about lengths isnt
harder than dealing with expressions in any other
situation. The main thing is to remember how to
make an expression in the first place.
Once youve mastered that, you should be ready to
tackle area...