Book cover for Prealgebra

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

ISBN #9781938168994

1st Edition

5,729 Questions

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210,912 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces integers, which include both positive and negative numbers as well as zero. Students learn how to locate and order these numbers on a number line, find opposites, evaluate absolute values, and simplify expressions. Real-world contexts such as temperature, elevation, and banking overdrafts help illustrate these concepts. The chapter also initiates the addition of integers, using visual models like colored counters to demonstrate the operations.

Learning Objectives

1

Locate and order positive and negative integers on the number line.

2

Identify and find opposites and absolute values of integers.

3

Simplify expressions involving absolute value and translate word phrases into integer expressions.

4

Model arithmetic operations, particularly addition, with integers using real-world contexts and manipulative strategies.

Key Concepts

CONCEPT

DEFINITION

Integer

A whole number that can be positive, negative, or zero.

Number Line

A visual representation of numbers arranged in order, where positive numbers lie to the right of zero and negative numbers to the left.

Absolute Value

The distance of a number from zero on the number line; always nonnegative. Represented by |n|.

Opposite

A number that is the same distance from zero as a given number, but on the opposite side of zero. For any number a, its opposite is −a.

Manipulative Mathematics

A teaching approach that uses physical objects (like colored counters) to model abstract mathematical concepts such as adding integers.

Example Problems

Example 1

Locate and label the given points on a number line. a. 2 b. ?2 c. ?5

Example 2

Locate and label the given points on a number line. a. 5 b. ?5 c. ?2

Example 3

Locate and label the given points on a number line. a. ?8 b. 8 c. ?6

Example 4

Locate and label the given points on a number line. a. ?7 b. 7 c. ?1

Example 5

order each of the following pairs of numbers, using < or >. a. 9__4 b. ?3__6 c. ?8__?2 d. 1__?10
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Step-by-Step Explanations

QUESTION

How do you plot the numbers 3, -3, and -2 on a number line?

STEP-BY-STEP ANSWER:

Step 1: Draw a horizontal line and mark a central point labeled 0.
Step 2: To plot 3, move three equal units to the right of 0 and place a point.
Step 3: To plot -3, move three equal units to the left of 0 and place a point.
Step 4: To plot -2, move two equal units to the left of 0 and place a point.
Final Answer: The points representing 3, -3, and -2 are correctly placed on the number line according to their distance from zero.

Plotting Integers on a Number Line

QUESTION

Find the opposite of 10 and -10.

STEP-BY-STEP ANSWER:

Step 1: Identify the given number.
Step 2: The opposite is the number with the same magnitude but reversed sign.
Step 3: For 10, the opposite is -10.
Step 4: For -10, the opposite is 10.
Final Answer: The opposite of 10 is -10 and the opposite of -10 is 10.

Finding Opposites

QUESTION

Simplify the expression |9 − 5| − |7 − 6|.

STEP-BY-STEP ANSWER:

Step 1: Simplify inside each absolute value symbol: 9 − 5 = 4 and 7 − 6 = 1.
Step 2: Take the absolute value of each result: |4| = 4 and |1| = 1.
Step 3: Subtract the second absolute value from the first: 4 − 1 = 3.
Final Answer: The simplified expression equals 3.

Simplifying an Absolute Value Expression

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Common Mistakes

  • Placing negative numbers on the wrong side of zero on the number line.
  • Confusing the operation symbol ‘?’ for subtraction with the negative sign indicating a negative number or the opposite.
  • Neglecting to simplify expressions inside the absolute value symbols before applying the absolute value.
  • Incorrectly reversing the sign when finding the opposite of a number.