Thinking About Place Value in Grade 2 Introduction: Building an Understanding of Place Value as a Foundation for Addition and Subtraction Students in Grade 2 need a firm understanding of place value. Addition and subtraction are a major emphasis at this grade level, and it is important for students to have a strong conceptual understanding of the base ten system, positional notation, and how bundling and unbundling of numbers can assist them in adding and subtracting . In particular, the Common Core State Standards for Grade 2 dealing with Number and Operations in Base Ten call for students to both understand place value and then use place value understanding and properties of operations to add and subtract. The foundation work for place value begins in Kindergarten and first grade. The Common Core State Standards for Kindergarten expect students to work with numbers 11-­‐19 to gain foundations for place value. Students at this grade level are expected to be able to compose and decompose numbers in this range into ten ones and some further ones (K.NBT.1). The Grade 1 Common Core State Standards call for students to extend their understandings of place value more broadly to two-­‐digit numbers (1.NBT.2-­‐
6) by not only understanding that the two digits of a two-­‐digit number represent amounts of tens and ones but also by comparing two-­‐digit numbers and using place value understanding to add within 100, mentally find ten more or ten less than a number, and subtract multiples of 10 in the range 10-­‐90 from multiples of 10 in the range of 10-­‐90. For students in Grade 2, the Common Core State Standards set the expectation that they be able to understand the three digits of a three-­‐digit number as representing the amounts of hundreds, tens, and ones. They are also expected to be able to read and write numbers to 1000 using base ten numerals, number names, and expanded form. They are expected to be able to compare two three-­‐digit numbers and to use place value understanding to add and subtract. A solid understanding of place value is critical as students in Grade 2 are expected to be able to apply these understandings to fluently add and subtract with 100, add up to four two-­‐digit numbers, add and subtract within 1000, mentally add 10 or 100 to a given number 100-­‐900, mentally subtract 10 or 100 from a given number 100-­‐900, and explain why addition and subtraction strategies work (2.NBT.1-­‐9). The following unit is designed to help students attain a deep understanding of place value concepts and how they are related to the VALUE of numbers to 1000 (2.NBT.1, 2.NBT.3, and 2.NVT.4) It is the foundation for subsequent studies of operations based on place value, including addition and subtraction at Grade 2 (2.NBT.5-­‐9). In subsequent grades, the place value understandings that students have acquired in Grade 2 will provide the foundation for generalizing place value understanding for multi-­‐
digit whole numbers and decimals and using such understandings to perform operations 1 with multi-­‐digit whole numbers and decimals. For example, in Grades 4 and 5 students extend place value to include reading, writing, and comparing decimal numbers (4.NF.6-­‐
7 and 5.NBT.2, 5.NBT.3, 5.NBT3a, and 5.NBT 3b). Overview UNIT TITLE: Thinking About Place Value in Grade 2 NUMBER OF INSTRUCTIONAL DAYS: at least 7 (7-­‐10) TARGETED CONTENT STANDARDS: Understand place value 2.NBT.1 Understand that the three digits of a three-­‐digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: •
100 can be thought of as a bundle of ten tens — called a “hundred.” •
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). 2.NBT.3 Read and write numbers to 1000 using base-­‐ten numerals, number names, and expanded form. 2.NBT.4 Compare two three-­‐digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. TARGETED MATHEMATICAL PRACTICES: MP.6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. MP.7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. They see complicated things as single objects or as being composed of several objects. UNDERSTANDINGS: Understanding place value is having number sense of the base ten units, how these units are bundled and unbundled at will, and connecting this understanding to the positional notation system. This understanding leads to discovery of the STRUCTURE of numbers and how we can use the place value system’s structure to quickly and easily read and compare numbers and perform operations. ESSENTIAL QUESTIONS: 1. What do numbers do for us? 2. How does a place value system make numbers easier to use? 2 3. How do we decide which numbers are larger/smaller? BACKGROUND INFORMATION: Students are not likely to meet the standards for adding and subtracting within 1,000 without a solid understanding of place value to 1,000.1 There are two main advantages to the place value system. One is that it allows us to express very large numbers compactly. In just a few seconds, we can type a number larger than the estimated number of atoms in the entire Milky Way galaxy2: 9872459087300872309870932871098327409187439817509348918729870239813121 Writing this number using hash marks would take many sheets of paper—but it fits on a single line using place value notation. The second advantage of the place value system is that it allows for efficient computation of sums, differences, products, and quotients. If a minister tells the Roman emperor that there are CLXXVII legions in Gaul, and that each legion needs MMCCXCIV loaves of bread, then the minister had better hope that the emperor doesn’t demand a fast answer to the question of how many loaves of bread are needed altogether. But in the place value system, there is a pretty fast algorithm for 2294 × 177.) How does the place value system work? The fundamental components (or structure) of the system are: • Base-­‐ten units: Ones, tens, hundreds, thousands, and so on. Students have to get a feel for the sizes of these units (up to hundreds in Grade 2). • Flexible bundling and unbundling: Base-­‐ten units can be broken down and built back up in different ways. By definition, ten ones make a larger unit called “a ten.” By definition, ten tens make a larger unit called “a hundred.” And by definition, ten hundreds make a larger unit called “a thousand.” It is equally true that a hundred ones make a hundred; a thousand ones make a thousand; and a hundred tens also make a thousand. Bundling and unbundling are central ideas in developing computation algorithms. Rods, flats, and cubes are good for illustrating the ways to bundle and unbundle base-­‐ten units. (Students should understand that a one is equal to 1 small cube, a ten is equal to one rod, a hundred is equal to one flat, and a thousand is equal to one large cube.) 1 To say that students won’t meet standard Z without meeting standard Y is not to say that they must demonstrate that they have met standard Y before they go on to learn the math that will allow them to meet standard Z. The curriculum might be structured so that students gradually work simultaneously toward meeting both Y and Z. 2 This is an estimate based on the following information: 10^80 estimated atoms in the Universe divided by 10^11 estimated number of galaxies = 10^69 estimated atoms in our galaxy (assuming the Milky Way is of average size. 3 • Positional notation: According to convention, the location of each digit in a multi-­‐digit number corresponds to a particular base-­‐ten unit. Also, the digit itself tells how many copies3 of that unit are in the number. So, 908 is 9 hundreds and 8 ones. Understanding place value means understanding that all three of these things in connection with one another form the basic structure of our numbering system. Working with cubes, rods, and flats will not, by itself, teach place value because these manipulatives do not have any connection to positional notation. The place value system is connected to communication of numbers and their values. Working with place value requires writing and saying numbers with an understanding of how they name quantities. Students must be encouraged to use precise terms and symbols when they express themselves quantitatively in writing or verbally. Assessing Conceptual Understanding of Place Value The Common Core State Standards present a balanced approach to mathematics that stresses equally the goals of conceptual understanding, fluency, and application. Many in the field welcomed this development in mathematics education while expressing uncertainty as to how conceptual understanding in particular could be assessed. Students should be assessed in a variety of ways including, but not limited to, informal questions/discussion, math journaling, pair-­‐and-­‐share, and more formal summative assessments, such as quizzes. Early assessments should be done with partners or in groups but should progress to independent assessments at the end of the unit. Scaffolding should be plentiful initially but slowly removed as students become more comfortable with the concepts. LEARNING PLAN: Day 1 – Use counting exercises to convince students of the need for a system. If your second graders are able to grasp the concept of the number of atoms in the Milky Way, you might use the example provided in the BACKGROUND INFORMATION as a way to introduce the need for place value when representing very large numbers. Otherwise you might introduce the concept with a counting experiment using hash marks, one hash mark for each item. MATERIALS / RESOURCES • Items to count: Pencils, Unix cubes, coins, cubes, shapes, beans, M&M’s, jellybeans, etc ACTIVITIES 3 By using the word “copies,” we mean to describe an essentially multiplicative notion in terms appropriate to grade. 4 Have students work with a partner and move among 3 to 5 stations set up with varying numbers of objects, at least one of which is a large number (e.g. 24 pencils, 115 pennies, 703 beans). Students need to first decide on which partner will be the recorder. They need to count the items together using a hash mark for each item. They will have to decide how to group the hash marks. (Most will likely use the typical system of grouping by 5’s. Note that this exercise is not intended to require that students count by 5’s but some may be able to do so when finding their final answer.) Have students compare their findings and discuss how their answers are the same or different and some possible reasons for the differences. You should know the actual counts to divulge when the counting has be completed but take into account the possibility that some smaller items may be lost in the process. Discuss with the class the need for a system for expressing the larger numbers more efficiently. OPTIONAL – An additional station might contain a large jar of jellybeans, and students can guess at the number of jellybeans in the jar, possibly after looking at the “size” of groups of 1, 10, and 100 jellybeans. [This hint at estimation based on the size of the groups will be helpful when students must estimate measurements based on the size of the measuring unit. (See 2.MD.3)] ASSESSMENT Observe students as they group the hash marks and discuss their findings with others. Ask probing questions such as, “How did you decide to group the hash marks?” “How will you find the final number?” “Why is your answer different from your neighbor’s?” Watch for students who are struggling. Allow extra time, support, and possibly some smaller numbers of items. If some students need additional challenge and seem to understand these basic concepts they might be allowed to play a place value game: [See http://www.aaamath.com/B/g21c_px1.htm] Day 2 – Expose students to the term “value” and how it applies to numbers. Students need to be exposed and understand the term “value” and how that term applies to “Hundreds”, “Tens”, and “Ones” places. MATERIALS / RESOURCES •
Cubes, rods, and flats ACTIVITIES Use cubes, rods, and flats to represent numbers. Have students work in pairs and explain to their partner how they decided to express numbers including some that are close to 1000, and how they know their groups of cubes, rods, and flats are equal to the number. They should be encouraged to arrange the numbers in different ways to represent the same total value. For example, 200 might be represented with 2 flats or 5 20 rods or 200 cubes. Discussion of the advantages and disadvantages of each should take place with the class. [Also see http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html] ASSESSMENTS Watch and listen to students as they work with partners. Ask probing questions if they seem to be confused. Have students use journaling to explain what they learned about the definition of “value” as it related to the placement of digits in a number. Day 3 – Play the game “Place the Value.” (NOTE: This day might be repeated if students do not appear to feel comfortable with the place value concepts addressed in the game.) MATERIALS/RESOURCES • One gameboard is needed for each pair of students. A gameboard is an opened file folder (made so that the labels allow students to face each other as they play): draw a box in the middle for card deck; on each end of the folder, draw and label three boxes for the “Hundreds”, “Tens”, and “Ones” place value. • Each pair of students will need one deck of cards with the numbers 0 through 9 printed on them. ACTIVITIES • Divide students into pairs. Each pair will receive a gameboard and a “deck” of cards. Cards are to be shuffled and placed face down on the gameboard. • The object of the game is to create the largest number value using the cards drawn. (Ask students to name the largest possible number that can be created with the numbers #0-­‐9. Note: It is 987.) Students alternate turns by drawing a card and placing it in the “Hundreds”, “Tens”, or “Ones” box. For example, if students are creating the largest number, the best place to put the number “9″ if drawn, would be in the hundreds place. Once a card has been placed in a value box, it cannot be changed to another one. The student who creates the largest value wins the round. When students are successful in creating the largest number, the object can then be changed to creating the smallest value number. When creating the smallest number, the number “0″ would work best in the “Hundreds” place value. The student that creates the smallest number wins the round. ASSESSMENT • Classroom monitoring of students as they play “Place the Value” • Give the students numbers 0-­‐9 and on a sheet of paper ask them to: o Write the largest number they can create using 3 of those numbers o Write the smallest number they can create using 3 of those numbers o Give students pairs of numbers and ask them to write the numbers on their paper and circle the largest number (for example: 258 and 401, 967 and 854, 6 etc. o Give students pairs of numbers and ask them to circle the smallest number (for example: 153 and 246, 750 and 698, etc.) [See: http://www.oercommons.org/courses/place-­‐the-­‐value-­‐place-­‐value/view] Also see other place value games and resources: http://education.jlab.org/placevalue/index.html http://education.jlab.org/placevalue/index.html http://www.aaamath.com/B/g21c_px1.htm Day 4 – Expanded Notation MATERIALS / RESOURCES Digit cards (0-­‐9) Print the words HUNDREDS, TENS, and ONES in large letters on paper or cardstock to be cut out. ACTIVITIES Introduce expanded notation by having students write 3-­‐digit numbers using words and digits. Print the words HUNDREDS, TENS, and ONES in large letters and have the students cut them out. Write numbers on the board with 1 to 3 digits. Have students use the 0-­‐9 cards and the place value words to lay the number on their desks. Then check with other students to verify they are in agreement. Ask students to consider whether there are other ways to place the cards and words but still have an equal number. ASSESSMENT On the next page are some short questions that are meant to suggest ways in which teachers can assess place value understanding among their students. Questions like these could be part of an in-­‐class quiz or unit test. An expanded notation worksheet like the one below can be used as introduction, practice, or intervention. However it should be noted that this represents the most basic of conceptual understanding. Deeper understanding requires more complex questions, like those on the summative assessment at the end of this unit. 7 Solutions: a. 2-­‐-­‐-­‐3-­‐-­‐-­‐4 b. 8-­‐-­‐-­‐0-­‐-­‐-­‐9 c. 5-­‐-­‐-­‐7-­‐-­‐-­‐1 d. 1-­‐-­‐-­‐6-­‐-­‐-­‐0 e. 0-­‐-­‐-­‐6-­‐-­‐-­‐7 f. 348 g. 602 h. 5 i. 70 j. 999 [For more suggestions see http://www.superteacherworksheets.com/place-­‐value.html] 8 Day 5 – Play “Place Value to Thousandths” game. (NOTE: This day might be repeated if students do not appear to feel comfortable with the place value concepts addressed in the game.) MATERIALS/RESOURCES • Each student will need a place value chart that shows thousands, hundreds, tens, and ones. • They will also need a set of cards that show the numbers 0-­‐9 (digit cards). These numbers should be small enough to fit on the place value chart. ACTIVITIES • Assign students a partner to work with, making sure to put students together who can work compatibly. Tell the students that they can work together and discuss their answers. • Next, call out four numbers and ask the students to make the smallest number possible on their charts. For example, with the digits 2, 7, 4, and 9 the smallest number would be 2,479. Then ask them to find the largest number. (9,742) You can also ask them to make a number with a 2 in the hundreds place. Walk around while they are working and look on their mats to see who is having trouble. Next, let the students take turns giving four numbers for the rest of the class to put on their charts. • Here is a list of additional challenges that you can use with this activity: Make a 4-­‐digit number that is less than 2000. (i.e.using the digits: 9,2,5,0) Make a 4-­‐digit number that is greater than 1500. (i.e.using the digits: 5,7,1,9) Make a 3-­‐digit number that is greater than 650. (i.e.using the digits: 368) Make a 3-­‐digit number that has a 0 in the tens place. (i.e.using the digits 0,6,3) Make a 4-­‐digit number that is an even number, and is greater than 4000. (i.e.using the digits: 3,7,4,1) Make a 4-­‐digit number that is an odd number, and is less than 2500. (i.e.using the digits: 1,4,2,6) What number is closest to 4000? (i.e. using digits 2,9,3,4) • You can add to this list of challenges to make the activity longer if the students are asking for more. ASSESSMENT As the students are working on this activity, walk around, meeting with each set of partners, asking them questions. You can tell by what they put on their charts, if they understand the concept of place value. The students that are struggling should be pulled out in a small group at a later time for targeted work. Go back to the place value mats with the numeration sets, and ask the students to show a number, such as 153, on their mats. Keep practicing until the students feel more comfortable with this concept. When the students can play this game with place value, they are generally more motivated to learn this challenging concept. 9 NOTE: This hands-­‐on activity is designed to help students gain a better understanding of the abstract concept of place value.. Teachers can pull out these charts and cards periodically to practice place value. [See http://www.oercommons.org/courses/place-­‐value-­‐to-­‐the-­‐thousands-­‐place-­‐review-­‐
activity/view] Day 6 – Compare and Order Numbers to 1000 MATERIALS / RESOURCES • Blackboard • Digit cards (0-­‐9) ACTIVITIES Students should start by considering how to use place value to determine the relative size of numbers. Practice with students on the board or overhead and have them determine which numbers are larger/smaller. Start with a few 1-­‐digit numbers, then increase to 2-­‐digits (including some that are near 100), and then 3-­‐digits (including some that are near 1000). Working in pairs, have one of the partners use three number cards (0-­‐9) to create a number less than 1000. Then have the partner decide if s/he can use the same numbers to make a smaller number. Have them both write the number sentence using the symbols: < or >. The structure of the number, and how it can be used to determine the relative size of a number, should be emphasized. (NOTE: This activity might be presented on one day and the assessment the next with a short review beforehand.) ASSESSMENTS The exercises on the following pages should follow this practice and should be done independently as a summative assessment of understanding ordering numbers to 1000.
10 11 SOLUTION:
Answers: 742 247 Explanation should include information regarding the value of each place and each digit: The smallest digit is the 2 so it should be placed in the largest place vale position, followed by the next smallest (4) and then the largest (7). Students should be encouraged to use precise mathematical language in their explanations. 4
4 This exercise was borrowed from New York City DOE and is available as an open resource on OERCommons.org: http://www.oercommons.org/courses/carol-­‐s-­‐numbers/view 12 NOTE: The following is a released item for place value from a statewide test with some commentary. This task relates to the first cluster in 2.NBT: “Understand place value.” Within that cluster, it relates to 2.NBT.1, 2.NBT.3, and 2.NBT.4. In isolation, this multiple-­‐choice item will not provide information on whether students really understand place value. Students can solve a problem like this by learning a rote strategy of comparing digits from left to right. This is the same strategy used to alphabetize words. It is interesting that an alphabetization strategy can be used to order numbers in base-­‐ten notation. But such a strategy should come as the endpoint of one’s study of place value—a result of one’s understanding, rather than a substitute for it. SOLUTION: 9103 is the largest number so Oak Park (B) had the most visitors. 13 Day 7 – Summative Assessment The following page is a summative assessment for this unit. It should be accomplished independently but might be preceded by a review exercise. Remind students that in the directions “one number” would include numbers with one or more digits. The assessment is followed by solutions and commentary, which identify some of issues highlighted by each group of questions. The following game could be used for review and/or intervention: http://www.toonuniversity.com/flash.asp?err=496&engine=5 14 Make true equations. Write one number in every space. Draw a picture if it helps. 1) 1 hundred + 4 tens = ______ 2) 4 tens + 1 hundred = ______ 3) 14 tens = 10 tens + _____ tens = _____ hundred + 4 tens = ______ 4) 7 ones + 5 hundreds = ______ 5) 8 hundreds = ______ 6) 106 = 1 hundred + _____tens + _____ones 7) 106 = _____tens + _____ones 8) 106 = _____ones 9) 90+300+4= ______ Are these comparisons true or false? 10) 2 hundreds + 3 ones > 5 tens + 9 ones ______ 11) 9 tens + 2 hundreds + 4 ones < 924 ______ 12) 456 < 5 hundreds ______ 15 This task relates to the first cluster in 2.NBT: “Understand place value.” Within that cluster, it relates to 2.NBT.1, 2.NBT.3, and 2.NBT.4. ASSESSMENT SOLUTIONS: Make true equations. Write one number in every space. Draw a picture if it helps. 1) 1 hundred + 4 tens = 140 2) 4 tens + 1 hundred = 41 3) 14 tens = 10 tens + 4 tens = 1 hundred + 4 tens = 140 4) 7 ones + 5 hundreds = 507 5) 8 hundreds = 800 6) 106 = 1 hundred + 0 tens + 6 ones 7) 106 = 10 tens + 6 ones 8) 106 = 106 ones 9) 90+300+4= 394 Are these comparisons true or false? 10) 2 hundreds + 3 ones > 5 tens + 9 ones TRUE 11) 9 tens + 2 hundreds + 4 ones < 924 FALSE 12) 456 < 5 hundreds TRUE 16 ASSESSMENT COMMENTARY: 17