© 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

Addition 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 addition facts. The researched-based 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: Doubles Part 3: Make 10 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 Addition 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. Let’s look at some examples. What is 5 + 5? 5 + 5 = 10 So, if = 10, how could we use that fact to solve 5 + 6? 5 10 Students should draw the models in their math journals/notebooks. © DMTI (2017) | Resource Materials

Part 1 So, if = 10, how could we use that fact to solve 5 + 6? 5 10 1 6 5 + 5 = 10 = 11 5 + 6 = 11 11 Students should draw the models in their math journals/notebooks. How did we change 6 so that we could use to solve 5 + 6? We decomposed 6 in to 5 and 1 so we could use to solve © DMTI (2017) | Resource Materials

Part 1 Let’s use the sentence frame to explain the way we used the fact to solve 5 10 1 6 5 + 5 = 10 = 11 5 + 6 = 11 11 It is crucial for students to restate the target fact I know = 10 and = 11. So, that means = 11. © DMTI (2017) | Resource Materials

Part 1 We used as an anchor fact. 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. © DMTI (2017) | Resource Materials

Part 1 There are 3 anchor facts strategies that can be helpful with addition facts. These are: Doubles Make 10 Compensation Let’s look at each of these three more closely. © 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. How could you use to solve 6 + 7? Use a model to explain your thinking. I know = 12 and 12 + ____ = ____. So, that means = ____. 1 13 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 13 © DMTI (2017) | Resource Materials

Part 1 Another useful anchor fact for addition facts is the Make 10 strategy. We can Make 10 with facts that have one number that is close to 10. For example, What combination of 10 would be helpful if we started at 8? 8 + 2 = 10 How could we use = 10 to solve 8 + 5? Students may think of as a Make 10 fact. While the sum of is 10, it is probably best for students to categorize this fact as a double more so than a Make 10. © DMTI (2017) | Resource Materials

Part 1 8 + 2 = 10 How could we use = 10 to solve 8 + 5? Use a model to help explain your thinking. I know = 10 and 10 + ____ = ____. So, that means = ____. 3 13 13 2 10 3 8 5 13 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. © DMTI (2017) | Resource Materials

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

Part 1 3 + 9 How could we use to solve 3 + 9? Use a model to help explain your thinking. 3 9 1 10 12 13 I know = 13 and 13 - ____ = ____. So, that means = ____. 1 12 12 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 often is used in connection with either Doubles or Make 10. For this example, notice how we still used the number 10, but we just used it a little differently than when we added on to Make 10. Compensation is a strategy that means we will add more than we needed to and then subtract that extra amount to find our final sum. 3 + 9 = – 1 = 12 © 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 change one of these addends (Compensation) to make this problem easier to add? 8 + 6 I know ____ + ____= ____ and ____ - ____ = ____. So, that means = ____. 2 8 16 14 Students may decompose the numbers instead of compensating. Remind them that decomposing is a very useful way to manipulate numbers but that they are being asked to use Compensation specifically for the problem. 14 8 6 2 16 © 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. 8 + 6 = To help students better understand how Compensation differs from Make 10 and Doubles strategies have some students use cubes to model how Compensation requires new cubes to be put into the sets and then accounted for after joining the sets. For example, if you have 8 red cubes and 6 blue cubes, you add 2 more blue cubes to use a double with Compensation. But, these 2 cubes must be removed at the end to find the correct sum. 3 + 9 = © DMTI (2017) | Resource Materials

Part 2 Addition Doubles © DMTI (2017) | Resource Materials

Part 2: Doubles Let’s practice using Doubles to solve basic addition facts. We will start drawing visual models of these facts using dot patterns and then we will use bar models. If you need to build the numbers in the problems with cubes or counters, you may. You can also use the cubes or counters to show your doubles strategy if you want. Just make sure you also draw a visual model that matches the problem and your doubles strategy. Materials needed: Each student will need paper (scrap paper or math journals) to draw the models. It may be necessary for students to build the models with cubes or counters prior to drawing. © DMTI (2017) | Resource Materials

Doubles Dot patterns © DMTI (2017) | Resource Materials

Part 2 Draw this model. If this is a model for 3, what would a double of 3 look like? Now draw models that show how changes to become: 3 + 3 = 6 3 + 4 = = 7 3 + 2 = – 1 = 5 Students should share their models before being shown the example. Students should read each equation aloud to a partner. 3 + 4 3 + 2 © DMTI (2017) | Resource Materials

Part 2 Draw this model. Now change the model so it shows How would the model change to show 4 + 5? Use the sentence frame to describe how you used the double to solve 4 5 I know = 8 and = 9, so that means = 9. 4 + 5 = = 9 © DMTI (2017) | Resource Materials

Part 2 Now we are going to practice using doubles and modeling them with dot patterns. For each example, draw a dot pattern for the double and then change the pattern when the next fact is shown. Use the sentence frame to describe your double strategy to a partner. © DMTI (2017) | Resource Materials

Part 2 3 + 3 I know ____+ ____= ____ and ____ + ____ = ____, so that means ____ + ____ = ____. 4 3 Present each part to students and have them draw models, follow the sentence frame and read the equation before proceeding on to the next example. 4 + 3 = = 7 4 + 3 © DMTI (2017) | Resource Materials

Part 2 4 + 4 I know ____+ ____= ____ and ____ + ____ = ____, so that means ____ + ____ = ____. 4 6 4 + 6 = = 10 Present each part to students and have them draw models, follow the sentence frame and read the equation before proceeding on to the next example. 4 + 6 © DMTI (2017) | Resource Materials

Part 2 Now let’s practice using the doubles strategy with addition and drawing dot patterns to match each fact. For each fact you are shown, draw a dot pattern and then describe how you could use a doubles addition fact to solve the problem with a partner. Students may have a variety of doubles variations they choose to use. Encourage students to share the different ways in which the doubles anchor facts can be used instead of favoring one choice over others. Not all of the facts provided in the following slides are ideal for using doubles to solve them. However, the additional challenge of applying doubles facts to a variety of facts will help students develop a better sense of what facts they prefer to use doubles for after they learn additional anchor facts. © DMTI (2017) | Resource Materials

Part 2 3 + 4 © DMTI (2017) | Resource Materials

Doubles Bar models © DMTI (2017) | Resource Materials

Part 2: Doubles Let’s use bar models to show how doubles can be helpful in learning our basic addition facts. What is 5 + 5? 5 + 5 = 10 So, if = 10, how could we use that fact to solve 5 + 6? How would the bar model change? Draw the changes on your model. Draw this model. 5 10 Materials needed: Each pair of students should have clear writing/drawing space, something to write with and 10 connecting cubes of three different colors (30 total). Students should draw the models and share their ideas. © DMTI (2017) | Resource Materials

Part 2 I know = 10 and = 11. So, that means = 11. 5 10 1 6 5 + 5 = 10 = 11 5 + 6 = 11 11 Define decompose as, “…to break apart.” Students can model the concept by putting their hands together and then saying “decompose” as they pull their hands apart. How did we change 6 so that we could use to solve 5 + 6? We decomposed 6 in to 5 and 1 so we could use to solve © DMTI (2017) | Resource Materials

Part 2 Now that we have seen an open bar model used to model basic facts, let’s use the cubes to see how the model is built. This will help you use bar models with future problems. Build this with one color of cubes from your set. What number did we compose? 5 6 5 10 1 6 11 Define compose as, “…to put together or build.” When students move to drawing a copy of the cube model, some may need to trace the stick of cubes and then slide the stick above their traced outline. They can then use the connected edges of the cubes as a guide to partition the bar into 5 units. They can also use a single cube and trace the cube 5 times. Draw a copy of the cube model you built on your paper. Now build and draw the rest of the model to show = 10 using a different color of cubes. Show what changes for the model to become © DMTI (2017) | Resource Materials

Part 2 We can build bar models with cubes and draw the models to show all of the cubes, but some times it can be faster to just draw an open bar model like the example shown below. We will use open bar models from now, unless you want to draw the other type. 5 10 1 6 11 5 6 10 © DMTI (2017) | Resource Materials

Part 2 Draw a model that shows a doubles fact that would help you solve Use the word decompose to describe any time you have taken numbers apart and follow the sentence frame. Notice that you may either add or subtract depending on what double fact you use. Students can build the models with cubes if needed. Let students work with this slide presented and then have them share their models and strategies before presenting the examples. I know ____ + ____ = _____ and ____ + ____ = _____. So, that means = 13. © DMTI (2017) | Resource Materials

Part 2 7 + 6 6 12 1 7 13 Students can build the models with cubes if needed. They should read the sentence frame together while one or more students point to each part of the model the sentence frame is describing. I know = 12 and = 13. So, that means = 13. © DMTI (2017) | Resource Materials

Part 2 7 + 6 6 12 1 7 13 13 1 7 14 6 Students can build the models with cubes if needed. Remind them to use the word “decompose” to describe situations in which they broke a number apart. Even in the compensation example it is possible to describe the adjustment made at the end as “decomposing” the 7 into 6 and 1. I know = 14 and 14 – 1 = 13. So, that means = 13. © DMTI (2017) | Resource Materials

Part 2 Now let’s practice using doubles to solve addition facts and bar models to show our thinking. For each of the addition fact presented, use a doubles strategy of your choice and draw a model. Share your ideas with someone and see if you had the same strategy. © DMTI (2017) | Resource Materials

Part 3 Addition Make 10 © DMTI (2017) | Resource Materials

Part 3: Make 10 Let’s practice using Make 10 facts to solve basic addition facts. We will start drawing visual models of these facts using ten frames and then we will use bar models and number lines. If you need to build the numbers in the problems with cubes or counters, you may. You can also use the cubes or counters to show your Make 10 strategy if you want. Just make sure you also draw a visual model that matches the problem and your strategy. © DMTI (2017) | Resource Materials

Part 3: Make 10 Before we learn to use the Make 10 strategy, let’s make sure we know our combinations of 10. We will use the Ten Frame Mat and cubes or counters to practice our addition facts that have a sum of 10. A sum is the result of adding. For each example, place the counters on your Ten Frame Mat to match the ten frame you are shown. Then, find what number needs to added to compose a sum of 10. Finally, write an equation (number sentence) that matches the combination of 10. Let’s look at an example together. Materials needed: Each student needs a copy of the Ten Frame Mat (provided on the next slide) and 10 cubes or counters. They can also use scrap paper to notate the equations. The Ten Frame Mat could also be placed inside a transparent sleeve with students using dry erase markers to write their equations. Define compose as, “…to put together or build.” © DMTI (2017) | Resource Materials

Ten Frame Mat Print this for students. © DMTI (2017) | Resource Materials

Part 3 Fill in your ten frame to match this example. How many units are shown? How many more units de we need to compose 10? What equation matches this combination of 10? We composed 10 with 9 and 1. 9 + 1 = 10 10 = 9 + 1 Remind students that equations are math sentences that often use an = sign. (Note: in later grades students will learn about new equations called inequalities so avoid saying that all equations have an = sign). Students may find the order of the 10 fact to be “unusual.” They can certainly rewrite the equation in the order = 10, however developing some flexibility with the use of the = sign is an additional benefit of this task. Taking the opportunity to develop this flexibility while developing fact fluency can be beneficial to students’ later learning. 9 units are shown. We need 1 more unit to compose 10. 10 = 9 + 1 © DMTI (2017) | Resource Materials

Part 3 Now let’s practice using ten frames and equations to match the combinations of 10. © DMTI (2017) | Resource Materials

Part 3 We composed 10 with _____ and _____. Students should answer each question and restate the combination of 10 in pairs or as a class. How many units are shown? How many more units de we need to compose 10? What equation matches this combination of 10? © DMTI (2017) | Resource Materials

Make 10 Bar Models and number lines © DMTI (2017) | Resource Materials

Note to Teachers Several of the following activities come from DMTI Unit Modules in Grades K, 1 and 2. However, in these unit modules, the concept of Make 10 is extended into the Make the Next Unit strategy. This means that number sets beyond the basic addition facts (e.g ) are addressed. In this fluency module, only basic facts are addressed. © DMTI (2017) | Resource Materials

Part 3 Martina says she has found an easy strategy to add numbers if one of the numbers is close to 10. Here is how she solved 9 + 3: I know = 10, and I know = 12. So, that means that = 12. A) Use your cubes to model Martina’s thinking. B) Explain to a partner why her strategy works. Use the word decomposed in your explanation. Materials needed: 20 connecting cubes for each student in two colors (10 of each color), drawing paper Have students share their models and explanations before presenting the subsequent slides. © DMTI (2015) | Resource Materials

Part 3 9 + 3 I know = 10, and I know = 12. So, that means that = 12. A) Use your cubes to model Martina’s thinking. 9 +3 1 2 © DMTI (2015) | Resource Materials

Part 3 9 + 3 I know = 10, and I know = 12. So, that means that = 12. B) Explain to a partner why her strategy works. Use the word decomposed in your explanation. Martina’s strategy works because she decomposed the 3 into 1 and 2. That way, she can make 10 by adding It is easier to add on to 10 than it is to add on to 9 so her strategy makes the problem easier. © DMTI (2015) | Resource Materials

Part 3 We can draw open bar models, like those shown in the Doubles section, to represent what was modeled with cubes on the number line. 2 10 1 9 3 12 © DMTI (2017) | Resource Materials

Part 3 Now, use a number line to model Martina’s strategy. Where is the 3 from the problem, 9 + 3? The 3 is decomposed into 1 and 2 so that we can make a 10 with + 3 This strategy is called Make 10 because we add part of one number to make a 10. © DMTI (2015) | Resource Materials

Part 3 Use Martina’s “Make 10” strategy to solve the following problems. Model each problem and strategy with cubes and then show the same strategy on a bar model or number line. Use the sentence frame to describe your strategies. 9+4 9+5 9+6 I know _____ + _____ = 10 and 10 + _____ = ______. So, that means ______ + ______ = ______ 9+7 9+8 9+9 8+3 8+4 8+5 Students should share their ideas and models with partners and as a class. © DMTI (2015) | Resource Materials

Part 3 Now let’s practice using the Make 10 strategy to solve many more addition facts. The Make 10 fact will be suggested for you after you have had a chance to think about your strategy. Each student needs writing space to draw models. © DMTI (2017) | Resource Materials

Part 3 7 + 4 Use = 10. Have students think briefly about the Make 10 fact they could use and what the correct sum for the target fact would be before presenting the Make 10 suggestion. © DMTI (2017) | Resource Materials

Part 4: Compensation Sometimes it is easiest to change numbers in addition facts so that we actually add 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? 7 + 7 – 1 = 14 – 1 = – 1 = 13 – 1 = 12 Students should share their different strategies before being shown the examples. © 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 added more than we needed to. 7 + 7 – 1 = 14 – 1 = – 1 = 13 – 1 = 12 © DMTI (2017) | Resource Materials

Part 4 You have used Compensation before with Doubles facts. 7 + 6 = – 1 But, Compensation is a very useful strategy for facts that we have not practiced much. These are facts that involve adding a larger number to a very small number. 3 + 9 We can think of as and then use a Make 10 strategy, but we can also use Compensation and change the problem to – 1 = = 12. © DMTI (2017) | Resource Materials

Part 4 Let’s practice using Compensation by modeling some problems with cubes and bar models. Build this with one color of cubes and then draw a bar model to match your cubes. Now use the other color of cubes to show that your are adding 8 to 4. This will be a model of Materials needed: Each student needs 10 cubes of one color and 10 cubes of another color. Each student also needs writing materials. 4 8 © DMTI (2017) | Resource Materials

Part 4 Sometimes problems such as can be difficult to remember. Many people like to rethink of the problem as 8 + 4, but if you chose to keep the problem in the original order, how could you use Compensation to make the problem easier to solve? 4 + 8 = – 2 What would this Compensation strategy look like with the cubes or in your bar model? = 14 – 2 = 12 Materials needed: Each student needs 10 cubes of one color and 10 cubes of another color. Each student also needs writing materials. 10 4 8 X X © DMTI (2017) | Resource Materials

Part 4 Because Compensation involves adding or subtracting too much, the number line is a very helpful model to match this strategy. You can show that you “jumped” too far on the number line and then need to jump back the amount you compensated. Think about using to solve What would this look like on a number line? Students should explain each part of the equation and number line to a partner and as a class. 6 + 6 – 1 = 12 – 1 = 11 = 6 + 5 © DMTI (2017) | Resource Materials

Part 4 Now let’s practice using Compensation to solve basic addition 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 added too much and then subtracted the extra amount you added. Students need materials to draw their models. Note that the given problems will imply students either compensate with a double or by using 10. Student may occasionally be tempted to use the Commutative Property to change the order of the addends. Remind them that the purpose of this task is to practice compensation and ask that after they have solved the problem using their preferred strategy, they solve the problem again using compensation. © DMTI (2017) | Resource Materials

Part 5: Practice Now that we have good strategies to solve basic addition 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 addition facts, you will likely have started to memorize some facts. Try to see if you can remember the sum of each given fact as quickly as possible. Then, describe your favorite strategy you could use to solve the fact, even if you just remembered the sum. 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

Strategy Menu Doubles Make Compensation Part 5 I know = ____ because I used a _____________ strategy. 4 + 5 © 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 = ). 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 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 (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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