Explanation of the Standard


 Jeffery Hubbard
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1 Jenny Kerola CCLM Project 2 July 15, 2011 Grade level: 2 Domain: Operations and Algebraic Thinking (OA) Cluster: Add and Subtract within 20 Standard: 2.OA.2 2. Fluently add and subtract within 20 using mental strategies. 2 By end of Grade 2, know from memory all sums of two onedigit numbers. 2 See standard 1.OA.6 for a list of mental strategies. Explanation of the Standard Assumptions of Student and Teacher Background Knowledge Students Background Knowledge It s important to consider what this standard assumes students already know. The superscript 2 above refers to standard 1.OA.6 and mental strategies students should be fluent with from first grade. Those strategies they should know include the following: counting on making ten decomposing a number leading to a ten using the relationship between addition and subtraction creating equivalent but easier or known sums Please see the table below in this section for examples of some of mental strategies students should know and use. In addition to being able to use those strategies within 20, students should be fluent with addition and subtraction combinations with a total up to 10 from first grade (know all of the make 10 combinations). Part of standard 1.OA.6 also means that students will have experiences in first grade with adding and subtracting within 20, even though fluency is only expected within 10. Vocabulary from the Standard Fluent means being able to solve problems fast (within 35 seconds) and accurately in one of the following ways: by just knowing it (from memory), by using patterns (i.e. adding 0 always gives the same number), and by using strategies. Fluency is a mixture of the three ways and the ability to be flexible with using the various strategies. This does NOT mean drilling basic facts with flashcards or using traditional timetests. It is okay, however, to practice facts by strategy (i.e. practice all facts where students would make 10 to solve them: = ). I know = 10, so 2 more is 12.
2 Vocabulary continued Strategy Purposeful manipulations that may be chosen for specific problems. This means my strategies are based on context and specific numbers for a given problem, and they may not work for every problem. Example: = Instead of solving with an algorithm, I might think about money. I know 3 quarters + 1 quarter is 100 cents, so is one less, or 99. So, I think 75 + (251). The above example is a strategy because it may not work with every problem. In other words, I would not automatically relate = to money to help me mentally solve that problem. Level 1, 2, and 3 Strategies The following table gives examples of the three different levels of strategies students might use with their addition combinations (and the reciprocal equations). In order to be fluent, students should be using level three strategies mentally for some of the singledigit addition combinations up to 20. A level two (count on) strategy is considered to be fluent if it is used efficiently (meaning only counting on about 3 numbers within the 35 seconds). Notice that level 1 (counting all) strategies are not considered proficient in second grade. Taken from: Draft, 5/29/11, comment at commoncoretools.wordpress.com DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin Milwaukee. ( )
3 Putting it Altogether and Making Sense of the Standard Standard This means that students can 2. Fluently add and subtract within 20 using mental strategies. 2 By end of Grade 2, know from memory all sums of two onedigit numbers. See standard 1.OA.6 for a list of mental strategies. 1. Use a variety of strategies to fluently add and subtract within 20 (first by using objects, diagrams, tens frames, 100 chart and then transferring to performing the strategies mentally). They should be able to choose an efficient strategy based on many experiences adding and subtracting within Know or figure out (within 35 seconds) addition combinations of two singledigit numbers up to a total of 20 (and the subtraction problems that are the opposite of the addition combinations). Counting On = I said 9 and went 2 more 10, 11. Examples of Student Strategies Doubles I know = 8. Doubles Plus = = I know = 10, so is just o more, 11. Making Ten = Students might picture tens frames in their minds. Decomposing a Number Leading to 10 (Backing down through 10) Students can use their experience with number lines to picture this strategy in their mind = Counting On to Solve Subtraction Problems 13 9 = I took away 9, so 10, 11, 12, 13. I counted on 4 times, so 13 9 = 4. I know I need 2 more to make 10 (fill the first tens frame) and then I would have 4 left, so = First I went back 4 to get to 10, then I had to go back 1 more, so altogether I went back
4 School Mathematics Textbook Program a. Textbook Development How Fluency is Introduced The Investigations curriculum begins developing the concept of fluency in kindergarten with activities to develop subitizing (just knowing how many objects are in a set without needing to count) through 5 and composing and decomposing numbers through five. It continues developing the concept in first grade, focusing on subitizing through 10 and building fluency with addtion and subtraction of singledigits within 10. How Fluency is Developed in Grade 2 Fluency is developed through both daily routines and lessons including activities. The lesson(s) where activities can be found are included so teachers can see the progression and frequency of concepts in the textbook. Routines Today s Number (unit through end of year) Students practice composing and decomposing a number by finding as many ways as they can to make the number. Throughout the year the guidelines for how to do that get more specific. In the beginning of the year single digit numbers are used, and by the end of the first unit, students add and subtract within 20. Quick Images (unit through end of year) This routine is continued from first grade and begins in grade 2 with dot patterns in ten frames. The image is flashed for three seconds and students recreate it in a blank ten frame. Then double ten frames are used showing make ten combinations. By the end of the first unit, three ten frames are used with the first two showing a ten combination, and students find the total number of dots. The routine continues to progress throughout the year. Pocket Day (unit through end of year) Students use various methods to record as a class how many pockets each person is wearing. The first time, as a class, students add up the total number of pockets first by looking for make 10 combinations. Throughout the year, the strategies students use in this routine progress to eventually adding tens and ones. Activities Guess My Number on the Number Line (unit ) This is a game that introduces students to the number line, a tool they might use later on to mentally picture level 3 strategies. A number is chosen and kids make guess about what the number is. After each guess, the person who chose the number gives a clue (i.e. the number is greater than 10). This game is also important in developing number sense about the magnitude of whole numbers compared to other whole numbers, which is important in addition and subtraction. Plus 1 or 2 Bingo (unit ) Given a game board with singledigit numbers through 12, students take a card and figure out the combinations for plus 1 and plus 2 of the number. Then, they strategically mark one of the totals off on the game board, trying to get bingo. The activity reviews combinations through ten. Make 10 (unit ) This is a game where students lay out number cards face up in a 5 x 4 array and look for combinations that make 10. Tens Go Fish (unit ) A card game mirrored after Go Fish where students look for pairs that make 10.
5 *Kids think, How many more would I need to make 10? These games push kids to the next level of reasoning, either from level 1 to 2 (counting all to counting on) or from level 2 to level 3. After multiple days playing these games, students are given cards with the make 10 and plus one or plus 2 combinations and use them to continue to practice developing fluency. Magic Pot (doubling) (unit 14.6) A story is read aloud about a magic pot that doubles everything that is put into it to introduce the concept of doubling. Students write story problems about doubling. Games About Doubling (unit ) Students play 2 different games called Double It and Double Arrays where students practice counting on to double a number and seeing doubles as 2 rows of the same number of squares. They also solve and write story problems to develop their understanding and fluency around doubles. Problems about Combining Three Groups/Number Strings (unit , 1.5, 1.6) Students solve story problems about three different groups in two different ways, to explore the associative property. Then they decontextualize the concept with number strings and use cubes to explore combining more than two singledigit numbers. Students use combinations they just know (make ten, doubles). Near Doubles (Unit 31.2, 1.5, 1.6) Students discuss how knowing doubles combinations can help them figure out combinations plus or minus 1. Beat the Calculator (unit 31.3, 1.5, 1.6) This is a game where students have a partner and take turns solving number strings on paper and on the calculator. The person without the calculator tries to beat the calculator by using combinations he/she knows. Students quickly see that using level 3 strategies is much more efficient than counting all or counting on. Close to 20 (unit ) From five cards, students use three to get as close to 20 as possible. Students record their total and how far away it was from 20. This game lets them practice level 3 strategies to be more efficient. Addition and Subtraction Story Problems (unit ) Students use strategies to solve addition and subtraction problems in context. Discussions model those strategies and push children to see subtraction as the opposite of addition. Cover Up (unit ) Students have a certain number of counters and hide some of them underneath a paper. Their partners figure out how many they re hiding by using their strategies. Counting Bags (unit ) Students take a bag with an unknown total number of counters and decide how to count them (by 1s, 2s, 5s, 10s). Observing students do this activity would reveal which numbers they are able to subitize. Grouping by 10s (unit 34.2) This is a discussion based on previous activities with cubes and what students noticed when making towers of 10 cubes. As a class, a table is created by breaking down a number into how many towers of 10 could be created and how many cubes would be leftover. This visual helps kids see plus 10 combinations, which are addressed directly in following lessons. How Fluency Is Reviewed The remaining two units on addition, subtraction and the number system (units 6 and 8) focus on solving addition and subtraction story problems. Students get practice in using the combinations they know and come to know additional combinations through repeatedly solving story problems and playing games. The concepts they have learned leading to fluency are revisited throughout the year during the daily routines mentioned above.
6 b. Conclusions In the second grade curriculum, Investigations includes many activities and routines that help build students conceptual reasoning and number sense to support attaining fluency with addition and subtraction of two singledigit numbers within twenty. The first two number and operations units are filled with activities, games, and daily routines that support this standard. There are sufficient activities to build fluency with the make ten combinations in the first unit (although according to the CCSSM, more of this work should be done in first grade), plus one or two combinations, and doubles combinations. However, the traditionally more challenging combinations (near doubles) and remaining combinations that do not fit in a category (i.e. 8+5, 9+4, etc.) are not addressed frequently enough for all kids to develop proficiency and fluency with them. Another area where the Investigations curriculum needs more supplementation is in the work of developing fluency with subtraction within twenty. There is not specific work with subtraction until the end of the first unit and then it is in the context of stories. The next number and operations unit is the third unit. Again, subtraction work is minimal compared to the work students do around addition. There needs to be more opportunities for kids to see the relationship between addition and subtraction and to practice subtraction both in and out of context to gain fluency. c. Suggestion Investigations includes many routines and activities that help students attain fluency in knowing or efficiently finding the sum of two singledigit numbers. Teachers need to know and remember the definition of fluency and help students reach proficiency not through drill and kill or timetests but through many meaningful experiences adding and subtracting within 20. Some of the work done in the first unit with the make 10 combinations will need to be done in first grade to meet the CCSSM. To make the program stronger and help students also reach fluency in the reciprocal equations (subtraction), teachers need to take out some other sessions that do not directly relate to the CCSSM or that do not need to be addressed at this level and replace them with sessions where students are engaged in activities that allow them to compose and decompose numbers and in games where they can practice using those subtraction equations. Teachers can also help supplement through additional homework where students are asked to play some of the games at home and do activities to support their work with the standard that s going on at school. A Check Point Formative Assessment that Reveals Student Thinking For the formative assessment, I have adapted one from Investigations. It is a checklist that is meant as a way to collect information for the teacher, so there is no example of student work. However, I have included an example of what it might look like when filled out. This assessment is normally done at the end of the first unit. Instead, teachers can use the checklist toward the beginning of the unit to get information about what students retained from first grade. Although the CCSSM dictates students already be fluent within 10 and shouldn t necessarily need this assessment in grade 2, using this assessment in second grade will help teachers shift content to better align with the CCSSM. It will help teachers know which sessions they can eliminate for the whole group and replace with other content. I have also included a similar checklist that would be used later on and would be done in the same way. The checklist I ve created allows teachers to record more detailed information about how their students are thinking about the combinations. Teachers give students a set of the make 10 combination cards in random order and students go through the cards, providing the answers. Teachers check which strategy the student used or write a comment to record observations. To show they are fluent, students need to be fast and accurate with the combinations. There can be a mixture of just knowing the answers, knowing some answers from patterns, and knowing some answers by using strategies.
7 Student Name Assessment Checklist for Make 10 Combinations Combination Just know Count on Use an equivalent or a known combination Comments
8 A mixture of just knows and level 2 and 3 strategies = fluency The student understands how to count on efficiently by starting with the largest number and can do so in 3 seconds. This shows he used a level 3 strategy to solve He changed it into a doubles combination he knew. DRAFT DOCUMENT, UNEDITED COPY. This material was developed for the Common Core Leadership in Mathematics (CCLM) project at the University of Wisconsin Milwaukee. ( )
9 Student Name Assessment Checklist for NearDoubles Combinations Combination Just know Count on Use an equivalent or a known combination Comments
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