Learn on PengiSaxon Math, Course 1Chapter 2: Problem Solving with Number and Operations

Lesson 11: Problems About Combining & Separating

In this Grade 6 Saxon Math Course 1 lesson, students learn to identify and solve word problems using the addition pattern for combining (some + some more = total) and the subtraction pattern for separating (beginning amount − some went away = what remains). Students apply a four-step problem-solving process to write equations, find unknown values, and check their answers using real-world contexts. This lesson is part of Chapter 2 and builds foundational skills for translating word problem plots into algebraic equations.

Section 1

📘 Problems About Combining • Problems About Separating

New Concept

We can use the plot of a word problem to write an equation for the problem.

Why it matters

Translating word problems into equations is the fundamental skill of algebra, turning stories into solvable models. This allows you to predict outcomes and analyze systems in everything from engineering to economics.

What’s next

Next, you will apply this by identifying 'combining' and 'separating' plots to build and solve your first algebraic equations.

Section 2

Problems About Combining

Property

Problems about combining have an addition pattern:

Some+some more=total \text{Some} + \text{some more} = \text{total}
s+m=t s + m = t

Examples

Leo had 15 comics and his friend gave him some more. Now he has 40 comics. How many did he get? 15+c=4015 + c = 40, so c=40−15=25c = 40 - 15 = 25 comics.
You saved 50 dollars for a new game. After your birthday, you have 95 dollars. How much did you get? 50+m=9550 + m = 95, so m=95−50=45m = 95 - 50 = 45 dollars.
A baker had 8 cups of flour and bought more. Now she has 20 cups. How much flour did she buy? 8+f=208 + f = 20, so f=20−8=12f = 20 - 8 = 12 cups.

Explanation

Think of this like leveling up in a game! You start with some experience points, earn more on a quest, and end up with a new total. To figure out how many points you earned, you just subtract your starting amount from your final total. This pattern helps you find the missing piece in any combining story.

Section 3

Problems About Separating

Property

Problems about separating have a subtraction pattern:

Beginning amount−some went away=what remains \text{Beginning amount} - \text{some went away} = \text{what remains}
b−a=r b - a = r

Examples

A bus started with 35 passengers. At a stop, some got off, leaving 22. How many got off? 35−p=2235 - p = 22, so p=35−22=13p = 35 - 22 = 13 people.
You had 40 dollars and bought a gift. You now have 18 dollars left. How much was the gift? 40−g=1840 - g = 18, so g=40−18=22g = 40 - 18 = 22 dollars.
A jar contained 50 cookies. After a party, 15 cookies were left. How many cookies were eaten? 50−c=1550 - c = 15, so c=50−15=35c = 50 - 15 = 35 cookies.

Explanation

Picture a pizza with 8 slices. After you and your friends eat, only 2 slices are left. The separating pattern is your detective tool to find how many slices vanished! By knowing the starting amount and what remains, you can always solve for the part that went away. It makes you a master of subtraction mysteries.

Book overview

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Chapter 2: Problem Solving with Number and Operations

  1. Lesson 1Current

    Lesson 11: Problems About Combining & Separating

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  2. Lesson 2

    Lesson 12: Place Value Through Trillions

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  3. Lesson 3

    Lesson 13: Problems About Comparing

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  4. Lesson 4

    Lesson 14: The Number Line: Negative Numbers

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  5. Lesson 5

    Lesson 15: Problems About Equal Groups

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  6. Lesson 6

    Lesson 16: Rounding Whole Numbers

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  7. Lesson 7

    Lesson 17: The Number Line: Fractions and Mixed Numbers

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  8. Lesson 8

    Lesson 18: Average

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  9. Lesson 9

    Lesson 19: Factors

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  10. Lesson 10

    Lesson 20: Greatest Common Factor (GCF)

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  11. Lesson 11

    Investigation 2: Investigating Fractions with Manipulatives

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Lesson overview

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Section 1

📘 Problems About Combining • Problems About Separating

New Concept

We can use the plot of a word problem to write an equation for the problem.

Why it matters

Translating word problems into equations is the fundamental skill of algebra, turning stories into solvable models. This allows you to predict outcomes and analyze systems in everything from engineering to economics.

What’s next

Next, you will apply this by identifying 'combining' and 'separating' plots to build and solve your first algebraic equations.

Section 2

Problems About Combining

Property

Problems about combining have an addition pattern:

Some+some more=total \text{Some} + \text{some more} = \text{total}
s+m=t s + m = t

Examples

Leo had 15 comics and his friend gave him some more. Now he has 40 comics. How many did he get? 15+c=4015 + c = 40, so c=40−15=25c = 40 - 15 = 25 comics.
You saved 50 dollars for a new game. After your birthday, you have 95 dollars. How much did you get? 50+m=9550 + m = 95, so m=95−50=45m = 95 - 50 = 45 dollars.
A baker had 8 cups of flour and bought more. Now she has 20 cups. How much flour did she buy? 8+f=208 + f = 20, so f=20−8=12f = 20 - 8 = 12 cups.

Explanation

Think of this like leveling up in a game! You start with some experience points, earn more on a quest, and end up with a new total. To figure out how many points you earned, you just subtract your starting amount from your final total. This pattern helps you find the missing piece in any combining story.

Section 3

Problems About Separating

Property

Problems about separating have a subtraction pattern:

Beginning amount−some went away=what remains \text{Beginning amount} - \text{some went away} = \text{what remains}
b−a=r b - a = r

Examples

A bus started with 35 passengers. At a stop, some got off, leaving 22. How many got off? 35−p=2235 - p = 22, so p=35−22=13p = 35 - 22 = 13 people.
You had 40 dollars and bought a gift. You now have 18 dollars left. How much was the gift? 40−g=1840 - g = 18, so g=40−18=22g = 40 - 18 = 22 dollars.
A jar contained 50 cookies. After a party, 15 cookies were left. How many cookies were eaten? 50−c=1550 - c = 15, so c=50−15=35c = 50 - 15 = 35 cookies.

Explanation

Picture a pizza with 8 slices. After you and your friends eat, only 2 slices are left. The separating pattern is your detective tool to find how many slices vanished! By knowing the starting amount and what remains, you can always solve for the part that went away. It makes you a master of subtraction mysteries.

Book overview

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

Continue this chapter

Chapter 2: Problem Solving with Number and Operations

  1. Lesson 1Current

    Lesson 11: Problems About Combining & Separating

    LearnPractice
  2. Lesson 2

    Lesson 12: Place Value Through Trillions

    LearnPractice
  3. Lesson 3

    Lesson 13: Problems About Comparing

    LearnPractice
  4. Lesson 4

    Lesson 14: The Number Line: Negative Numbers

    LearnPractice
  5. Lesson 5

    Lesson 15: Problems About Equal Groups

    LearnPractice
  6. Lesson 6

    Lesson 16: Rounding Whole Numbers

    LearnPractice
  7. Lesson 7

    Lesson 17: The Number Line: Fractions and Mixed Numbers

    LearnPractice
  8. Lesson 8

    Lesson 18: Average

    LearnPractice
  9. Lesson 9

    Lesson 19: Factors

    LearnPractice
  10. Lesson 10

    Lesson 20: Greatest Common Factor (GCF)

    LearnPractice
  11. Lesson 11

    Investigation 2: Investigating Fractions with Manipulatives

    LearnPractice