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1 Horizons Pre-Algebra, Teacher’s Guide

Pre-Algebra Teachers Guide

Author:

Shelly Chittam, M.S.

Managing Editor:

Alan Christopherson, M.S.

Editors:

Laura Messner, B.A.

Rachelle Wiersma, M.A.

Graphic Design & Illustration:

Ron A. Hartmann, B.A.

Alpha Omega Publications, Inc. • Rock Rapids, IA

Horizons Pre-Algebra, Teacher’s Guide 2

©MMX by Alpha Omega Publications, Inc.®

804 N. 2nd Ave. E.

Rock Rapids, IA 51246-1759

All rights reserved.

No part of this publication may be reproduced, stored in an electronic retrieval system, or transmitted in any form by any means—electronic, mechanical, photocopy, recording

or otherwise—without the prior written permission of Alpha Omega Publications, Inc. Brief quotations may be used in literary review.

Printed in the United States of America

ISBN 978-0-7403-2243-3

All rights reserved.


Course Introduction ......................................................................................... 5

Readiness Evaluation ....................................................................................... 12

Preparing a Lesson .......................................................................................... 17

Scope & Sequence ........................................................................................... 21

Where to Use Mathematics Worksheets .............................................................. 25

Appearance of Concepts ................................................................................... 27

Teacher’s Lessons ............................................................................................ 38

Contents

Pre-Algebra

Teachers Guide



Purpose

This pre-algebra course has a two-fold purpose. First, students have a thorough review of math concepts taught in elementary school that are vital for success in upper-level math courses. These concepts include basic math operations with whole numbers, decimals, fractions, percents, roots, and exponents. Emphasis is placed on practical application of the concepts.

The second purpose of the course is to introduce the students to concepts of algebra, trigonometry, and geometry in preparation for upper-level math courses. After completing this

course of study, students should be well prepared for a high school level course in Algebra I.

Materials

Materials available for this course include the Teacher’s Guide, the Student Book, and the Tests and Resources Book. The students will have to supply notebook paper, as well as a scientifi c calculator, colored pencils, a ruler, and graph paper. Often the Student Workbook will not have suffi cient space for working out all of the steps to the problems. Notebook paper should be used for these situations. Graph paper should have no more than fi ve squares per inch, although quad-rule paper is recommended. The Student Workbook and the Tests and Resources Book were designed to be consumables. Both have perforated pages for easy tear out but it is recommended that the Student Workbook remain intact to serve as a resource when students wish to review previously covered concepts.

Course Introduction


Layout

Each Lesson in the Student Text has a teaching box in the upper left side of the fi rst page and a Classwork section in the upper right side of the fi rst page. The teaching box is intended

for use by both the teacher and the students as an aid to understanding the lesson. New concepts are presented here in detail so students who miss a lesson in class should be able to catch up any missed work with minimal outside help. The Classwork section is intended for the

class to do together, with individual students explaining the problems for the class.

213Horizons Pre-Algebra, Student Book

Draw a net for a square pyramid.

ACTIVITIES

CLASSWORK1

2 Draw a net for a cube that is different from the one pictured above.

Lesson 97Nets of Solid Figures

A net !" # $!%&' ( )*+, &$ # -.!/'&0,1$&!1#% +,2+,$,1-#-&!1 !" # -3+,,/'&0,1$&!1#% ( )*+,4 5- is what a solid would look like if it were opened #- -3, ,'),$ #1' 2%#6,' 7 #- !1 # $*+"#6,4

For example, the net of a cube could look like -3, ( )*+, 8,%!.4

When cut out and folded on the lines to make a $!%&'9 &- .!*%' %!!: %&:, -3, ( )*+,$ 8,%!.4

Some other common nets are those of octahedrons (8 faces), dodecahedrons (12 faces), and icosahedrons (20 faces).

Teaching Box Classwork



ACTIVITIES

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CLASSWORK1

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Activities

Layout continued:

Following the Classwork section is the Activities section. The fi rst problem set in each Activities section is for reinforcement of the concept taught in that lesson. The remaining Activities sections are for review of previously taught concepts. The Activities sections are part of the assignment for each lesson.


Lesson Plans

Each Lesson Plan lists all concepts taught and reviewed for that individual lesson. The Learning Objectives always relate to the new material taught in that lesson. Each Lesson Plan contains Teaching Tips to aid the teacher in presenting the new material. As often as possible, new material is introduced following a review of related, previously-taught material. The Lesson Plans give detailed helps for the teacher, including sample problems, illustrations, and visual

aids. The solution keys for the student activities are also part of each lesson plan.

109Horizons Pre-Algebra, Teacher’s Guide

Lesson 33

Concepts ! Fraction-decimal equivalents ! Absolute value ! Divisibility tests ! Prime/Composite numbers ! Multiplying by powers of 10 ! Dividing by powers of 10 ! Real world math

Learning ObjectivesThe student will be able to:

! Use division to convert a fraction to a decimal

! Write a decimal as a fraction ! Reduce fractions to simplest form ! Memorize common fraction-decimal

equivalents

Materials Needed ! Student Book, Lesson 33 ! "#$%&'()*+,%'-$.!/$01%$#+0

! Worksheet 17

Teaching Tips Ask students how many quarters

are in a dollar. (4) Ask students what fraction of a dollar is one

quarter. ( )14

Ask students what the value of a quarter is. ($0.25) Repeat this procedure with 2

quarters ( 12 , $0.50), and 3

quarters ( 34 , $0.75).

Show students the relationship between the fraction and decimal representations in the above examples.

Introduce the fraction-decimal equivalents in the student book. Students should memorize these values. They are not expected to memorize fractions that are not on this list.

Concepts

Learning Objectives

Teaching Tips

Materials Needed

Solution Keys


Lesson Plans continued:

Some Lesson Plans will include a Worksheet. These are found in the Tests and Resources Book. Some Worksheets are for additional practice of a new concept while others are for review or a quiz grade. The Lesson Plan will indicate which case applies for each Worksheet. Those intended for additional practice will appear in the Assignments section at the end of the Lesson Plan.

163Horizons Pre-Algebra, Test Booklet

Worksheet 54

1

2

Identify each angle as acute, right, obtuse, or straight. Find the measure of the complementary and supplementary angles, if applicable, for each given angle.

Use the diagram below to answer the questions.

Use the diagram below to answer the questions.

List each straight angle using three letters.

and

19°

167°

71°

136°

180°

2°

A F

D

G

H

BJ

CE

AB CD GH JF

Horizons Pre-Algebra, Teacher’s Guide76

Teaching Tips, Cont. Explain that it is not necessary

to prime factor the radicand, but rather look for perfect squares that are factors.

Have students express each of the following numbers as the product of a perfect square and another factor: 12 = (4 × 3), 18 = (9 × 2), 20 = (4 × 5), 24 = (4 × 6).

Show students that the process for !"#!$%&!'%()*+,%,(()%#-%)*+%-&.+%&-% !"#!$%)*+%-/0&,+%,(()1%+23+4)%the number of equal factors in the radicand must equal the number in the index. For example, the cube root of a number must have the same factor appearing three times in the radicand. The 4th root of a number must have the same factor appearing 4 times in the radicand. Encourage students to prime factor the radicand if they are having "#5 306)'% !"#!$%)*+%,(()7

Complete the Classwork exercises. Have some students work the problems on the board for the class. All students should work the problems in their books.

Assignment ! Complete Lesson 18, Activities 2-6.

Worksheet(In Tests & Resources)

Worksheet Solution

(In Teacher’s Guide)


Learning Styles

Students learn in different ways. Some students can master a concept by listening to instructions or watching someone else do it while others are very “hand-on” and must physically do something to learn a new concept. This book addresses the various learning styles by using a lecture-demonstration method to teach new concepts and review old concepts, and manipulatives are used where appropriate to aid in the understanding of new

concepts.

Algebra Tiles

Algebra tiles are located in the Tests and Resources Book. Students should cut these out the fi rst time the Lesson Plan calls for them, and store them in a zip-top bag for future use. These manipulatives will assist both visual and kinesthetic learners in mastering algebraic concepts. Details on their use are given in the Lesson Plans where needed.

112 Horizons Pre-Algebra, Student Book

What is your occupation? I am a store owner and operator.

Where do you work? I work at my store.

Did you attend college? If so, what was yourmajor? No, I did not attend college.

What parts of your job require the use ofmath? Just about everything I do involves math.

For example, purchasing, invoicing, sales, !"#$%$&'($)"**+$ !"#$%$,&!-.'*+$%&/+$&'($ &0!"))$all involve math. I also make deliveries for my )"1&)$12*%",3!*4$5$'33($%"$#$-2!3$.'$(3).63!0$expenses, calculate the estimated time for each delivery, the best order for delivering the items, and the best route to each destination.

What is the biggest “problem” you havefaced that required the use of math to solve?

My biggest challenge is inventory control. I need to know what items sell well at different times during the year. I need to order my inventory in time to have enough of each item in stock when it is most needed. Also, I do not want to order too much of an item and have extras left in storage.

Are there any other interesting math uses you have experienced? It is impossible to run a business without a good basic math background. Math is an exact science. It will always tell you if you are right or wrong in your business decisions.

A Math Minute with...

Denny S. – Store Owner/Operator

A Math Minute with . . .

At the beginning of each set of 10 lessons the students will read an interview from a Christian who is using pre-algebra in his or her daily life. The word problems that appear in the section will be based on the career of the individual that has been interviewed. Each of the 16 sections of material in this course utilizes a different individual and career. None of these people were college math majors and some of them have successful careers without earning a college degree.

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College Test Prep

As your students progress through their high school years, they will take a number of standardized tests that measure their skills in math, grammar, writing, vocabulary, and reading comprehension. Most colleges use the scores on these tests to determine whether or not to grant students admission to their colleges. Many scholarships are also based on the test scores, so it is important that students do as well as they can.

At the close of each set of 10 lessons, the students will be given a section of multiple choice questions. These questions are the same style and format as questions that are likely to appear on the math sections of standardized tests. They are also the same diffi culty level as the pre-algebra questions that appear on the tests.

Evaluation

This course has 16 tests, 4 exams, and 80 worksheets. One test follows each set of 10 lessons, and one exam follows every 40 lessons. Exam 4 is also a fi nal exam. You have the option of administering the fi rst two pages as a fourth quarter exam, or all six pages as a cumulative fi nal exam. Many of the worksheets are used as quizzes at the teacher’s discretion. Worksheets that are appropriate for quizzes are identifi ed in the corresponding Lesson Plans.


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Test Skills 9It’s College Test Prep Time!

21Horizons Pre-Algebra, Test Booklet

1

2

3

4

5

6

Use the divisibility tests to tell if 2, 3, 4, 5, 6, and 9 are factors of each number. 12 points

Express the prime factorization of each number using exponents. 3 points

Solve, following the order of operations. 2 points

Solve. 3 points

Add, subtract, multiply, or divide as indicated. 4 points

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Pre-Algebra Scope and Sequence

1. Number Theory*Number terminologyPrime and compositePrime factorizationFinding all factors of a numberFactor treesExponentsScientifi c notationOrder of operationsSquare roots*Absolute value

2. Integers>, <, =Comparing integersInversesAdding integersSubtracting integersMultiplying integersDividing integers

3. Addition and SubtractionAddition properties and termsSubtraction properties and termsAddition with 4, 5, and 6 digitsSubtraction with 4, 5, and 6 digitsAdding and subtracting equationsSolving for missing addendsEstimation

4. MultiplicationProperties, terms, and factsMultiplesTwo digit times two digitThree digit times three digitFour digit times four digitFive digit times fi ve digitSolving for missing factors*Exponents: meaning of a negative exponent, powers of negative numbers, product of powers, power of a power, power of a product

5. DivisionProperties and termsOne, two, three, and four digit divisorsDivisibility tests


6. GeometryPerimeter*Circumference of circleArea of parallelogram, triangle, and circle*Area of trapezoidVolume of cube and cylinder*Volume of pyramid, cone, and sphere*Nets of 3-D shapes*Surface area of prism, cylinder, sphere, and pyramidShapes and solidsSymmetryCongruent polygons*Congruent trianglesPoints, lines, planes, rays, line segmentsParallel, intersecting, perpendicular lines*TransversalsAngles: acute, right, obtuse*Angles: adjacent, complementary, supplementaryAngles: ray, vertexCircles: radius, diameter, chord, central angle*Circles: tangentTriangles: scalene, isosceles, equilateralTriangles: right, equiangular, acute, obtusePolygons: sides, vertices, diagonals*Pythagorean formula*Informal Transformations: fl ips, turns, slides, scaling *Transformations: refl ection, rotation, translation, dilation

7. FractionsEquivalent fractionsGreatest common factorLeast common multipleReducing fractionsComparing fractionsAdding fractions and mixed numbersSubtracting fractions and mixed numbersMultiplying fractions and mixed numbersDividing fractions and mixed numbersImproper fractionsReciprocals

8. DecimalsDecimal-fraction equivalentsDecimal-percent equivalentsWord numbers to hundred thousandthsComparing decimalsAdding decimalsSubtracting decimalsMultiplying decimalsDividing decimalsPowers of 10Repeating decimals

Horizons Pre-Algebra Scope and Sequence, continued:


9. Ratio and proportionWriting simple ratiosFinding equal ratios*Writing proportionsCross multiplyingSolving proportionsRatio as a percent*Scale drawings

10. MeasurementEnglish-metric linear equivalentsEnglish-metric liquid equivalentsEnglish-metric weight equivalentsFahrenheit-Celsius conversionsCelsius-Fahrenheit conversions*Mass*Density*VelocityTime

11. GraphsBarLinePictographCircleCoordinate graphs*Using graphs to solve equations

12. PercentFinding the percent of a numberPercent-decimal equivalentsDiscount*Mark-up*Percent increase*Percent decrease*Percents less than 1*Percents more than 100

13. ApplicationsSimple interest*Compound interest*Commission

14. Problem solvingChoosing an operationMultiple step problemsWriting equationsReasonable and unreasonable answers



15. Probability and statisticsMeasures of center: mean, median, modeMeasures of dispersion (spread): range*Frequency distribution*Histogram*Stem-and-leaf plots*Box-and-whisker plots*Probability and odds*Odds in favor*Odds against*Multiplication principle of counting*Combined probabilities*Permutations*Combinations*Mutually exclusive events*Independent and dependent events*Expected value

16. Algebra*Like termsWriting expressions with one variableEquations with one variable*Inequalities with one variable*Two variable equations*Slope and y-intercept*Graphs of equations*Graphing inequalities*Functions*Graphs of functions*Systems of equations*Using algebra tiles*Polynomial expressions*Adding polynomials*Subtracting polynomials*Multiplying monomials*Dividing monomials*Multiplying polynomials by monomials*Dividing polynomials by monomials*Multiplying polynomials*Dividing polynomials*FOIL method*Simplifying expressions

17. Trigonometry*Trigonometric ratios: sine, cosine, tangent*Trigonometric relationships in right triangles

*New concepts



Lesson 1

Concepts• Number terminology• Properties of addition• Addition with regrouping• Subtraction with regrouping


• Defi ne terms related to numbers• Identify numbers as natural,

whole, integer, rational, irrational, and real

• Apply the properties of addition• Add sets of 4-digit addends• Subtract two 4-digit numbers

Materials Needed• Student Book, Lesson 1• A Math Minute with Dan D.

Teaching Tips Ø Administer the Readiness Test.

This test is not to be graded as part of the course grade, but rather as an aid in determining individual student readiness for pre-algebra. Worksheets may be assigned as necessary to assist students who need further help.

Ø Emphasize that math is necessary for life, not just for those who pursue a career in a math-related fi eld. Introduce Math Minutes. These features will appear throughout the book at the beginning of every 10-lesson segment. Each word problem in the 10 lessons following a Math Minute will relate to the individual featured in the Math Minute. Introduce Dan D. – Missionary Radio, Orphanage.


Teaching Tips, Cont. Ø Defi ne the terms in the teaching

box of Lesson 1. Ask students to give other examples of each type of number. They may fi nd it diffi cult to think of other examples of irra-tional numbers. That is fi ne at this point in time. Some students may give the square root of other num-bers. This is a correct answer UN-LESS the student gives the square root of a perfect square. This concept will be discussed further in Lesson 18.

Ø Complete the Classwork exercises orally with the students supply-ing the answers. Have students mark the correct answers in their books. Explain that the value of πis a decimal that never ends and never repeats. In math, it is ac-ceptable to use the value 3.14 or

22

7

for π when an exact answer is not required.

Ø The fi rst 100 digits of pi: 3.141592653589793238462643 38327950288419716939937510 58209749445923078164062862 08998628034825342117067…. (Neither you nor the students are expected to know or memorize this. Often, students will ask, just to see if you know!)

Ø Review addition and subtraction based on student performance on the Readiness Test.

Assignment• Complete Lesson 1, Activities 2-3.


Lesson 10

Concepts• Exponents• Prime factorization• Signed numbers• Multiplication


• Defi ne exponent and base• Use exponents to express products• Write exponential notations in

expanded form• Solve exponential expressions

Materials Needed• Student Book, Lesson 10• Calculator

Teaching Tips Ø Many older calculators will calculate

exponential numbers when you repeatedly press the [=] key. Try this on your calculator before class to make sure it works! Have a student press [2] [x] [2] [=] [=] [=] . . . and read the numbers as they appear. The students should get 4, 8, 16, 32, etc. Note: This will not work on the new scientifi c calculators or those with multiple display lines.

Ø Defi ne exponent and base from the

teaching box. Tell students that the base is the number on the bot-tom. (This concept will carry over in later years when they are learn-ing logarithms with different bas-es.) It will also help to remember that the exponent is elevated.


Teaching Tips, Cont. Ø Demonstrate the proper form for

writing numbers with exponents, using the numbers from the calculator as an example.

Ø Complete the Classwork exercises. Have some students work the problems on the board for the class. All students should work the problems in their books.

Ø Review for Test 1 using worksheets 1-5. These worksheets were all assigned in previous lessons.

Assignments• Complete Lesson 10, Activities

2-5.• Study for test (Lessons 1-7)


Lesson 15

Concepts• Divisibility tests• Prime/Composite numbers• Order of operations• Prime factorization• Exponents• Math in the real world


• Memorize the divisibility tests• Apply the divisibility tests to

natural numbers• Identify 3-digit numbers as

multiples of 2, 3, 4, 5, 6, 9, or 10.

Materials Needed• Student Book, Lesson 15• Worksheet 8

Teaching Tips Ø Have students complete Worksheet

8 in class. This may be for added practice of earlier topics, or graded as a quiz, if desired.

Ø Teach the divisibility tests in the

teaching box. Tell the students that there is no simple divisibility test for 7. Trying to divide a number by 7 as a test is faster and easier than the applicable divisibility test. The same is true for a divisibility test for 8, although if a number is divisible by 4, you can divide by 4 and apply the divisibility test for 2 to the quotient. (The divisibility tests for 7 and 8 are provided on the following page for your reference. Students are not expected to memorize the tests for 7 and 8.)


Teaching Tips, Cont. Ø Explain that the divisibility tests

should be memorized so the students can apply them when needed. They will be used extensively beginning in Lesson 18.

Ø Complete the Classwork exercises. Have some students work the problems on the board for the class. All students should work the problems in their books.

Assignment• Complete Lesson 15, Activities 2-5.

Notes on Divisibility Tests

Divisibility test for 7:Multiply the digit in the ones place by two. Subtract this number from the remaining digits in the original number. If the difference is divisible by 7 (or equal to 0), then the original number is also divisible by 7. If the number is still too big to tell, repeat the above steps until you get a number that you know is or is not a multiple of 7.For example, is 392 divisible by 7?2 × 2 = 439 – 4 = 3535 is divisible by 7; (7 × 5), so 392 is also divisible by 7; (7 × 56)

Divisibility test for 8:If the last three digits of a number are divisible by 8, then the number is divisible by 8. This is because 1000 is divisible by 8, so you only have to check the last three digits.

Order of Operations


Teaching Tips, Cont. Ø Have students read the Math

Minute interview for Lessons 81-90.

Ø If you plan to administer Exam 2, review as time permits when all students have fi nished Test 8.

Assignments• Complete It’s College Test Prep

Time!• Read A Math Minute with…

Mike L. – Professional Landscaper• Study for Exam 2 (Lessons 1-77)


Lesson 83

Concepts• Perimeter and area of rectangles• Perimeter and area of squares• Simple interest• Probability and odds• Math in the real world


• Defi ne rectangle and square• Calculate the perimeter and area of

rectangle• Calculate the perimeter and area of

a square


Teaching Tips Ø Review parallelograms and

rhombuses. (See Lesson 82.)

Ø Teach rectangles and squares from the teaching box. Ask students if rectangles are parallelograms. (Yes.) Why? (The opposite sides are parallel.) Ask students if squares are parallelograms. (Yes.) Why? (The opposite sides are parallel.)

Ø Ask students if rectangles are rhombuses. (Sometimes.) Why or why not? (The opposite sides are not always equal.) Ask students if squares are rhombuses. (Yes.) Why? (They are parallelograms and the opposite sides are equal.)


Teaching Tips, Cont. Ø Teach the formulas for fi nding the

perimeter and area of rectangles and squares. Show students that these are essentially the same formulas as those for rectangles and rhombuses.

Ø Ask the students why the lengths of the sides are used in the area formulas for rectangles and squares but not in the area formulas for parallelograms and rhombuses. (The sides are perpendicular, so the side is also the height.)

Ø Complete the Classwork exercises. Have some students work the problems on the board for the class and explain their answers. All students should work the problems in their books.

Assignments• Complete Lesson 83, Activities 2-5.• Worksheet 42

Perimeter and Area of Rectangles and Squares


Lesson 148

Concepts• Area of regular polygons• Apothem• Perimeter• Math in the real world


• Calculate the area of regular polygons

• Defi ne regular polygon• Defi ne apothem• Use the apothem and perimeter to

fi nd the area of a regular polygon• Determine the area of irregular

shapes


Teaching Tips Ø Have students complete Worksheet

74 in class. This may be for added practice of earlier topics or graded as a quiz, if desired.

Ø Review 30-60-90 triangles. (See Lesson 123.)

Ø Teach regular polygons from the teaching box. Explain that it does not matter how many sides a polygon has, as long as all sides are congruent and all angles are congruent.

Ø Ask the students what congruent means. (Having the same size and shape.)


Teaching Tips, Cont. Ø Teach the defi nition of apothem

from the teaching box. This word is pronounced with the emphasis on the fi rst syllable, like apple. Many students try to pronounce it so it sounds like “a possum,” but this is incorrect. Stress the correct pronunciation from the beginning.

Ø Teach the formula for fi nding the area of a regular polygon using the apothem and perimeter. Students are not expected to memorize this formula, but they must know what each variable represents.

Ø Ask the students how they would fi nd the area of a fi gure that looked like part of it was missing. (Find the area of the entire piece and the area of the missing section. Subtract the area of the missing section from the area of the whole thing.)

Ø Complete the Classwork exercise. Have one student work the problem on the board for the class and explain the answer. All students should work the problem in their books.

Assignment• Complete Lesson 148, Activities

2-4.

Refl ections, Rotations, Scaling

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