Problem Solving ( Integers, Rational Numbers, and Exponents)

Polya's Problem Solving Strategy

Target:

1. Recall concepts and operations of integers, rational numbers and exponents

2. Use Polya's 4 step process in problem solving

Review:

• Integers

• Rational Numbers

• Exponents

Integers

• Addition

• Subtraction

• Multiplication and Division

Rules for Adding Integers

Like Signs = Find the sum of their absolute values and use the sign common to both integers.

Unlike Signs = Find the difference of their absolute values and use the sign of the integer with the greater absolute value.

Multiple Choice

Add.

5 + (-9)

• 4

• -4

• 14

• -14

Multiple Choice

Add.

(-11)+(-23)

• 12

• -12

• -34

• 34

Multiple Choice

Add.

16+6

• 22

• -22

• -6

• 6

Multiple Choice

Add.

(-10)+16

• 26

• -26

• -6

• 6

Rules for Subtracting Integers

• To subtract an integer, simply add its opposite or

• For all numbers a and b, a - b = a + (-b)

Multiple Choice

Subtract.

16 - (-10)

• 6

• -26

• 26

• -6

Multiple Choice

Subtract.

10 - 6

• -16

• 16

• -4

• 4

Rules in Multiplying and Dividing Integers

When you multiply/divide two integers with the same sign, the product is positive/quotient is positive.

When you multiply/divide numbers with different signs, the product/quotient is negative.

Multiple Choice

Find the product.

(-6)(-8)

• -86

• 68

• -48

• 48

Multiple Choice

Find the product.

3 x (-15)

• -45

• 45

• 12

• -12

Multiple Choice

Find the quotient.

60/(-4)

• -15

• 15

• -64

• 64

Multiple Choice

Find the quotient.

(-21)/(-7)

• -3

• 3

• 28

• -28

Ratonal Numbers

• Four Basic Operations (Add, Subtract, Multiply, and Divide)

Multiple Choice

Solve.

 $\frac{1}{2}+\frac{3}{4}-\frac{1}{4}\ =\ ?$ 

• 5/4

• 1

• 3/4

• 1/2

Multiple Choice

Solve.

 $\frac{2}{3}\ x\ \frac{4}{5}\ x\ \frac{15}{11}\ =\ ?$  

• 8/13

• 8/11

• 13/8

• 11/8

Multiple Choice

Solve.

 $\frac{3}{4}\div\frac{4}{5}=?$  

• 12/20

• 16/15

• 15/16

• 20/12

Exponents

Multiple Choice

Simplify.

 $14^2$  

• 196

• 169

• 28

Multiple Choice

Simplify.

 $\left(-3\right)^4$  

• -27

• 27

• 81

• -81

Multiple Choice

 $\left(\frac{2}{3}\right)^2$  

• 4/9

• 4/3

• 2/9

• 2/6

Multiple Choice

Simplify.

 $\left(4\right)^3-\left(-1\right)^3$  

• 63

• 64

• 65

Polya's Problem Solving Strategy

Step 1. Understand the problem

Step 2. Device a plan (translate)

Step 3. Carry out the plan (solve)

Step 4. Look back (check and interpret)

Problem 1 (Integers)

A frog in a well is 12 meters below the ground level. if it jumps 5 meters and then falls back 3 meters, how far is the from from the ground level?

A frog in a well is 12 meters below the ground level. if it jumps 5 meters and then falls back 3 meters, how far is the from from the ground level?

Given:

12 m below the ground = location of the frog (-12)

5m = distance covered by the frog when it jumps

3m = distance ( the frog falls back when it jumps)

What is asked in the problem?

Plan:

Add 5 to -12 when the frog jumps and subtract 3 after falling back.

Solve:

(-12)+5-3 = -10

Check/Interpret:

The frog should be 10 meters below the ground after jumping.

Multiple Choice

A submarine is 65 meters below sea level. It rises 26 meters and then dives 15 meters. How far is the submarine below sea level?

• 54

• 55

• 56

• 57

Problem 2 (Integers)

Jorge was charged P1,000.00 at his favorite clothing boutique, so the balance in his account is -P1,000.00. If he goes shopping and triples the amount he owes, what will be the present balance of his account?

Jorge was charged P1,000.00 at his favorite clothing boutique, so the balance in his account is -P1,000.00. If he goes shopping and triples the amount he owes, what will be the present balance of his account?

Given: -P1,000.00 = Jorge's balance in his account

What is asked in the problem?

Plan:

"triples" = three times

multiply -P1,000.00 by 3

Solve:

(-1,000)x(3) = -3,000.00

Check/Interpret:

Jorge's present balance in his account is -P3,000.00 after shopping.

Multiple Choice

Jorge was charged P1,000.00 at his favorite clothing boutique, so the balance in his account is -P1,000.00. He goes shopping and doubles the amount he owes. If his monthly salary is P18,000.00, how much money will he receive after deducting the amount he bought in the boutique?

• P20,000.00

• P19,000.00

• P16,000.00

• -P20,000.00

Let's go!!

Problem 3 (Rational Numbers)

Human bones are 3/10 living tissues, 9/20 minerals, and the rest is water. What fraction of the human bone is water?

Human bones are 3/10 living tissues, 9/20 minerals, and the rest is water. What fraction of the human bone is water?

Given:

3/10 = part of human bones which are living tissues

9/20 = part of human bones which are minerals

x = water

What is asked in the problem?

Plan:
Human bones - (Living tissues + minerals) = water
Solve:
 $x\ =\ 1\ -\ \left(\frac{3}{10}+\frac{9}{20}\right)\ =\ 1\ -\ \left(\frac{6}{20}+\frac{9}{20}\right)$  $x\ =1-\frac{15}{20}\ =\ \ \frac{5}{20}\ =\ \frac{1}{4}$  

Check: $\frac{1}{4}+\frac{3}{10}+\frac{9}{20}\ =\ \frac{5}{20}+\frac{6}{20}+\frac{9}{20}\ =\ \frac{20}{20}\ =\ 1$  

 $\therefore$  A human bone is composed of 1/4 water.

Multiple Choice

Last week, Crissa spent 1/3 of her weekly allowance on snacks, 2/5 on school materials, and 1/4 of the remainder on transportation. If she saved the rest of her allowance, what fraction of her allowance did she save?

• 59/60

• 4/12

• 8/12

• 1/60

Problem 4 (Rational Numbers)

One half of the students in a school are girls, 3/5 of these girls are studying in lower classes. What fraction of these students are girls who are also studying in lower classes?

One half of the students in a school are girls, 3/5 of these girls are studying in lower classes. What fraction of these students are girls who are also studying in lower classes?

Given:

1/2 = fraction of girls studying in school

3/5 = fraction of the girls who are studying in lower classes

What is asked in the problem?

Plan:

3/5 x 1/2 = ??

Solve:

3/5 x 1/2 = 3/10

Check/Interpret:

3/10 of the students are girls who are also studying in lower classes

Multiple Choice

Balsi reads three-fifth of 75 pages of his lesson. How many more pages he need to complete the lesson?

• 45

• 40

• 35

• 30

Problem 5 (Exponents)

An asteroid travels at a speed of   $6^8$  miles per day, how many miles will it travel in   $6^3$  days? Leave your answer in the form a^b

Given:

6^8 mi/day = speed of the asteroid

6^3 days = number of days for the asteroid to travel

What is asked in the problem?

Plan:

(6^8)(6^3) = the distance an asteroid can travel

Solve:

(6^8)(6^3) = 6^11

Check/Interpret:

The asteroid can travel 6^11 miles.

Multiple Choice

You are enlarging a photo on your computer screen. The photo starts at 3 centimeters wide. Each time you enlarge the photo, its width is doubled. You enlarge the photo four times. What is the final width of the photo on your screen?

• 6

• 12

• 24

• 48

Multiple Choice

On Tuesday, you invited 2 friends to your party. On Wednesday, each of these friends invited 2 other friends. This pattern continued Thursday and Friday. How many people were invited on Friday? Write the answer as a power. How many people were invited in all?

• 8 and 14

• 8 and 30

• 16 and 30

• 16 and 46