Pre-Algebra Out Loud: Learning Mathematics Through Reading and Writing Activities (2013)

Chapter 3. Topics in Integers

WHAT? Introduction

Studying the process of ancient Egyptian multiplication helps pre-algebra students understand the development of the method we use today for multiplying. The first mini-lesson in this chapter introduces this ancient method of multiplying integers. The second mini-lesson discusses integral exponents and the concept of repeated multiplication.

WHY? Objectives

Using the activities in this chapter, pre-algebra students will:

· Determine and collect the important concepts from a lesson for display

· Create concept cards that they can use as study or homework aids

· Use the Frayer method and charts to display the important features, examples, and nonexamples of concepts and processes related to integers or exponents

· Work in small groups to discuss and solve problems and come up with an appropriate rule for solving these types of problems

· Compare and contrast related mathematical concepts or processes and display this knowledge in a premade matrix

· Assess their own knowledge or understanding of integers

Mini-Lesson 3.1 Egyptian Multiplication

CCSS Standards 6.NS: Number Systems

Compute fluently with multi-digit numbers and find common factors and multiples.

The first number systems were developed over four thousand years ago by the Egyptians and Babylonians. This is an ancient Egyptian number:

In our number system this would be 12,425. Each symbol (not necessarily read from left to right) stands for 5 ones, 2 tens, 4 hundreds, 2 thousands, and 1 crooked finger for 10,000. The Egyptians focused on tens because of the number of human fingers and toes, allowing them to use each digit as a tally mark.

The Egyptians understood the usefulness of multiples of numbers as demonstrated in the doubling process used in their multiplication. Let's explore this multiplication system by looking at an example: 18 × 24.

Consider the first two columns in the table exploring the example. First, we choose 1 and 24 as the multiplicands (a number to be multiplied by another number) and begin doubling each. In the second row, we double 1 to get 2 and double 24 to get 48. Then we double 2 to get 4 and 24 to get 48. This is much like solving equations where if you multiply one side by a number like 2, you must multiply the other side, too.

Continue this doubling until we have some numbers, not necessarily all of them, in the first column that add up to 18. We place a check mark by 2 and 16. Add the two corresponding numbers (across from 2 and 16), 48 and 384, which gives the solution 432. Multiplying, we see 18 × 24 = 432.

We could just as well have chosen 1 and 18 to start. Each process gives the same solution. The last two columns in the table show the work using 18 as the multiplier.

Example: 18 × 24 = 432

Note the following partial sums: 2 × 24 = 48 and 16 × 24 = 384. Adding the two partial sums gives 48 + 384 = 432.

Next, multiply 52 × 86 using the doubling process:

Find and add up values from the left column to get 86: .

Then add the numbers from the right that correspond to arrows checked on the numbers from the left side; for example, 2 corresponds to 104, and 16 corresponds to 832.

Adding gives the solution.

Checking, .

The Egyptians knew that they needed a number system to display their inventories of food and animals, measure their land, and record taxes that were paid. However, it was only Egyptian men who learned to read and write numbers and words. Egyptians believed that women's heads were constructed in such a way that they would explode if too much knowledge was put into them!

Teaching Tip

Teaching students to multiply as the Egyptians did is a supplementary mathematical activity. Wait until students have mastered two- or three-digit multiplication and have a good understanding of doubling numbers. Asking students to compare our method with the Egyptian method reveals their understanding of multiplication and leads to some lively classroom discussion.

Activity 3.1: The Writing Is on the Wall

WHAT? Description

Elementary teachers often use a word wall or bulletin board to help students learn the spelling and vocabulary of math terms. Your wall will be more useful to students by grouping terms into, for example, the Top Ten, Fabulous Five, or Foremost Four mathematics concepts from each lesson or chapter. Students should be asked to give the key concepts.

Students collect significant pre-algebra or algebra words that they then use to construct their math glossaries and add to their mathematical vocabulary. These concepts are the same words that students will see in all later algebra courses.

WHY? Objectives

During this activity, pre-algebra students:

· Choose key concepts from a section in their textbook or that day's lesson

· Collect words to help construct an algebra glossary over the entire course

HOW? Example

Here are five terms to add to the Fabulous Five Number Forms on the math wall:

Real

Rational

Integer

Whole

Natural

Worksheet 3.1: The Writing Is on the Wall

Name _____ Date _____

Directions: Choose ten concepts that involve integers. Look for terms that are related in the same way to the integers. You may use today's lesson or a chapter from your textbook on integers. Then try finding six symbols that may be used with the integers. Write these words under the following categories. Be creative!

Top Ten Terms

Sensational Six Symbols

Activity 3.2: Concept Cards

WHAT? Description

Similar to the concept circle activity described in Chapter One, concept cards are an excellent way for students to create study aids for tests or other assessments. Index cards work well for this activity.

After reading the assigned text and doing any assigned problems, students review the content looking for key concepts. They write the concept on one side of an index card and the definition, features, facts, or theorems regarding the concept on the other side.

Students can use these concept cards to construct a glossary for the entire course or as a study guide for tests or exams. The cards can also become part of a game in which each student reads the definition of a concept and his or her partner guesses the concept described.

WHY? Objectives

During this activity, pre-algebra students:

· Identify the key concepts and terms from the text and use these terms to create concept cards

· Define the concepts in their own words, promoting ownership of the knowledge

· Create a useful study guide for upcoming assessments

HOW? Example

Have students read the chapter on multiplication in their textbooks, paying special attention to the key concepts. When they finish, ask them to construct at least four concept cards as in the example on the next page.

Make sure students remember to write out the definition and important facts or features about the concept on the reverse side of the card.

Activity 3.3: Frayer Model

WHAT? Description

The Frayer model (Frayer, Frederick, & Klausmeier, 1969) is a writing strategy that stresses word categorization, including defining a concept and considering attributes, nonattributes, examples, nonexamples, and other important features. The model presents an excellent reading strategy that requires students to review, reflect on, and study the key concepts of a unit. The four-square model is explained in the example.

WHY? Objectives

During this activity, pre-algebra students:

· Read to search for the facts or features of the concept

· Analyze and write out the attributes and nonattributes of the concept

· Complete a graphic that they can use as a study aid

HOW? Example

Worksheet 3.2: Frayer Model

Name _____ Date _____

Directions: Fill in this sheet by defining “multiplying signed numbers,” giving any important rules, and providing examples and nonexamples of this concept. There is not just one right answer for each category.

MULTIPLYING SIGNED NUMBERS

Your definition of:

Rules:

Worksheet 3.3: Frayer Model

Name _____ Date _____

Directions: Choose a mathematical concept from the chapter on integers. Then fill in the worksheet by defining your concept, giving any important rules, and providing examples and nonexamples of this concept. There is not just one right answer for each category.

Concept = _____

Your definition:

Rules:

Mini-Lesson 3.2 Integral Exponents

CCSS Standard 8.EE: Expressions and Equations

Work with radicals and integer exponents.

This lesson explores integers and exponents.

Exponents show repeated multiplication. For example, consider the following exponential expression:

In the expression, is first expressed in expanded form. An exponential expression is also called a power. A power has two parts: a base (2 in the example) and an exponent (5 in the example). The expression is therefore read as “two to the fifth power.” Note that 2 is written out five times, then multiplied to arrive at the value of 32.

Teaching Tip

· Any number to the first power is the number itself:

,

· Any number raised to the 0 power is 1:

,

This table shows the different forms of certain exponential expressions, including how the exponential expression should be read.

Some students may confuse the expression −n² with (−n)². The n in −n² represents the base, and therefore n is the value squared, while the negative or opposite sign remains. In (−n)², the value (−n) is the base and is squared, making the solution a positive number. This will be true for all exponents that are integers. For example,

Remember that exponents are used when simplifying certain arithmetic and algebraic expressions. To evaluate these expressions requires following a certain order, referred to as the order of operations:

1. Do the inside parentheses first.

2. Do the exponents from left to right.

3. Do the multiplication and division from left to right.

4. Do the addition and subtraction from left to right.

A popular and easy way to remember the order with which to do the operations is to remember the acronym PEMDAS:

P PLEASE (parentheses)

E EXCUSE (exponents)

M MY (multiplication)

D DEAR (division)

A AUNT (addition)

S SALLY (subtraction)

The operations are completed from left to right. Multiplication and division are done left to right as either comes up; for example, . The same applies to addition and subtraction.

Many expressions can be simplified and equations solved by using the order of operations and combining like terms when using variables as in the following examples:

1.

2.

3. . Note here that x² and x³ are not like terms and cannot be combined.

(simplify using PEMDAS on both sides of the equal sign first)

(combine like terms)

(subtract 3x from both sides)

(divide both sides by 21)

Activity 3.4: What's My Rule?

WHAT? Description

For this guided discovery activity, students, working alone or in groups, are presented with several examples of a certain mathematical concept or rule. Then they brainstorm and write the rule or conjecture (that is, what they think the rule might be). Next, they solve problems using their conjecture.

This activity is best completed with a teacher or teacher's aide available to check for the accuracy of rules.

WHY? Objectives

During this activity, pre-algebra students:

· Discover mathematical rules while observing and completing several problems that use the rule

· Practice writing and using certain mathematical conjectures

· Work cooperatively to arrive at a well-expressed and accurate rule

HOW? Examples

The rule here is “Raising to a power means repeated multiplication of the exponential notation”:

1.

2.

3.

4.

5.

Worksheet 3.4: What's Your Rule? Scientific Notation

Name _____ Date _____

Directions: After studying the examples, try solving the unfinished problems. Then write a rule in a complete and clearly constructed sentence that explains how you solve scientific notation problems.

Worksheet 3.5: What's Your Rule? Exponent Laws

Name _____ Date _____

Directions: After studying the examples, try solving the unfinished problems. Then write a rule in a complete and clearly constructed sentence that explains how you simplify each type of exponential problems.

Activity 3.5: Comparison-and-Contrast Matrix

WHAT? Description

Students can use the comparison-and-contrast matrix (Vacca & Vacca, 1999) to compare and contrast related features of related concepts. Choose the concept and features, and ask students to fill in the blank squares. They can then use the completed matrix as a study guide or turn it in as an assignment. Students' answers may be objective or subjective depending on the concepts or features to be compared.

WHY? Objectives

During this activity, pre-algebra students:

· Research and reflect on the similarities and differences of related mathematical concepts

· Receive feedback from their teacher about the validity of the facts in their matrices

HOW? Example

This matrix shows the different forms for an exponential expression—how they differ and how they are alike. The tables can be used in different ways to compare and contrast the features of an algebraic concept.

Worksheet 3.6: Comparison-and-Contrast Matrix: Integral Exponent

Name _____ Date _____

Directions: Fill in the matrix (the first row has been filled in as an example). You and your peers might find different answers.

Worksheet 3.7: Comparison-and-Contrast Matrix: Exponent Laws

Name _____ Date _____

Directions: Fill in the matrix (the first row has been filled in as an example). You and your peers might find different answers.

Worksheet 3.8: Comparison-and-Contrast Matrix: Multiplication

Name _____ Date _____

Directions: Fill in the matrix. You and your peers might find different answers.

Activity 3.6: Knowledge Ratings

WHAT? Description

Knowledge ratings allow students to assess their prior understanding of a topic (Blachowicz, 1986). Using this activity, your students will be able to see the value in measuring their own progress and determining where they need to study more. The survey headings for knowledge rating charts may take various forms depending on the topic of the chart, as the example in this section shows.

WHY? Objectives

During this activity, pre-algebra students:

· Fill out knowledge ratings charts

· Target problem areas and make study plans

· Point out personal problem areas to teachers

HOW? Example

Rate how much you know about integer operations (positive and negative) by placing an X in the cell under the heading that best describes how you would rate your understanding of the subjects in the first column.

Worksheet 3.9: Knowledge Rating Activity for Exponent Rules

Name _____ Date _____

Directions: How much do you know about these terms? Place an X in the spaces for which you agree.