**[tippy title=”Problems 1-8″]Problems 1-8**

There are many problems throughout the text labeled IN YOUR OWN WORDS. Just relax; do not be afraid to give your opinion. For the most part, these questions do not have “right” or “wrong” answers. You should spend about five minutes in answering each of these questions.[/tippy]

**[tippy title=”Problems 9-14″]Problems 9-14**

Addition is accepted as an *undefined term*, but multiplication is defined as repeated addition. You need to carefully apply this definition for these problems. For example, the answer to **Problem 9a** is not 6, but rather is 2*3 = 3 + 3.[/tippy]

**[tippy title=”Problems 15-26″]Problems 15-26**

Review the definition of the associative and commutative properties. Remember that when the commutative property is used, the order in which the elements appear from left to right is changed, but the grouping is not changed. When the associative property is used, the elements are grouped differently, but the order in which they appear is not changed.[/tippy]

**[tippy title=”Problems 27-30″]Problems 27-30**

These problems are designed to test your understanding the commutative and associative properties in interesting everyday settings.[/tippy]

**[tippy title=”Problems 31-32″]Problems 31-32**

There are many problems throughout the text labeled IN YOUR OWN WORDS. Just relax; do not be afraid to give your opinion. For the most part, these questions do not have “right” or “wrong” answers. You should spend about five minutes in answering each of these questions.[/tippy]

**[tippy title=”Problems 33-34″]Problems 33-34**

These problems are discussing operations defined by table. Remember to read the column at the left first. In Problem 34, (-1) x *i * is located in row 2 column 3. On the other hand *i* x (-1) is located in row 3 column 2.[/tippy]

**[tippy title=”Problems 35-36″]Problems 35-36**

See Examples 1 and 2. A set is not closed for an operation if you can find one example of an answer that is not in the set. To show that a set is closed for an operation, you must look at the table of all outcomes.[/tippy]

**[tippy title=”Problems 37-40″]Problems 37-40**

These problems are defining a variety of operations, with focus on the commutative and associate properties. Remember, commutative is a property of *order* and associative is a property of *grouping*. Make sure you understand each defined operation before attempting to answer the question in the problem.[/tippy]

**[tippy title=”Problems 41-42″]Problems 41-42**

Make sure you understand the operation before proceeding.[/tippy]

**[tippy title=”Problem 41″]Problem 41**

A set is not closed for an operation if you can find one example of an answer that is not in the set. To show that a set is closed for an operation, you must look at the table of all outcomes.[/tippy]

**[tippy title=”Problem 42″]Problem 42**

Remember, the associate property is a grouping property and the commutative property is a property of order.[/tippy]

**[tippy title=”Problems 43-44″]Problems 43-44**

Make sure you understand the operation before proceeding. Carry out the requested operations and answer the questions.[/tippy]

**[tippy title=”Problem 45″]Problem 45**

*a(b + c) = ab + ab* is the distributive property for multiplication over addition.

We apply this property to the given operations:

**a.** Right over down would be:

*a right (b down c) = (a right b) down (a right c)*

Apply this for at least three sets of three natural numbers.

**b.** Down over right would be:

*a down (b right c) = (a down b) right (a down c)*

Apply this for at least three sets of three natural numbers.[/tippy]

**[tippy title=”Problems 46-47″]Problems 46-47**

Use the distributive property (see Example 4) for mental operations as in:

6 * 82 = 6*(80 + 2) = 6*80 + 6*2 = 480 + 12 = 492

You can do the above steps mentally.[/tippy]

**[tippy title=”Problem 48″]Problem 48**

Construct a multiplication table for the given set, and then look to see if the set is commutative or associative for this table. For the commutative property, there are four possibilities and for the associative property, there are eight possibilities.[/tippy]

**[tippy title=”Problem 49″]Problem 49**

Construct a multiplication table for the given set, and then look to see if the set is commutative or associative for this table. For the commutative property, there are nine possibilities and for the associative property, there are 27 possible ordered triplets. If you try several and look for patterns, you should be able to form a conclusion without trying all of them. However, to be absolutely certain, you will need to try all possibilities.[/tippy]

**[tippy title=”Problems 50-51″]Problems 50-51**

Remember, the even numbers are numbers of the form 2*n*; namely, {2, 4, 6, … }.

The odd numbers are of the form 2*n* + 1; namely, {1, 3, 5, … }.[/tippy]

**Note:** Homework Hints are given only for the Level 1 and Level 2 problems.

However, as you go through the book be sure you look at all the examples in the text. If you need hints for the Level 3 problems, check some sources for help on the internet (see the LINKS for that particular section). As a last resort, you can call the author at (707) 829-0606.

On the other hand, the problems designated “Problem Solving” generally require techniques that do not have textbook examples.

There are many sources for homework help on the internet.

**Algebra.help**

Here is a site where technology meets mathematics. You can search a particular topic or choose lessons, calculators, worksheets for extra practice or other resources.

http://www.algebrahelp.com/

**Ask Dr. Math**

Dr. Math is a registered trademark. This is an excellent site at which you can search to see if your question has been previously asked, or you can send your question directly to Dr. Math to receive an answer.

http://mathforum.org/dr.math/

**Quick Math**

This site provides online graphing calculators. This is especially useful if you do not have your own calculator.

http://www.quickmath.com/

**The Math Forum @ Drexel**

This site provides an internet mathematics library that can help if you need extra help. For additional homework help at this site, click one of the links in the right-hand column.

http://mathforum.org/