Learn on PengiSaxon Math, Intermediate 4Chapter 5: Lessons 41–50, Investigation 5

Lesson 46: Relating Multiplication and Division, Part 1, Activity Using a Multiplication Table to Divide

In this Grade 4 Saxon Math Intermediate 4 lesson, students learn how multiplication and division are inverse operations by using a multiplication table to find missing factors and solve division problems. They practice writing division equations with a division box and apply the concept of "undoing" multiplication to divide numbers such as 32 ÷ 4 and 18 ÷ 2. The lesson builds foundational understanding of factors, products, and the relationship between the two operations.

Section 1

📘 Relating Multiplication and Division, Part 1

New Concept

Multiplication and division are inverse operations. One operation undoes the other.

Why it matters

Understanding inverse operations is your first step toward mastering algebraic thinking, where you'll solve for unknown values in complex equations. This single principle of 'undoing' operations is the key to manipulating formulas in physics, finance, and computer science.

What’s next

Next, you’ll use this inverse relationship to solve division problems by thinking of them as finding a missing factor in multiplication.

Section 2

Factors and product

Property

Multiplication involves three key numbers. The numbers you multiply together are called factors, and the result is the product. So, if you know your two factors, your mission is to multiply them to find the final product. The fundamental relationship is always expressed as:

Factor×Factor=Product\text{Factor} \times \text{Factor} = \text{Product}

Example

In the equation 4×3=124 \times 3 = 12, the numbers 4 and 3 are the factors, and 12 is the product. If you multiply the factors 8 and 5, you get the product 40, as shown by 8×5=408 \times 5 = 40. For 7×6=427 \times 6 = 42, the factors are 7 and 6, while the product is 42.

Explanation

Think of factors as secret ingredients and the product as the cake you bake. When you multiply the ingredients, you get the final creation. Every multiplication problem is just a recipe for finding the tasty result. It’s as simple as combining what you have to see what you get!

Section 3

Division

Property

Division is the tool you use to find a missing factor when you already know the other factor and the product. It’s the reverse process of multiplication, effectively 'undoing' it to solve for an unknown. If you have a problem like 4×w=124 \times w = 12, you use division, written as

43)12‾\frac{4}{3\overline{)12}}
, to find the missing factor is 4.

Example

To find the missing factor in n×6=54n \times 6 = 54, you perform the division 54÷6=954 \div 6 = 9. If a teacher divides 20 students into 4 teams, you calculate 4)20‾4\overline{)20} to find there are 5 students per team. The problem 8×?=728 \times ? = 72 is solved by division: 72÷8=972 \div 8 = 9.

Explanation

Imagine you have a completed puzzle (the product) but one piece is missing (a factor). Division is your detective tool to figure out exactly what that missing piece looks like. It works backward from the answer to find the parts that made it up in the first place.

Section 4

Inverse operations

Property

Multiplication and division are a dynamic duo known as inverse operations. This means that one operation can completely undo the other. If you perform a multiplication, you can use division to get right back to where you started, and vice versa. It’s a mathematical round trip that always brings you back home, demonstrating their perfectly balanced relationship.

Example

The inverse of the multiplication fact 5×9=455 \times 9 = 45 is the division fact 45÷9=545 \div 9 = 5. If you solve 28÷7=428 \div 7 = 4, the inverse operation to check your work is 4×7=284 \times 7 = 28. Starting with 12, multiplying by 3 gives 36; its inverse, 36÷336 \div 3, returns you to 12.

Explanation

Think of them as a 'do' and 'undo' button. Multiplication builds a tower with blocks, and division is the 'undo' button that takes the tower apart to show you the original blocks. They are perfect opposites that always keep each other in check, making math predictable and fun!

Book overview

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Chapter 5: Lessons 41–50, Investigation 5

  1. Lesson 1

    Lesson 41: Subtracting Across Zero, Missing Factors

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  2. Lesson 2

    Lesson 42: Rounding Numbers to Estimate

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  3. Lesson 3

    Lesson 43: Adding and Subtracting Decimal Numbers, Part 1, Activity Adding and Subtracting Decimals

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  4. Lesson 4

    Lesson 44: Multiplying Two-Digit Numbers, Part 1

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  5. Lesson 5

    Lesson 45: Parentheses and the Associative Property, Naming Lines and Segments

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  6. Lesson 6Current

    Lesson 46: Relating Multiplication and Division, Part 1, Activity Using a Multiplication Table to Divide

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  7. Lesson 7

    Lesson 47: Relating Multiplication and Division, Part 2

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  8. Lesson 8

    Lesson 48: Multiplying Two-Digit Numbers, Part 2

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  9. Lesson 9

    Lesson 49: Word Problems About Equal Groups, Part 1

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  10. Lesson 10

    Lesson 50: Adding and Subtracting Decimal Numbers, Part 2, Activity Adding and Subtracting Decimals

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  11. Lesson 11

    Investigation 5: Percents, Activity Percent

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Relating Multiplication and Division, Part 1

New Concept

Multiplication and division are inverse operations. One operation undoes the other.

Why it matters

Understanding inverse operations is your first step toward mastering algebraic thinking, where you'll solve for unknown values in complex equations. This single principle of 'undoing' operations is the key to manipulating formulas in physics, finance, and computer science.

What’s next

Next, you’ll use this inverse relationship to solve division problems by thinking of them as finding a missing factor in multiplication.

Section 2

Factors and product

Property

Multiplication involves three key numbers. The numbers you multiply together are called factors, and the result is the product. So, if you know your two factors, your mission is to multiply them to find the final product. The fundamental relationship is always expressed as:

Factor×Factor=Product\text{Factor} \times \text{Factor} = \text{Product}

Example

In the equation 4×3=124 \times 3 = 12, the numbers 4 and 3 are the factors, and 12 is the product. If you multiply the factors 8 and 5, you get the product 40, as shown by 8×5=408 \times 5 = 40. For 7×6=427 \times 6 = 42, the factors are 7 and 6, while the product is 42.

Explanation

Think of factors as secret ingredients and the product as the cake you bake. When you multiply the ingredients, you get the final creation. Every multiplication problem is just a recipe for finding the tasty result. It’s as simple as combining what you have to see what you get!

Section 3

Division

Property

Division is the tool you use to find a missing factor when you already know the other factor and the product. It’s the reverse process of multiplication, effectively 'undoing' it to solve for an unknown. If you have a problem like 4×w=124 \times w = 12, you use division, written as

43)12‾\frac{4}{3\overline{)12}}
, to find the missing factor is 4.

Example

To find the missing factor in n×6=54n \times 6 = 54, you perform the division 54÷6=954 \div 6 = 9. If a teacher divides 20 students into 4 teams, you calculate 4)20‾4\overline{)20} to find there are 5 students per team. The problem 8×?=728 \times ? = 72 is solved by division: 72÷8=972 \div 8 = 9.

Explanation

Imagine you have a completed puzzle (the product) but one piece is missing (a factor). Division is your detective tool to figure out exactly what that missing piece looks like. It works backward from the answer to find the parts that made it up in the first place.

Section 4

Inverse operations

Property

Multiplication and division are a dynamic duo known as inverse operations. This means that one operation can completely undo the other. If you perform a multiplication, you can use division to get right back to where you started, and vice versa. It’s a mathematical round trip that always brings you back home, demonstrating their perfectly balanced relationship.

Example

The inverse of the multiplication fact 5×9=455 \times 9 = 45 is the division fact 45÷9=545 \div 9 = 5. If you solve 28÷7=428 \div 7 = 4, the inverse operation to check your work is 4×7=284 \times 7 = 28. Starting with 12, multiplying by 3 gives 36; its inverse, 36÷336 \div 3, returns you to 12.

Explanation

Think of them as a 'do' and 'undo' button. Multiplication builds a tower with blocks, and division is the 'undo' button that takes the tower apart to show you the original blocks. They are perfect opposites that always keep each other in check, making math predictable and fun!

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 5: Lessons 41–50, Investigation 5

  1. Lesson 1

    Lesson 41: Subtracting Across Zero, Missing Factors

    LearnPractice
  2. Lesson 2

    Lesson 42: Rounding Numbers to Estimate

    LearnPractice
  3. Lesson 3

    Lesson 43: Adding and Subtracting Decimal Numbers, Part 1, Activity Adding and Subtracting Decimals

    LearnPractice
  4. Lesson 4

    Lesson 44: Multiplying Two-Digit Numbers, Part 1

    LearnPractice
  5. Lesson 5

    Lesson 45: Parentheses and the Associative Property, Naming Lines and Segments

    LearnPractice
  6. Lesson 6Current

    Lesson 46: Relating Multiplication and Division, Part 1, Activity Using a Multiplication Table to Divide

    LearnPractice
  7. Lesson 7

    Lesson 47: Relating Multiplication and Division, Part 2

    LearnPractice
  8. Lesson 8

    Lesson 48: Multiplying Two-Digit Numbers, Part 2

    LearnPractice
  9. Lesson 9

    Lesson 49: Word Problems About Equal Groups, Part 1

    LearnPractice
  10. Lesson 10

    Lesson 50: Adding and Subtracting Decimal Numbers, Part 2, Activity Adding and Subtracting Decimals

    LearnPractice
  11. Lesson 11

    Investigation 5: Percents, Activity Percent

    LearnPractice