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Lesson excerpt:

NC Mathematics Standard:

Add and subtract within 20.

NC.2.OA.2 Demonstrate fluency with addition and subtraction, within 20, using mental strategies.

Standards for Mathematical Practice:

1.   Make sense of problems and persevere in solving them.

4.   Model with mathematics.

5.   Use appropriate tools strategically.

6.   Attend to precision.

7.   Look for and make use of structure.

Student Outcomes:

• I can chart number facts I have memorized on an addition chart.

• I can use mental strategies to add numbers within 20 with ease.

• I can recall from memory all sums of two one-digit numbers.

Materials:

• Addition chart for each student

• Highlighter and pencil for each student

• Optional- poster size chart or other way to display chart

Advance Preparation:

Thinking:

• Students have had instruction in many types of strategies at a conceptual level before this task is introduced. (See Special Notes at the bottom of this task for first grade standard.)

• A focus on number relationships is important in building upon this task as student work toward fluency.

• Teacher will need to read the attached article Developing Number Sense including the Basic Facts

Materials:

• Addition charts will need to be copied for each student and highlighters provided.

• A poster of the Addition chart can be created if desired.

Directions:

1. For this task, the teacher must set up an environment so that students feel comfortable looking at what they know and what they need to learn.  Making student’s responsible for their own learning rather than comparing them to other students requires discussions and respect for each other and for building confidence.

2. Students store this chart in a math folder or inside a math journal and use it to record facts that they are comfortable in knowing “mentally”. 

3. As teachers work with mental strategies, students are given an opportunity to record the facts they know mentally and can recall without the use of fingers or extended thinking.  When they feel confident that they know the facts, they highlight the box to create a visual that shows the facts they know and the facts they need to learn.

4. Student addition charts help teachers know where to focus instruction of strategies.  For example after doubles are taught and practiced, students can fill in the doubles on their charts.  When near doubles are taught they can fill in the facts they know “mentally.”  Keeping a record of their thinking and the facts they know will help students see the chart beginning to fill. 

5. A logical progression of strategy instruction that can be used to help students become fluent with the facts is one more and two more than facts, facts with zero, doubles facts, doubles plus one facts, facts that are left over after the strategies listed above. Also, always being aware of the relationship between the facts such as 7+4 and 4+7 being the same.  This understanding will help students see that the number of total facts they must learn in really only half of what is showing on the chart.

6. This addition chart can also be an assessment task to guide instruction as teachers assess to see which facts students know “mentally” and which ones they need to learn.  When working with students individually or in small groups teachers can note if students are using their fingers or needing more think time rather than “mentally” knowing the sums.  If used this way, it would be given to students after all strategy instruction has been taught rather than spaced out and used after individual lessons.  Additional assessment tasks can be found at http://commoncoretasks.wikispaces.com/. 

Questions to Pose:

Before:

What do we know about the relationship between two addends?

How does that relationship help us know more facts?

During:

What strategy are you using to recall these facts?

Which strategies are most helpful to you in recalling facts?

What could you tell your classmates that would help them recall facts faster?

How is the addition chart helpful to you?

After:

How will knowing my facts help me in other areas of math?

Possible Misconceptions/Suggestions:

Special Notes:

This second grade standard refers to the strategies listed in first grade.  The first grade standard is listed below with the appropriate strategies that students will use in first grade and will continue to build upon in second as they work to build fluency to 20.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows
12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Some students may already be fluent with their facts and use of strategies and can spend this time doing tasks that are more appropriate for them.  For example, this would be a good time to work on Marcy Cook task cards (www.marcycookmath.com), the Illuminations website (www.illumination.nctm.org) or other problem solving tasks.

Also, it is very important that strategy instruction be paced over the course of the year.  Integration of these strategies with problem solving tasks will help students see the importance of being fluent with number.

Solutions: N/A

Article from: Partners for Mathematics Learning, 2009

Developing Number Sense including the Basic Facts

Composition of numbers is the foundation of computational fluency. Students must know all the parts that make up a number in order to be fluent with basic facts (Postlewait, Adams, Shih 2003, p. 354). These number relationships play a significant role in fact mastery. Children should master the basic facts of arithmetic that are essential components of fluency with paper-and-pencil and mental computation and with estimation. At the same time, however, mastery should not be expected too soon. Children will need many exploratory experiences, and the time to identify relationships among numbers and efficient thinking strategies to derive answers to unknown facts from known facts. Practice to improve speed and accuracy should be used but only under the right conditions; that is, practice with a cluster of facts should be used only after children have developed an efficient way to derive answers from those facts. (NCTM 1989, 47)

According to John Van de Walle there are three components essential to promoting meaningful addition and subtraction fact mastery. These components are;

1.   Help children develop a strong understanding of number relationships and of the operations.

2.   Develop efficient strategies for fact retrieval through practice.

3.   Provide drill in the use and selection of those strategies once they have been developed. (Van de Walle, 2006, p. 95) Strategy practice must directly relate to one or more number relationships. Van de Walle suggests several number relationships that help children develop an understanding of basic facts. These strategies should be made explicit in the classroom. Strategies for addition facts are:

a.   one-more-than and two-more-than facts or counting up

b.   facts with zero

c.   doubles

d.   near doubles

e.   make ten facts

f.    commutative property

g.   compensation

Van de Walle suggests using “think-addition” as a powerful strategy for developing fluency with subtraction facts. An example of the “think-addition” strategy is when solving 8-5, think “five and what makes 8?” Other strategies for subtraction mastery are:

a.   counting back

b.   counting up

c.   doubles

d.   fact families

e.   subtracting from ten (Buchholz, 2004, p. 365)

Using strategies to solve problems develops over time. It is through class discussions that students begin to match strategies to numbers in problems. Helping students make the connections is a key objective of the classroom teacher. “Students do not immediately see these connections and may not see them at all unless they are examined and discussed.” (Huinker, 2003 p.352). Van de Walle writes that teachers need to plan lessons in which specific strategies are highlighted. These lessons include simple story problems designed to make certain strategies explicit. The second type of lesson revolves around a collection of facts for which a specific type of strategy is appropriate. (Van de Walle, p. 96). An example of this type of lesson is a series of problems where using doubles would help solve the problems.

Knowledge of the addition combinations (facts) should be judged by fluency in use, not necessarily by instantaneous recall. Through repeated use and familiarity, students will come to know most of the addition combinations quickly and a few others by using some quick and comfortable strategy that is based on reasoning about numbers. (Russell and Economopoulos, 2008, p. 192)

As students are working to develop understanding of the number combinations they are working on the part-part-whole relationship. They understand that there are parts within a number (7 include 6+1, 4 + 3, etc.). They also begin decomposing larger numbers. Teachers can develop number talks that focus on the connection between knowing “number facts” and knowing larger number combinations. For example a teacher could pose these problems (one at a time) on the board:

4 + 5 = ___ 4 + 2 = ___

40 + 50 = ___ 40 + 20 = ___

3 + 3 = ___ 6 + 2 = ___

30 + 30 = ___ 60 + 20 = ___

After the class solves the first equation show the second related equation. They can solve with cubes until the connection is made. Do several similar problems so the children can start making the connection between knowing number combinations for one digit number and how they relate to two digit numbers.

Sources:

Buchholz, Lisa. “Learning Strategies for Addition and Subtraction Facts: The Road to Fluency and the License to Think.” Teaching Children Mathematics (March 2004): 362-367.

Huinker, DeAnn, Janis L. Freckman, and Meghan B. Steinmeyer. “Subtractions Strategies from Children’s Thinking: Moving toward Fluency with Greater Numbers.” Teaching Children Mathematics (February 2003): 347-353.

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM.

Postlewait, Kristian B., MichelleR. Adams, and Jeffrey C. Shih. “Promoting Meaningful Mastery of Addition and Subtraction.” Teaching Children Mathematics (February 2003): 354-357.

Russell, Susan Jo and Karen Economopoulos. Investigations in Number, Data, and Space, : Counting, Coins and Combinations, grade 2. Pearson Education, Inc. 2008.

Van de Walle, John A. and LouAnn H. Lovin. Teaching Student-Centered Mathematics, Grades K-3. Boston: Pearson Education, Inc., 2006.