Open Number Line Mats

Transcription

1 Open Number Line Mats Congratulations on your purchase of these Really Good Stuff Open Number Line Mats a versatile and open-ended tool for solving a variety of addition and subtraction problems. This Really Good Stuff product includes: 6 Open Number Line Mats, Write Again wipe-off laminate This Really Good Stuff Activity Guide Introducing the Open Number Line Mats Before introducing the Open Number Line Mats, make copies of this Really Good Stuff Activity Guide and file the pages for future use. Or, download another copy of it from our Web site at Always use a dry erase marker on the Mats in order to preserve their Write Again wipe-off laminate surface. Explain to students that they will be using a number line in a new way to help them solve addition and subtraction problems. Remind students that they first used number lines to help them skip count and to add, and then they learned to go backward along a number line to help them subtract. Indicate that now that they are working with bigger numbers (or fractions, or decimals, or integers, and so forth), they will use a number line to help them solve problems with addition and subtraction, but in a different way. Ask students what the highest and lowest numbers along a number line are. On the board, draw a line with an arrow on each end, and show students how a number line goes on forever in both directions. Explain to students that because of this, they can create number lines that start anywhere along the continuum. Tell students these are called open number lines, because when they label them they can begin at any number. Display the following problem: By twelve o clock, Brian had collected $150 from the can drive. He collected another $132 by the end of the drive. How much money did Brian collect altogether? Ask students what number sentence would reflect this problem. Write = on the board, above your demo number line. Explain to students that instead of starting to label your number line at 0 you are going to begin the number line at 150 the bigger of the two numbers then label your demo open number line with 150. Indicate that you are using the larger number in the problem as your starting point. Tell students that you are going to solve an addition problem, and ask students what direction along the number line you will be moving. After students respond up, mark an exaggerated arrow on the right-hand end of the open number line. Explain to students that you are going to use landmark numbers or rounded friendly numbers to help you solve the problem. Ask students to break apart the addend 132. And write the numbers 100 and 32, beneath 132. Explain to students that you can use the 100, to jump up to 250 along the open number line. Draw a large jump along the number line, cross out the 100, and label the jump you made with 100. All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159667

2 Open Number Line Mats Ask students to tell you what they think the next step should be. If necessary, explain that you can use friendly numbers to break the problem down further, by counting by tens to solve the remainder of the problem. Starting at the 250, draw a smaller jump up the line, and label it with 10. Ask students where you are now along the number line, and record 260 below the line. Make two more jumps of 10, recording 270 and 280 beneath the line. Make a final, even smaller jump, label it with 2, and record 282 beneath the line. Circle 282 and complete the number sentence. students that they should record the bigger number from the problem first and that because they will be making jumps down the number line, they are to record the 364 on the right side of the open number line. Ask students to use friendly numbers to help them find the easiest way to jump 128 down the open number line. Tell students to break apart 128, explaining that 100 and 28 are one way, then have them record this under their number sentence. Ask students where they would land if they made a jump of 100 down the number line. Instruct students to record the 264 beneath the number line and to cross out the 100 beneath the number sentence. Practicing with the Open Number Line Mats Distribute an Open Number Line Mat and dry erase marker to each student, and display the following problem: Lara needed 364 inches of ribbon for her party favors. She found she only had 128 inches. How many more inches of ribbon would she need to get? Ask students to tell you what the question is asking. If necessary, explain that the question is asking them to find the difference between 364 and 128. Tell students to record = in the white box on their Mats. Ask students which direction they need to move along the open number line if the problem involves subtraction. If necessary, remind them that they are to move down the number line. Instruct students to use their dry erase markers to emphasize the arrow on the lefthand side of their open number lines. Explain to Show students how 28 would equal two groups of 10 and another 8. You may find students using different strategies as they work with the open number line. For example, as in the example above, students may find it easier to break apart the 8 into two 4 s, because it will get them to the friendly number 240 by only making the first jump of 4. Use these opportunities to encourage students to share their strategies with each other to open up their problem-solving options. Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159667

3 Open Number Line Banner - Decimals Congratulations on your purchase of this Really Good Stuff Open Number Line Banner - Decimals an introduction to the open number line strategy for solving addition and subtraction problems with decimals. This Really Good Stuff product includes: Open Number Line Banner - Decimals, laminated This Really Good Stuff Activity Guide Displaying the Open Number Line Banner - Decimals Before displaying the Open Number Line Banner - Decimals, make copies of this Really Good Stuff Activity Guide, cut apart the reproducibles, and file the pages for future use. Or, download another copy of it from our Web site at Hang the Banner where students will be able to see it easily. Introducing the Open Number Line Banner - Decimals As you draw a horizontal line on the board and label it with the numbers 1, 2, 3, and 4, remind students that the number line shows the numbers that we count. Then draw arrows on each end and review that the arrows indicate that the number line goes on forever in both directions. Explain that in between each whole number there are an infinite number of places that we can label with fractions or decimals because they are part of a whole number. For example, as you mark 1.5 on your number line, remind students that 1.5 is halfway between 1 and 2. Then ask students where the number 2.8 should be marked along the number line. Show additional decimals along the number line, if necessary. Emphasize that the open number line is a powerful problem-solving tool because it will let them use whatever part of the number line they need. Pointing to the Open Number Line Banner - Decimals, indicate that you are going to use the Banner to show them how to use an open number line to solve problems that require addition and subtraction of decimals. Read the first problem on the Banner. Say, From this, I know that Tory starts with 4.8 meters and needs 0.64 meters more. I need to find how much fabric she needs altogether. First, I need to find 4.8 on the number line and then add or move up the number line by The whole number of 4 tells me that 4.8 is greater than 4 and less than 5 and the.8 tells me that it is between 4 and 5. Point to 4.8 on the Banner. Next, I need to think about getting to a friendly number. When I work with decimals, I try to get to whole numbers for friendly numbers. I know 4.8 is eight tenths away from 4, so it is two tenths away from 5. My first jump will be to 5, and I know two tenths can be written as 0.2. Point to the +0.2 arrow and say, I have moved two tenths of the 0.64 that I need to move up the number line, so I need to figure out the difference between two tenths (0.2) and sixty-four hundredths (0.64). Think aloud and say, Two tenths is equal to twenty hundredths and the difference between 20 and 64 is 44, so I need to move forty-four hundredths (0.44) farther up the number line. Because I am starting from a friendly whole number, it is easy to make this jump. Move your finger from 5 to 5.44 and say, Tory needed a total of 5.44 meters of fabric. Read the next problem on the Banner. Think aloud and say, I know Samantha had a race time of seconds and that she wants to shave 3.1 seconds off her time. That means she wants to be 3.1 seconds faster than she is right now. To solve this, I need to start at and subtract 3.1 or jump 3.1 down the number line. Since I m subtracting and moving down the number line, the larger number in the problem (55.28) is labeled at the right-hand end of the line. As I think about jumping 3.1 down the number line, I need to get to a friendly number. In this problem, the closest friendly number is 55, so I am going to subtract 3. I know 55 3 = 52, therefore, = I need to move one All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159714

4 Open Number Line Banner - Decimals tenth (0.1) farther down the number line, so I need to think that twenty-eight hundredths (.28) minus ten hundredths (.10) equals eighteen hundredths, so I end at We know Samantha s goal is seconds. Introducing the Adding and Subtracting Decimals Along an Open Number Line Reproducible Copy and distribute the Adding and Subtracting Decimals Along an Open Number Line Reproducible. Write the first problem = on the board. Draw a number line and say, The addition sign tells us that we will be making jumps up the number line. Make an exaggerated arrow at the right-hand end of the line. Say, Let s start with the bigger number of 6.8, and write it on the left-hand side of the number line. Say, The closest whole number, or friendly number, is 7, so let s jump there. That s a jump of Label the jump. Then say, Because we jumped an extra two tenths to reach the whole number of 7, we need to subtract it from 2.35, that is = That means, we have 2.15 left to jump, and 7 plus 2.15 is Label 9.15 and the jump of Answers: 1) 9.15, 2) 4.12, 3) 44.58, 4) 14.13, 5) 30.2, 6) Introducing the Decimals Problem Solving Along an Open Number Line Reproducible Copy and distribute the Decimals Problem Solving Along an Open Number Line Reproducible. Read the first problem together. Then say, This problem is asking us to find how much Felix weighs now, so we need to add 57.8 and 3.6. Label 57.8 along the number line, and explain that the class will need to jump 3.6 up the number line. Lead students to identify 58 as the closest friendly number and say, Let s get to 58 because that is the closest whole number and from there we can jump whole numbers: from 58 to 59 to 60 to 61. This means we have already jumped 3.2 up the number line. How much do we still need to jump? (Guide them, if necessary, to do: =.4.) That s right, from 61, we need to jump.4 more. So we know that Felix weighed 61.4 at the end of fifth grade. Answers: 1) 61.4 pounds, 21.8 points, seconds, 9.95 feet Thinking Flexibly At times, it may make sense to over jump. For example, write the problem = on the board. Say, This problem asks us to jump up 2.98, which is very close to the whole number 3, so we can jump up 3 to 6.4 and then back.02 to Challenge students to come up with multiple ways to solve the same problem. Have students compare their work with a partner and point out the similarities and differences in their strategies. Making Connections to Other Strategies Have students solve one of the problems using other addition or subtraction strategies. Help students to recognize that there are multiple ways to solve the same problem. Encourage them to make connections between the two methods. Urge them to notice things about the order they solve the problem and the place value. Remind students that the open number line can help them keep track of their jumps a little easier. All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159714

5 Adding and Subtracting Decimals Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159714

6 Decimals Problem Solving Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159714

7 Open Number Line Banner - Fractions Congratulations on your purchase of this Really Good Stuff Open Number Line Banner - Fractions an introduction to the open number line strategy for solving addition and subtraction problems with fractions. This Really Good Stuff product includes: Open Number Line Banner - Fractions, laminated This Really Good Stuff Activity Guide Displaying the Open Number Line Banner - Fractions Before displaying the Open Number Line Banner - Fractions, make copies of this Really Good Stuff Activity Guide, cut apart the reproducibles, and file the pages for future use. Or, download another copy of it from our Web site at Hang the Banner where students will be able to see it easily. Introducing the Open Number Line Banner - Fractions As you draw a horizontal line on the board and label it with the numbers of 1, 2, 3, and 4, remind students that a number line shows the numbers that we count. Then draw arrows on each end of the number line and review with students that the arrows indicate that the number line goes on forever in both directions. Explain that in between each whole number there are an infinite number of places that we can label with fractions or decimals because they are part of a whole number. For example, as you mark on your number line, remind students that is halfway between 2 and 3. Then ask students where the number should be marked on the number line. Show additional fractions on the number line, if necessary. Emphasize that the open number line is a powerful problem-solving tool because it helps them to visualize the parts of the number line that they need to use. Pointing to the Open Number Line Banner - Fractions, indicate that you are going to use this Banner to show them how to use the open number line to solve problems that require addition and subtraction of fractions. Explain that adding and subtracting fractions can be confusing because you are dealing with parts of numbers and that sometimes they have different denominators. But the open number line can help them keep track of all the information in a problem. Read the first problem on the Banner. Think aloud and say, I know Lindsay started with cups and added 3 4 cup more. I need to figure out how much there is in all, so I need to add the two numbers, which means that I move up the number line. Pointing to 1 1 2, say I ll start at 1 1 2, then think about getting to a friendly number. When I work with fractions, it makes the most sense to try to use whole numbers for friendly numbers. So 2 is the next whole number, and it is 1 2 away from My first jump will be Pointing to the jump, say, I have moved 1 2 of the 3 4 that I need to move up the number line, so I need to figure out the difference between 1 2 and 3 4 to know how much more I need to jump. Think aloud and say, 1 2 is equal to 2 4, and = 1 4, so I need to jump 1 4. Since I got to a friendly whole number, it is easy to make this jump. Move your finger from 2 to on the number line. Conclude by saying, Lindsay has cups of dry ingredients in the bowl. Read the next problem on the Banner. Thinking aloud, say, I know Tyler had a whole pizza and that he and his friends ate 3 8. On the number line, a whole is written as 1. I need to subtract 3 8, because they ate that much pizza, and I need to find how much he had left. Point to the 1 on the number line. This time, I All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159780

8 Open Number Line Banner - Fractions am starting at a friendly number, so now I m going to think about making small jumps. As you say the following, move your finger down the number line and over the arrows: I know that 3 8 is the same as I also know that 1 whole is the same as 8 8. Now, is 7 8, is 6 8, and is 5 8. Conclude with, Tyler has 5 8 of the pizza left. Introducing Practice Adding and Subtracting Fractions Along an Open Number Line Reproducible Copy and distribute the Practice Adding and Subtracting Fractions Along an Open Number Line Reproducible. Write the first problem of = on the board. Draw a number line, and label 5 8. Say, The addition sign tells us to move up the number line, and the 7 8 tells us how far. Let s start at 5 8. Mark an exaggerated arrow on the right-hand end of the number line to show that you are going to jump up the number line. Say, The closest friendly number is 1, so let s jump there. Because = 1, we jump up Label the jump with Think aloud and say, This means that we have 4 8 left to jump (because = 4 8). I know that 1 plus 4 8 is Label and the jump of When we solve fraction problems, we need to make sure the fraction is in its lowest terms. The number of can be written in its lowest terms as 1 1 2, so = Answers: 1) 1 1 2, 2) 2 1 2, 3) 2 1 3, 4) 3 1 4, 5) 3 1 2, 6) Introducing the Adding and Subtracting Fractions Along an Open Number Line Reproducible Copy and distribute the Adding and Subtracting Fractions Along an Open Number Line Reproducible. Read the first problem together. Then say, This problem is asking us to find how much soil Henry has left. We need to start at 4 and subtract Label 4 on the number line, and say, We are already at a friendly number, so let s subtract the whole number first: 4 1 = 3. Label the jump of -1. Now we can jump 1 4 more: 3 minus 1 4 is Label the jump of - 1 4, and write beneath the number line. Finish with, Henry has cups of dirt left. Answers: 1) cups, 2) inches, 3) cups, 4) feet Thinking Flexibly At times, it may make sense to over jump. For example, write the problem = on the board. Say, This problem asks us to jump up 7 8. We know that 7 8 is very close to the number 1, so we can jump up to 1. We also know that 1 and 5 8 is the same as Then we jump back, or subtract, the 1 8 to 1 4 8, which is written in its lowest terms as Challenge students to come up with multiple ways to solve the same problem. Have students compare their work with a partner and point out the similarities and differences in their strategies. Making Connections to Other Strategies Have students solve one of the problems using other addition or subtraction strategies. Help students to recognize that there are multiple ways to solve the same problem. Encourage them to make connections between the methods. Remind students that the open number line can help them to keep track of their jumps a little easier. All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159780

9 Practice Adding and Subtracting Fractions Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159780

10 Adding and Subtracting Fractions Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159780

11 Open Number Line Banner - Integers Congratulations on your purchase of this Really Good Stuff Open Number Line Banner - Integers an introduction to the open number line strategy for solving addition and subtraction problems with integers. This Really Good Stuff product includes: Open Number Line Banner - Integers, laminated This Really Good Stuff Activity Guide Displaying the Open Number Line Banner - Integers Before displaying the Open Number Line Banner - Integers, make copies of this Really Good Stuff Activity Guide, cut apart the reproducibles, and file the pages for future use. Or, download another copy of it from our Web site at Hang the Banner where students will be able to see it easily. Introducing the Open Number Line Banner - Integers As you draw a horizontal line on the board and label it with the positive numbers of 1, 2, 3, and 4, remind students that the number line shows the numbers that we count. Then draw arrows on each end of the number line, and review with students that the arrows indicate that the number line goes on forever in both directions. Label the number line with 0, -1, -2, -3, -4. Ask students what they notice about the number line: If necessary, encourage them to be specific about positive or negative numbers; for example, when we count up negative numbers on the number line, the numbers get smaller. Pointing to the Open Number Line Banner - Integers, indicate that you are going to use the Banner to show them how to use an open number line to solve problems that require addition and subtraction of integers. Explain that adding and subtracting integers can be confusing because of the positive and negative numbers, but the open number line can help them to keep track of all the information in a problem. Read the first problem on the Banner. Say, I know that Owen hiked from 210 feet below sea level, or -210 feet, to 468 feet. Point to -210 and 468 on the Banner. So I need to find the difference between -210 and 468, or how far -210 is from 468 along the open number line. First, I want to think about getting to a friendly number. When I m working with integers, 0 is very important because it marks where positive numbers end and negative numbers begin, so 0 will usually be the best friendly number to use. Point from -210 to 0 and then 468. This is a jump of Then I jump To find the answer, I need to add the two jumps: = 678, so Owen s altitude changed by 678 feet. Read the next problem on the Banner. Say, I know Lauren started with $44 and then wrote a check for $67. The problem says that Lauren overdrew her account, which means that she didn t have enough money in her account to write a $67 check. I need to figure out how much money she overdrew it by, so I need to start at 44 and jump down 67 along the number line. Point to the 44 on the number line. Remember that 0 is a friendly number when dealing with integers, so I ll jump from 44 to 0. Point to this All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159819

12 Open Number Line Banner - Integers jump. That s a jump of -44 because I went down the number line. I need my jumps to equal -67, so I need to keep jumping down. I ll start with a jump of -10. That takes me to -10 on the number line, and I need to remember that now I have jumped down a total of -54. I can jump down 10 more. Point to the -10 and -10 jumps, or -20. Now I have jumped a total of -64. So in order to jump a total of -67 (the amount of Lauren s check), I need to jump -3 more to -23. Point to the remaining jumps and say, I ended at -23, so Lauren overdrew her account by $23. Introducing Adding and Subtracting Integers Along an Open Number Line Reproducible Copy and distribute the Adding and Subtracting Integers Along an Open Number Line Reproducible. Write the first problem =. Draw a number line, and label -16. Say, The addition sign tells us to move up the number line and the 28 tells us how far. Let s start at -16. We are going to need to go past 0, so let s jump there first. That s a jump of +16. Label 0 and the jump with +16. Then say, We need to jump up 28 in all, so let s jump up 10 more. Label the jump of +10 to 10. Say, Now we ve jumped up 26, (and we know that = 2), so we need to jump 2 more to total 28. Label the jump of +2 to 12. Conclude by saying, So = 12. Answers: 1) 12, 2) -13, 3) -34, 4) -100, 5) 80, 6) -166 Introducing the Integers Problem Solving Along an Open Number Line Reproducible Copy and distribute the Integers Problem Solving Along an Open Number Line Reproducible. Read the first problem together. Then say, This problem is asking us to find the difference between -12 and 9. First, we need to start at -12 and jump up to 0. Label -12 and 0 on the number line as well as the jump of +12. Next, we need to get to 9, so we jump up 9. Label the +9 jump. Now we need to figure out how far we jumped: =21, so the difference in temperature is 21. Answers: 1) 21, 2) -25 points, 3) 180 feet, 4) $35 Thinking Flexibly Challenge students to come up with multiple ways to solve the same problem. Have students compare their work with a partner and point out the similarities and differences in their strategies. Making Connections to Other Strategies Have students solve one of the problems using other strategies for adding and subtracting integers. Help students to recognize that there are multiple ways to solve the same problem. Encourage them to make connections between the methods. Remind students that the open number line can help them to keep track of their jumps a little more easily. All activity guides can be found online: Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159819

13 Adding and Subtracting Integers Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159819

14 Integers Problem Solving Along an Open Number Line Reproducible Helping Teachers Make A Difference 2011 Really Good Stuff Made in USA #159819