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

Lesson 41: Subtracting Across Zero, Missing Factors

In this Grade 4 Saxon Math lesson from Intermediate 4, Chapter 5, students learn how to subtract across zeros by regrouping across multiple place values, such as exchanging hundreds for tens before subtracting. They also practice finding missing factors in multiplication equations like 5n = 40 by using known multiplication facts to identify the unknown value. Both skills are developed through real-world money and measurement problems to build conceptual understanding.

Section 1

📘 Subtracting Across Zero, Missing Factors

New Concept

Numbers that are multiplied are called factors and the answer is the product.

factor×factor=product \text{factor} \times \text{factor} = \text{product}

What’s next

Next, you'll find missing factors in equations like 5n=405n = 40. This is your first step into algebra, a skill for solving puzzles and unlocking unknown values.

Section 2

Subtracting Across Zero

Property

When subtracting from a number with a zero, you must regroup from a higher place value. For example, to solve 405−126405 - 126, you can regroup one hundred into ten tens, making it possible to borrow for the ones place.

Examples

Example: Solve 503−279503 - 279. Regroup the 50 tens into 49 tens and 10 ones, leaving you with 224.

−4913503−279224\begin{align*} &\phantom{-}49_{13} \\ &5\cancel{0}3 \\ - &279 \\ \hline &224 \end{align*}

Example: Janet has 600 dollars and spends 184 dollars. She has 416 dollars left.

−5910600−184416\begin{align*} &\phantom{-}59_{10} \\ &6\cancel{0}0 \\ - &184 \\ \hline &416 \end{align*}

Example: Calculate 8.00 dollars−3.45 dollars8.00 \text{ dollars} - 3.45 \text{ dollars}. This equals 4.55 dollars.

−7.9108.00−3.454.55\begin{align*} &\phantom{-}7.9_{10} \\ &8.\cancel{0}0 \\ - &3.45 \\ \hline &4.55 \end{align*}

Explanation

Stuck with a zero when you need to borrow? Just skip over to the next place value and break it down! It’s like trading a big bill for smaller ones, so you have plenty to share. Now you can finish subtracting with ease!

Section 3

Regrouping In One Step

Property

You can regroup across zero in a single step. Instead of borrowing twice, view the higher place values as a single number. For example, in 405405, see the '40' in the hundreds and tens place as '40 tens'.

Examples

Example: Solve 405−126405 - 126. View 40 tens, borrow 1, leaving 39 tens and 15 ones.

−3915405−126279\begin{align*} &\phantom{-}39_{15} \\ &\cancel{40}5 \\ - &126 \\ \hline &279 \end{align*}

Example: Solve 900−442900 - 442. View 90 tens, borrow 1, leaving 89 tens and 10 ones.

−8910900−442458\begin{align*} &\phantom{-}89_{10} \\ &\cancel{90}0 \\ - &442 \\ \hline &458 \end{align*}

Example: Solve 602−345602 - 345. View 60 tens, borrow 1, leaving 59 tens and 12 ones.

−5912602−345257\begin{align*} &\phantom{-}59_{12} \\ &\cancel{60}2 \\ - &345 \\ \hline &257 \end{align*}

Explanation

Why take two steps when you can take one? Just group the digits together! Think of 405 not as 4-0-5, but as 'forty tens' and 5 ones. Borrowing one ten leaves you with 39 tens, making subtraction super quick and slick!

Section 4

Missing Factors

Property

Recall that numbers that are multiplied are called factors and the answer is the product. If we know one factor and the product, we can find the other factor.

factor×factor=product \text{factor} \times \text{factor} = \text{product}

Examples

Example: Find the missing factor in 5n=405n = 40. Think: '5 times what number equals 40?' Since 5×8=405 \times 8 = 40, the missing factor is n=8n=8.
Example: Find the missing factor in a×4=36a \times 4 = 36. Think: 'What number times 4 equals 36?' Since 9×4=369 \times 4 = 36, the missing factor is a=9a=9.
Example: Find the missing factor in 8w=328w = 32. Think: '8 times what number equals 32?' Since 8×4=328 \times 4 = 32, the missing factor is w=4w=4.

Explanation

Solving for a missing factor is like being a math detective! You know who one of the culprits is (a factor) and what they did together (the product). You just need to figure out who the mystery partner is. Think: 'What times this equals that?'

Book overview

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Chapter 5: Lessons 41–50, Investigation 5

  1. Lesson 1Current

    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 6

    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

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Section 1

📘 Subtracting Across Zero, Missing Factors

New Concept

Numbers that are multiplied are called factors and the answer is the product.

factor×factor=product \text{factor} \times \text{factor} = \text{product}

What’s next

Next, you'll find missing factors in equations like 5n=405n = 40. This is your first step into algebra, a skill for solving puzzles and unlocking unknown values.

Section 2

Subtracting Across Zero

Property

When subtracting from a number with a zero, you must regroup from a higher place value. For example, to solve 405−126405 - 126, you can regroup one hundred into ten tens, making it possible to borrow for the ones place.

Examples

Example: Solve 503−279503 - 279. Regroup the 50 tens into 49 tens and 10 ones, leaving you with 224.

−4913503−279224\begin{align*} &\phantom{-}49_{13} \\ &5\cancel{0}3 \\ - &279 \\ \hline &224 \end{align*}

Example: Janet has 600 dollars and spends 184 dollars. She has 416 dollars left.

−5910600−184416\begin{align*} &\phantom{-}59_{10} \\ &6\cancel{0}0 \\ - &184 \\ \hline &416 \end{align*}

Example: Calculate 8.00 dollars−3.45 dollars8.00 \text{ dollars} - 3.45 \text{ dollars}. This equals 4.55 dollars.

−7.9108.00−3.454.55\begin{align*} &\phantom{-}7.9_{10} \\ &8.\cancel{0}0 \\ - &3.45 \\ \hline &4.55 \end{align*}

Explanation

Stuck with a zero when you need to borrow? Just skip over to the next place value and break it down! It’s like trading a big bill for smaller ones, so you have plenty to share. Now you can finish subtracting with ease!

Section 3

Regrouping In One Step

Property

You can regroup across zero in a single step. Instead of borrowing twice, view the higher place values as a single number. For example, in 405405, see the '40' in the hundreds and tens place as '40 tens'.

Examples

Example: Solve 405−126405 - 126. View 40 tens, borrow 1, leaving 39 tens and 15 ones.

−3915405−126279\begin{align*} &\phantom{-}39_{15} \\ &\cancel{40}5 \\ - &126 \\ \hline &279 \end{align*}

Example: Solve 900−442900 - 442. View 90 tens, borrow 1, leaving 89 tens and 10 ones.

−8910900−442458\begin{align*} &\phantom{-}89_{10} \\ &\cancel{90}0 \\ - &442 \\ \hline &458 \end{align*}

Example: Solve 602−345602 - 345. View 60 tens, borrow 1, leaving 59 tens and 12 ones.

−5912602−345257\begin{align*} &\phantom{-}59_{12} \\ &\cancel{60}2 \\ - &345 \\ \hline &257 \end{align*}

Explanation

Why take two steps when you can take one? Just group the digits together! Think of 405 not as 4-0-5, but as 'forty tens' and 5 ones. Borrowing one ten leaves you with 39 tens, making subtraction super quick and slick!

Section 4

Missing Factors

Property

Recall that numbers that are multiplied are called factors and the answer is the product. If we know one factor and the product, we can find the other factor.

factor×factor=product \text{factor} \times \text{factor} = \text{product}

Examples

Example: Find the missing factor in 5n=405n = 40. Think: '5 times what number equals 40?' Since 5×8=405 \times 8 = 40, the missing factor is n=8n=8.
Example: Find the missing factor in a×4=36a \times 4 = 36. Think: 'What number times 4 equals 36?' Since 9×4=369 \times 4 = 36, the missing factor is a=9a=9.
Example: Find the missing factor in 8w=328w = 32. Think: '8 times what number equals 32?' Since 8×4=328 \times 4 = 32, the missing factor is w=4w=4.

Explanation

Solving for a missing factor is like being a math detective! You know who one of the culprits is (a factor) and what they did together (the product). You just need to figure out who the mystery partner is. Think: 'What times this equals that?'

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 1Current

    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 6

    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

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  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