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© DMTI (2018) | Resource Materials | www.dmtinstitute.com
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 (2018) | Resource Materials |

About this Supplemental Module
This module can be used by teachers at many different grade levels to support students’ mental strategies for addition and subtraction of multi-digit numbers. While there are no specified grade levels for this module, it is advised that these strategies are best practiced by students who are solving much larger and more difficult problems as the focus of this module is mental strategies, not paper and pencil procedures. The tasks in this module should can be used as 5-15 minute daily warmups, small group intervention lessons or as independent practice for students working in centers. Many of the instructions found in these lessons come from Grade 2 Units 3,5 and 7. See those modules for additional lessons if needed, regardless of students’ grade levels. © DMTI (2018) | Resource Materials |

Mental Strategies for Addition and Subtraction
Supplemental Module © DMTI (2018) | Resource Materials |

Module Sequence Note to Teachers: This supplemental module may be used with a variety of grade levels and in many different instructional settings. It may be unnecessary to proceed from lesson to lesson depending on students’ prior knowledge or the intended purpose of using the module. Lesson 1: Decomposing by Place value – Addition Lesson 2: Decomposing by Place Value – Subtraction Lesson 3: Making the Next Unit – Addition Lesson 4: Making the Next Unit – Subtraction Lesson 5: Compensation – Addition Lesson 6: Compensation – Subtraction Lesson 7: Mixed Practice Supplemental Materials: Strategy Cards for Basic Facts © DMTI (2018) | Resource Materials |

Lesson 1 Mental Strategies Decomposing by place value - addition © DMTI (2018) | Resource Materials |

Lesson 1: Decomposing by Place Value - Addition
When we add, we can decompose numbers by place value to make the addition easier. Let’s look at an example for Think about decomposing 42 in to tens and ones. Then, do the same for 19. Now add the units of ten and ones and then add the partial sums together. This is what you would have done in your head to solve the problem. = 50 2 + 9 = 11 = 61 © DMTI (2018) | Resource Materials |

Here is a visual model that shows what decomposing by place value looks like for Think about how you could use this strategy to solve other problems like in your head. 1. This model shows the units of place value in 40 +10 50 2 + 9 11 2. When you add like units together, you see the partial sums of 50 and 11. 3. Sometimes, you will compose a new unit of place value. In this case, the 11 ones compose 1 unit of ten with 1 unit of one left over. 50 +11 61 © DMTI (2018) | Resource Materials |

For the following problems: 1. Decompose the numbers by place value. 2. Add like units of place value. 3. Add the partial sums to find the sum of the larger problem. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Lesson 1 Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. © DMTI (2018) | Resource Materials |

Complete the Lesson 1 practice worksheet by using a decomposing strategy focused on place value. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 1 Worksheet: Decomposing by Place Value with Addition Name:__________________________ Problem Equations or Models Lesson 1 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 2 Mental strategies Decomposing by place value - Subtraction © DMTI (2018) | Resource Materials |

Lesson 2: Decomposing by Place Value - Subtraction
We can subtract mentally by decomposing by place value. Below are models that show how to use this strategy to solve 42 – 13. 40 -10 2 - 3 1. Decompose the numbers in the problem by place value. 30 - 10 20 12 - 3 9 2. Subtract like units of place value using expanded form and draw what this looks like in a second model. Because we are trying to subtract more units of one than we have (2 -3 ) we will decompose 1 unit of ten into 10 units of one. 42 - 13 29 3. We then combine the partial differences together to find the difference for the original subtraction problem. © DMTI (2018) | Resource Materials |

For the following problems: 1. Decompose the numbers by place value. 2. Subtract like units of place value. 3. You may need to decompose some of the numbers when you are trying to subtract more units than you have of some place values. 4. Remember to compose the partial differences to find the difference of the original problem. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Complete the Lesson 2 practice worksheet by using a decomposing strategy focused on place value. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 2 Worksheet: Decomposing by Place Value with Addition Name:__________________________ Problem Equations or Models Lesson 2 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 3 Mental Strategies Making the next Unit - Addition © DMTI (2018) | Resource Materials |

Lesson 3: Making the Next Unit - Addition
2. The strategy of making the next unit is helpful for this problem. Why? 17 + 9 = 20 = 26 = 26 Students should share their ideas and take notes. After the examples are provided, students should describe where each step is in the model and how the model matches the equations. They should also be able to explain how the strategy might work for larger problems. It is helpful to make the next unit because 17 is very close to 20. We can decompose 9 into 3 and 6 so we can make 20 starting at 17. © DMTI (2018) | Resource Materials |

3. How could you use the strategy of making the next place value unit for these problems? (Try to solve them without writing anything down and using only mental math.) (a) (b) (c) Students should share their strategies before being presented with the examples given in the following slides. © DMTI (2018) | Resource Materials |

Students should describe each part of the process and why making the next unit of place value is a useful strategy. © DMTI (2018) | Resource Materials |

(b) Students should describe each part of the process and why making the next unit of place value is a useful strategy. © DMTI (2018) | Resource Materials |

Each time we added on to the first number so we could compose the next unit of place value. Lesson 3: Making the Next Unit - Addition (c) 4. What is the pattern you notice in the last three problems? Students should describe each part of the process and why making the next unit of place value is a useful strategy. Some students may not be clear how this problem relates to the previous examples. Remind them that in this case, you are still composing the next unit of place value but because they started at 97 the next unit would be 100. It may be helpful to remind students that 100 is the same as 10 units of ten. © DMTI (2018) | Resource Materials |

For the following problems: 1. Look at the first number in each problem. 2. Decide what next unit of place value the number is close to. 3. Decompose the second number to get to the next unit. 4. Finish solving the problem by adding the remaining part of the second number. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Lesson 3 17 + 8 Use = 20. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to compose the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Lesson 3 Use = 20. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to compose the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Complete the Lesson 3 practice worksheet by using the strategy of making the next unit of place value. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 3 Worksheet: Making the Next Unit - Addition Name:__________________________ Problem Equations or Models Lesson 3 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 4 Mental strategies Making the next unit - subtraction © DMTI (2018) | Resource Materials |

Lesson 4: Making the Next Unit - Subtraction
We can also use the strategy of Making the Next Unit with subtraction. How could we use 13 – 3 = 10 to help solve 13 – 4? I know 13 – 3 = 10 and 10 – 1 = 9 so that means 13 – 4 = 9. We decomposed 4 into 3 and 1 so we could make 10 starting at 13. - 1 - 3 © DMTI (2018) | Resource Materials |

This subtraction strategy is exactly the same as it was with addition problems, but instead of adding up we are subtracting backward. Think of what the next unit would be for each of these numbers if you had to count backwards: 52 83 104 50 Now we will look at how to use these ideas to solve subtraction problems. 80 100 © DMTI (2018) | Resource Materials |

How can you make the next unit to solve 62 – 23? Use the Word Bank to help your explanation. Word Bank decomposed units of one units of ten We decomposed 23 so that we could subtract the units of one first and then the units of ten. We subtracted 2 from 62 to make the next unit of ten and then subtracted the remaining unit of one. Students can present their ideas and discuss them as a class or with a partner before being shown the example model and explanation. © DMTI (2018) | Resource Materials |

For the following problems: 1. Look at the first number in each problem. 2. Decide what the next unit of place value the number is close to. 3. Decompose the second number to get to the next unit. 4. Finish solving the problem by subtracting the remaining part of the second number. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Lesson 4 23 - 8 Use = 20. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to get to the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Lesson 4 Use = 50. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to get to the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Lesson 4 Use 61 – 1 = 60. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to get to the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Lesson 4 Use = 150. Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. The method to get to the next unit is provided as a suggestion if needed. © DMTI (2018) | Resource Materials |

Complete the Lesson 4 practice worksheet by using the strategy of making the next unit of place value. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 4 Worksheet: Making the Next Unit - Subtraction Name:__________________________ Problem Equations or Models 22 - 8 Lesson 4 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 5 Mental Strategies Compensation - addition © DMTI (2018) | Resource Materials |

Lesson 5: Compensation - Addition
We have looked at several strategies to add and subtract numbers. Now let’s look at one more strategy that can sometimes make adding and subtracting easier. This strategy is called Compensation. Here is Monica’s strategy to use compensation to add She modeled her thinking with a number line and equations. © DMTI (2018) | Resource Materials |

Here is Monica’s strategy to use compensation to add She modeled her thinking with a number line and equations. = 27 Have students describe each step and why they think the steps in the strategy work. © DMTI (2018) | Resource Materials |

Here is Monica’s strategy to use compensation to add She modeled her thinking with a number line and equations. = 27 27 – 1 = 26 = 26 Have students describe each step and why they think the steps in the strategy work. Compensation is a strategy we can use when we are adding or subtracting a number that is very close to a ten. We change that number into a ten and then change our answer to match the original problem. © DMTI (2018) | Resource Materials |

Where do you see the + 9 in this model of the compensation strategy? 10 – 1 = 9 Adding 10 and subtracting 1 is the same as adding 9. © DMTI (2018) | Resource Materials |

For the following problems: 1. Look at the second number in each problem. 2. Decide what the next unit of place value the number is close to. 3. Change the second number and add it to the first number. 4. Subtract the amount you changed the second number by from the sum to find the sum of the original problem. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Lesson 5 15 + 9 Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. © DMTI (2018) | Resource Materials |

Complete the Lesson 5 practice worksheet by using the compensation strategy. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 5 Worksheet: Compensation - Addition Name:__________________________ Problem Equations or Models Lesson 5 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 6: Compensation - Subtraction
When we use compensation with subtraction it is just like compensating with addition. The only difference is we are going backwards first and then have to add back the extra we subtracted. How do you think we could use compensation to solve this subtraction problem? Model your thinking with a number line. 26 – 10 = 16 = 17 26 – 9 = 17 Why did we add 1 here? We added 1 because we subtracted 10. We were only supposed to subtract 9. So, we took away 1 too many. If we subtract 10 and add 1 it is the same as subtracting 9. © DMTI (2018) | Resource Materials |

For the following problems: 1. Look at the second number in each problem. 2. Decide what the next unit of place value the number is close to. 3. Change the second number and subtract it from the first number. 4. Add the amount you changed the second number by from the sum to find the difference of the original problem. Students may write down their strategies on paper if needed (e.g. in a math journal), but try to encourage them to solve each problem mentally. The following slides provide a series of example problems to provide students. The slides can be copied and modified to increase the number of problems with similar number sets for more repetitious practice. © DMTI (2018) | Resource Materials |

Lesson 6 16 - 9 Present this slide to students and give them a short moment to try to solve the problem mentally. Students can record their thinking on paper in math journals/notebooks. After working independently students should discuss their strategies with a partner and possibly, as a class. © DMTI (2018) | Resource Materials |

Complete the Lesson 6 practice worksheet by using the compensation strategy. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 6 Worksheet: Compensation - Subtraction Name:__________________________ Problem Equations or Models 28 - 9 Lesson 6 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Lesson 7: Mixed Practice
Complete the Lesson 7 practice worksheet by using any strategy you feel is most appropriate for the given problems. Use the space provided to write equations or draw models to show your strategy is correct. © DMTI (2018) | Resource Materials |

Lesson 7 Worksheet: Mixed Practice Name:__________________________ Problem Equations or Models Lesson 7 Worksheet The numbers provided can be modified to create multiple practice sheets or to adjust the difficulty level of each problem. © DMTI (2018) | Resource Materials |

Strategy Cards for Basic Facts
Print the following slides for individual or pairs of students. Students may not need to practice all of these facts. Provide them with cards that match the facts they are the least fluent with. There is a blank template as well that can be used to create additional facts students need to practice. 1. Students cutout the cards (horizontally), fold in half and then shuffle the cards. 2. On the back of the card, students draw models and/or write equations that match their preferred strategy. 3. If desired, students can include a second strategy or model that could be useful for the same problem. The space provided could be any of the following combinations: 2 equations, 1 equation and 1 model, 2 models, 2 equations and 2 models 4. Students may use strategies not previously learned provided they are efficient. Students can use place value models, number lines and equations to communicate their thinking. The Strategy Cards can be used by pairs of students or can be taken home for additional practice. If the cards are sent home, provide the above directions and possibly copies of some of the informational slides in this module so parents are informed about how to best use the cards as well as the purpose for developing fact fluency with the DMTI approach. These cards can also be found in the much lengthier Addition and Subtraction Fact Fluency supplemental modules. © DMTI (2018) | Resource Materials |

Strategy Card Example 4 + 5 Strategy or Model 4 + 5 = 9 = = 9 Fold here. Students should be presented with the front of the card (showing the fact) with back facing their partner. © DMTI (2018) | Resource Materials |

When solving basic addition facts there are typically three helpful strategies: Doubles 7 + 8 = Make 10 7 + 8 = Compensation 7 + 8 = © DMTI (2018) | Resource Materials |

When solving basic subtraction facts the same strategies we used for addition are also helpful: Doubles = 8 because = 15 Make 10 = Compensation = 15 – © DMTI (2018) | Resource Materials |

Strategy Card Example 13 - 5 Strategy or Model 13 – 5 = 8 13 – 3 – 2 = 10 – 2 = 8 Fold here. Students should be presented with the front of the card (showing the fact) with the back facing their partner. © DMTI (2018) | Resource Materials |

© DMTI (2018) | Resource Materials |

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