problem solving subtraction lesson 5 9

Lesson 5.9 Problem Solving-Subtraction

By betsy schuman on feb 17, 2014.

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Subtraction Lesson

• Demonstrate an understanding of place value
• Learn how to subtract multi-digit whole numbers
• Learn how to subtract multi-digit whole numbers with regrouping (borrowing)

How to Subtract Multi-Digit Whole Numbers

Vertical subtraction - multi-digit subtraction without regrouping (borrowing).

• Arrange the numbers vertically and by place value, the minuend will be on top and the subtrahend will be on the bottom. Remember, subtraction is not commutative, so the order does matter here.
• Draw a horizontal line underneath the bottom number and place a "-" to the left of the bottom number
• Subtract the digits in the rightmost column, this will contain the ones' place digit for each number. We subtract the bottom number away from the top number.
• Place the result directly below the horizontal line
• Repeat the subtraction process in each column moving left until there are no more columns to subtract

Vertical Subtraction - Multi-Digit Subtraction with Regrouping (Borrowing)

• In this case, the subtraction in the ones' column provides an issue. We can't subtract 1 - 5 since 1 is not large enough to take 5 away. When this occurs, we can borrow from the next digit left. 71 = 7 tens + 1 ones. If we borrow 1 ten or 10 ones from the 7, we could rewrite the number 71 as: 6 tens + 11 ones. This would allow us to subtract in the ones' column since 11 is larger than 5.
• Now we can subtract 11 - 5

Skills Check:

Find each difference.

Please choose the best answer.

5,616 - 2,796

Congrats, Your Score is 100 %

Better Luck Next Time, Your Score is %

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Addition and Subtraction  - Subtracting Two- and Three-Digit Numbers

Addition and subtraction  -, subtracting two- and three-digit numbers, addition and subtraction subtracting two- and three-digit numbers.

Addition and Subtraction: Subtracting Two- and Three-Digit Numbers

Lesson 5: subtracting two- and three-digit numbers.

/en/additionsubtraction/introduction-to-subtraction/content/

Subtracting larger numbers

In Introduction to Subtraction , we learned that counting and using visuals can be useful for solving basic subtraction problems. For instance, say you have 9 apples and you use 6 to make a pie. To find out how many apples are left, you could represent the situation like this:

It's easy to count and see that 3 apples are left.

What if you need to solve a subtraction problem that starts with a large number? For instance, let's say instead of making an apple pie, you want to pick apples from an apple tree. The tree has 30 apples and you pick 21 . We could write this as 30 - 21 .

You might see why counting to solve this problem isn't a good idea. When you have a subtraction problem that starts with a large number, it could take a long time to set up the problem. Imagine the time it would take to count out 30 objects and then take away 21! Also, it would be easy to lose track as you counted. You could end up with the wrong answer.

For this reason, when people solve a subtraction problem with large numbers, they set up the problem in a way that makes it easy to solve one step at a time. Let's see how this works with another problem: 79 - 13 .

In the last lesson, we learned how to write expressions. However, subtracting with larger numbers is easier when the expressions are written in a different way.

Instead of writing the numbers side by side…

Place the numbers so they are stacked — one number on top and one number on the bottom.

With a stacked subtraction expression, the larger number is always written on top. Here, that number is 79 .

Write the amount being subtracted underneath the top number. That's 13 .

Put the minus sign to the left of the numbers.

Instead of an equals sign, put a line underneath the bottom number.

When you stack a subtraction expression, make sure the numbers are lined up correctly. They are always lined up on the right. Here, we lined up 9 and 3 .

Here's another problem, 576 - 2 . With this problem, see how we lined up the numbers to the right?

No matter how many digits are in the numbers, always line up the numbers to the right.

Solving Stacked Subtraction Problems

If you feel comfortable with the subtraction skills from Introduction to Subtraction , you're ready to start solving stacked subtraction problems.

Let's try to solve 49 - 7 .

With all stacked subtraction problems, we start with the digits that are farthest to the right. Here, we'll begin with 9 and 7 .

9 - 7 = 2 . The difference is 2 . It's important to write 2 directly beneath the digits we just subtracted.

Now let's find the difference of the digits to the left. The top digit is 4 , but there's nothing beneath it.

4 minus nothing is 4 , so we'll write 4 beneath the line.

Our result is 42 . 49 - 7 = 42 .

Let's see how this works with another problem: 88 - 62 .

As always, start with the digits that are farthest to the right. Here, they are 8 and 2 .

8 - 2 = 6 . Make sure to write 6 below the line.

Next, find the difference of the digits to the left, 8 and 6 .

8 - 6 is 2 . Write 2 below the line.

The answer is 26 . 88 - 62 = 26 .

In the slideshow, you saw that stacked subtraction problems are always solved from right to left . The expressions below are solved the same way. First, the bottom right digit is subtracted from the top right digit. Then, the bottom left digit is subtracted from the top left digit.

Stack these subtraction problems and solve them. Then, check your answer by typing it into the box.

Subtracting Larger Numbers

Stacked subtraction can also be used for finding the difference of larger numbers. No matter how many digits there are, you subtract the same way every time — from right to left.

These subtraction problems have larger numbers. Solve them, and then check your answer by typing it into the box.

Sometimes when you subtract, you will notice that the top digit is smaller than the bottom. For example, take a look at this problem:

Normally, we'd start on the right with 5 - 9. However, since 9 is bigger than 5, we can't subtract normally. Instead, we have to use a technique called borrowing .

Let's see how it works.

First, we'll make sure the expression is set up correctly. The larger number is stacked on top of the smaller number.

As with all stacked subtraction problems, begin with the digits farthest to the right. Here, they are 5 and 9 .

5 is smaller than 9 , so we'll need to borrow to make 5 larger.

We'll borrow from the digit to the left of 5 . Here, it's 7 . We'll take 1 from it....

7 - 1 = 6 . To help us remember that we subtracted 1, we'll cross out the 7 and write 6 above it.

Then, we'll place the 1 we took next to the 5 ...

5 becomes 15 . See how it looks like 15?

15 is larger than 9, which means we can subtract. We'll solve for 15 - 9 .

15 - 9 = 6 . We'll write 6 beneath the line.

Next, find the difference of the digits to the left: 6 - 2 .

6 - 2 = 4 . We'll write 4 beneath the line.

Our answer is 46 . 75 - 29 = 46 .

As you borrow, always cross out the digit you borrow from and write the new value above it. Remember to always place the 1 next to the smaller digit.

Try these problems to practice borrowing. Check your answer by typing it into the box.

Borrowing More Than Once

Sometimes the top number might have two or more digits that are smaller than the digits beneath them. In that case, you'll need to borrow more than once. It will always work the same way. You'll always subtract 1 from the digit to the left and place 1 next to the smaller digit.

In some cases, you might notice that the number to the left is zero. Check out the slideshow below to see an example of what to do.

Let's look at the example 300 minus 54. We would begin on the right with 0 minus 4 . However, zero is smaller than 4, so we would need to borrow from the next digit to the left.

The next digit to the left, however, is zero ! We can't borrow if nothing is there. So what do we do?

We have to go to the next digit to the left. Think of it like asking your neighbor for a cup of sugar. If the first neighbor doesn't have any, you would move to the next neighbor over to ask for some to borrow.

Since the next number over is 3 , we'll borrow from that.

Just like when we borrow normally, we'll subtract 1 from 3 to make it 2 . We'll place the 1 next to the number on the right to make it 10 .

Remember though, we originally needed to borrow in order to do 0 minus 4 . Now that we have 10 in the middle, we can borrow from it.

Cross out the 10 and subtract 1 to make it 9 .

Then, place the 1 next to the 0 in order to make it 10 . Now you're ready to subtract.

10 minus 4 is 6.

9 minus 5 is 4.

There is nothing to subtract from the 2, so we just bring it down, and we're finished!

The answer is 246 .

Try solving these subtraction problems to practice borrowing more than one time. Check your answer by typing it in the box.

Checking Your Work

In the last few lessons, you learned how to solve addition and subtraction problems. As you practice these math skills, it's a good idea to get into the habit of checking your work . Checking will help you know if your answers are correct. When you're ready to check the answer to subtraction problems, you'll need to use addition.

Let's look at this problem: 9 - 7 = 2 .

How do we know that 2 is the correct answer? We can check by adding.

Let's set up our addition problem. First, we'll write the subtraction problem's answer. That means we'll write 2 .

Next, we'll add the amount that was subtracted, 7 .

Time to add. 2 + 7 = 9 .

If we subtracted correctly, the answer will match the larger number in our subtraction problem.

They match — 9 and 9 . Our answer was correct.

Let's try using addition to check the answer to another subtraction problem: 54 - 21 = 33 .

Let's set up our addition problem. First write the answer to the subtraction problem, 33 .

Then add back the number that was subtracted, 21 .

Now it's time to add. 33 + 21 = 54 .

Finally, we'll check to see if 54 matches the larger number in our subtraction problem. It does!

Practice subtracting these problems. You'll have to use borrowing to solve some of the problems. There are 4 sets of problems with 3 problems each.

/en/additionsubtraction/video-subtraction/content/

Activity: Addition and subtraction

2. Try the rounding challenge

Rounding is an important skill your child will be expected to use to estimate and to check answers to calculations.

You could help your child to practise rounding with the calculations below. For each sum, you and your child could round the numbers to the nearest 10, 100, or 1000 in order to estimate the answer. Which one of you can get the estimate fastest? Then ask your child to work out the correct answer. Whose estimate was closest to the real answer? You can add an extra element of challenge by using a stopwatch or timer!

3567 + 2332 6102 + 2923 1345 + 3348 8799 – 4889 9833 – 5541

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Course: 5th grade   >   Unit 3

Subtracting decimals: 9.57-8.09.

• Subtracting decimals: 10.1-3.93
• Subtraction strategies with hundredths
• More advanced subtraction strategies with hundredths
• Subtract decimals < 1 (hundredths)
• Subtract decimals and whole numbers (hundredths)
• Subtract decimals (hundredths)
• Subtract decimals: FAQ

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Video transcript

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Curriculum  /  Math  /  5th Grade  /  Unit 4: Addition and Subtraction of Fractions/Decimals  /  Lesson 9

Addition and Subtraction of Fractions/Decimals

Lesson 9 of 15

Criteria for Success

Tips for teachers, anchor tasks.

Problem Set

Target Task

Additional practice.

Subtract fractions from fractions greater than 2 with unlike denominators.

Common Core Standards

Core standards.

The core standards covered in this lesson

Number and Operations—Fractions

5.NF.A.1 — Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

5.NF.A.2 — Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Foundational Standards

The foundational standards covered in this lesson

4.NF.A.1 — Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

4.NF.A.2 — Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Find common units for fractions with unlike denominators by finding equivalent fractions using multiplication or division. 
• Understand that there is more than one possibility for the common unit used, and use that to optionally find the least common denominator. 
• Subtract two fractions, including mixed numbers, with unlike denominators that require regrouping whose whole is greater than 2, simplifying and writing the sum as a mixed number, if applicable.
• Assess the reasonableness of an answer using number sense and estimation (MP.1).
• Solve one-step word problems involving the subtraction of two fractions with unlike denominators whose whole is more than 2 (MP.4).

Suggestions for teachers to help them teach this lesson

• “Calculations with mixed numbers provide opportunities for students to compare approaches and justify steps in their computations (MP.3)” (NF Progression, p. 13). In general, given the Grade 4 instruction on this content, it’s unlikely that students will rewrite mixed numbers as “improper” fractions and subtract but instead will regroup just one whole and subtract (which is “an analogue of what students learned when…subtracting numbers...: decomposing a unit of the minuend into small used… Instead of decomposing a ten into 10 ones…, a one [is] decomposed into” fractional units, such as 3 thirds) (NF Progression, p. 13). For the problems that require regrouping, you should at least go through the strategies of (1) regrouping a whole to subtract (e.g., $$9\frac{1}{12}-\frac{7}{12}=8\frac{13}{12}-\frac{7}{12}=8\frac{6}{12}$$ ) and (2) subtracting the wholes, then regrouping to subtract (e.g., $$14\frac{7}{18}-12\frac{13}{18}=2\frac{7}{18}-\frac{13}{18}=1\frac{23}{18}-\frac{13}{18}=1\frac{12}{18}$$ ) since these are universal strategies. Students may also use computation-specific strategies, which are listed below.  
• For some problems in this lesson, students may use a computation-specific strategy. For example, students might think of a computation as an unknown-addend problem and use an addition strategy to solve (including the mental strategy of making a whole, e.g., to solve $$2\frac{1}{5}-1\frac{1}{2}=2\frac{2}{10}-1\frac{5}{10}$$ , a student might add  $$1\frac{5}{10}+\frac{5}{10}=2$$ and $$2+\frac{2}{10}=2\frac{2}{10}$$ , so the difference is $$\frac{5}{10}+\frac{2}{10}=\frac{7}{10}$$ ). They may also subtract like units, but then use the mental strategy of going down over a whole, e.g. $$2\frac{1}{5}-1\frac{1}{2}=2\frac{2}{10}-1\frac{5}{10}=1\frac{2}{10}-\frac{5}{10}=1\frac{2}{10}-\frac{2}{10}-\frac{3}{10}=1-\frac{3}{10}=\frac{7}{10}$$ .

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Tasks designed to teach criteria for success of the lesson, and guidance to help draw out student understanding

a.   Estimate which two whole numbers each of the following differences will be in between.

• $${2{1\over2}-1{1\over5}}$$
• $${2{1\over5}-1{1\over2}}$$

b.   Solve for the actual differences in Part (a) above.

Guiding Questions

Grade 5 Mathematics > Module 3 > Topic C > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds . © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US  license. Accessed Dec. 2, 2016, 5:15 p.m..

a.   Estimate the following differences. Determine whether the actual difference will be more or less than the estimated difference. Be prepared to explain your reasoning.

• $${6{1\over5}-5{11\over12}}$$
• $${5{3\over10}-2{1\over2}}$$
• $$8\frac{7}{9}-3\frac{11}{12}$$

b.   Solve for the actual differences in Part (a) above. Are your answers reasonable? Why or why not?

Anton and Emmy are competing in the long jump. Anton has a long jump of  $$6\frac{1}{4}$$  feet. This is $$1\frac{5}{6}$$ feet longer than Emmy’s long jump. How far, in feet, is Emmy’s long jump?

Unlock the answer keys for this lesson's problem set and extra practice problems to save time and support student learning.

Discussion of Problem Set

• Look at #5. What fractions did you come up with that had a difference of $${2{1\over5}}$$ ?
• Look at #7. What is the difference of your two fractions? Was anyone able to come up with a smaller difference? What if you used fractions greater than 1 for the fractional part of each mixed number? Why do you think it is that we don’t usually write numbers in that way?

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Solve. Show or explain your work.

$${5{1\over2}-3{1\over7}}$$

It takes Joseph  $$5\tfrac{5}{6}$$ hours to finish his book. It takes Annette  $$3\tfrac{7}{8}$$ hours to finish her book. What is the difference in time between how long it took Joseph and Annette to finish their books?

Student Response

An example response to the Target Task at the level of detail expected of the students.

The Extra Practice Problems can be used as additional practice for homework, during an intervention block, etc. Daily Word Problems and Fluency Activities are aligned to the content of the unit but not necessarily to the lesson objective, therefore feel free to use them anytime during your school day.

Extra Practice Problems

Answer keys for Problem Sets and Extra Practice Problems are available with a Fishtank Plus subscription.

Word Problems and Fluency Activities

Help students strengthen their application and fluency skills with daily word problem practice and content-aligned fluency activities.

Topic A: Addition and Subtraction of Fractions

Recognize and generate equivalent fractions.

Add fractions with like denominators.

Subtract fractions with like denominators.

Add fractions with unlike denominators whose sum is less than 1.

5.NF.A.1 5.NF.A.2

Subtract fractions from fractions less than 1 with unlike denominators.

Add fractions with unlike denominators whose sum is less than 2.

Subtract fractions from fractions less than 2 with unlike denominators.

Add fractions with unlike denominators whose sum is greater than 2.

Use benchmark fractions and number sense to estimate mentally and assess the reasonableness of answers.

Add and subtract more than two fractions. 

Solve two- and multi-step word problems involving addition and subtraction of fractions.

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Topic B: Addition and Subtraction of Decimals

Add decimals.

Subtract decimals.

Solve two- and multi-step word problems involving addition and subtraction of decimals.

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Learning the Relationship Between Addition and Subtraction - Lesson Plan

In this engaging math lesson, students will dive into the relationship between addition and subtraction. through various activities, they will discover how these two operations are connected and learn strategies to solve subtraction problems using addition. the lesson includes a warm-up, hands-on tasks, an exit slip, and opportunities for further practice and challenges..

Know more about Learning the Relationship Between Addition and Subtraction - Lesson Plan

The main focus of this lesson is to explore and understand the relationship between addition and subtraction.

Students practice solving subtraction problems using addition by thinking of counting forward or using addition sentences.

Yes, there are optional sections in the lesson plan that provide additional practice problems for students who need more reinforcement or want to challenge themselves.

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