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Becky Berg Strategy Based Method for Fact Fluency 1

A Strategy Based Method for Developing Fluency of Addition & Subtraction Facts

By: Becky Berg CAUTION: Conceptual Understanding Comes First Memorizing facts is not the first step to fact fluency. Proceed with caution! It is essential that students have lived and breathed the addition and subtraction strategies through a variety of concrete, pictorial, and abstract experiences. Student must not simply remember these important math facts. (O'Connell & SanGiovanni) They need to “walk the walk” and “talk the talk” (Do it and talk about their thinking and strategies!) This must be done before we link these strategies to fact fluency. We must build their conceptual understanding first and foremost. If I told you to memorize ~ + @ = !, it would be meaningless and quite difficult because I have no conceptual understanding of ~ and @ equals ! Remember…. it is the deep conceptual understanding that will take them through algebra and beyond! Let’s take a look at the connections to Common Core State Standards that leads to fact fluency. (CCSSO)

Kindergarten Common Core Connections In Kindergarten, within the Operations and Algebraic Thinking domain (K.OA. 1 through K.OA.5), students experience representing addition and subtraction with objects, fingers, drawings, sounds, and kinesthetic movements. They use objects and drawings to solve problems within 10. They decompose numbers less than or equal to 10 into pairs in more than one way using objects, drawings, and/or equations. In standard K.OA.5, it states that students can fluently add and subtract within 5. Students in Kindergarten also gain experience with our Base Ten system while engaging in K.NBT.1. This standard has them “live and breathe” the meaning of the teen numbers and realize that they are composed of 1 ten and ____ ones. This strong understanding of the teen numbers and 10 + n is helpful as we move into fluency of all facts within 20 in second grade. 1st Grade Common Core Connections In 1st Grade, the work of conceptual understanding continues. Common Core Standard 1.OA.6 strongly states the importance of a multitude of strategies for adding and subtracting within 20. Common Core Standard 1.OA.6 states: “Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as…

• Counting on • Making ten; decomposing a number leading to 10 • Using the relationship between addition and subtraction • Creating equivalent but easier or known sums (Adding 6 + 7 by using doubles “I know 6 +

6 is 12, so I need to add 1 more which is 13.) Fluently using these strategies and giving students time to experience and share their thinking regarding these strategies is essential. I repeat!...Memorizing facts is meaningless if there is not a deep conceptual understanding.


2nd Grade Common Core Connections So, as you can see, the vision is that once students get to 2nd Grade, they have already had LOTS of experiences with the strategies relating to addition and subtraction facts within 20. Therefore, it is now time for Standard 2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. To successfully do this, we must link to the strategies.

Linking Strategies to Fact Fluency It seems as though, in many cases, we work so very hard on the various strategies: counting on, doubles, doubles plus one, combinations of ten and near ten, etc. However, then when it’s time to work on fact fluency…we sequence these by first doing +1, then +2, +3, +4, and so forth. Why would we do this, when we know the importance of connecting to what they already know…the strategies! (O'Connell & SanGiovanni). What we should be doing is working on fact fluency in a way that directly connects to these strategies we experienced since Kindergarten! Inverse Operations: Connecting Addition and Subtraction In the Common Core State Standards, it does not suggest that we teach addition and subtraction strategies in isolation of one another. For example, in 2.OA.1 it states “Use addition and subtraction within 100 to solve one-‐ and two step word problems….”. CCSS 2.OA.2 states “Fluently add and subtract within 20 using mental strategies (CCSSO). Common Core expects our students to conceptually understand that part-‐part-‐whole relationship, which lends itself to understanding the inverse relationships. “Since knowledge of the addition facts can be applied to subtraction, there is no need to learn these facts as isolated procedures. HOWEVER, this assumes that the relationships between inverse operations have been thoroughly explored, discussed, and internalized.” (Chapin & Johnson, 2006) We want them to recognize the inverse and the Part-‐Part-‐Whole relationship within addition and subtraction. We want them to use what they know about quantity and the relationships of these quantities to lead them to conceptual understanding and eventually fact fluency. One operation undoes the other. Once again, first and foremost, this is NOT something that you “tell students” and then move into memorizing the addition and subtraction facts. Much has to do with the questions, the math discourse, the hands on and pictorial experiences they are provided. These fact fluency sets you will find relate addition and subtraction facts based on a specific strategy. Common FAQ: What is considered fluency? How fast? A little research relating to this question should be helpful. Hasselbring, the developer of FASTT Math, feels students have automatized math facts when response times are “down to around 1 second”. (Hasslebring & Goin, 1987) Other researchers consider facts to be automatic when a response comes in 2 or 3 seconds. (Isaacs & Carroll) Based on this research, I suggest that students should be able to say the fact within approximately 2 seconds. Basically, I should hold up the flashcard and the student should say it . If they have to take time to figure it out, they have a strategy but they have not linked this to automaticity (just knowing). The idea is that they should know these facts as readily as they can read sight words such as this, the and when. When they see these words…they don’t have to figure them out, they just know them automatically. The same is true with knowing facts to fluency. Students must be linking the strategies to automaticity.


Printing and Organizing the Fact Fluency Sets There are 8 fact sets based on strategies. Each set includes 12-‐19 fact cards. The fact cards are saved as separate PDF files and also in one combined file that includes all of the fact sets. The pages are set up for duplex printing (back to back) so that the answer is printed on the back of the card. Then, simply cut the cards apart and place each into a small manila envelope or small container. I have also attached labels that you can print on 5160 Avery labels to put on the outside of the envelopes. Scheduling It would be ideal that students spend approximately 10 minutes, 3-‐5 times a week to effectively implement this system. You can do it at the start of your math block or integrate it as one of your workstations or centers while doing guided math groups. Research supports this, as according to Marzano, scattered practice-‐five to ten minutes a day, spread throughout the school year—yields powerful results.(Marzano, 2001) Implementing the Fact Fluency Sets The concept of paired practice has been done by a variety of programs. However, it is the K-‐5 Math Resources website that did a nice job of combining paired practice and the research on expanding recall of facts. Here are the steps for implementing paired practice: (K5MTR)

1. Students work in pairs. Partner A makes sure the fact set is in a random order. Partner A holds the deck of Fact Fluency Cards with the problem facing Partner B. Partner B reads the entire problem aloud and must give the answer without error. If the answer given is correct the card is put in “correct pile” on the table. If the answer given is incorrect, Partner A reads the problem and answer aloud and Partner B repeats it twice.

Example: Kim: Eight plus seven equals thirteen. Noah: (reads from back of card) Eight plus seven equals fifteen. Say it. Kim: Eight plus seven equals fifteen. Noah: Again Kim: Eight plus seven equals fifteen.

2. What do you do with fact cards that a student gets incorrect? After Partner B repeats the fact correctly two times, Partner A puts the ‘error’ card back into the deck 2 -‐ 4 spaces behind the front card. Partner A then proceeds with the next card as usual. Placing the “error card” close to the front provides the student with practice with this fact in a few seconds, which is an effective strategy based on brain-‐research.

3. Student B is finished “being the student” when he/she completes all the cards correctly. The students then switch roles.


Ready to Be Assessed: Conference Needed! When a student is successfully able to fluently answer all of the cards within a set , without having any error cards and repeats during practice…the student is ready to be assessed. So, let’s say Noah successfully gets through Set C with their partner. Noah would then put his name and date on the front of the sticky note. He would put Set C on the back of the note. (This way, it’s not announced to the world as to which set a student is currently on.) Noah would then put the sticky note on a chart or space on the whiteboard that simply says, “Fluency Conference Needed”.

Teacher Assessment & Recording on Mastery Chart While pairs of students are practicing their facts, the teacher’s role is to have fluency conferences with students that are ready to be assessed. Within this 10 minute time period, a teacher could easily assess and conference with several students. So, for example, the teacher has a fluency conference with Noah using Set C. If he has mastered (with fluency) all of the facts in that set, the teacher would put her signature and comments on Noah’s Fact Fluency Mastery Chart. Noah would then put the date and his own comments regarding his mastery of this set. These recording sheets can be put into their math journals or folders. In the resources, I am also including a recording sheet for teachers, so at a glance you can easily see which sets each student has mastered.


Bibliography CCSSO, N. C. (n.d.). (N. C. CCSSO, Producer, & NGA Center and CCSSO ) Retrieved Nov 10, 2012, from Common Core State Standards: http://www.corestandards.org/ Chapin, S. H., & Johnson, A. (2006). Math Matters (Vol. 2nd Edition). Sausalito, CA, USA: Math Solutions. Hasslebring, T., & Goin, L. &. (1987). Effective Math Insgruction: Developing Automaticity. Teaching Exceptional Children , 30-‐33. Isaacs, A. C., & Carroll, W. M. (n.d.). Strategies for basic-‐facts instruction. Teaching Children Mathetmatics , 508-‐515. K5MTR, K.-‐5. M. (n.d.). Computational Fluency. (K5MTR, Producer, & K5MTR) Retrieved Nov 10, 2012, from K-‐5 Math Teaching Resources: http://www.k-‐5mathteachingresources.com/ Marzano, P. P. (2001). Classroom Instruction That Works. Alexandria, Virginia: ASCD. O'Connell, S., & SanGiovanni, J. Mastering the Basic Math Facts in Addition and Subtraction. Portsmouth, NH: Heinemann.


Addition Fact Sets Based on Strategy

Note: In some cases, if the facts have repeatedly been done in other sets, the fact may not be included in a set. (Ex.: 0 + 0, 1 + 1 were done in Set A & Set B. So, therefore these 2 facts were not included in Set E.) Also, on the fact cards, I reversed some of the addends. So, when working on adding 2, sometimes the 2 is the first addend and other times it is the second addend. This is to encourage students to use the commutative property when solving.

Set (Total of 7 Sets) Description Facts in This Set

Set A -‐16 Cards Adding 0, Subtracting 0, Subtracting All

Set A focuses on not only adding and subtracting zero, but subtracting all (which results in zero).

0 + 0 1 + 0 0 + 2 3 + 0 0 + 4 5 + 0 0 + 6 7 + 0 0 + 8 9 + 0 0 + 10

4 – 0 6 – 0 9 – 0 5 – 5 8 – 8

Set B – 18 Cards Adding 1, Subtracting 1, Difference of 1

Set B focuses on 1 more, 1 less and a difference of 1. (Example: What is the difference between 7 and 6? Relate this to a number line.)

1 + 0 1 + 1 2 + 1 1 + 3 1+ 4 5+ 1 1 + 6 7 + 1 1 + 8 9 + 1 1 + 10

6 – 1 8 – 1 4 – 3 5 – 4 7 – 6 9 – 8 10 -‐ 9

Set C -‐ 17 Cards Adding 2, Subtracting 2, Difference of 2

Set C relates to counting on by 2, counting back by 2, and a difference of 2. (What do we notice about 7 and 5, 8 and 6? How are they different and what is their difference?)

2 + 0 1 + 2 2 + 2 2 + 3 4 + 2 2 + 5 6 + 2 2 + 7 2 + 8 9 + 2 2 + 10

5 – 2 6 – 2 8 -‐ 2 7 – 5 8 -‐ 6 9 – 7

Set D – 18 Cards Friendly Ten & Near Ten

Set D relates to knowing combos of ten and using that knowledge to help with “close to 10”. If I know that 5 + 5 = 10, then 6 + 5 is 1 more, or 11. If I know that 10 -‐3 = 7 , then what would 11 – 3 = ____ (Think: 3 + 7 = 10, so 3 + 8 =

0 + 10 9 + 1 8 + 2 3 + 7 4 + 6 5 + 5 6 + 5

10 – 7 10 – 6 10 – 4 11 – 3 11 – 7 9 – 5 9 – 3


11) If 10 – 3 = 7, then 9 – 3 would be one less or 6. Or, if I know 3 + 7 = 10, then 3 + 6 = 9.

7 + 4 8 + 3 5 + 4 6 + 3

Set E – 15 Cards Adding & Subtracting Doubles

Doubles are visually easy to picture in our head and memorize as we are working with the same quantity.

2 + 2 3 + 3 4 + 4 5 + 5 6 + 6 7 + 7 8 + 8 9 + 9 10 + 10

6 – 3 8 – 4 10 – 5 14 – 7 16 – 8 18 -‐ 9

Set F – 12 Cards Adding and Subtracting Near Doubles

Use doubles facts for near doubles. If 2 + 2 = 4, then 2 + 3 = 5. When seeing 7 – 3, “I know that 3 + 3 = 6, so 3 + 4 equals one more , which is 7. “

2 + 3 3 + 4 4 + 5 5 + 6 6 + 7 7 + 8 8 + 9

5 – 2 7 – 3 9 – 5 11 – 5 15 -‐ 8

Set G – 19 Cards Connecting to Base 10 Adding & Subtracting 10 and 9

NBT domain helps with this strategy. For example, in K.NBT.1 (Kindergarten), students decompose teen numbers into tens and one. So, based on knowing 10 + 8 = 18, then what is 9 + 8?

10 + 3 10 + 4 5 + 10 10 + 6 7 + 10 10 + 8 9 + 10 9 + 3 4 + 9 5 + 9 9 + 6 7 + 9 9 + 8

12 – 2 16 -‐ 6 18 – 8 13 – 10 14 – 9 15 – 9 16 -‐ 9

Set H Leftovers 13 Cards

These are the extras that don’t quite fit as nicely under a specific strategy. However, kids can still use strategies to get them to fluency. For example, they might think of 5 + 7 as 5 + 5, and then add 2 more. 6 plus 8, could be thought of as the double 6 + 6 and then add two more.

8 + 6 3 + 5 4 + 8 5 + 7 5 + 8 6 + 8 8 + 5 7 + 5

8 – 5 12 -‐ 5 12 – 8 13 – 5 14 -‐ 8


Set A + 0, - 0, Sub. All Set B + 1, - 1, Diff. of 1 Set C + 2, -2, Diff. of 2 Set D Friendly 10/Near 10 Set E Add & Sub. Doubles

Set F Add & Sub. Near Doubles

Set G Connecting to Base 10 Set H Leftovers

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A Strategy Based Method for Developing Fluency of Addition ...gcamath.com/uploads/9/1/4/0/9140392/fact_fluency_-_conceptual... · A Strategy Based Method for Developing Fluency of