© DMTI (2017) | Resource Materials

© DMTI (2017) | Resource Materials
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

Subtraction Fact Fluency
Supplemental Module © DMTI (2017) | Resource Materials

About this Supplemental Module
This module can be used by teachers at many different grade levels to support students’ fluency with basic subtraction facts. The process for developing fact fluency used in the DMTI modules focuses on practicing number relationships and making use of visual models and guided language structures. Students who lack fact fluency frequently do so because they are attempting to memorize facts individually and are not able to connect related facts in ways that help them recall the facts fluently. The DMTI approach to fact fluency has students progress gradually through a series of tasks that will not have students working particularly quickly in the early part of the module. Over time, though, the tasks accelerate students’ experiences and will support their recall of basic facts. The tasks in this module should be used for 5-15 minutes per day, 3-5 days per week for a period of weeks before a short break is taken. Then, coming back to tasks that students need more work with on a similar schedule will ultimately increase fact fluency. © DMTI (2017) | Resource Materials

Module Sequence Part 1: Anchor Facts Part 2: Make 10 Part 3: Addition Doubles Part 4: Compensation Part 5: Practice 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. Parts 2, 3 and 4 develop fluency concepts in a gradual progression from very informal to more complex and explicit understandings. You may find it useful to start at different points within the lessons depending on the needs of students. © DMTI (2017) | Resource Materials

Part 1 Subtraction Anchor Facts © DMTI (2017) | Resource Materials

Note to Teachers Part 1 of this module serves as an introduction to the concept of Anchor Facts strategies which will be used throughout to support students’ development of basic fact fluency. For younger students who may not have much background with fact fluency, it may be best to begin with Part 2 and proceed through the module sequentially. The information in Part 1 may not be helpful to students who have never practiced facts in this way. However, the information in Part 1 can be useful for older students and teachers in order to focus and bring coherence to future learning and the DMTI approach to building fact fluency. © DMTI (2017) | Resource Materials

Part 1: Anchor Facts We can use facts we know, or that are easy to remember, to help us become more fluent with more difficult facts. What is 11 – 6? How did you solve the problem? If you have the fact memorized, how could you have solved it quickly if you forgot 11 – 6? Did you use or 6 + 6? Did you solve the problem by decomposing the 6 and thinking of 11 – 6 as 11 – 1 – 5 = 5? Perhaps you had a different strategy. Lets look at some strategies for subtraction facts now. Students should describe their strategies in pairs and as a class. Define decomposing as, “…breaking apart.” © DMTI (2017) | Resource Materials

Part 1 Some of you used 5 + 5, or 11 – 1 as anchor facts. Anchors are heavy metal hooks boats use to keep them in place in the water. The anchor is hooked to the ground under the water. Anchor facts are used in the same way. We connect facts that are difficult to remember to the easier anchor fact. Students need writing materials and should copy the example model in math journals/notebooks. Students should focus on where each step in each strategy is in the model. Describe how each of these two strategies for solving 11 – 6 match the model. Draw this model. 11 – 6 = 11 – 1 – 5 = 5 because 10 – 5 = 5 11 = so 11 – 6 = 5 © DMTI (2017) | Resource Materials

Part 1 This example introduces two of the three Anchor Facts strategies we will use to build our fluency with basic subtraction facts. These are Make 10 and using Addition Doubles. 11 – 6 = 11 – 1 – 5 = 5 because 10 – 5 = 5 Make 10 11 = so 11 – 6 = 5 Addition Doubles © DMTI (2017) | Resource Materials

Part 1 Here are the 3 anchor facts strategies that can be helpful with subtraction facts. Make 10 Addition Doubles Compensation Let’s look at each of these three more closely. © DMTI (2017) | Resource Materials

Part 1 One very useful anchor fact for subtraction facts is the Make 10 strategy. We can Make 10 with facts that have one number that is close to 10. For example, How could we get to 10 if we start at 13? = 10 What do you notice about the units of one in this subtraction problem that might help us subtract from any teen number? With a teen number, we can just subtract the units of one and that will always leave 10 left over. Students should discuss and share their ideas. Now how could we use 13 – 3 to help us solve 13 – 5? © DMTI (2017) | Resource Materials

Part 1 13 – 3 = 10 We are using 13 – 3 to help us solve 13 – 5. Use a model to help explain your thinking. I know = 10 and 10 - ____ = ____. So, that means = ____. 2 8 8 2 10 3 8 5 13 What happened to the 5? Students should describe their strategies and models before being shown the examples. They should draw a copy of the model if they were unsuccessful generating a similar model. We decomposed 5 into 3 and 2 so we could make 10 starting at 13. This is the Make 10 strategy for subtraction. © DMTI (2017) | Resource Materials

Part 1 Doubles are facts in which you add the same number to itself. For example, = 12. Doubles are often easy to remember and many more difficult facts can be anchored to a doubles fact. Let’s look at an addition example first. How could you use to solve 6 + 7? Use a model to explain your thinking. Students should try to follow the sentence frame before being presented with the model. They should draw a copy of the model if they were unsuccessful generating a similar model. 6 12 1 7 13 I know = 12 and 12 + ____ = ____. So, that means = ____. 1 13 13 © DMTI (2017) | Resource Materials

Part 1 We can use Addition Doubles to solve subtraction facts, too. Describe how helps you think about what 13 – 6 or 13 – 7 will be. Use the model to guide your explanations. 13 – 6 = 7 because = 12 and = 13. This means 13 = , so subtracting 6 leaves 7. 6 12 1 7 13 13 – 7 = 6 because = 12 and = 13. This means 13 = , so subtracting 7 leaves 6. You may notice it can be a little difficult to explain how to use addition doubles to solve subtraction facts in words. It is most important that you can use a model to prove you are correct, and that you work to be able to use this strategy mentally even if describing your thinking is not easy. © DMTI (2017) | Resource Materials

Part 1 Compensation is the final anchor fact strategy we will use for subtraction facts. Compensation is when you change one of the numbers to make the subtraction easier, but then have to change your answer to match the original problem. Let’s look at an example of compensation with the fact © DMTI (2017) | Resource Materials

Part 1 12 - 9 How could we use to solve ? Use a model to help explain your thinking. 2 9 10 3 12 1 I know 12 – 10 = 2 and 2 + ____ = ____. So, that means = ____. 1 3 3 Students should share their strategies and models prior to examining the given examples. Students should be asked to describe where each step and part of the strategy can be found in either model. The number line is used for compensation here because it lends itself well to the idea of “jumping/moving too much” which is at the heart of compensation strategies. It is less clear as a model for the previous anchor fact strategies. © DMTI (2017) | Resource Materials

Part 1 The reason we are looking at Compensation last is that it is a strategy that often uses the number 10 or a double. For this example, notice how we used the number 10, but we just used it a little differently than when we subtracted with the Make 10 strategy. 12 -9 = = 3 Compensation is a strategy that means we will subtract more than we need to and then add that extra amount to find our final difference. Define difference as the result of subtracting. © DMTI (2017) | Resource Materials

Part 1 We can also use Compensation with a doubles fact. You can think of this as either a doubles strategy or Compensation. The label you give your strategy is less important than your ability to use the strategy accurately and in a way you understand. How could you use the double to help you solve 14 – 6? 8 7 6 1 14 I know = 14 so 14 – 7 = 7. But, 7 is 1 more than 6 so 14 – 6 = 8, not 7. The language students use to describe their strategies can vary. The example provided in red text is designed to include all relevant information for the compensation strategy in an organized manner. However, the strategy can be somewhat complicated to describe. Keep the students’ focus on compensation as a strategy that involves subtracting “more” than was intended and how they need to make an adjustment at the end to account for this. 14 – 6 = 14 – 7 + 1 © DMTI (2017) | Resource Materials

Part 1 Remember that when we use Compensation, we are changing the numbers in ways that mean we need to do something to account for this change. When we use a Doubles or Make 10 strategy, we are decomposing the numbers we already have. We are not changing the numbers like we do when we use Compensation. To help students better understand how Compensation differs from Make 10 and Doubles strategies have some students use cubes to model how Compensation requires more cubes to be removed and then added back in at the end.. For example, if you have 12 cubes and you take way 10, you need to add 1 cube back in to show 12 – 9. 12 – 9 = 12 – 14 – 6 = 14 – 7 + 1 © DMTI (2017) | Resource Materials

Part 2 Subtraction Make 10 © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 Students should share their ideas before being shown the examples. A number line template is provided after the examples. It can be printed for students and they can either write directly on the paper or the page can be placed in a transparent sleeve and students can use dry erase markers to write on the number lines. © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 - 2 2 © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 - 3 2 3 © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 - 4 2 3 4 © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 - 5 2 3 4 5 © DMTI (2017) | Resource Materials

Part 2: Make 10 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 - 6 2 3 4 5 6 © DMTI (2017) | Resource Materials

Part 2 Solve the following problems using the number line. Do you notice any patterns? = 10 = 10 = 10 = 10 = 10 2 3 4 Whenever you have a number with 1 unit of ten, you can subtract the units of one and get to 10. 5 6 © DMTI (2017) | Resource Materials

Number Line Template Number line template for the previous examples. © DMTI (2017) | Resource Materials

Part 2 Let’s use this pattern of making 10 with subtraction to help build our subtraction fact fluency. 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 (2017) | Resource Materials

Part 2 The Make 10 strategy can help with many subtraction problems. Let’s practice using the Make 10 strategy for subtraction facts. For each problem you are given: 1. Think about how many units of one must be subtracted to get to 10 and decompose the number you are subtracting to take these units of one away. 2. Subtract the remaining units of one to find the final answer to the problem. 3. Follow the sentence frame to describe your strategy. The first example is completed for you. 4. Use the number line to make sure your answer is correct. Students can redraw the number line, use the previous template or can use the provided example as a visual support. Students may need support decomposing the subtracted number. Use cubes or models to support their understanding of the decompositions. If students appear to not be ready for this level of decomposing fluency, use the supplemental Quick Image module to build their number sense and their understanding of composing and decomposing numbers. © DMTI (2017) | Resource Materials

Part 2 I know _____ - _____ = 10 and 10 - _____ = _____. So, that means _____ - _____ = _____. 11 1 1 9 11 2 9 11 - 2 © DMTI (2017) | Resource Materials

Part 2 I know _____ - _____ = 10 and 10 - _____ = _____. So, that means _____ - _____ = _____. 11 - 3 © DMTI (2017) | Resource Materials

Part 3 Subtraction Addition doubles © DMTI (2017) | Resource Materials

Part 3: Addition Doubles
For some subtraction facts, we can rethink of them as addition doubles and use our knowledge of doubles to solve the fact quickly. Let’s use bar models to show these strategies. We can also use cubes to first show how addition doubles can help us with subtraction problems. If you use cubes, make sure you draw bar models to match what you did with the cubes. © DMTI (2017) | Resource Materials

Part 3: Addition Doubles
If you had to think of addition doubles that are related to 11 – 6, what doubles would you choose? 5 + 5 = 10 6 + 6 = 12 How could you use these addition doubles to solve 11 – 6? What would a model look like that shows your strategy? Materials needed: Each pair of students should have clear writing/drawing space, something to write with and (if needed) 10 connecting cubes of three different colors (30 total). Students should draw the models and share their ideas even if they use cubes to initially represent their strategies. © DMTI (2017) | Resource Materials

Part 3 5 + 5 = 10 so 11 – 6 = 5 because 11 is 1 more than 10. 5 10 1 6 11 If 10 – 5 is 5 then 11 – 6 must also be 5 because both 11 and 6 are one more than 10 and 5. © DMTI (2017) | Resource Materials

Part 3 6 + 6 = 12 so 11 – 6 = 5 because 12 is 1 more than 11. 5 11 1 6 12 If 12 – 6 is 6 then 11 – 6 must be 5 because 11 is one less than 12. © DMTI (2017) | Resource Materials

Part 3 Let’s practice using Addition Doubles to help solve subtraction problems. For each of the given examples: 1. Solve the subtraction fact. 2. Use cubes or a model drawing to show how an Addition Double could solve the subtraction fact. 3. You will be given Addition Doubles facts to try to use after you solve the problem with your own Addition Doubles. Students should complete all of their work before being shown the addition doubles facts. These facts are presented to 1) improve students’ use of addition doubles to solve subtraction facts and 2) provide additional repetitious practice with addition doubles facts. Using addition doubles for subtraction facts is sometimes difficult for students. The purpose of having them practice this strategy is to gradually build their flexible thinking and to lay a foundation of fact fluency that will be refined later when students are given the choice of anchor facts. © DMTI (2017) | Resource Materials

Part 3 7 - 3 How could you use = 6? How could you use = 8? For this example and the following examples you may to encourage students to just think of addition until they have determine the related facts. In this case students might say: “I know = 6 so is 7. That means 7 – 3 = 4.” “I know = 8 so is 7. That means 7 – 3 = 4.” Use this idea with the remaining problems if it is helpful for students. © DMTI (2017) | Resource Materials

Part 3 9 - 5 How could you use = 10? How could you use = 8? © DMTI (2017) | Resource Materials

Part 4 Subtraction Compensation © DMTI (2017) | Resource Materials

Part 4: Compensation Sometimes it is easiest to change numbers in subtraction facts so that we actually subtract more than we should. Think about these two examples: If you didn’t know these facts, what would be some easier facts you could use? 14 – = 14 – 6 = = 13 – 9 = 4 Students should share their different strategies before being shown the examples. Students should also be asked to explain and demonstrate where the – 6 and – 9 are found in the number lines. These will be the distances between 14 and 8 and 13 and 4 but the quantities of 6 and 9 are not notated in the models because of the compensation strategy. © DMTI (2017) | Resource Materials

Part 4 When we change numbers in the way we have in these two examples, we are using a strategy called Compensation. Compensation means that we change one or both of the numbers to make the problem easier to solve and then must adjust our answer to get back to our original problem. Notice how both examples have an extra step at the end because we subtracted more than we needed to. = 14 – 6 = = 13 – 9 = 4 Students can be shown examples from the Addition fact fluency module that highlights how to use compensation with addition to support their familiarity with the strategy. © DMTI (2017) | Resource Materials

Part 4 Let’s practice using Compensation by modeling some problems with cubes and bar models. Build this model of 12 with your cubes or counters. Now, if you had to subtract 5 from 12 you would be solving 12 – 5. But, 12 – 6 is even easier because we know 12 – 6 = 6. Model 12 – 6 with cubes. Materials needed: Each student needs 12 cubes of one color. © DMTI (2017) | Resource Materials

Part 4 Let’s practice using Compensation by modeling some problems with cubes and bar models. Build this model of 12 with your cubes or counters. Now, if you had to subtract 5 from 12 you would be solving 12 – 5. But, 12 – 6 is even easier because we know 12 – 6 = 6. Model 12 – 6 with cubes. Now that we have modeled 12 – 6, how do we change the cubes to show 12 – 5? Materials needed: Each student needs 12 cubes of one color. - 6 X X X X X X © DMTI (2017) | Resource Materials

Part 4 We actually subtracted 1 more than we needed to ( 6 = 5 + 1) so we must add this extra cube back. So, if 12 – 6 = 6, then 12 – 5 is 1 more. 12 – 5 = 7 We can also model this with a number line. - 5 7 - 6 - 5 X X X X X X © DMTI (2017) | Resource Materials

Part 4 Let’s model a Compensation strategy for 12 – 9 with cubes and a number line. What would be an easy way to use compensation with 12 -9? 12 – 10 = 2 Materials needed: Each student needs 12 cubes of one color. - 10 X X X X X X X X X X © DMTI (2017) | Resource Materials

Part 4 Let’s model a Compensation strategy for 12 – 9 with cubes and a number line. What would be an easy way to use compensation with 12 -9? 12 – 10 = 2 If 12 – 10 = 2, how do we changed to problem back to 12 -9? 12 – = 12 – 9 = 3 Materials needed: Each student needs 12 cubes of one color. - 10 - 9 X X X X X X X X X X © DMTI (2017) | Resource Materials

Part 4 What would this strategy look like when modeled on a number line? 12 – = 12 – 9 = 3 3 - 9 X X X X X X X X X © DMTI (2017) | Resource Materials

Part 4 Now let’s practice using Compensation to solve basic subtraction facts. For each given problem, think about how you could change the numbers in a way that matches the Compensation strategy. Then, model your thinking with a number line. The number line will help show how you have subtracted too much and need to add back the extra amount you took away. Students need materials to draw their models. Note that the given problems will imply students either compensate with a double or by using 10. Students should be able to explain why they chose the related facts to use. © DMTI (2017) | Resource Materials

Part 5: Practice Now that we have good strategies to solve basic subtraction facts, let’s practice our facts to see if we can become more fluent. The word fluent means that you know or find the sum very quickly and accurately. Because we have been practicing our subtraction facts, you will likely have started to memorize some facts. Try to see if you can remember the difference for each given fact as quickly as possible. Then, describe your favorite strategy you could use to solve the fact, even if you just have it memorized. You will be given a Strategy Menu to select from. Make sure students try to recall the fact as quickly as possible but avoid giving the impression that recalling facts is always better than using a strategy. Students will ultimately become fluent with their facts regardless of whether they have the facts memorized or need a brief moment to use a strategy. If they feel comfortable using strategies, over time students will memorize a much larger number of facts than if they intentionally try to memorize them. An over-emphasis on memorization can inadvertently have a negative effect on fact fluency for some students, particularly those who find math difficult or do not memorize well (e.g. names, directions, phone numbers). © DMTI (2017) | Resource Materials

Part 5 Strategy Menu Make Addition Doubles Compensation I know = ____ because I used a _____________ strategy. 6 - 4 © DMTI (2017) | Resource Materials

Strategy Cards 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 fact. The space provided could be any of the following combinations: 2 equations, 1 equation and 1 model, 2 models, 2 equations and 2 models Note: students may have strategies for specific facts that do not fit any of the practiced Anchor Facts. At this time in their learning, encourage students to use their favorite strategy provided it is accurate and increases fluency. (e.g. 12 – 8 = 12 – 4 – 4) Students can use bar models, number lines, dot patterns 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. © DMTI (2017) | 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 (2017) | Resource Materials

© DMTI (2017) | Resource Materials | www.dmtinstitute.com
Brendefur and Strother (2017). DMTI Inc. © DMTI (2017) | Resource Materials |

© DMTI (2017) | Resource Materials

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