Grade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand


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1 Grade 2: Using a Number Line to Order and Compare Numbers Place Value Horizontal Content Strand Texas Essential Knowledge and Skills (TEKS): (2.1) Number, operation, and quantitative reasoning. The student understands how place value is used to represent whole numbers. The student is expected to: (A) use concrete models of hundreds, tens, and ones to represent a given whole number (up to 999) in various ways; (B) use place value to read, write, and describe the value of whole numbers to 999; and (C) use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<, =, >). TEKS Connections to other Grade Level Strands. (2.5) Patterns, relationships, and algebraic thinking. The student uses patterns in numbers and operations. The student is expected to: (A) find patterns in numbers such as in a 100s chart; (2.8) Geometry and spatial reasoning. The student recognizes that a line can be used to represent a set of numbers and its properties. The student is expected to use whole numbers to locate and name points on a number line. (2.11) Probability and statistics. The student organizes data to make it useful for interpreting information. The student is expected to: (C) use data to describe events as more likely or less likely such as drawing a certain color crayon from a bag of seven red crayons and three green crayons. (2.13) Underlying processes and mathematical tools. The student communicates about Grade 2 mathematics using informal language. The student is expected to: (A) explain and record observations using objects, words, pictures, numbers, and technology; and (B) relate informal language to mathematical language and symbols. (2.14) Underlying processes and mathematical tools. The student uses logical reasoning. The student is expected to justify his or her thinking using objects, words, pictures, numbers, and technology. Purpose: In addition to concrete materials, such as Base 10 blocks, students will use a number line as a tool to order and compare numbers to 999. A number line is a geometric visual designed to magnify the magnitude of numbers by emphasizing the distances between them. The lesson outlined below also integrates probability by having the students randomly draw cards with 3digit numbers and then plot the cards on a number line in order to determine the likelihood of choosing one greater than, or less than a target number selected by the teacher. The objective is to help students focus on these 3digit numbers as whole quantities, instead of a random arrangement of digits. Mathematics TEKS Connections: Grades K2 290
2 Suggested Vocabulary: hundreds more ones less more likely tens less likely number line Materials: Base 10 blocks Interlocking cubes Advanced Preparation: Number Line: 10 sentence strips pieced together For Each Student: Handout 21: Hundreds, Tens, and Ones Number Cards Handout 22: 3Digit Representation Handout 23: 3Digit Comparison Game Suggested Pacing: 3 days Mathematics TEKS Connections: Grades K2 291
3 Procedures Engage: 1. Distribute interlocking cubes, Base 10 blocks, and one deck containing a complete set of Handout 21: Hundreds, Tens, and Ones Number Cards per student. Teacher Notes Materials Run the Handout 21: Hundreds Cards on red cardstock, the Tens Cards on blue, and the Ones Cards on yellow. Each child should have only one of each card e.g., one 100, one 200, one 300, etc. 2. Show students a 10s long. What is the relationship between this long and the other Base 10 blocks? How can you tell what the relationship is? It takes ten of these longs to make the big square [100s block], or It takes 10 of the small cubes to make the rod because the long is divided into 10 squares. How could you construct this long with the interlocking cubes? You need 10. Vocabulary Use the correct terminology to label the Base 10 blocks. (See Figure A below:) Figure A Flat Long Single 3. Have the students recreate the 10s (long) with the connecting cube. When students have finished making the 10s (long) with the interlocking cubes, show a 100s flat. (Refer to the Research section in the Teacher Notes.) Mathematics TEKS Connections: Grades K2 292
4 How many of the 10s (longs) you made would we need in order to make this flat? How do you know? You need 10 of these longs to make one flat because the flat is divided into 10 strips [columns] or the flat is make up of 100 squares, so you need 10 sets of 10 to make When the students are finished, call on ten volunteers to come forward with the longs they made with the interlocking cubes. As you join each long together to construct one flat, direct the class to count with you by 10s: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Research John Van De Walle (2004) cautions teachers about assuming that students will inherently make sense of the place value concept intended within the design of pregrouped Base 10 materials: No model will guarantee that children are reflecting on the tentoone relationship in the materials. With pregrouped materials we need to make sure that children understand that a ten piece really is the same as 10 ones (p.184). 5. Instruct the students to take the 300 red cards, the 40 blue cards, and the 8 yellow cards. Use the Base 10 blocks at your table to create the numbers you just drew from your deck of number cards. Students should get 3 flats, 4 longs, and 8 singles. 6. Explain the following scenario about how a first grader might count the Base 10 blocks to figure what number they represent: (Refer to the Research segment in the Teacher Notes.) I once saw a first grader combine this same set of Base 10 blocks and then count them like this: Point to the flats and say 100, 200, 300. Then move to the longs and continue counting by 100s 400, 500, 600, 700. Continue to the singles and start counting by ones 701, 702, 703, 704, 705, 706, 707, 708. Do you think that this first grader counted the blocks correctly? Why? No, because he kept on counting by hundreds instead of changing to 10s when he was pointing to the longs. Research Clements, D. H., Samara, J., and DiBiase, A.M. (2002) cite evidence gathered by a panel of researchers indicating that most typically developing 7 and 8 yearolds are just learning how to skip count meaningfully. A child would demonstrate this ability if he or she could change his or her counting rule e.g., counting by 10s and then switching to 1s to accommodate different groupings, as demonstrated in the lesson e.g., a group of 3 flats, 4 longs, and 8 singles. Mathematics TEKS Connections: Grades K2 293
5 7. Lead the class in the correct skip counting sequence for enumerating the set of base ten blocks they made to represent the cards (300, 40, 8) drawn from the pile:100, 200, 300, 310, 320, 330, 340, 341, 342, 343, 344, 345, 346, 347, and Discuss how to represent in numerical form: (Refer to the Scaffolding section in the Teacher Notes.) Why do you think that the 3s digit on the 300 card, the 4 on the 40 card, and the 8 on the 8 card are highlighted? Explain your thinking. Responses may vary. Possible responses include: Each digit stands for the number of base ten blocks you use. Since there are 3 flats, the 3 shows how many hundreds. Since there are 4 longs, the 4 on the 40 card stands for how many tens. The 8 on the 8 cards shows how many ones. How could we write three hundred fortyeight so that we only use 3 digits? You write a 3 first, then 4, and then 8: Display a number line. (See the Materials section in the Teacher Notes for information on how to assemble this display.) Discuss how this tool could be used to represent 3digit numbers: Where would I place 300 on this number line if I wanted to show its distance and relationship to other numbers? Above the dot where it says 300 If we skip count by 100s on this number line, how many numbers (Refer to the red cards) would be greater than 300? You would have 6 numbers that are greater and 2 that are less than 300. How could you tell? All of the numbers that are to the right of 300 are bigger [greater], and all of the numbers to the left are smaller [less than]. Scaffolding To help students transfer from expanded to digitized notation, use a flip chart visual in which one digit representative of the corresponding place value can be superimposed upon the number of greatest value. (See the example below) 2 2 Materials Piece together a series of 10 sentence strips each with a line drawn down the middle and labeled with a 100s number. Place a large red dot marking the location of the end number. For example, the first strip would be labeled with a 0 at the far left edge and a 100 at the far right edge with a red dot above each numeral. After the 100s strip, adhere the 200s strip, again with the 200 labeled to the far right. Continue in this fashion to Mathematics TEKS Connections: Grades K2 294
6 10. Introduce the following probability exercise so that the students will concentrate more in depth on the relationship (the distance as it relates to magnitude e.g., greater than/ less than) between the numbers. (See the TEKS Correlation explained in the Teacher Notes.) Since we have already removed the 300 card from our stack of red cards, what is a more likely possibility if each student in the class were to select randomly another card from the red deck Would most of the selected numbers be greater than or less than 300? Why? More students will draw a number that is greater than 300 because more of the cards in the red deck are above 300. There are only two 100 and 200 that are less than 300. Direct each student to select a card from 100s cards (the red stack). Then, instruct the class to bring their cards to the number line and tape them where they belong. Count how many cards are greater than/ less than 300. Discuss the results. (See Figure B in the Teacher Notes section.) TEKS Correlation This probability component of the lesson allows the students to apply TEKS 2.11 (C): The student is expected to use data to describe events as more likely or less likely such as drawing a certain color crayon from a bag of seven red crayons and three green crayons Figure B Results of the Number Draw What can we say about the results of our probability exercise? Responses may vary. If students from another class were to draw from among the same set of cards that you did with a 300 removed from the pile, what would be more likely to happen? There would more likely be more students who chose a card greater than 300 Vocabulary Make sure students are familiar with the vocabulary term probability. Mathematics TEKS Connections: Grades K2 295
7 Explore: 1. Distribute Handout 22: 3Digit Representation Recording Sheet to students. (See the Materials explanation in the Teacher Notes section.) Give direction to the class about what they will be doing about what they will be doing as they work independently to complete the Representation Recording Sheet: 3Digit. First, draw one card from each stack of cards one from the red stack (100s), one from the blue stack (10s), and one from the yellow stack (1s). Once you choose your number, represent it with Base 10 blocks. Materials Handout 22: 3Digit Representation Recording Sheet requires that students first make a concrete model using Base 10 blocks. Then the students represent that model pictorially in the indicated column on the recording sheet. The expanded notation is an example of an abstract representation. The number line is a transition from pictorial to abstract. Draw a picture of your Base 10 block representation on 3Digit Representation Recording Sheet in the upper lefthand box. Write the expanded notation in the right hand box. Write the actual number in numeric notation inside the circular box. Then, show where the number would be located on the number line. Concrete to pictorial Pictorial to abstract Abstract 2. Ask students to work independently on the 3Digit Representation Recording Sheet. As the class is working, move about from student to student, monitoring each child s progress and asking questions either to redirect or clarify the students thinking. Select from the questions below to scaffold instruction: Why do you not record the 0 digits on the 100s and 10s cards? The 10s card tells you how many sets of ten to add to the hundred, and the 1s card tells you how many singles to add to the ten. How did you count your Base 10 blocks to be sure that you are correctly representing the number you drew from the pile of cards? Responses may vary. Scaffolding Relate students experience counting money to help them figure out how to count the Base 10 blocks. Use a dollar bill to represent a flat, a dime to represent a long, and a penny to represent the singles. If you were going to count money, what would you start counting first, dollar bills, dimes, or pennies? Why? Responses may vary. Possible responses include: The dollar bills because they are bigger and are worth more. Refer to the scaffolding section in the Teacher Notes for information on how to assist students who have difficulty changing their skip counting from 100s, to 10s, and then 1s. Mathematics TEKS Connections: Grades K2 296
8 Would you place your number on the number line before or after 500? Responses may vary 3. As the students finish, allow them to compare their numbers to a classmate s number. Explain: 1. Gather the class together and inform the students that they will form a human number line using the 3digit numbers they created and modeled during the Explore portion of the lesson. (See Figure C in the Teacher Notes.) 2. Instruct students to consult their 3digit number and recording on Handout 22: 3Digit Representation Recording Sheet and the number line display you created with sentence strips to determine how to line themselves up at the front of the room from least to greatest. Say, When you find where you think you belong in reference to the number line, compare your number with your neighbor s to see if you or your neighbor are where you need to be. Justify your reasoning to the person standing to your left and right. (See the Scaffolding section in the Teacher Notes.) If your number is less than your neighbor s, where should you be in reference to that classmate to the right or to the left? Explain. I should be to the left because the numbers get larger as you move from left to right or If your neighbor has a 600 and you have a 500, you should be to the left because 500 is less than 600. Figure C Scaffolding The authors of the Principles and Standards for School Mathematics (NCTM, 2000) specify reasoning and proof as one of five important process standards. They also point out how young children will often need to use concrete models in order to explain their thinking Young children will express their conjectures and describe their thinking in their own words and often explore them using concrete materials and examples (p. 57). Allow children who have difficulty using words to justify their reasoning to use the Base 10 models to compare numbers. 3. As the students are positioning themselves at the front of the room from least to greatest, instruct each child to justify how he or she knows he or she is where he or she should be. (Again, consult the Teacher Notes about how to scaffold this activity.) Mathematics TEKS Connections: Grades K2 297
9 4. Highlight comparison vocabulary to help the students make sense of the concept of greater than/ less than. (Refer to the Vocabulary section in the Teacher Notes.) Have student partner and become a human balance. If each student were holding his or her Base 10 blocks where he or she is standing, which side do you think would be heavier? Right or Left? Explain? The left side would be heavier because the further you move to the left where 999 is, the more Base 10 blocks each student will have. Elaborate: 1. Help students make sense of how to use place value to compare and order numbers. Select two numbers created by different students that were within the same century e.g., 345 and 356. Say, As some of you were trying to figure out how to position yourselves within the number line, I noticed that [student s name] and [student s name] both have the numbers with the same digit in the 100s place. How could we figure out which of these two numbers is greater? Responses may vary. Refer to the Scaffolding explanation and Figure D in the Teacher Notes. What would we do if the digits in the 10s place were the same? Compare the digits in the 1s place. Vocabulary To help make the concept of greater than/ less than more explicit and visual for struggling learners, gather a set of base ten blocks from the student standing at the furthest left of the number line and another set of blocks from the student standing at the far right. Place both sets of blocks on a balance scale (the set with the greatest number of blocks in the left hand bin, and the set with the least amount in the right hand bin). Scaffolding If students fail to point out how they could move to and check the digit in the tens place to determine which of the two numbers is greater, display the colored cards the two selected classmates used to make the highlighted numbers. These cards show the quantities in expanded notation. Then, compare and contrast. Which of the two 100s cards that [student s name] and [student s name] used is greater? They are the same. Which of the two 10s cards is greater? The card with 50 Figure D Student A Student B Greater Mathematics TEKS Connections: Grades K2 298
10 2. Once the students have an understanding of how to use place value to compare numbers, bring the class s attention back to the sentence strip number line display. Remove from the wall display one of the sentence strips representing a distance of 100 e.g., the strip with all of the 300 numbers. Refer to Figure E in the Teacher Notes. Say, On a scratch sheet of paper, show me how you would represent sets [increments] of ten on this 300s number line. Explain to a partner how you would position the numbers 10, 20, 30, etc. Refer to the Scaffolding explanation in the Teacher Notes. 3. When students finish making their number line sketching, have them share their partitioning strategies. Discuss criterion for incorporating and recording the students suggestions onto the classroom model: Is there the same amount of space between each increment of ten to include the single digits 19? Responses may vary. See the Materials explanation in the Teacher Notes. Which 10s number is closest to the end of the number line? 90 What would this number be called on the actual number line in reference to 400? 390 What would this same number look like if translated onto the next sentence strip to the right [the 400s sentence strip]? 490 Figure E The sentence strip with all of the 300 numbers will actually be on the sentence strip ending with Scaffolding If students have difficulty partitioning their number line into increments of ten, give them a benchmark number (e.g., 50), which is located in the middle. Materials Show students a ruler to help them visualize the equal distances between marks on a number line. Connections Lead the students to notice the pattern that unfolds as you move up by 100s from any number e.g., 490, 590, 690, etc. Mathematics TEKS Connections: Grades K2 299
11 Evaluate: 1. Distribute Handout 23: 3Digit Comparison Game Recording Sheet to students. Explain how to use the sheet: Look at the five digit cards located in the left hand column. Select three of the cards to make a 3digit number. Try to make the greatest number possible. Record that number in the middle column on the recording sheet. See Figure F in the Teacher Notes. Compare your number with you partner s result. Use the correct symbol to express greater than [>], less than [<], or equal to [=]. The student with the greatest number wins that round. 2. As the class is working on the 3Digit Comparison Game Recording Sheet, move about from group to group observing to see which strategies students use to compare the magnitude of the numbers they create with the sample digits cards. Refer to the Informal Assessment section in the Teacher Notes for more information on what type of understandings or misconceptions to document. Figure F: Select a student volunteer to play against for a demonstration round. Model how to record the results on Handout 23: 3Digit Comparison Game Recording Sheet Optional Game Can also play Place Value Target Game to create the largest number. Use three different geometry shapes in which to record the numbers. Roll die, place numeral in shape. Repeat twice to complete number. Compare results. Largest number created wins. Informal Assessment: Do the students use place value (looking at the position of the digits) to determine if their number is greater? Do students need to use concrete models (Base 10 blocks) or visuals (a number line) in order to make comparisons between numbers? Mathematics TEKS Connections: Grades K2 300
12 Handout 21: Hundreds Cards Red Cards Mathematics TEKS Connections: Grades K2 301
13 Handout 21: Tens Cards Blue Cards Mathematics TEKS Connections: Grades K2 302
14 Handout 21: Ones Cards Yellow Cards Mathematics TEKS Connections: Grades K2 303
15 Handout 22: 3Digit Representation Recording Sheet Base 10 Model (Pictorial) Expanded Notation 3digit Number Number Line Mathematics TEKS Connections: Grades K2 304
16 Handout 23: 3Digit Comparison Game Recording Sheet Select 3 numeral cards from among the following to make a 3digit number: My 3digit number <, > or = My partner s number Mathematics TEKS Connections: Grades K2 305
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