open number line mats - really good stuff · good stuff® open number line mats—a versatile ......

Congratulations on your purchase of these ReallyGood Stuff® Open Number Line Mats—a versatileand open-ended tool for solving a variety of additionand 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 MatsBefore introducing the Open Number Line Mats,make copies of this Really Good Stuff® ActivityGuide and file the pages for future use. Or, downloadanother copy of it from our Web site atwww.reallygoodstuff.com. Always use a dry erasemarker on the Mats in order to preserve their WriteAgain® wipe-off laminate surface.

Explain to students that they will be using a numberline in a new way to help them solve addition andsubtraction problems. Remind students that theyfirst used number lines to help them skip count andto add, and then they learned to go backward alonga number line to help them subtract. Indicate thatnow that they are working with bigger numbers (orfractions, or decimals, or integers, and so forth),they will use a number line to help them solveproblems with addition and subtraction, but in adifferent way.

Ask students what the highest and lowest numbersalong a number line are. On the board, draw a linewith an arrow on each end, and show students howa number line goes on forever in both directions.Explain to students that because of this, they cancreate number lines that start anywhere along thecontinuum. Tell students these are called “opennumber lines,” because when they label them theycan begin at any number.

Display the following problem:

By twelve o’clock, Brian had collected $150 from thecan drive. He collected another $132 by the end ofthe drive. How much money did Brian collectaltogether?Ask students what number sentence would reflectthis problem. Write 150 + 132 = on the board, aboveyour demo number line. Explain to students thatinstead of starting to label your number line at 0you are going to begin the number line at 150—thebigger of the two numbers—then label your demoopen number line with 150. Indicate that you areusing the larger number in the problem as yourstarting point.

Tell students that you are going to solve an additionproblem, and ask students what direction along thenumber line you will be moving. After studentsrespond “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” numbersto help you solve the problem. Ask students to breakapart the addend 132. And write the numbers 100and 32, beneath 132. Explain to students that youcan use the 100, to jump up to 250 along the opennumber line. Draw a large “jump” along the numberline, cross out the 100, and label the jump you madewith 100.

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Open Number Line Mats


Ask students to tell you what they think the nextstep should be. If necessary, explain that you canuse friendly numbers to break the problem downfurther, by counting by tens to solve the remainderof the problem. Starting at the 250, draw a smallerjump up the line, and label it with 10. Ask studentswhere you are now along the number line, and record260 below the line. Make two more jumps of 10,recording 270 and 280 beneath the line. Make afinal, even smaller jump, label it with 2, and record282 beneath the line. Circle 282 and complete thenumber sentence.

Practicing with the Open Number Line MatsDistribute an Open Number Line Mat and dry erasemarker to each student, and display the followingproblem:

Lara needed 364 inches of ribbon for her partyfavors. She found she only had 128 inches. Howmany more inches of ribbon would she need to get?

Ask students to tell you what the question isasking. If necessary, explain that the question isasking them to find the difference between 364 and128. Tell students to record 364 – 128 = in thewhite box on their Mats. Ask students whichdirection they need to move along the open numberline if the problem involves subtraction. If necessary,remind them that they are to move down thenumber line. Instruct students to use their dryerase markers to emphasize the arrow on the left-hand side of their open number lines. Explain to

students that they should record the bigger numberfrom the problem first and that because they will bemaking jumps down the number line, they are torecord the 364 on the right side of the open numberline. Ask students to use “friendly numbers” to helpthem find the easiest way to jump 128 down theopen number line. Tell students to break apart 128,explaining that 100 and 28 are one way, then havethem record this under their number sentence. Askstudents where they would land if they made a jumpof 100 down the number line. Instruct students torecord the 264 beneath the number line and tocross out the 100 beneath the number sentence.

Show students how 28 would equal two groups of 10and another 8. You may find students usingdifferent strategies as they work with the opennumber line. For example, as in the example above,students may find it easier to break apart the 8into two 4’s, because it will get them to the friendlynumber 240 by only making the first jump of 4. Usethese opportunities to encourage students to sharetheir strategies with each other to open up theirproblem-solving options.

Open Number Line Mats

Congratulations on your purchase of this ReallyGood Stuff® Open Number Line Banner -Decimals—an introduction to the open number linestrategy for solving addition and subtractionproblems 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 -DecimalsBefore displaying the Open Number Line Banner -Decimals, make copies of this Really Good Stuff®

Activity Guide, cut apart the reproducibles, and filethe pages for future use. Or, download another copyof it from our Web site at www.reallygoodstuff.com.Hang the Banner where students will be able to seeit easily.

Introducing the Open Number Line Banner -DecimalsAs you draw a horizontal line on the board and labelit with the numbers 1, 2, 3, and 4, remind studentsthat the number line shows the numbers that wecount. Then draw arrows on each end and reviewthat the arrows indicate that the number line goeson forever in both directions. Explain that in betweeneach whole number there are an infinite number ofplaces that we can label with fractions or decimalsbecause they are part of a whole number. Forexample, as you mark 1.5 on your number line, remindstudents that 1.5 is halfway between 1 and 2. Thenask students where the number 2.8 should bemarked along the number line. Show additionaldecimals along the number line, if necessary.Emphasize that the open number line is a powerfulproblem-solving tool because it will let them usewhatever 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 numberline to solve problems that require addition andsubtraction of decimals. Read the first problem onthe Banner. Say, “From this, I know that Tory startswith 4.8 meters and needs 0.64 meters more. Ineed to find how much fabric she needs altogether.First, I need to find 4.8 on the number line and thenadd or move up the number line by 0.64. The wholenumber of 4 tells me that 4.8 is greater than 4 andless than 5 and the .8 tells me that it is between 4and 5.” Point to 4.8 on the Banner. “Next, I need tothink about getting to a ‘friendly’ number. When Iwork with decimals, I try to get to whole numbers for‘friendly’ numbers. I know 4.8 is eight tenths awayfrom 4, so it is two tenths away from 5. My firstjump will be to 5, and I know two tenths can bewritten as 0.2.” Point to the +0.2 arrow and say, “Ihave moved two tenths of the 0.64 that I need tomove up the number line, so I need to figure out thedifference between two tenths (0.2) and sixty-fourhundredths (0.64).” Think aloud and say, “Twotenths is equal to twenty hundredths and thedifference between 20 and 64 is 44, so I need tomove forty-four hundredths (0.44) farther up thenumber line. Because I am starting from a friendlywhole number, it is easy to make this jump.” Moveyour finger from 5 to 5.44 and say, “Tory needed atotal of 5.44 meters of fabric.”

Read the next problem on the Banner. Think aloudand say, “I know Samantha had a race time of 55.28seconds and that she wants to shave 3.1 secondsoff her time. That means she wants to be 3.1seconds faster than she is right now. To solve this, Ineed to start at 55.28 and subtract 3.1 or jump 3.1down the number line. Since I’m subtracting andmoving down the number line, the larger number inthe problem (55.28) is labeled at the right-hand endof the line. As I think about jumping 3.1 down thenumber line, I need to get to a friendly number. Inthis problem, the closest friendly number is 55, so Iam going to subtract 3. I know 55 – 3 = 52,therefore, 55.28 – 3 = 52.28. I need to move one

Open Number Line Banner - Decimals



tenth (0.1) farther down the number line, so I needto think that twenty-eight hundredths (.28) minusten hundredths (.10) equals eighteen hundredths, soI end at 52.18. We know Samantha’s goal is 52.18seconds.”

Introducing the Adding and Subtracting DecimalsAlong an Open Number Line ReproducibleCopy and distribute the Adding and SubtractingDecimals Along an Open Number Line Reproducible.Write the first problem 6.8 + 2.35 = on the board.Draw a number line and say, “The addition sign tellsus that we will be making jumps up the number line.”Make an exaggerated arrow at the right-hand end ofthe line. Say, “Let’s start with the bigger number of6.8,” and write it on the left-hand side of thenumber line.” Say, “The closest whole number, orfriendly number, is 7, so let’s jump there. That’s ajump of +0.2.” Label the jump. Then say, “Because wejumped an extra two tenths to reach the wholenumber of 7, we need to subtract it from 2.35, thatis 2.35 – .2 = 2.15. That means, we have 2.15 left tojump, and 7 plus 2.15 is 9.15.” Label 9.15 and thejump of +2.15.

Answers: 1) 9.15, 2) 4.12, 3) 44.58, 4) 14.13, 5)30.2, 6) 14.82

Introducing the Decimals Problem Solving Along anOpen Number Line ReproducibleCopy and distribute the Decimals Problem SolvingAlong an Open Number Line Reproducible. Read thefirst problem together. Then say, “This problem isasking us to find how much Felix weighs now, so weneed to add 57.8 and 3.6.” Label 57.8 along thenumber line, and explain that the class will need tojump 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 wholenumber and from there we can jump whole numbers:from 58 to 59 to 60 to 61. This means we havealready jumped 3.2 up the number line. How much dowe still need to jump? (Guide them, if necessary, todo: 3.6 – 3.2 = .4.) That’s right, from 61, we need tojump .4 more. So we know that Felix weighed 61.4 atthe end of fifth grade.”

Answers: 1) 61.4 pounds, 21.8 points, 38.88seconds, 9.95 feet

Thinking FlexiblyAt times, it may make sense to “over jump.” Forexample, write the problem 3.4 + 2.98 = on theboard. Say, “This problem asks us to jump up 2.98,which is very close to the whole number 3, so we canjump up 3 to 6.4 and then back .02 to 6.38.”

Challenge students to come up with multiple waysto solve the same problem. Have students comparetheir work with a partner and point out thesimilarities and differences in their strategies.

Making Connections to Other StrategiesHave students solve one of the problems using otheraddition or subtraction strategies. Help students torecognize that there are multiple ways to solve thesame problem. Encourage them to make connectionsbetween the two methods. Urge them to noticethings about the order they solve the problem andthe place value. Remind students that the opennumber line can help them keep track of their jumpsa little easier.

Open Number Line Banner - Decimals



Adding and Subtracting Decimals Along an Open Number Line Reproducible


Decimals Problem Solving Along an Open Number Line Reproducible


Congratulations on your purchase of this ReallyGood Stuff® Open Number Line Banner -Fractions—an introduction to the open number linestrategy for solving addition and subtractionproblems with fractions.

This Really Good Stuff® product includes:• Open Number Line Banner - Fractions,


Displaying the Open Number Line Banner -FractionsBefore displaying the Open Number Line Banner -Fractions, make copies of this Really Good Stuff®

Activity Guide, cut apart the reproducibles, and filethe pages for future use. Or, download another copyof it from our Web site at www.reallygoodstuff.com.Hang the Banner where students will be able to seeit easily.

Introducing the Open Number Line Banner -FractionsAs you draw a horizontal line on the board and labelit with the numbers of 1, 2, 3, and 4, remindstudents that a number line shows the numbersthat we count. Then draw arrows on each end of thenumber line and review with students that thearrows indicate that the number line goes on foreverin both directions. Explain that in between eachwhole number there are an infinite number of placesthat we can label with fractions or decimals becausethey are part of a whole number. For example, as youmark 21⁄2 on your number line, remind students that21⁄2 is halfway between 2 and 3. Then ask studentswhere the number 31⁄4 should be marked on thenumber line. Show additional fractions on thenumber line, if necessary. Emphasize that the opennumber line is a powerful problem-solving tool

because it helps them to visualize the parts of thenumber line that they need to use.

Pointing to the Open Number Line Banner -Fractions, indicate that you are going to use thisBanner to show them how to use the open numberline to solve problems that require addition andsubtraction of fractions. Explain that adding andsubtracting fractions can be confusing because youare dealing with parts of numbers and thatsometimes they have different denominators. Butthe open number line can help them keep track of allthe information in a problem. Read the first problemon the Banner. Think aloud and say, “I know Lindsaystarted with 11⁄2 cups and added 3⁄4 cup more. I needto figure out how much there is in all, so I need toadd the two numbers, which means that I move upthe number line.” Pointing to 11⁄2, say “I’ll start at 11⁄2,then think about getting to a ‘friendly’ number. WhenI work with fractions, it makes the most sense totry to use whole numbers for ‘friendly’ numbers. So 2is the next whole number, and it is 1⁄2 away from 11⁄2.My first jump will be +1⁄2.” Pointing to the +1⁄2 jump,say, “I have moved 1⁄2 of the 3⁄4 that I need to move upthe number line, so I need to figure out thedifference between 1⁄2 and 3⁄4 to know how much more Ineed to jump.” Think aloud and say, “1⁄2 is equal to 2⁄4,and 3⁄4 – 2⁄4 = 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 21⁄4 on the number line.Conclude by saying, “Lindsay has 21⁄4 cups of dryingredients in the bowl.”

Read the next problem on the Banner. Thinking aloud,say, “I know Tyler had a whole pizza and that he andhis friends ate 3⁄8. On the number line, a whole iswritten as 1. I need to subtract 3⁄8, because they atethat much pizza, and I need to find how much he had left.” Point to the 1 on the number line. “This time, I

Open Number Line Banner - Fractions



am starting at a friendly number, so now I’m goingto think about making small jumps.” As you say thefollowing, move your finger down the number line andover the arrows: “I know that 3⁄8 is the same as 1⁄8 +1⁄8 + 1⁄8. I also know that 1 whole is the same as 8⁄8.Now, 8⁄8 – 1⁄8 is 7⁄8, 7⁄8 –1⁄8 is 6⁄8, and 6⁄8 – 1⁄8 is 5⁄8.”Conclude with, “Tyler has 5⁄8 of the pizza left.”

Introducing Practice Adding and SubtractingFractions Along an Open Number Line ReproducibleCopy and distribute the Practice Adding andSubtracting Fractions Along an Open Number LineReproducible. Write the first problem of 5⁄8 + 7⁄8 = onthe board. Draw a number line, and label 5⁄8. Say, “Theaddition sign tells us to move up the number line,and the 7⁄8 tells us how far. Let’s start at 5⁄8.” Markan exaggerated arrow on the right-hand end of thenumber line to show that you are going to jump upthe number line. Say, “The closest friendly number is1, so let’s jump there. Because 5⁄8 + 3⁄8 = 1, we jump up+3⁄8.” Label the jump with +3⁄8. Think aloud and say,“This means that we have 4⁄8 left to jump (because 7⁄8– 3⁄8 = 4⁄8). I know that 1 plus 4⁄8 is 14⁄8.” Label 14⁄8 andthe jump of +4⁄8. “When we solve fraction problems, weneed to make sure the fraction is in its lowestterms. The number of 14⁄8 can be written in its lowestterms as 11⁄2, so 5⁄8 + 7⁄8 = 11⁄2.”

Answers: 1) 11⁄2, 2) 21⁄2, 3) 21⁄3, 4) 31⁄4, 5) 31⁄2, 6) 21⁄4

Introducing the Adding and Subtracting FractionsAlong an Open Number Line ReproducibleCopy and distribute the Adding and SubtractingFractions Along an Open Number Line Reproducible.Read the first problem together. Then say, “Thisproblem is asking us to find how much soil Henry has

left. We need to start at 4 and subtract 11⁄4.” Label 4on the number line, and say, “We are already at afriendly number, so let’s subtract the whole numberfirst: 4 – 1 = 3.” Label the jump of -1. “Now we canjump 1⁄4 more: 3 minus 1⁄4 is 23⁄4.” Label the jump of -1⁄4,and write 23⁄4 beneath the number line. Finish with,“Henry has 23⁄4 cups of dirt left.”

Answers: 1) 23⁄4 cups, 2) 113⁄8 inches, 3) 41⁄4 cups, 4)11⁄8 feet

Thinking FlexiblyAt times, it may make sense to “over jump.” Forexample, write the problem 5⁄8 + 7⁄8 = on the board.Say, “This problem asks us to jump up 7⁄8. We knowthat 7⁄8 is very close to the number 1, so we can jumpup to 1. We also know that 1 and 5⁄8 is the same as15⁄8. Then we jump back, or subtract, the 1⁄8 to 14⁄8,which is written in its lowest terms as 11⁄2.”

Challenge students to come up with multiple waysto solve the same problem. Have students comparetheir work with a partner and point out thesimilarities and differences in their strategies.

Making Connections to Other StrategiesHave students solve one of the problems using otheraddition or subtraction strategies. Help students torecognize that there are multiple ways to solve thesame problem. Encourage them to make connectionsbetween the methods. Remind students that theopen number line can help them to keep track oftheir jumps a little easier.

Open Number Line Banner - Fractions



Practice Adding and Subtracting Fractions Along an Open Number Line Reproducible


Adding and Subtracting Fractions Along an Open Number Line Reproducible


Congratulations on your purchase of this ReallyGood Stuff® Open Number Line Banner -Integers—an introduction to the open numberline strategy for solving addition andsubtraction problems with integers.

This Really Good Stuff® product includes:• Open Number Line Banner - Integers,


Displaying the Open Number Line Banner -IntegersBefore displaying the Open Number LineBanner - Integers, make copies of thisReally Good Stuff® Activity Guide, cut apartthe reproducibles, and file the pages for futureuse. Or, download another copy of it from ourWeb site at www.reallygoodstuff.com. Hang theBanner where students will be able to see iteasily.

Introducing the Open Number Line Banner -IntegersAs you draw a horizontal line on the board andlabel it with the positive numbers of 1, 2, 3, and4, remind students that the number line showsthe numbers that we count. Then draw arrowson each end of the number line, and review withstudents that the arrows indicate that thenumber line goes on forever in both directions.Label the number line with 0, -1, -2, -3, -4. Askstudents what they notice about the numberline: If necessary, encourage them to be specificabout positive or negative numbers; for example,when we count up negative numbers on thenumber line, the numbers get smaller.

Pointing to the Open Number Line Banner -Integers, indicate that you are going to use theBanner to show them how to use an opennumber line to solve problems that requireaddition and subtraction of integers. Explainthat adding and subtracting integers can beconfusing because of the positive and negativenumbers, but the open number line can helpthem to keep track of all the information in aproblem. Read the first problem on the Banner.Say, “I know that Owen hiked from 210 feet belowsea level, or -210 feet, to 468 feet.” Point to -210and 468 on the Banner. “So I need to find thedifference 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 wherepositive numbers end and negative numbersbegin, so 0 will usually be the best ‘friendly’number to use.” Point from -210 to 0 and then468. “This is a jump of +210. Then I jump +468.To find the answer, I need to add the two jumps:210 + 468 = 678, so Owen’s altitude changedby 678 feet.”

Read the next problem on the Banner. Say, “Iknow Lauren started with $44 and then wrote acheck for $67. The problem says that Laurenoverdrew her account, which means that shedidn’t have enough money in her account towrite a $67 check. I need to figure out how muchmoney she overdrew it by, so I need to start at44 and jump down 67 along the number line.”Point to the 44 on the number line. “Rememberthat 0 is a ‘friendly’ number when dealing withintegers, so I’ll jump from 44 to 0.” Point to this

Open Number Line Banner - Integers



jump. “That’s a jump of -44 because I went downthe number line. I need my jumps to equal -67,so I need to keep jumping down. I’ll start with ajump of -10. That takes me to -10 on the numberline, and I need to remember that now I havejumped down a total of -54. I can jump down 10more.” Point to the -10 and -10 jumps, or -20.“Now I have jumped a total of -64. So in orderto jump a total of -67 (the amount of Lauren’scheck), I need to jump -3 more to -23.” Point tothe remaining jumps and say, “I ended at -23, soLauren overdrew her account by $23.”

Introducing Adding and Subtracting IntegersAlong an Open Number Line ReproducibleCopy and distribute the Adding and SubtractingIntegers Along an Open Number LineReproducible. Write the first problem -16 +28 = .Draw a number line, and label -16. Say, “Theaddition sign tells us to move up the number lineand the 28 tells us how far. Let’s start at -16.We are going to need to go past 0, so let’s jumpthere first. That’s a jump of +16.” Label 0 andthe jump with +16. Then say, “We need to jumpup 28 in all, so let’s jump up 10 more.” Label thejump of +10 to 10. Say, “Now we’ve jumped up26, (and we know that 28 – 26 = 2), so we needto jump 2 more to total 28.” Label the jump of+2 to 12. Conclude by saying, “So -16 + 28 = 12.”

Answers: 1) 12, 2) -13, 3) -34, 4) -100, 5) 80,6) -166

Introducing the Integers Problem Solving Alongan Open Number Line ReproducibleCopy and distribute the Integers Problem SolvingAlong an Open Number Line Reproducible. Readthe first problem together. Then say, “Thisproblem is asking us to find the differencebetween -12 and 9. First, we need to start at -12 and jump up to 0.” Label -12 and 0 on thenumber line as well as the jump of +12. “Next, weneed to get to 9, so we jump up 9.” Label the +9jump. “Now we need to figure out how far wejumped: 12 + 9 =21, so the difference intemperature is 21°.”

Answers: 1) 21°, 2) -25 points, 3) 180 feet, 4)$35

Thinking FlexiblyChallenge students to come up with multipleways to solve the same problem. Have studentscompare their work with a partner and point outthe similarities and differences in theirstrategies.

Making Connections to Other StrategiesHave students solve one of the problems usingother strategies for adding and subtractingintegers. Help students to recognize that thereare multiple ways to solve the same problem.Encourage them to make connections betweenthe methods. Remind students that the opennumber line can help them to keep track of theirjumps a little more easily.

Open Number Line Banner - Integers



Adding and Subtracting Integers Along an Open Number Line Reproducible


Integers Problem Solving Along an Open Number Line Reproducible


open number line mats - really good stuff · good stuff® open number line mats—a versatile ......

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