Download TargetFundamentals™ Essential Math Skills Pre

TargetFundamentals™
Essential Math Skills
Pre-Algebra
Skill 3
The student will simplify computations using the inverse relationships of squaring
and finding square roots to solve problems.
Skill:
INSTRUCTIONAL PREPARATION
Duplicate the following (one per student unless otherwise indicated):
• To Square or Not to Square reference sheet
• Where’s the Root? worksheet
Prepare an overhead transparency of the following:
• To Square or Not to Square reference sheet
• Where’s the Root? worksheet
RECALL
Before beginning the Review component, facilitate a discussion based on the following
questions:
¾How is an equation solved? (The variable is isolated using inverse operations.)
¾What is the inverse operation of addition? (Subtraction)
¾What is the inverse operation of multiplication? (Division)
REVIEW
1.
To begin, write the fraction 2 on the transparency and ask the following questions:
4
¾What does it mean to simplify a fraction? (Remove common factors larger than 1
from the numerator and denominator.)
¾Can this fraction be simplified? If so, to what? (Yes, by dividing 2 out of the
numerator and denominator, leaving 1 )
2
¾Why do we bother to simplify fractions? (To make it easier to solve problems
containing fractions.)
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-1
Last printed 8/19/04
Explain that expressions containing square roots can often be simplified, for the same
reason—to make it easier to solve problems. Today they will be simplifying expressions
containing either the square root or the square of a number, and they will be using the same
concept of inverse operations with which they are already familiar.
2.
Distribute the To Square or Not to Square reference sheet and display the transparency.
Have the students read through the “Words to Know” section located at the bottom of the
reference sheet. Point out the difference between the terms, “radical” and “radicand,”
emphasizing that the radical is the square root symbol, and the radicand is the number inside
the radical. Answer any questions about these terms.
Direct attention to the “Inverse Operations” box and explain how squaring and taking the
square root are inverse operations, so long as the numbers are nonnegative. Squaring the
square root will return the original number, as will taking the square root of the square of the
original number. Work through the first example in the section, showing how both cases
return the original number (3), because it is positive. Emphasize that care must be taken
when there is a negative number in the radicand. Ask the following questions:
¾If you multiply a positive number by itself, what kind of number results? (A positive
number)
¾What do you get when you multiply a negative number by itself? (A positive number)
¾Will you always get a positive result when squaring a number? (Yes, unless the
number is 0, which squared is still 0)
Explain to the students that there are numbers that will give negative numbers when
squared, and they may learn about them in later math courses. For now, whenever the
radicand is negative, they can conclude that the answer will not be a real number. Work
through the second example, asking these questions:
¾According to the order of operations, what do we do first? (Square the -4)
¾After we do that, what is the result? ( 16 )
¾What’s the answer? (4) (Point out the list of perfect squares on the right of the
reference sheet, and tell the students that they can refer to it when working today’s
problems.)
¾Why isn’t the answer the same as the original number? (Because there are two
numbers whose square is 16, 4 and -4.)
Remind them that a radical by itself means to take the positive of the two numbers whose
square is the radicand, +4 in this case. To get -4, the expression would be - 16 .
Have the students work in pairs to evaluate the four expressions in the next box. When they
have finished, have volunteers write their answers on the transparency. The answers are: 7,
5, 1, and “not a real number.” Answer any questions.
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-2
Last printed 8/19/04
3.
Direct attention to the “Simplifying Square Roots” section. Explain that the key to this is
factoring, like with simplifying fractions. The difference is that here, they’re looking for
factors that are perfect squares. If the radicand has a perfect square factor, then they can
break up the root into the product of two roots, one with the perfect square radicand. The
radicand of the other root should have no perfect square factors larger than 1. Then, the
perfect square root can be simplified. Work through the two examples, showing how the
original radicand was factored, and how the perfect square root was simplified. Have
student pairs work the two sample problems. When they have completed, have volunteers
write their answers on the transparency. The answers are: 7 2 and 3 11 . If you feel that
the students need more practice, then dictate additional problems for them to solve on the
reverse side of their reference sheets.
4.
Direct attention to the “Estimating Square Roots” section. Explain that it’s important to be
able to estimate an answer to a problem, to verify that you’re on the right track and to see if
the final answer makes sense. One way of estimating square roots is to find the two
consecutive perfect squares that are on either side of the radicand. The value of the square
root is then between the square roots of the two perfect squares. Work through the example
and then have student pairs solve the two sample problems as before. The answers are:
41 is between 6 and 7; 83 is between 9 and 10. Answer any questions about the lesson.
5.
Distribute the Where’s the Root? worksheet, to be completed independently. Allow
adequate time for task completion. Ask volunteers to display their solutions on the
transparency, and verify their results using the Teacher’s Answer Key.
WRAP-UP
•
To conclude this lesson, have the students write a response to the following prompts, using
complete sentences, in their math journals or on a sheet of notebook paper. Allow adequate
time for task completion and then ask volunteers to share their responses with the class.
¾When simplifying a radical, what are you looking for? (Two factors one of which is a
perfect square)
¾What is one way to estimate a square root? (Find the two consecutive perfect squares
between which it lies; the square root lies between the roots of the two perfect
squares.)
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-3
Last printed 8/19/04
To Square or Not to Square
Inverse Operations
•
If x ≥ 0, then
•
If x ≥ 0, then
•
If x < 0, then
d xi
2
Perfect Squares
12 = 1, 1 = 1
62 = 36, 36 = 6
x2 = x
2 2 = 4, 4 = 2
7 2 = 49, 49 = 7
x2 = -x
32 = 9, 9 = 3
82 = 64, 64 = 8
=3
4 2 = 16, 16 = 4
9 2 = 81, 81 = 9
b g
52 = 25, 25 = 5
102 = 100, 100 = 10
32 = 9 = 3,
b-4g
2
=x
d 3i
2
= 16 = 4 = - -4
Use the inverse properties to evaluate these expressions:
49
52
Simplifying Square Roots
•
•
•
Find a perfect square factor in the radicand
Write the root as a product of two roots
Simplify the root of the perfect square
50 = 25 2 = 5 2
24 = 4 6 = 2 6
Estimating Square Roots
•
•
If the radicand is not a perfect square, then
find the two perfect squares surrounding
the radicand.
The square root is between the roots of the
perfect squares
53 is between 49 and 64
53 is between 7 ( 49 ) and 8 ( 64 )
Simplify:
98
-1
(-1) 2
Estimate:
99
41
83
WORDS TO KNOW
Perfect square – The product of an integer and itself 1, 4, 9, 16, 25, 36, …
Radical sign (
) – A symbol that indicates the square root
Radicand – The number under the radical sign
Square root of a number – The number when multiplied by itself will equal
the number in the radical 36 = 6, 64 = 8
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-4
Last printed 8/19/04
Name ________________________________________________________________________
Where’s the Root?
Directions: For Problems 1-8, simplify each expression. If the answer is not an integer, then
express your answers in simplified radical form and give the two integers between which the
answer lies.
2.
112
4.
- 72
28
6.
d2 31i
-4
8.
(-3) 2
1.
81
3.
d 7i
5.
7.
2
2
Directions: For Problems 9 and 10, read each problem carefully and solve.
9.
What is the area of a square with side lengths measuring
3 units long?
10. Substitute the values a = -3 and b = 4 into this equation and solve for c: c2 = a2 + b2.
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-5
Last printed 8/19/04
TEACHER’S ANSWER KEY
Where’s the Root?
1.
2.
3.
4.
9
11
7
-6 2 , between -9 and -8
5. 2 7, between 5 and 6
6. 4 • 31 = 124
7. Not a real number
8. 3
9. 3 square units
10. c2 = (-3)2 + 42 = 9 + 16 = 25
c2 = 25
c = ±5
TargetFundamentals™
 2004 Evans Newton Incorporated
MP.3-6
Last printed 8/19/04