Grade 1 and 2: Basic Addition and Subtraction Facts Series 2: First and Second Grade

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1 Grade 1 and 2: Basic Addition and Subtraction Facts Series 2: First and Second Grade Texas Essential Knowledge and Skills (TEKS): (1.3) Number, operation and quantitative reasoning. The student recognizes and solves problems in addition and subtraction situations. (B) use concrete and pictorial models to apply basic addition and subtraction facts (up to = 18 and 18 9 = 9) (2.3) Number, operation, and quantitative reasoning. The student adds and subtracts whole numbers to solve problems. (A) recall and apply basic addition and subtraction facts (to 18) Purpose: The second concept that will be covered is Basic Addition and Subtraction Facts. This lesson is written for first and second grade. To begin, participants will experience a myriad of organizers to build and reinforce understanding and create visual or mental images to aid in recall of basic facts. They will use strategies such as Counting On, Counting Back, Doubles, Near Doubles, Make Ten, Related Facts and Compensation to learn and remember basic facts up to the sum of 18. The activities will encourage the use of specific strategies to help acquaint the students with a variety of strategies. After repeated experience with organizers, students generalize which organizer works the best for particular facts. After repeated experience with organizers, students should efficiently recall basic facts by creating a visual or a mental image. Suggested Vocabulary: add doubles minus 1 matching patterns addends doubles plus 1 minuends plus compensation even numbers minus problem counting on fact Families near doubles solution counting back inverse Operation numbers 1-18 subtract difference make Ten odd numbers subtrahends doubles mat pair sum Materials: Linking cubes, such as Unifix cubes (2 sets of 10 in 2 different colors) Linking cubes, such as Unifix cubes (1 bag of 100 per group of 4) Proportional fraction manipulatives, such as Cuisenaire rods Teddy Bear counters (optional) Handouts 3-68 to 3-88 Blank recording paper Mathematics TEKS Connections: Grades K-2 692

2 For Each Student: Handout 3-85 : Three In A Row Game Board For Each Group of Students: Handout 3-69: Doubles and Near Doubles Cards - cardstock and cut. Handout 3-70: Doubles and Near Doubles Linking Cubes Mat 1 cardstock, laminate, and tape together. Handout 3-71: Doubles and Near Doubles Linking Cubes Mat 2 cardstock, laminate, and tape together. Handout 3-72: Doubles and Near Doubles Linking Cubes Mat 3 cardstock, laminate, and tape together. Handout 3-73: Doubles and Near Doubles Linking Cubes Mat 4 cardstock, laminate, and tape together. Handout 3-74: Doubles and Near Doubles Connecting Cubes Mat 1 - cardstock, laminate, and tape together. Handout 3-75: Doubles and Near Doubles Connecting Cubes Mat 2 - cardstock, laminate, and tape together. Handout 3-76: Doubles and Near Doubles Connecting Cubes Mat 3 - cardstock, laminate, and tape together. Handout 3-77: Blank Ten Frames - cardstock Handout 3-78: Make Ten Organizer - cardstock Handout 3-80: Part-Part-Whole Mat 1 - cardstock Handout 3-81: Basic Facts Headings 1 Handout 3-82: Basic Facts Headings 2 Handout 3-83: Basic Facts Headings 3 Handout 3-84: Basic Facts Cards Handout 3-86: Three in a Row Picture Cards - two sets in color and cut Handout 3-87: Three in a Row Numeral Cards - two sets in color and cut Handout 3-88: Three in a Row Ten-Frame Numeral Cards - two sets in color and cut (optional) Literature Connection: Math Fables by Greg Tang Double the Ducks by Stuart Murphy Mathematics TEKS Connections: Grades K-2 693

3 Procedures Engage: 1. Tell the students that you are going to give them a problem to solve. There are manipulatives available should they need to use them. The problem is: There are eight girls and nine boys in line for the basketball game. How many children are in the line? There are 17 children in line. The purpose of this activity is to see how the students solved this problem. Do not let them say, I just know it. Notes TEKS Connections Have manipulatives available for the students to be able to use whenever needed. First grade is still on the concrete and pictorial level while building understanding of the basic facts; therefore, manipulatives should always be available. Second grade should have manipulatives available while they are developing strategies to learn their basic facts. This will allow the students to develop mental pictures that they can later recall when applying their basic facts. Present the problem to the students and have them solve it any way they can. Have them record how they solved it. 2. Have the students record their strategies on Handout 3-68: Problem and post their recordings when they are finished. Group the recordings by similar strategies. Use Handout 3-68: Problem as recording sheet. The purpose of this Engage activity is to assess what strategies the students already know. You could also ask the students to solve this problem in two different ways or in as many ways as they know. Handout 3-68: Problem Mathematics TEKS Connections: Grades K-2 694

4 Research Fluency might be manifested in using a combination of mental strategies and jottings on paper or using an algorithm with paper and pencil, particularly when the numbers are large, to produce accurate results quickly. Regardless of the particular method used, students should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general (NCTM, Principles and Standards, 2000, p 32). In the Mathematics TEKS Refinements Professional Development Materials (MTRs) some of these strategies were mentioned. Some strategies were also mentioned in Session 1 of the workshop. The intent is to have the students see that several strategies could be used to solve basic facts. Some strategies the students will not know will be demonstrated in the Explain section. 3. Have the students make generalizations of each grouping of strategies. The intent is for the students to see similarities and to label these similarities with the names of the strategies such as Near Doubles, Doubles, Make Ten, Splitting Strategies, Counting On, Counting Back, Compensation, and Related Facts. If the students cannot make the generalizations, the instructor should help the students make this connection. Can you tell me how these recordings are similar? Responses may vary. This question asks for the similarity in the strategy used. Research Children should see relationships that exist among the facts. There are two major benefits of this emphasis on relationships. The emphasis on relationships improves retention because it is easier to remember things that are related to other things that we already know. Also, recognition of those relationships drastically reduces the amount of memorization needed (Tucker, Singleton, and Weaver, 2006, p. 92). Mathematics TEKS Connections: Grades K-2 695

5 Explore: 1. Have the students explain which strategies they used to solve the problem. For example: a. Doubles Plus 1 strategy may be explained as: is close to 9+9 =18; so take one from 18 and the result is 17. b. Related Facts strategy may be explained as: I know that = 17, so 17-9 equals 8. c. Make Ten strategy may be explained as: 8 + 9= 18 because if you take one from 8 and add it to the 9, you have =17. Teacher Note This explore section is designed to allow children to share how they solved the problems using different strategies and developing their own generalizations of which strategies work the best for this problem. Use the organizers with careful questioning in order to help students experience the strategies in their own words first, then add the next label of the next strategy. Scaffolding This section has several organizers to use as tools for the students. This is an effective form of instructional scaffolding. Caution Introduce one organizer at a time and be certain that the students can demonstrate understanding with it before introducing another organizer. Organizers may be introduced periodically but not all at once. Research The importance of using graphic organizers is addressed by Ashlock: Our assessment tasks should be varied in format so that students with different intelligences and learning styles can demonstrate what they understand and are able to do. A graphic organizer often provides a useful format for an assessment task: it can focus on relationships while requiring fewer verbal skills. Also, tasks based on graphic organizers can usually be administered to a group of children at one time. Graphic organizers that have been used during instruction are especially useful for diagnosis (Ashlock, 2002, p. 35). Mathematics TEKS Connections: Grades K-2 696

6 Note The following strategies, Counting On and Counting Back are more effective when used when only having to count on or back at the most three or four numbers. 2. If a student chooses Counting On or Counting Back strategy, have him or her explain it to the class. How can we use the Counting On strategy to solve this problem? Responses may vary. Would this strategy be best for solving this problem, 8 + 9, or a problem like 8 + 3? Responses may vary. Possible responses include: This strategy is best for solving because it is easy to lose where you are in the counting on process when the number you are counting on is too large. 3. If a student solved the problem using near doubles, have the student explain his or her strategy. Show the students the Doubles and Near Doubles mat. How can we use this mat to solve this problem? Responses may vary. Let s fill all of the doubles on the mat with their matching number sentences first. What do you notice about the sums? Responses may vary. They are all even numbers. What do you notice about the addends? Responses may vary. They are either both odd or both even. If you had what would the sum be? The sum would be 6. Is six an even or odd number? Six is an even number. Looking at this mat, if you had the sum of 8, what could the addends be? The addends could be 4 and 4. Counting On: Start with the large number and count on the smaller number so that adding would count on 8 numbers starting from 9: 10, 11, 12, 13, 14, 15, 16, 17. Counting Back: Use this strategy for subtraction. For example, 9 3 would count back as 8, 7, 6. A graphic organizer effectively used for this strategy is a number line or a hundred chart. Have the mat available to review. Handout 3-69: Doubles and Near Doubles Cards will be the facts for either mat. If the students are using linking cubes, such as Unifix cubes, then tape together Handouts 3-70: Doubles and Near Doubles Linking Cubes Mat 1 to 3-73: Doubles and Near Doubles Linking Cubes Mat 4. If the students are using proportional connecting cubes, such as Cuisenaire rods, then use Handout 3-74: Doubles and Near Doubles Connecting Cubes Mat 1 to Handout 3-76: Doubles and Near Doubles Connecting Cubes Mat 3. Doubles: = 18 Near Doubles: Doubles +1: can be renamed as Doubles -1: can be renamed as Have the students understand plus one and minus one by sliding one of each of the doubles to one of the neighboring white spaces. Again, have the students identify the number sentences and sums that match the doubles plus or doubles minus one. Mathematics TEKS Connections: Grades K-2 697

7 If the sum is an even number, you could use doubles to solve the problem. Since doubles are two of the same thing, the numbers would have to be the same numbers. Since has an odd number and an even number addend, then we know just doubles would not work since the addends are not both odd or both even. We either need to add one to 8+8, or subtract one from 9+9. This is an example of near doubles. If we slide one of the stacks of 8 from 8+8 to the right and one of the stacks of 9 from the 9+9 to the left and join the two stacks, what would be the number sentence for a near doubles example? The number sentence would be = 17 How will knowing this strategy help you with your basic facts? Responses may vary. Possible responses include: Doubles will have even number sums and Near Doubles will have odd number sums. Also, if you only have the minuend, then if the minuend is even, the subtrahend and difference would be doubles. If the minuend is odd, then the subtrahend and the difference would be one odd and one even number. Therefore, knowing this strategy will be a way to see if your answer is reasonable. Handout 3-72: Doubles and Near Doubles Mat 3 Research Exploring thinking strategies like these or realizing that is the same as will help students see the meaning of the operations. Such explorations also help teachers learn what students are thinking (NCTM, 2002, p. 34). Research Researchers have found that the doubles (6+6, 8+8, and so on) tend to be easier for children to commit to memory than are other basic addition facts. A reasonable approach might be to get the doubles memorized and then build on them to find answers to other facts. In fact, the doubles strategy is one of the more successful ones (Tucker, Singleton and Weaver, 2006, p. 105). Research Tucker, Singleton and Weaver believe that mats help develop a mental picture which helps retain the basic facts: The algorithms should be taught with effective physical or pictorial models. Modeling a concept or skill lets the children see what the concept or skill looks like and helps the children to develop clear mental imagery. Modeling gives meaning to the algorithms... the rules arise out to patterns observed by the children as they use appropriate models that allow them to visualize the mathematical procedures (Tucker, Singleton and Weaver, p. 93). Mathematics TEKS Connections: Grades K-2 698

8 4. If a student chooses the Make Ten strategy, have he or she explain it to the group. One way to make ten is to use ten frames with counters. The students will get eight counters to place in the first ten frame to represent the number of girls. Next, the students will place nine counters in the second ten frame to represent the number of boys. How many spaces are still open in the ten frames? Responses may vary. Students should then physically move 1 counter from the 8 counters to make a full 10-frame or move 2 counters from the 9 counters to make the 2 nd frame a full 10-frame. This will show one full ten frame and 7 remaining counters. 10+7=17. If no one used that strategy say: Another mat that we could use is the Make Ten mat. Show the students the Make Ten mat. How can this mat be used to make ten? Responses may vary. Possible responses include: Place 8 counters and then 9 counters on the mat. If the students fill one column with 10 and then adds the remaining counter to the other column of ten, the result would be = 17. Making a ten with the counters makes it easier to count. The students could have also filled the bottom or top half of the columns using 5 as a benchmark number on the Make Ten mat. This could also demonstrate a splitting strategy of =17 which could be another way to compute to 17. Make Ten: can be renamed as = Use Handouts 3-78: Make Ten Organizer and 3-79: Make Ten Organizer Recording Sheet to have the Unifix cube recording sheet to make 10 demonstrated to the class. (Handout 3-77: Blank Ten Frames can be used if students are struggling and are familiar with ten frames. This handout can be used with manipulatives such as teddy bear counters.) Handout 3-78: Make Ten Organizer Research Tucker, Singleton and Weaver (2006) believe that Make Ten is another effective strategy when used with a device to provide mental imagery for the process: The children should practice using the 10- frame to find answers until established. Then children are able to get answers by visualizing how they would use the 10- frame. Mathematics TEKS Connections: Grades K-2 699

9 5. If a student chooses the Splitting Strategy, have him or her explain it to the class. If a student did not use the splitting strategy, ask the students: Did anyone break numbers apart to combine them again to make friendlier numbers? Responses may vary. This is also called the splitting strategy. 6. Another mat we could use is the Part-Part-Whole Mat. Have you ever used a Part-Part-Whole mat before? Responses may vary. How did you use it? The Part-Part-Whole Mat was used to understand addition or subtraction and how to write number sentences How do we label the parts and the total? Make sure students label girls and boys and total number of children. The Part-Part-Whole Mat can also show the fact families for = 17 or = 17. However, if you also hide one of the parts, then 17 9 = 8, or hide another part for 17 8 = If a student chooses the compensation strategy, have him or her explain it to the class. If not, ask the students if anyone could think of a way to combine the strategies to solve the problem. Splitting Strategy (some books call this recomposition) The nine is split into so that the 2 could be used with the 8 to make 10. Then the 7 could be added to the 10 making a simpler number sentence of = 17. Related Facts (Fact Families) Fact families help students to understand = 17, so = 17, 17 8 = 9, and 17 9 =8. All four facts can be demonstrated using the Part-Part-Whole mat. Looking at the Part-Part-Whole mat, Handout 3-80: Part-Part-Whole Mat. 8 girls 9 boys 17 children The students could see that = 17 and = 17. However, if you also hide one of the parts, then 17 9 = 8 or 17 8 = 9. Related Facts (Inverse Operations): The student may use the Part-Part-Whole mat and say: I saw 17 8 =? and thought to myself, I do know that = 17; so 17 8 = 9. Compensation can also be solved with a combination of the strategies above. For example, the splitting strategy of decomposing 9 to 7 + 2, then make tens by joining (8+2) + 7. Or (8+8) + 1 = 17 or 4+ (5+5) +3 = 17. Mathematics TEKS Connections: Grades K-2 700

10 Explain 1. Have the students sit in table groups. Give the students 5 facts on cards Have the students look at the facts and think of all the strategies that could be used to solve that fact. Give the students the following headings that they could sort with the facts: Near Doubles, Doubles, Make Ten, Splitting Strategies, Counting On, Counting Back, Compensation, and Related Facts. 3. Have the students tell which headings the facts would go under and justify their answer. Elaborate 1. Have Doubles and Near Doubles, Make Ten, and Part-Part-Whole mats or organizers available with a variety of manipulatives including Unifix cubes, and Cuisenaire rods available. When two numbers are added, the total amount (their sum) is 13. What might the two numbers be? Responses may vary. Possible responses include: 0, 13; 1, 12; 2, 11; 3, 10; 4, 9; 5, 8; 6, 7. The students will need one set of the headers Handouts 3-81: Basic Facts Headings 1 to 3-83: Basic Facts Headings 3 of the strategies and one set of the basic facts Handout 3-84: Basic Facts Cards. There is also a card for Other Strategies (should the students have solved the problem differently or if they are confused about the identity of their strategy). In the MTRs some of these strategies were mentioned. The intent is to have the students see that there are several strategies that could be used to solve basic facts. Some strategies that the students do not know will be demonstrated in the explain section. Have the students justify their answers. For example: Doubles Doubles Make Ten 5 +5 =10 plus one more is 11; or take four from five and add to six to make ten then add the one left over from the five. Related facts by using Fact Families: I know = 11; so = 11. Related facts by using Inverse Operations: I know 11 6 = 5 so = 11 Counting On: Start with 6 and count on 5 times (6... 7, 8, 9, 10, 11). Scaffolding This section has several organizers to use as tools for the students. This is a type of instructional scaffolding (Echevarria, Vogt, and Short, 2004). Mathematics TEKS Connections: Grades K-2 701

11 What strategies did you use to find the solutions? Responses may vary. Possible responses include: students may use organizers; they may use organizers differently than discussed; they may make a list such as above; or they make a table of the numbers above. + = 13 Evaluate 1. Practice addition and subtraction by playing 3 in a row using Handout 3-85: Three in Row Game Board. Have the students play with a partner and use the 3 by 3 grid and playing cards with numbers one through nine. Research The importance of using graphic organizers is addressed by Ashlock: Our assessment tasks should be varied in format so that students with different intelligences and learning styles can demonstrate what they understand and are able to do. A graphic organizer often provides a useful format for an assessment task: it can focus on relationships while requiring fewer verbal skills. Also, tasks based on graphic organizers can usually be administered to a group of children at one time. Graphic organizers that have been used during instruction are especially useful for diagnosis (Ashlock, 2002, p. 35). Give each student 2 sets of cards 1-9, of both Handout 3-86, 3-87, or 3-88 and Handout 3-85: Three in Row Game Board. Have the students shuffle the cards before playing. Handout 3-88: Three in a Row Ten-Frame Numeral Cards can be used for first grade as ten frames are used to help the students use the make ten strategy easier to apply when recalling the facts. Handout 3-86 has a pictorial model of the number and can be used for first grade as well as second grade. For most of the cards on this handout, the pictorial model depicts the quantity on the card by the number of sides the object has on the card. Handout 3-87 is for a second grade class if the teacher would prefer to focus on recalling basic facts without a pictorial model. Mathematics TEKS Connections: Grades K-2 702

12 Handout 3-86: Three-In-A-Row Picture Cards Handout 3-88: Three In a Row Ten-Frame Numerical Cards 2. Tell the students that they will form their own game board by choosing 9 numbers from 0 to 18. They will place one of each of the nine numbers in the squares. They may not write a number more than once. Place the cards between the two players (to be seen easily). The students cannot write a number on the game board more than once on Handout Probability could also be introduced with this activity. The probability of getting a particular sum or difference is: for a 0-9 times, 1 16 times, 2 15 times, 3 14 times, 4-13 times, 5 12 times, 6-11 times, 7 10 times, 8-9 times, 9 8 times, 10-9 times, 11-8 times, 12 7 times, 13 6 times, 14 5 times; 15 4 times; 16-3 times; 17 2 times; and 18 1 time. Therefore, the most probable numbers to choose would be 1, 2, 3, 4, 5, 6, 7, then choose 2 numbers from 8, 10, or Mathematics TEKS Connections: Grades K-2 703

13 3. Each partner draws a card and places it face up. 4. The partners will use both cards to form an addition or subtraction problem that will give them either the sum or difference on their card. If the sum and difference can be formed from the two cards, students may claim both numbers on their Three in a Row Game Board. If the number is not on the board, then the student will not mark a space on the game board (similar to Bingo). The students may put an x on the number, they may use 2 color counters to cover the numbers, or they may also color in the square of the numbers in order to claim the sum or difference or both. Ex: = = 2 Research Why use a game format? The best games will be games involving only a few students, preferably students with rather comparable abilities. In such games a student can feel secure enough to try simple recall. We should choose games that provide immediate or early verification; students should learn promptly if they recalled correctly. Games are also useful for retention once the number combinations are mastered, and research suggests that relatively infrequent use of games can maintain skill with number combinations (Ashlock, 2002, p. 74). Mathematics TEKS Connections: Grades K-2 704

14 Handout 3-68: Problem Grades 1-2 There are eight girls and nine boys in line for the basketball game. How many children are in the line? Mathematics TEKS Connections: Grades K-2 705

15 Handout 3-69: Doubles and Near Doubles Cards Grades = = = = = = = = = = = = = = = = = = = = Mathematics TEKS Connections: Grades K-2 706

16 Handout 3-70: Doubles and Near Doubles Linking Cubes Mat 1 Grades 1-2 Mathematics TEKS Connections: Grades K-2 707

17 Handout 3-71: Doubles and Near Doubles Linking Cubes Mat 2 Grades 1-2 Mathematics TEKS Connections: Grades K-2 708

18 Handout 3-72: Doubles and Near Doubles Linking Cubes Mat 3 Grades 1-2 Mathematics TEKS Connections: Grades K-2 709

19 Handout 3-73: Doubles and Near Doubles Linking Cubes Mat 4 Grades 1-2 Mathematics TEKS Connections: Grades K-2 710

20 Handout 3-74: Doubles and Near Doubles Connecting Cubes Mat 1 Mathematics TEKS Connections: Grades K-2 711

21 Handout 3-75: Doubles and Near Doubles Connecting Cubes Mat 2 Mathematics TEKS Connections: Grades K-2 712

22 Handout 3-76: Doubles and Near Doubles Connecting Cubes Mat 3 Mathematics TEKS Connections: Grades K-2 713

23 Handout 3-77: Blank Ten-Frames Grades 1-2 Mathematics TEKS Connections: Grades K-2 714

24 Handout 3-78: Make Ten Organizer Grades 1-2 = Mathematics TEKS Connections: Grades K-2 715

25 Handout 3-79: Make Ten Organizer Recording Sheet Grades 1-2 Mathematics TEKS Connections: Grades K-2 716

26 Handout 3-80: Part-Part-Whole Mat 2 Grades 1-2 Mathematics TEKS Connections: Grades K-2 717

27 Handout 3-81: Basic Facts Headings 1 Grades 1-2 Doubles Near Doubles Make Ten Counting On Mathematics TEKS Connections: Grades K-2 718

28 Handout 3-82: Basic Facts Headings 2 Grades 1-2 Counting Back Splitting Strategy Compensation Related Facts Mathematics TEKS Connections: Grades K-2 719

29 Handout 3-83: Basic Facts Headings 3 Grades 1-2 Other Strategy Mathematics TEKS Connections: Grades K-2 720

30 Handout 3-84 Basic Facts Cards = 18 9 = = 15 7 = = 14 9 = = 13 6 = = 12 7 = = 14 8 = Mathematics TEKS Connections: Grades K-2 721

31 Handout 3-85: Three in a Row Game Board Grades 1-2 Mathematics TEKS Connections: Grades K-2 722

32 Handout 3-86: Three in a Row Picture Cards Mathematics TEKS Connections: Grades K-2 723

33 Handout 3-87: Three in a Row Numeral Cards Mathematics TEKS Connections: Grades K-2 724

34 Handout 3-88: Three in a Row Ten-Frame Numerical Cards Mathematics TEKS Connections: Grades K-2 725

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