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The Developing Mathematical Thinking Institute (DMTI) is dedicated to enhancing students’ learning of mathematics by supporting educators in the implementation of research-based instructional strategies through high-quality professional development. For more information contact Dr. Brendefur at © DMTI (2017) | Resource Materials |

Addition and Subtraction
Supplemental Module Number and Operations in Base 10 Teaching the Standard Addition and Subtraction algorithms for Understanding © DMTI (2017) | Resource Materials |

About the DMTI Modules These modules are designed to guide classroom instruction and formative assessment for teachers implementing the DMTI curricular materials. You will notice that the number of lessons in the DMTI modules will be somewhat less than the typical number of school days suggested for the unit. For example, a unit intended to require 3 weeks of instruction may not have an associated module that is comprised of 15 lessons. This is for three reasons: 1) Teachers may desire to collect student data from the DMTI Common Assessments prior to the end of the unit (e.g. during the last week) in order to have time to provide intervention based on findings in the students’ performance on the Common Assessment; 2) Many teachers have lessons or activities they have successfully used in the past for certain math topics that may fit well into some part of the unit but are not found in the DMTI modules; 3) While most lessons in the DMTI modules can be completed in one daily lesson period, it may be necessary to provide students with a slower pace at times or accelerate the pace if a lesson is too easy. In the latter case, the next lesson in the module may be started. © DMTI (2017) | Resource Materials |

About this Supplemental Module
Teaching the standard addition and subtraction algorithms for understanding requires students to have a deep conceptual knowledge of place value as well as procedural fluency with basic addition and subtraction facts. This module is meant to be used if Grade 3 teachers choose to teach these algorithms at some point in the year as well as at a time of the teacher’s choosing for Grade 4 teachers. Other grade levels may find the lessons useful in teaching or reviewing the algorithms. Fourth grade students should master both of these algorithms from both a procedural and conceptual standpoint. However, the timing of when to teach these algorithms is ideally at the teacher’s discretion and should be determined by examining ample assessment data regarding students’ understanding of place value as well as other place value-based methods for adding and subtracting base 10 numbers (e.g. partial sums/differences in expanded form). During fourth grade, students must also learn these two algorithms with such a deep level of understanding they can make sense of common errors and misconceptions students typically struggle with when learning the algorithms. Students should also be able to explain the algorithms using structural language and visually model the concepts embedded within the algorithms if needed. This module also includes practice templates that can be modified to give students ample practice opportunities. © DMTI (2017) | Resource Materials |

Module Sequence Lesson 1: The Standard Addition Algorithm Lesson 2: Addition Practice Template Lesson 3: The Standard Subtraction Algorithm Lesson 4: Subtraction Practice Template Lesson 5: Addition and Subtraction Practice Template © DMTI (2017) | Resource Materials |

Lesson 1 Addition The Standard Addition Algorithm © DMTI (2017) | Resource Materials |

Lesson 1: Standard Algorithm for Addition
We can use place value to add numbers in a special method called an algorithm. An algorithm is a set of steps that follow the same order every time. We will learn what is called the standard algorithm for addition. This algorithm will help us add numbers quickly and easily even when the numbers become very large. But first, let’s start with an example: © DMTI (2017) | Resource Materials |

Let’s review what we know and what we are going to learn…. When you compose a new unit of a larger place value it is often called regrouping. It is also called regrouping when you decompose a unit of place value into 10 of the next smaller units of place value. We will investigate this concept later when we learn more about subtraction. For now, let’s look at regrouping in addition problems using the Standard Algorithm and Partial Sums in Expanded Form. Student should share the task of reading the different statements. © DMTI (2017) | Resource Materials |

What would look like using Partial Sums in Expanded Form and modeling it with a Bar Model? 40 +10 50 2 + 9 11 5 units of ten 11 units of one © DMTI (2017) | Resource Materials |

What would look like using Partial Sums in Expanded Form and modeling it with a Bar Model? 40 +10 50 2 + 9 11 10 units of one composed a new unit of ten. There is still 1 unit of one left. 1 unit of ten and 1 unit of one. 5 units of ten 11 units of one © DMTI (2017) | Resource Materials |

What would look like using Partial Sums in Expanded Form and modeling it with a Bar Model? 40 +10 50 2 + 9 11 50 +11 61 = 61 6 units of ten 1 unit of one © DMTI (2017) | Resource Materials |

If we just used Partial Sums without a model, it would look like this: Now let’s look at the Standard Algorithm and how each step is followed. 42 +19 11 +50 61 42 +19 © DMTI (2017) | Resource Materials |

If we just used Partial Sums without a model, it would look like this: Now let’s look at the Standard Algorithm and how each step is followed. 1 42 +19 11 +50 61 42 +19 1 © DMTI (2017) | Resource Materials |

If we just used Partial Sums without a model, it would look like this: Now let’s look at the Standard Algorithm and how each step is followed. 1 What do you notice about how the Partial Sums method and the Standard Algorithm are the same or different? Share your ideas with a partner by completing this sentence starter: “When I compare Partial Sums and the Standard Algorithm, I notice…” 42 +19 11 +50 61 42 +19 61 Students should focus on how both methods use units of place value. They should also be able to describe how, in the standard algorithm, new units of place value composed of smaller place value units are “regrouped” or “carried” to the appropriate place value column that matches the new unit that has been composed. If students do not describe the process of regrouping using terms such as “unit” or “compose”, provide these terms and refer to previous lessons to ensure students can use these structural terms from this point forward. © DMTI (2017) | Resource Materials |

Now let’s look at how the Standard Addition Algorithm can be used with some larger numbers. Draw a place value model for © DMTI (2017) | Resource Materials |

100 30 40 Students may need Base 10 pieces to model some of these steps both enactively (physically) and to support their visual (iconic) models. 8 5 © DMTI (2017) | Resource Materials |

Now let’s look at how to find the sum using expanded form and models. 1. Rewrite the problem in expanded form. 100 30 40 100 30 +40 8 + 5 70 13 2. Add like units of place value to find the partial sums. 8 5 © DMTI (2017) | Resource Materials |

3. Let’s show how you found the partial sums by either building or drawing new models that have all of the like units of place value put together. Make sure your drawn model looks like this example before we continue. 100 30 40 30 +40 8 + 5 Encourage students to build the model and then draw a final version on paper. 100 70 13 8 5 © DMTI (2017) | Resource Materials |

4. Here is what the model would look like if we were using base-10 pieces. 100 70 It looks like we have composed a new unit of ten out of units of one. 30 +40 8 + 5 Students do not need to draw this intermediate step. You may also want pairs or small groups to have access to Base 10 pieces to model this step. 100 70 13 13 © DMTI (2017) | Resource Materials |

5. Replace the 10 units of one with 1 unit of ten using the base-10 pieces or in a new drawn model. 100 70 30 +40 8 + 5 100 70 13 13 © DMTI (2017) | Resource Materials |

5. Replace the 10 units of one with 1 unit of ten using the base-10 pieces or in a new drawn model. 100 70 10 30 +40 8 + 5 Students should not draw the model yet. They will able their final model differently. 100 70 13 3 © DMTI (2017) | Resource Materials |

This is what your final model should look after you draw and label it. 6. Add the partial sums and label your drawing to match the total sum. 100 70 80 10 30 +40 8 + 5 100 70 + 13 183 Students should draw the final example. 100 70 13 = 183 3 © DMTI (2017) | Resource Materials |

Now let’s look at what the Standard Algorithm looks like for this same process. Partial Sums in Expanded Form Standard Algorithm 138 +45 Students should be asked to describe each of the following steps using terms such as unit and compose. Focus on the key structural idea that when one unit of place value is added and a new larger unit of place value is composed, that newly composed unit is “regrouped” or “carried” in the standard algorithm whereas in partial sums it is not. Students should understand, and be able to state, that the standard algorithm requires all like units of place value to be added together. © DMTI (2017) | Resource Materials |

Now let’s look at what the Standard Algorithm looks like for this same process. Partial Sums in Expanded Form Standard Algorithm 1 138 +45 3 Students should say, “We have composed a new unit of ten by adding all of the units of one. That is why we have regrouped the ten into the tens’ column.” © DMTI (2017) | Resource Materials |

Now let’s look at what the Standard Algorithm looks like for this same process. Partial Sums in Expanded Form Standard Algorithm 1 138 +45 83 Students should say, “We have not composed a new unit of one hundred out of the units of ten so there is nothing to regroup or carry.” © DMTI (2017) | Resource Materials |

Now let’s look at what the Standard Algorithm looks like for this same process. Partial Sums in Expanded Form Standard Algorithm 1 138 +45 183 © DMTI (2017) | Resource Materials |

Based on what you have seen so far, how do you think the Standard Algorithm would be used to solve this problem? 178 +45 3 1 178 +45 23 1 178 +45 223 1 178 +45 We have composed a new unit of ten so we need to regroup that unit to the tens’ place. Have students try to follow the procedures in the algorithm based on their own intuitive notions prior to sharing the specific steps. We have composed a new unit of one hundred so we need to regroup that unit to the hundreds’ place. © DMTI (2017) | Resource Materials |

Now let’s continue to build on the concepts we have already seen. How would you use the Standard Algorithm to solve this problem, based on what we have already completed? We have composed a new unit of one hundred so we need to regroup that unit to the hundreds’ place. We have composed a new unit of ten so we need to regroup that unit to the tens’ place. 678 +145 3 1 678 +145 23 1 678 +145 823 1 Have students try to follow the procedures in the algorithm based on their own intuitive notions prior to sharing the specific steps. 678 +145 © DMTI (2017) | Resource Materials |

Now try to describe how Partial Sums in Expanded Form is similar and different to the Standard Algorithm for this same problem. Begin your statement by saying, “Something I notice is….” 678 +145 823 1 600 +100 700 70 +40 110 8 +5 13 Students should be guided to focus on how Partial Sums does not require regrouping units of place value as well as what the “carried” digits in the Standard Algorithm mean in relation to Partial Sums. For example, the digit 1 over the tens’ place indicates that, “…one unit of ten was composed after adding the units of ones.” © DMTI (2017) | Resource Materials |

Word Bank unit place value compose regroup Lesson 1: Standard Algorithm for Addition Now let’s look at one more example. Complete the Standard Algorithm for the following problem and describe each step to a partner using words from the Word Bank. 1678 + 745 3 1 1678 + 745 23 1 1678 + 745 423 1 1678 + 745 2423 1 1678 + 745 © DMTI (2017) | Resource Materials |

Lesson 2 Addition Addition practice template © DMTI (2017) | Resource Materials |

Lesson 2: Addition Practice
Let’s practice using the Standard Addition Algorithm for each of the problem strings on the practice worksheet. Try to think of each step in the algorithm as adding like units of place value and then regrouping, or carrying, any new units of place value you compose. Remember that you only write a digit indicating the unit you are regrouping. That is just a part of the algorithm that is intended to save space and allow you to think of adding digits instead of the actual values of each place value. Students should be familiar with the term digit. If they are not, explain that digits are the symbols used to write numbers just as letters are used to write words. © DMTI (2017) | Resource Materials |

Addition Algorithm Practice Template 2.1 Name:___________________________ A. B. C. 413 +157 473 +158 773 +458 2773 +1458 508 +253 678 +253 678 +553 4678 +3553 Each problem string is designed to progress in a sequential way. For additional practice, follow the same sequence shown in the examples but change the numbers in each problem. This gradual progression will help students make connections between related number sets, remember the steps in the algorithm correctly and be more accurate in their computation. It may also be important to let students know that in future problem solving situations when given expressions written horizontally such as , the standard algorithm requires that the larger addend be placed on top. 702 +178 762 +138 762 +538 7762 +3538 © DMTI (2017) | Resource Materials |

Lesson 3 Subtraction The Standard Subtraction Algorithm © DMTI (2017) | Resource Materials |

Lesson 3: Standard Algorithm for Subtraction
The Standard Algorithm for addition can actually help us learn the Standard Subtraction Algorithm. In the addition algorithm, we regrouped or “carried” new units of place value we composed by adding smaller units of place value. The standard algorithm for subtraction is essentially the same idea, but in the opposite direction. Instead of composing new units of place value and regrouping, we will decompose (or partition) larger units of place value when we find we need more units of a smaller place value. Let’s look at an example and begin by drawing a place value model for – 55. © DMTI (2017) | Resource Materials |

Let’s first look at how to solve this problem using models and Partial Differences in Expanded Form: 144 – 55. A) Solve using expanded form. B) What would the models look like for each step in your process? C) How can you use what you know about partitioning units of ten into units of one to help you regroup units of hundreds into units of ten? 4 40 100 Students should work independently and in pairs to model the problem. Have students present ideas to the class prior to going through the following examples. © DMTI (2017) | Resource Materials |

4 40 100 4 - 5 40 - 50 100 - 0 Students should be familiar with the sequence of procedures to solve and model these type of problems. As you present each step, ask students to explain what is happening and how it connects to their own ideas. “We are trying to subtract more units of one and more units of ten than we have.” “We can partition (or decompose) 1 unit of ten and 1 unit of one hundred to solve the problem.” © DMTI (2017) | Resource Materials |

100 30 40 14 4 - 5 40 - 50 100 - 0 14 - 5 30 - 50 100 - 0 Students should be familiar with the sequence of procedures to solve and model these type of problems. As you present each step, ask students to explain what is happening and how it connects to their own ideas. 4 © DMTI (2017) | Resource Materials |

130 100 30 14 30 - 50 100 - 0 130 - 50 14 - 5 Students should be familiar with the sequence of procedures to solve and model these type of problems. As you present each step, ask students to explain what is happening and how it connects to their own ideas. © DMTI (2017) | Resource Materials |

130 14 130 - 50 14 - 5 X X X X X Students should be familiar with the sequence of procedures to solve and model these type of problems. As you present each step, ask students to explain what is happening and how it connects to their own ideas. 80 9 X X X X X © DMTI (2017) | Resource Materials |

Now let’s compare Partial Differences in Expanded Form to the Standard Subtraction Algorithm. We need to partition (or decompose) a unit of ten into 10 units of one. 144 - 55 9 3 1 144 - 55 89 13 1 130 - 50 80 100 - 0 14 - 5 9 144 - 55 Students should describe the way in which each method requires decomposing/partitioning larger units of place value into 10 units of the next smaller place value in order to avoid negative numbers. We need to partition (or decompose) a unit of one hundred into 10 units of ten. = 89 © DMTI (2017) | Resource Materials |

Let’s look at another example of the Standard Subtraction Algorithm. How do you think the algorithm would be used to solve this problem? 341 - 153 8 3 1 341 - 153 88 2 13 1 341 - 153 188 2 13 1 341 - 153 Students should share and discuss their ideas before being presented the examples. We need to partition (or decompose) a unit of ten into 10 units of one. We need to partition (or decompose) a unit of one hundred into 10 units of ten. © DMTI (2017) | Resource Materials |

We are going to look at one of the most difficult types of problems to solve using the standard algorithm for subtraction. Notice how when we have the digit 0 in certain place values, we have to be very careful and thoughtful about how to use the algorithm. What do you think we can do to follow the steps in the standard algorithm for this problem? Have students discuss their ideas and share them with each other. 301 - 153 © DMTI (2017) | Resource Materials |

Here is how to use the standard algorithm for this problem. Partial Differences in Expanded Form is also shown. What connections can you make between these two methods? 300 - 100 100 90 - 50 40 11 - 3 8 200 301 - 153 8 2 9 1 301 - 153 48 2 9 1 301 - 153 148 2 9 1 301 - 153 Students should be able to describe how you must decompose/partition the units of one hundreds when you need more units of one and have no units of ten. = 148 © DMTI (2017) | Resource Materials |

Let’s think about how to use the standard algorithm to solve this problem. Let’s review the steps of the algorithm. 1. Start with the smallest unit of place value, this case this will be units of one. 2. If you are trying to subtract more units of one than you have in the top number, decompose/partition the next larger unit of place value to create 10 of the smaller units. 3. Continue this process from right to left as the place units increase. 642 - 257 Students should try to perform the algorithm prior to being shown the examples on the following slide. © DMTI (2017) | Resource Materials |

Word Bank unit place value decompose partition regroup Lesson 3: Standard Algorithm for Subtraction 642 - 257 5 3 1 642 - 257 85 5 13 1 642 - 257 385 5 13 1 642 - 257 Students should be asked to use the Word Bank as they discuss each step in the algorithm. © DMTI (2017) | Resource Materials |

Lesson 4: Subtraction Practice
Let’s practice using the Standard Subtraction Algorithm for each of the problem strings on the practice worksheet. Try to think of each step in the algorithm as subtracting like units of place value and then regrouping a larger unit of place value whenever you find that you need 10 units of a smaller place value. Remember that you only write a digit indicating the unit you are regrouping. That is just a part of the algorithm that is intended to save space and allow you to think of subtracting single digit or two-digit numbers instead of the actual values of each place value. © DMTI (2017) | Resource Materials |

Subtraction Algorithm Practice Template 4.1 Name:___________________________ A. B. C. 570 - 157 651 - 158 1231 - 458 4231 - 1458 701 - 253 1201 - 253 3431 - 1153 3001 - 1153 Each problem string is designed to progress in a sequential way. For addition practice, follow the same sequence shown in the examples but change the numbers in each problem. This gradual progression will help students make connections between related number sets, remember the steps in the algorithm correctly and be more accurate in their computation. It may also be important to let students know that in future problem solving situations when given expressions written horizontally such as , the standard algorithm requires that the larger number be placed on top. 831 - 209 1831 - 459 5831 - 1459 5003 - 1459 © DMTI (2017) | Resource Materials |

Lesson 5: Addition and Subtraction Practice
We are going to practice both the addition and subtraction standard algorithms by using what we know about the relationship between addition and subtraction. Addition and subtraction are inverse operations, which means that if we add two numbers we can then subtract one of the numbers from the sum and will find the other number as the answer. The same thing happens if we subtract two numbers. We can then add the difference and the number we subtracted to find the number we started with. Inverse operations can be thought of as operations that undo one another. Think of tying and untying your shoe laces. You follow the same steps, but in the opposite direction. © DMTI (2017) | Resource Materials |

Lesson 5: Addition and Subtraction Practice
For each given expression: 1. Solve using the standard algorithm. 2. Rewrite the problem using the inverse operation. 3. Solve the new problem. 4. You should find the answer to the new problem is one of the numbers from the original problem. If not, check your calculations in both of your methods. © DMTI (2017) | Resource Materials |

Addition and Subtraction Practice Template 5.1 Name:__________________________ Expression Inverse Operation 1, Modify the number sets for additional practice. © DMTI (2017) | Resource Materials |

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