Overview

Session Summary

This is the second of two sessions on Addition and Subtraction. It builds off of the K-2 session. In this session you will become familiar with the work that students do in grades 3-5 of Investigations that supports their developing understanding and fluency with addition and subtraction. You will use mental math to estimate and then solve an adult daily-life problem and then solve some estimation problems from the curriculum. You will work on several activities that focus on 100 as a landmark in the number system. In the last activity, you will examine several subtraction strategies by solving a starter problem set and also looking at student work.

In grades three through five students solve addition and subtraction problems involving larger numbers and practice the strategies they know to improve their fluency and efficiency. By discussing, refining, and comparing strategies for adding and subtracting, students continue to expand their understanding of these operations. The work in these grades also supports students’ understanding of place value and the base-ten number system.

While this session is focused on the work that third through fifth grade students do, it is important that teachers at all grade levels consider the mathematics that students are continuing to work on that begins in the earlier grades. As you work with children in grades K-2 on mathematical ideas, it is important to have an understanding of where those mathematical ideas arise in later grades, and how what you are doing supports their development.

Mathematics in this session

• Examine strategies for solving addition and subtraction problems
• Understand the relationship between addition and subtraction
• Visualize the number relationships and actions of a problem using number line representations
• Consider how students develop computational fluency adding and subtracting whole numbers

Common Core Alignment

The content in this session is aligned to the following Common Core State Standards and Math Practices:

MATH STANDARDS
Operations and Algebraic Thinking

• Use the four operations with whole numbers to solve problems [in this case, addition and subtraction (4)]

3.OA.8

4.OA.3

Number and Operations in Base Ten

• Use place value understanding and properties of operations to perform multi-digit arithmetic (3, 4 and 5)
• Generalize place value understanding for multi-digit whole numbers (4)
• Understand the place value system (5)

3.NBT.1

3.NBT.2

4.NBT.1-4

5.NBT.1

MATH PRACTICES
Subtraction strategies

MP1: Make sense of problems and persevere in solving them

MP2: Reason abstractly and quantitatively

MP5: Use appropriate tools strategically

MP6: Attend to precision

Understanding structure of 100 and 1000:

MP4: Model with mathematics

MP5: Use appropriate tools strategically

MP6: Attend to precision

You may also wish to review the full set of Common Core State Standards in Math and/or videos of the Math Practices enacted in Investigations

.

Estimating Sums

Overview

The purpose of this activity is to highlight the importance of developing and efficiently using mental strategies for computation. You will look at an estimation activity from Investigations and answer a set of More or Less Than problems from the third grade Ten-Minute Math activity. You will also consider the mathematical understanding that students are building by solving these estimation problems.

More or Less Than

“More or Less Than” is a third grade Ten-Minute Math activity in which students are asked to estimate sums of 2-digit numbers by using knowledge of place value and known combinations. By estimating these sums and discussing the estimates, students are developing addition strategies.

Consider the following problems:

1. 45 + 63
2. 68 + 35
3. 31 + 33 + 32

After considering your own strategies for solving “More or Less Than” problems, reflect on what mathematical understanding you think students are building by solving these problems. After doing your own thinking, click “Show” to read others’ reflections.

Understanding the Structure of 100 and 1,000

Overview

In this activity, you will solve a “How Many to 100?” problem and consider how the problem can be solved using either addition or subtraction. You will look at student work of this problem and examine the strategies and representations the students used. You will also look at a set of similar distance problems from grades three to five. Finally you will explore a fourth and fifth grade game that involves finding pairs of three-digit numbers that sum as close as possible to 1,000, and consider what students learn about numbers and operations by playing the game.

Understanding the Structure of 100 and 1,000

The number 100 is an important landmark in our number system. Below you will explore an activity from third grade that prepares students to use the relationship of 100 to other numbers to solve computation problems. You will then extend this idea to working with the number 1,000.

Early research in the development of Investigations showed that students need practice in making combinations of 100 and finding how far numbers are from 100. When third and fourth graders were asked to come up with pairs of numbers that summed to 100, they tended to suggest multiples of 10 (e.g. 60 + 40) or pairs with one number in the 90's (e.g., 91 + 9). They didn't have much sense of what you'd combine with 37 to get to 100 or how close numbers such as 87 or 113 are from 100 – and they didn't have good strategies for figuring this out.

How Many to 100?

This activity comes from the third grade unit, Trading Stickers, Combining Coins. It involves finding the difference between a 2-digit number and 100.

Consider the following problem:

Solve the problem and record your solution. Then click the Show link to see two possible ways to solve the problem.

As students solve “How Many to 100?” problems, they are developing their understanding of the structure of 100, the composition of two-digit numbers of 10s and 1s, and the relationship between addition and subtraction.

In the Addition and Subtraction Part 1 session you saw how students use models, such as the number line, to represent their addition strategies. To solve this sticker problem, students may use 100 Charts, number lines, or other representations to show their thinking.

Student Solutions

Next, take a look at how some students solved the Sticker Problem, noticing their strategies and use of representations. These student solutions include two important mathematical representations, the hundreds chart and the number line, which students use to visualize the number relationships and the action of the problem.

You may wish to review the DIET protocol for looking at student work before reviewing these examples.

Student Work #1: Misaki
Student Work #2: Nicholas and Dwayne
Student Work #3: Kelly and Gina
Student Work #4: Pilar and Adam

Once you have reflected on the questions yourself, click the Show link to review more information about the strategies students were using.

Problems for Grades 3-5

The “How Many to 100?” problem that we just worked on is from the first unit in grade three. Later, in grades four and five, students solve similar problems as their understanding of the base ten number system is extended to larger numbers. Students also solve problems where they “cross over” 100, or another landmark, to find the distance between two numbers (e.g., from 37 to 125). Investigations presents these distance problems in the context of travel. These distance problems naturally lend themselves to the use of number line representation.

In grades four and five students continue to solve distance problems with increasingly larger numbers, sometimes embedded in a multi-step problem such as this example from grade five.

NOTE: As they solve distance problems, students become more fluent with the use of landmarks in the number system as “stopping-off places.” They focus on the place value composition of numbers and develop their understanding of the relationship between addition and subtraction.

Close to 1,000

As students work on adding and subtracting 3-digit and 4-digit numbers, the number 1,000 is an important landmark to consider. You will now explore a game from fourth and fifth grade that helps students build an understanding of the structure of 1,000. The game, Close to 1,000, involves finding pairs of three-digit numbers that make a total as close as possible to 1,000.

Using the annotation tool, review three sample hands from the game Close to 1,000. You can navigate between sample hands by clicking on the letters A, B and C at the top of the page. Below each sample hand, respond to the prompts on the page and then click Submit Responses.

NOTE: A strategy that most adults and children will use when solving a problem mentally or developing a strategy on their own is to focus on the 100’s place first. It is useful to consider the largest part of the problem first and then work on the left over, smaller pieces. This approach also helps to keep the numbers as whole quantities. This is opposite of the way the carrying algorithm asks us to work through a problem. Using the traditional carrying algorithm, you begin with the ones place and move over place by place. While this is one way to keep track of the pieces of an addition problem, it is far less natural than considering the numbers as whole quantities and working with large chunks.

NOTE: This game highlights an understanding of Place Value as separating the 100s, 10s and the ones. This is certainly an important understanding for students to have – that each digit represents a certain amount in the number and if you separate the numbers you can add them back together. But it is important to remember that place value has a much deeper meaning. A deep understanding of place value allows you to see a number and the place it holds in the base ten number system. For example, the number 127 has one hundred, two tens and seven ones; but it also has 12 tens and 7 ones; five 25’s plus 2 ones; and one 100 and one 25 and two ones; it is between 120 and 130; it also has 12 tens and 7 ones. The ability to break numbers apart in the many, many ways that might be useful in solving a problem is dependent on a solid understanding of place value that extends further than the more traditional one.

Subtraction Strategies

Overview

In this activity, you will watch a video and solve a subtraction starter problem set. You will then name the strategies and see the solutions represented on a number line. You will then solve a new problem and look at student work with an eye for understanding the students’ strategies and what they understand about number relationships.

Subtraction Strategies

Throughout grades three, four, and five students practice addition and subtraction strategies, expand their repertoire of strategies, and become more efficient in solving these problems. As students examine and compare strategies, they focus on how each strategy starts and name the strategies according to the first steps of the strategies. In this activity you will work on subtraction and consider some of the strategies that students use to solve subtraction problems.

NOTE: This is the way Starter Problems are described in the Second Edition – “Starter Problems give students an opportunity to examine a variety of ways to solve a subtraction problem. This helps them build their repertoire of strategies to choose from when they approach a problem. Often the numbers in a problem will suggest one strategy more than another. Being able to make choices on the basis of the number relationships in a problem develops flexibility in problem solving, which is one of the important components of computational fluency.”

Work on your own to solve the set of problems below. You should be able to solve these three problems mentally.

• 150-70
• 150-80
• 78+22

After you solve these, choose one of them as your “first step” to solve the problem, 150 – 78. Keep track of the next steps you take and of how you decided that you had found the final answer to 150 – 78.

Fourth Grade Student Work: 1,405 – 619

The following student work samples from fourth grade will allow you to see more variety in the strategies that students use to solve subtraction problems. Before you look at the student work, solve this problem for yourself:

1,405 - 619

Solve it any way you want to. You may want to solve it in more than one way, and perhaps try out “subtracting in parts” or “adding up,” the strategies we just looked at in the starter problems for 150 – 78.

• Student Work #1: Steve
• Student Work #2: Anna
• Student Work #3: Enrique
• Student Work #4: Ursula

As you look at the examples, consider these questions:

• Does the student interpret the problem correctly?
• Does the student solve the problem efficiently?
• Does the student solve each piece accurately?
• Does the student keep track of the steps and determine the correct answer?
• Does the student use clear and concise notation?

Two of the solutions should look familiar:

• Steve solves the problem by subtracting in parts.
• Ursula solved the problem by adding up. (It is good practice to encourage students like Ursula to add up more efficiently using multiples of 10.)

The other two solutions will be less familiar:

• Anna solves the problem using a combination of strategies: she changes the number 1,405 to 1,400, then subtracts 619 in parts, and then adjusts her answer by adding 5 to 781.
• Enrique solves the problem by subtracting back, a strategy similar to—but the very opposite of – adding up. He has used a number line, but the left side represents the greater amount.

Please refer to Teacher Note, Representing Subtraction on the Number Line for more information about using a number line to represent subtraction strategies.

Discussion

What understandings about numbers and the operations of addition and subtraction do students need in order to solve subtraction problems using the strategies we discussed in this session?

What role can tools and representations such as the number lines, 100 charts and the sticker context play in students solving problems and showing their strategies?

Go to the Forum

Key Learning

• As students estimate sums to problems they develop strategies for mental computation and can use their estimates to judge the reasonableness of exact answer solutions.
• As students play games and solve problems focused on 100 and 1000, important landmarks in our number system, students develop an understanding of the structure of 100 and 1000, and the place value of 2 digit and 3 digit numbers, and learn combinations that make 100 and 1000. They can then use the relationship of 100 and 1000 to solve computation problems.
• Throughout grades three, four, and five students practice addition and subtraction strategies, expand their repertoire of strategies, and become more efficient in solving these problems.
• Describing, analyzing, and comparing strategies is an important part of the work students do in grades three, four, and five in order to develop strategies they can use efficiently and flexibly.

This session is aligned with the following K-2 Units in Investigations.