The purpose of this book is to help students develop basic math skills that serve as the building blocks for all mathematical equations, and problems solving concepts. If basic math skills are learned, adding new formulas and problem solving concepts simply become a matter of understanding when and what sequence to apply for each order of operation. If students do not learn the basic foundation of mathematics; addition, subtraction, multiplication, division, and the time table chart, math will always be a subject that students experience difficulty understanding and comprehending.

This book is intended to be a simple guide to help third grade students learn and become proficient in their application of basic math skills. Students will learn to use addition, subtraction, multiplication, division and the time table chart simultaneously while they learn the strategy of counting up.

"I have nothing against the usage of calculators, but when students are introduced to the calculator at the third grade level, they become dependent on the use of calculators for assistance with math computations. Once students learn to use the calculator it becomes unnecessary for them to master basic math skills, because they continue to rely on the use of calculators for assistance with math. The calculator becomes the math problem solving tool. Could this be part of the reason students are not performing at levels of proficiency required on standardized test?" It is important for students to learn basic math skills in its entirety, prior to being introduced to the calculator. Once students have developed a basic foundation of math skills then and only then should the calculator be introduced as a learning tool.

When students learn the strategy of "Counting Up" they will be able to rely on what they have learned to find the solutions to math problems.

Teach Multiplication, Division, and the Time Table All at the Same Time

An Instructional Guide for Learning Basic Math SkillsBy Andray McCuien

AuthorHouse

Copyright © 2011 Andray McCuien
All right reserved.
ISBN: 978-1-4567-2770-3

Contents

Title Page.....................................................1Content........................................................3Preface........................................................4How Students Learn.............................................5Reader Notes...................................................61 Multiplication and the Time Table............................112 Skip Counting................................................133 Introducing Long Division....................................154 Inverse Operation............................................295 Division Problems and Counting Up............................306 Counting-Up Strategy.........................................677 Counting-Up Strategy and Subtraction.........................69Conclusion.....................................................78Practice Pages for Instructors.................................79Scratch page...................................................82About the Author...............................................83

Chapter One

Multiplication and the Time Table Chart

At this point, students learn the rules for multiplying by 0 and 1. Rule for multiplying by 0: Any number multiplied by 0 is equal to 0.

0 × 1 = 0 0 × 4 = 0 0 × 7 = 0 0 × 10 = 0 0 × 2 = 0 0 × 5 = 0 0 × 8 = 0 0 × 11 = 0 0 × 3 = 0 0 × 6 = 0 0 × 9 = 0 0 × 12 = 0

Rule for multiplying by 1: Any number multiplied by 1 is equal to that same number.

1 × 1 = 1 1 × 4 = 4 1 × 7 = 7 1 × 10 = 10 1 × 2 = 2 1 × 5 = 5 1 × 8 = 8 1 × 11 = 11 1 × 3 = 3 1 × 6 = 6 1 × 9 = 9 1 × 12 = 12

To be sure students learn these two rules, have them write out the entire time table, multiplying each number from 1 to 12 by 0, showing the product of 0 for each set of two factors; have them repeat the same step to illustrate the second rule, multiplying each of the factors by 1 and showing the product.

These two rules are very simple and are usually easy for students to learn. It is usually not necessary to use the counting-up strategy to help students understand these two principles.

Now it is time to move on to the rest of the time table chart. Do not focus on how much of the time table chart students are able to remember at this point. This exercise is simply to help students become acquainted with the time table chart as a concept. Once students begin to apply the strategy of counting up, they will learn the time table chart in its entirety through repetition and practice.

Students should be given a time table chart to use when multiplication is introduced. This is simply to provide students with a visual aid to follow as the time tables are discussed.

Review each factor of the chart in its entirety. Because the rule of multiplying using the multiple of 1 has already been introduced, it should be reviewed briefly. Move on to the 2s, the 3s, and so forth, in sequence, until the entire time table chart from 1 to 12 has been introduced. Copies of the time table charts should not be used when students are applying the strategy of counting up.

Students tend to rely on the chart to determine the product of multiples rather than applying any type of problem-solving strategy. The charts should not be left in the possession of students, but passed out when the time tables are introduced and then collected at the conclusion of the lesson.

Time Table Chart and Basic Principles of Multiplication

Oral practice drills should be a 5- to 10-minute segment of daily instruction.

Chapter Two

Skip Counting

Now it is time to introduce students to the concept of skip counting by the numbers 2, 3, 5, and 10. Students must learn to add the number with which they are skip counting to each preceding product in sequence, using time table multiples, beginning with 1 and ending with 12. Students will learn to identify only the product of each set of multiples 1–12 multiplied by the table of 2, 3, 5, and 10.

Skip Counting by 2 Time Table of 2

2 × 1 = 2 2 × 4 = 8 2 × 7 = 14 2 × 10 = 20 2 × 2 = 4 2 × 5 = 10 2 × 8 = 16 2 × 11 = 22 2 × 3 = 6 2 × 6 = 11 2 × 9 = 18 2 × 12 = 24

Students will practice repeating the product of each set of multiples from the table of 2 in sequence until it is learned. Students are able to gesture using their fingers as they count aloud.

Practice Sequence

2 4 6 8 10 12 14 16 18 20 22 24

Skip Counting by 3 Time Table of 3

3 × 1 = 3 3 × 4 = 12 3 × 7 = 21 3 × 10 = 30 3 × 2 = 6 3 × 5 = 15 3 × 8 = 24 3 × 11 = 33 3 × 3 = 9 3 × 6 = 18 3 × 9 = 27 3 × 12 = 36

Students will practice repeating the product of each set of multiples from the table of 3 in sequence until it is learned. Students are able to gesture using their fingers as they count aloud.

Practice Sequence

3 6 9 12 15 18 21 24 27 30 33 36

Skip Counting by 5 Time Table of 5

5 × 1 = 5 5 × 4 = 20 5 × 7 = 35 5 × 10 = 50 5 × 2 = 10 5 × 5 = 25 5 × 8 = 40 5 × 11 = 55 5 × 3 = 15 5 × 6 = 30 5 × 9 = 45 5 × 12 = 60

Students will practice repeating the product of each set of multiples from the table of 5 in sequence until it is learned. Students are able to gesture using their fingers as they count aloud. While students may stop at 60, the illustrations continue up to 100.

Practice Sequence

5 10 15 20 25 30 35 40 45 50 55 60 65 70

75 80 85 90 95 100

Skip Counting by 10 Time Table of 10

10 × 1 = 10 10 × 4 = 40 10 × 7 = 70 10 × 10 = 100 10 × 2 = 20 10 × 5 = 50 10 × 8 = 80 10 × 11 = 110 10 × 3 = 30 10 × 6 = 60 10 × 9 = 90 10 × 12 = 120

Students will practice repeating the product of each set of multiples from the table of 10 in sequence until it is learned. Students are able to gesture using their fingers as they count aloud.

Practice Sequence

10 20 30 40 50 60 70 80 90 100 110 120

Chapter Three

Introducing Long Division

After students become familiar with multiplication, division can be introduced. I prefer to teach students long division using pencil and paper. It is very important to introduce students to key terms that are associated with division, such as inverse operations (operations that undo each other). The rules for dividing by 0 and 1 must also be introduced at this time.

Once students have become familiar with the terms associated with division, allow students to begin manual practice by assisting them as they solve several long-division problems using scratch paper and a pencil. Note the example below.

Long Division

6 [square root of 42]

Step 1: Help students determine the target number for the equation. The target number in this equation is 42.

Step 2: Students determine which multiplication table will be used to solve the equation. In this equation, the table of 6 will be used, because it must be determined how many times 6 will divide into 42.

Step 3: Students write out the first multiple of the table of 6, which will be 1. (This step process of writing out the table of 6 will continue until the missing multiple is identified that will produce a product of 42 when multiplied by 6.)

6 × 1 = 6

Students apply the strategy of counting up using their fingers to help them determine the next product, of 6 times 2.

6 × 2 =?

Students begin with the number 6 and count up 6 spaces, raising one finger as each number is counted aloud until 6 spaces have been counted.

6, 7 8 9 10 11 12

Students have counted up 6 spaces from the product of 6 x1 = 6, so they know that 6 × 2 = 12. Students manually list the new product.

6 × 1 = 6 6 × 2 = 12

Now students begin the process again to determine the product of the next factor, which is 3. To determine the new product, students will count 6 spaces up from the product of 12. Students will raise one finger as they count up each space.

6 × 3 =?

12, 13 14 15 16 17 18

Students have counted up 6 spaces from the product of 6 × 2 = 12, so they know that 6 × 3 = 18. Students manually list the new product.

6 × 1 = 6 6 × 2 = 12 6 × 3 = 18

Now students begin the process again to determine the product of the next factor, which is 4. To determine the new product, students will count up 6 spaces from the product of 18. Students will raise one finger as they count up each space.

6 × 4 =

18, 19 20 21 22 23 24

Students have counted up 6 spaces from the product of 6 × 3 = 18, so they know that 6 × 4 = 24. Students manually list the new product.

6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 6 × 4 = 24

Now students begin the process again to determine the product of the next factor, which is 5. To determine the new product, students will count up 6 spaces from the product of 24. Students will raise one finger as they count up each space.

6 × 5 = 19 24, 25 26 27 28 29 30

Students have counted up 6 spaces from the product of 6 × 4 = 24, so they know that 6 × 5 = 30. Students manually list the new product.

6 × 1 = 6 6 × 2 = 12 6 × 3 = 18 6 × 4 = 24 6 × 4 = 30

Now students begin the process again to determine the product of the next factor, which is 6. To determine the new product, students will count up 6 spaces from the product of 30. Students will raise one finger as they count up each space.

6 × 6 =

30, 31 32 33 34 35 36

Students have counted up 6 spaces from the product of 6 × 5 = 30, so they know that 6 × 6 = 36. Students manually list the new product.

6 × 1 = 6 6 × 5 = 30 6 × 2 = 12 6 × 6 = 36 6 × 3 = 18 6 × 4 = 24

Now students begin the process again to determine the product of the next factor, which is 7. To determine the new product, students will count up 6 spaces from the product of 36. Students will raise one finger as they count up each space.

6 × 7 =

36, 37 38 39 40 41 42

Students have counted up 6 spaces from the product of 6 × 6 = 36, so they know that 6 × 7 = 42. Students manually list the new product.

6 × 1 = 6 6 × 5 = 30 6 × 2 = 12 6 × 6 = 36 6 × 3 = 18 6 × 7 = 42 6 × 4 = 24

At this point, help students recognize that the target number of 42 has been identified. 6 × 7 = 42, so 6 will divide into 42, 7 times. Students will manually write out the entire step.

Now explain to students the entire process: 6 × 7 = 42; to determine the remainder, subtract the product from the target number. 42 – 42 is equal to 0 for a remainder.

It is very important to model this step on the chalkboard and with students individually until they grasp the concept

7/6| 42 – 42/0 R

Students will use scratch paper to manually write out the table of 6 as they identify each new product until the target number is reached. Once the target number is reached, students will manually write out each step in the process of solving the long-division equation. The classroom teacher should identify mistakes and areas of weakness by reviewing the scratch paper and reviewing concepts as necessary.

Long Division: Table of 8

72 ÷ 8 =

Step 1: Help students determine the target number for the equation. The target number in this equation is 72.

Step 2: Students determine which multiplication table will be used to solve the equation. In this equation, the table of 8 will be used, because it must be determined how many times 8 will divide into 72.

Step 3: Students write out the first multiple of the table of 8, which will be 1. (This process of writing out the table of 8 will continue until the missing multiple is identified that will produce a product of 72 when multiplied by 8.)

8 × 1 = 8

Step 4: Students apply the strategy of counting up using their fingers to help them determine the next product, of 8 times 2.

8 × 2 =

8, 9 10 11 12 13 14 15 16

Students have counted up 8 spaces from the product of 8 × 1 = 8, so they know that 8 × 2 = 16. Students manually list the new product.

8 × 1 = 8 8 × 2 = 16

Now students begin the process again to determine the product of the next factor, which is 3. To determine the new product, students count up 8 spaces from the product of 16. Students will raise one finger as they count up each space.

8 × 3 =

16, 17 18 19 20 21 22 23 24

Students have counted up 8 spaces from the product of 8 × 2 = 16, so they know that 8 × 3 = 24. Students manually list the new product.

8 × 1 = 8 8 × 2 = 16 8 × 3 = 24

Now students begin the process again to determine the product of the next factor, which is 4. To determine the new product, students count up 8 spaces from the product of 24. Students will raise one finger as they count up each space.

8 × 4 =

24, 25 26 27 28 29 30 31 32

Students have counted up 8 spaces from the product of 8 × 3 = 24, so they know that 8 × 4 = 32. Students manually list the new product.

8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32

Now students begin the process again to determine the product of the next factor, which is 5. To determine the new product, students count up 8 spaces from the product of 32. Students will raise one finger as they count up each space.

8 × 5 =

32, 33 34 35 36 37 38 39 40

Students have counted up 8 spaces from the product of 8 × 4 = 32, so they know that 8 × 5 = 40. Students manually list the new product.

8 × 1 = 8 8 × 5 = 40 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32

Now students begin the process again to determine the product of the next factor, which is 6. To determine the new product, students count up 8 spaces from the product of 40. Students will raise one finger as they count up each space.

8 × 6 =

40, 41 42 43 44 45 46 47 48

Students have counted up 8 spaces from the product of 8 × 5 = 40, so they know that 8 × 6 = 48. Students manually list the new product.

8 × 1 = 8 8 × 5 = 40 8 × 2 = 16 8 × 6 = 48 8 × 3 = 24 8 × 4 = 32

Now students begin the process again to determine the product of the next factor, which is 7. To determine the new product, students count up 8 spaces from the product of 48. Students will raise one finger as they count up each space.

8 × 7 =

48, 49 50 51 52 53 54 55 56

Students have counted up 8 spaces from the product of 8 × 6 = 48, so they know that 8 × 7 = 56. Students manually list the new product.

8 × 1 = 8 8 × 5 = 40 8 × 2 = 16 8 × 6 = 48 8 × 3 = 24 8 × 7 = 56 8 × 4 = 32

Now students begin the process again to determine the product of the next factor, which is 8. To determine the new product, students count up 8 spaces from the product of 56. Students will raise one finger as they count up each space

8 × 8 =

56, 57 58 59 60 61 62 63 64

Students have counted up 8 spaces from the product of 8 × 7 = 56, so they know that 8 × 8 = 64. Students manually list the new product.

8 × 1 = 8 8 × 6 = 48 8 × 2 = 16 8 × 7 = 56 8 × 3 = 24 8 × 8 = 64 8 × 4 = 32 8 × 5 = 40

(Continues...)

Excerpted from Teach Multiplication, Division, and the Time Table All at the Same Timeby Andray McCuien Copyright © 2011 by Andray McCuien. Excerpted by permission of AuthorHouse. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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