Learn on PengiSaxon Math, Course 1Chapter 6: Geometry and Number Operations

Lesson 54: Reducing by Grouping Factors Equal to 1

In this Grade 6 Saxon Math Course 1 lesson, students learn two key skills: reducing fractions by grouping factors equal to 1 using prime factorization, and dividing fractions by multiplying by the reciprocal of the divisor. The lesson walks through canceling common factors in numerators and denominators and applying the two-step process for fraction division, such as solving problems like three-fourths divided by one-half.

Section 1

📘 Reducing by Grouping Factors Equal to 1 and Dividing Fractions

New Concept

To divide by a fraction, you multiply by its reciprocal. This process answers the question of how many times the divisor fits into the dividend.

When the divisor is a fraction, we take two steps to find the answer. We first find how many of the divisors are in 1. This is the reciprocal of the divisor. Then we use the reciprocal to answer the original division problem by multiplying.

What’s next

You've just learned the core logic. Next, we’ll apply this two-step division process and use factor grouping in worked examples to simplify the results.

Section 2

Reducing by Grouping Factors Equal to 1

Property

To reduce a fraction, find common factors in the numerator and denominator. Since any number divided by itself is 1, these common factors can be grouped and canceled out.

Examples

2⋅2⋅3⋅52⋅2⋅7=2⋅2⋅3⋅52⋅2⋅7=157 \frac{2 \cdot 2 \cdot 3 \cdot 5}{2 \cdot 2 \cdot 7} = \frac{\cancel{2} \cdot \cancel{2} \cdot 3 \cdot 5}{\cancel{2} \cdot \cancel{2} \cdot 7} = \frac{15}{7}
3⋅5⋅72⋅3⋅5=3⋅5⋅72⋅3⋅5=72 \frac{3 \cdot 5 \cdot 7}{2 \cdot 3 \cdot 5} = \frac{\cancel{3} \cdot \cancel{5} \cdot 7}{2 \cdot \cancel{3} \cdot \cancel{5}} = \frac{7}{2}
2⋅3⋅3⋅113⋅3⋅5=2⋅3⋅3⋅113⋅3⋅5=225 \frac{2 \cdot 3 \cdot 3 \cdot 11}{3 \cdot 3 \cdot 5} = \frac{2 \cdot \cancel{3} \cdot \cancel{3} \cdot 11}{\cancel{3} \cdot \cancel{3} \cdot 5} = \frac{22}{5}

Explanation

Think of this as a math treasure hunt for the number 1! By breaking the top and bottom of a fraction into their smallest factors, you can spot pairs that are identical. Since any number divided by itself equals one, you can cancel them out. This simplifies the fraction to its core value, making big, scary fractions easy to handle.

Section 3

Dividing Fractions

Property

To divide by a fraction, multiply by its reciprocal. This is often remembered as 'invert and multiply'.

ab÷cd=ab×dc \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Examples

23÷14=23×41=83 \frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}
12÷35=12×53=56 \frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}
47÷23=47×32=1214=67 \frac{4}{7} \div \frac{2}{3} = \frac{4}{7} \times \frac{3}{2} = \frac{12}{14} = \frac{6}{7}

Explanation

Dividing fractions seems tricky, but it’s just asking 'how many of the second fraction fit into the first?' The secret is to flip the second fraction (find its reciprocal) and multiply. This works because you first find how many of your divisor-sized pieces fit into one whole, and then you multiply to see how many fit into your original amount.

Book overview

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Chapter 6: Geometry and Number Operations

  1. Lesson 1

    Lesson 51: Rounding Decimal Numbers

    LearnPractice
  2. Lesson 2

    Lesson 52: Mentally Dividing Decimal Numbers by 10 and by 100

    LearnPractice
  3. Lesson 3

    Lesson 53: Decimals Chart

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  4. Lesson 4Current

    Lesson 54: Reducing by Grouping Factors Equal to 1

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  5. Lesson 5

    Lesson 55: Common Denominators, Part 1

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  6. Lesson 6

    Lesson 56: Common Denominators, Part 2

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  7. Lesson 7

    Lesson 57: Adding and Subtracting Fractions: Three Steps

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  8. Lesson 8

    Lesson 58: Probability and Chance

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  9. Lesson 9

    Lesson 59: Adding Mixed Numbers

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  10. Lesson 10

    Lesson 60: Polygons

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  11. Lesson 11

    Investigation 6: Attributes of Geometric Solids

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Lesson overview

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Expand

Section 1

📘 Reducing by Grouping Factors Equal to 1 and Dividing Fractions

New Concept

To divide by a fraction, you multiply by its reciprocal. This process answers the question of how many times the divisor fits into the dividend.

When the divisor is a fraction, we take two steps to find the answer. We first find how many of the divisors are in 1. This is the reciprocal of the divisor. Then we use the reciprocal to answer the original division problem by multiplying.

What’s next

You've just learned the core logic. Next, we’ll apply this two-step division process and use factor grouping in worked examples to simplify the results.

Section 2

Reducing by Grouping Factors Equal to 1

Property

To reduce a fraction, find common factors in the numerator and denominator. Since any number divided by itself is 1, these common factors can be grouped and canceled out.

Examples

2⋅2⋅3⋅52⋅2⋅7=2⋅2⋅3⋅52⋅2⋅7=157 \frac{2 \cdot 2 \cdot 3 \cdot 5}{2 \cdot 2 \cdot 7} = \frac{\cancel{2} \cdot \cancel{2} \cdot 3 \cdot 5}{\cancel{2} \cdot \cancel{2} \cdot 7} = \frac{15}{7}
3⋅5⋅72⋅3⋅5=3⋅5⋅72⋅3⋅5=72 \frac{3 \cdot 5 \cdot 7}{2 \cdot 3 \cdot 5} = \frac{\cancel{3} \cdot \cancel{5} \cdot 7}{2 \cdot \cancel{3} \cdot \cancel{5}} = \frac{7}{2}
2⋅3⋅3⋅113⋅3⋅5=2⋅3⋅3⋅113⋅3⋅5=225 \frac{2 \cdot 3 \cdot 3 \cdot 11}{3 \cdot 3 \cdot 5} = \frac{2 \cdot \cancel{3} \cdot \cancel{3} \cdot 11}{\cancel{3} \cdot \cancel{3} \cdot 5} = \frac{22}{5}

Explanation

Think of this as a math treasure hunt for the number 1! By breaking the top and bottom of a fraction into their smallest factors, you can spot pairs that are identical. Since any number divided by itself equals one, you can cancel them out. This simplifies the fraction to its core value, making big, scary fractions easy to handle.

Section 3

Dividing Fractions

Property

To divide by a fraction, multiply by its reciprocal. This is often remembered as 'invert and multiply'.

ab÷cd=ab×dc \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Examples

23÷14=23×41=83 \frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}
12÷35=12×53=56 \frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}
47÷23=47×32=1214=67 \frac{4}{7} \div \frac{2}{3} = \frac{4}{7} \times \frac{3}{2} = \frac{12}{14} = \frac{6}{7}

Explanation

Dividing fractions seems tricky, but it’s just asking 'how many of the second fraction fit into the first?' The secret is to flip the second fraction (find its reciprocal) and multiply. This works because you first find how many of your divisor-sized pieces fit into one whole, and then you multiply to see how many fit into your original amount.

Book overview

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

Continue this chapter

Chapter 6: Geometry and Number Operations

  1. Lesson 1

    Lesson 51: Rounding Decimal Numbers

    LearnPractice
  2. Lesson 2

    Lesson 52: Mentally Dividing Decimal Numbers by 10 and by 100

    LearnPractice
  3. Lesson 3

    Lesson 53: Decimals Chart

    LearnPractice
  4. Lesson 4Current

    Lesson 54: Reducing by Grouping Factors Equal to 1

    LearnPractice
  5. Lesson 5

    Lesson 55: Common Denominators, Part 1

    LearnPractice
  6. Lesson 6

    Lesson 56: Common Denominators, Part 2

    LearnPractice
  7. Lesson 7

    Lesson 57: Adding and Subtracting Fractions: Three Steps

    LearnPractice
  8. Lesson 8

    Lesson 58: Probability and Chance

    LearnPractice
  9. Lesson 9

    Lesson 59: Adding Mixed Numbers

    LearnPractice
  10. Lesson 10

    Lesson 60: Polygons

    LearnPractice
  11. Lesson 11

    Investigation 6: Attributes of Geometric Solids

    LearnPractice