Zooming In On Number Lines:connecting whole numbers, decimals, fractionsCindy Carter

The Rashi SchoolNewton, Massachusetts

NCTM, San Diego, April 2010Created in PowerPoint 2008 Some parts of this presentation may not play perfectly on earlier versions of PowerPoint.Fonts may align differently on different machines.

http://thinkmath.edc.org

Children (all children, all programs, all times, in school and out) are theory buildersProvide environment, smart tools, opportunityStart with what they know Support multiple ways of thinking (Differentiating instruction without differentiating students!)

Mathematics (in school and out, all times, all topics) is the study of one disciplineAll topics related by underlying mathematicsSmart tools make that relationship clear

Dont reinvent the wheel

Better at basic skillsUnderstand underlying mathematics better, too

Todays talk Number Linea smart toolWhole numbers early grades building a strong and useful foundation

Decimals & Fractions look a lot the same using the number line to take advantage of knowledge about whole numbers

Unlike Cuisenaire rods, base 10 blocks, the number line is not just a school tool: mathematicians use it, too.All the numbers that students encounter through elementary and middle school live on the number line: counting numbers, zero, fractions, decimals, negative numbers.The Number Line is special

The Number Line Just another way to count?

Counting and measuring are different!We use numbers for counting and measuringCounting starts at 1. It uses only counting numbers (no fractions, decimals, negatives)Measuring starts at 0. It uses fractions, decimals, and negative numbers.Counters and base-10 blocks support counting, but not measuring.The number line supports measuring also.

Number lines represent measuringThe number line is like a ruler

Numbers are addressesThey also name their distance from zeroDistance is an important meaning of numbersAddition or subtraction? movingSign (+ or ) indicates direction

KindergartenVideo link to come

KindergartenVideo link to come

A child stands on the number line at 3Teacher (or another student) writes on the board+ 3 2Where did the child land?XXXOne image of + and Grade K

First gradeBuilding standard notation

2nd gradeBuilding mental math

Shannons Grade 2, pairs to 30Video link to come

Mental Number Linelearning to think mathematicallyMathematics takes place in the mind The right tools let young children develop that skill!229 +259 301,000,000 +1,000,030 30

Mental Number Linelearning to think mathematically2nd grade not ready for using the standard algorithm with such large numbersAre ready for mathematical competence and confidence Doing math is exciting!Building foundations for the standard algorithm

Folded Number Lines for bigger numbers

Number Line Hotelfolding is practical and Start at 47 and go down 10What is 10 higher than 23?

Number Line HotelHow are the numbers in one column alike?How are the numbers in one row alike?

Number Line HotelWhere are the 40s?

Conventional hundreds chartbigger numbers, but

Number LineSupports different models of how to do mathAlready talked about subtraction as moving backwards or down on the number lineSubtraction as distanceExample from 4th grade

Grade 4, revisiting subtractingVideo link to come

3rd GradeSubtraction of larger numbers using the distance model

Thinking about 76 and 47 as addressesSubtraction finding the distance between

How far is 47 from 76?293206Grade 3

What tens is it between?Buidling those 2 skillsWith 2nd graders

4047How far from the nearest tens?372nd graders get comfortable with tens neighbors and numbers betweenPairs to 10

Grade 2, finding 70 1/2Video link to come

How can we do 62 37?It means how far from 37 to 62?

Find the nearest multiples of 10 between them

So 62 37 = 25Another example of subtraction as distanceI have saved $37, but I need $62 to buy the

All numbers live on a number line!Numbers serve as location and as distanceAddition is adding distancesSubtraction measures the distance betweenDecimals and fractions live there, too.

Zooming InGrade 4

Zooming In MoreGrade 4Decimals are an extension of whole numbers

Zooming In Still MoreGrade 4

Fractions, too!4th graders building the notion ofzooming in on the number linenumbers are preprinted

Grade 4, zooming in on decimalsVideo link to come

Grade 4, zooming in on decimalsVideo link to come

And we zoomed in moreVideo link to come

Grade 4 of Think Math!

How far is 4.7 from 7.6?2.9.32.6Grades 4 & 5

4.7What ones is it between?

44.7How far from the nearest ones?0.70.3Pairs to 1

How far is 2 from 8 ?5

Same storyFind the nearest whole numbersFind the distance to eachSkill practice Pairs to 1

How far is 19 from 58?391308Addresses on the number lineZoom in DistancesTens betweenAdd it up

How far is 1.9 from 5.8?3.9.13.8Addresses on the number lineOnes betweenZoom inDistancesAdd it up

How far is 3 from 9 ?5Addresses on the number lineOnes betweenZoom inDistancesAdd it up

Adding/Subtracting: moving

Whats 16 + 32? (start: find the 10s row)Address: 16 Move: up 3 rows, right 2Number Line Hotel

Adding/Subtracting: moving

Whats 1.6 + 3.2? (start: find the 1s row)Address: 1.6 Move: up 3 rows, right .2 Decimal Number Line Hotel

Adding/Subtracting: moving

Whats 16 + 32? Whats 1.6 + 3.2?

How to start in late elementary (or even middle) schoolSame path start when you can Build the concepts with whole numbersLike the video of 4th graders working on subtraction as distance move naturally into fractions and decimalsZoom in!Skill practice pairs to 10, pairs to 1 (decimals), and pairs to 1 (fractions)

Ideas drawn from Think Math!http://thinkmath.edc.org/Learn more at Think Math! Information Exchange

****And you will see some of the ways*And you will see some of the waysChildren have plenty of experience with things that grow gradually: the height of water in the bathtub, their own height, the temperature of the bathwater as warm water enters. Those measurements do not step from 1 to 2 to 3, but move through all the values in between as well.

Counting starts when there is one object, and it matches the objects with whole numbers: seven animals, never seven and a half.***This is the moving right and left image. Here, numbers represent addresses (places) on the number line (where the child starts and stops) and also distances (how far the child moves, and which direction, for each instruction).

The addresses are also a kind of how far, in that they all say how far the child is from zero.Why start at 0 instead of 1?Why number from the bottom to the top?

On a conventional number chart that starts with 1, the sixties are not all in one row, nor are the 20s or 30s. Each decade is split, with the number that starts that decade at the end of one row, and the rest of the decade in the next row down. In a conventional number chart, 43 is not higher than 33; it is directly *below* 33, lower.

That doesnt necessarily confuse children, but it does make language more ambiguous. The question which is lower, 84 or 74 can be interpreted two ways. Yes, it is more precise to ask which is less, but we even mathematicians often use spatial imagery to talk about numbers. Low and high prices. Low and high temperatures. Low and high yield. And because that language is natural, and used, it is helpful to support it with the image.

Moreover, graphing uses a bottom-to-top increase as well as a left-to-right increase, so this image is consistent with general mathematical usage.*Why start at 0 instead of 1?Why number from the bottom to the top?

On a conventional number chart that starts with 1, the sixties are not all in one row, nor are the 20s or 30s. Each decade is split, with the number that starts that decade at the end of one row, and the rest of the decade in the next row down. In a conventional number chart, 43 is not higher than 33; it is directly *below* 33, lower.

That doesnt necessarily confuse children, but it does make language more ambiguous. The question which is lower, 84 or 74 can be interpreted two ways. Yes, it is more precise to ask which is less, but we even mathematicians often use spatial imagery to talk about numbers. Low and high prices. Low and high temperatures. Low and high yield. And because that language is natural, and used, it is helpful to support it with the image.

Moreover, graphing uses a bottom-to-top increase as well as a left-to-right increase, so this image is consistent with general mathematical usage.*Why start at 0 instead of 1?Why number from the bottom to the top?

On a conventional number chart that starts with 1, the sixties are not all in one row, nor are the 20s or 30s. Each decade is split, with the number that starts that decade at the end of one row, and the rest of the decade in the next row down. In a conventional number chart, 43 is not higher than 33; it is directly *below* 33, lower.

That doesnt necessarily confuse children, but it does make language more ambiguous. The question which is lower, 84 or 74 can be interpreted two ways. Yes, it is more precise to ask which is less, but we even mathematicians often use spatial imagery to talk about numbers. Low and high prices. Low and high temperatures. Low and high yield. And because that language is natural, and used, it is helpful to support it with the image.

Moreover, graphing uses a bottom-to-top increase as well as a left-to-right increase, so this image is consistent with general mathematical usage.*Why start at 0 instead of 1?Why number from the bottom to the top?

On a conventional number chart that starts with 1, the sixties are not all in one row, nor are the 20s or 30s. Each decade is split, with the number that starts that decade at the end of one row, and the rest of the decade in the next row down. In a conventional number chart, 43 is not higher than 33; it is directly *below* 33, lower.

That doesnt necessarily confuse children, but it does make language more ambiguous. The question which is lower, 84 or 74 can be interpreted two ways. Yes, it is more precise to ask which is less, but we even mathematicians often use spatial imagery to talk about numbers. Low and high prices. Low and high temperatures. Low and high yield. And because that language is natural, and used, it is helpful to support it with the image.

Moreover, graphing uses a bottom-to-top increase as well as a left-to-right increase, so this image is consistent with general mathematical usage.***And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

*Name a two digit number (e.g., 43) and ask what multiples of 10 are nearest to it. What tens is it between?

This question asks about place value and rounding/approximation, but the language is spatial! BETWEEN suggests position The number line again!

*In this context, it is natural to ask how far the number is from those nearest multiples of 10.

The pair of distances is, of course, a pair that makes 10, which theyve been studying to supercompetence since they started Think Math!

*This is the distance image of subtraction.

See FractionsStoryline.ppt and DecimalsStoryline.ppt for examples of subtraction of fractions and decimals on the number line.

Notice that this method does not involve borrowing or even notions of regrouping. It uses notions of rounding and distance from the nearest 10, and the shopkeepers notion of adding up from the smaller number to the larger.

37 + ___ = 62 Kids are *not* troubled by missing addends!

*Kids *not* troubled by missing addends!****To see fractional numbers between the whole numbers, we could zoom in on the number line with a magnifying glass. Depending on the strength of the magnifying glass, we might see one new number () or two new numbers ( and ) or four new numbers (, , , ), and so on. The magnifying glass that doubles what we can see, shows us the halves.

**And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

*And it generalizes perfectly to decimals, and explains why decimal algorithms for addition and subtraction are identical to the algorithms for whole numbers.

Do the same thing with 4.3 instead of 43 and ask which ones (instead of which tens) it is between.*How far from the nearest 1s. Notice how familiar these answers are. The whole world of decimals begins to feel strangely familiar.*And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

*And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

*And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

*And subtraction is the operation that answers how far one must move (and in which direction) to get from one number to another.

The question How much is 76 47? is identical with how far is 76 from 47 on a number line?

The number line image is useful for mental subtraction. Put a couple of useful rest stops along the way, at the nearest tens to each end, then add up the distances.

43 to 50 is 7 steps; 50 to 70 is 20 steps; 70 to 71 is 1 step. The total is 28 steps. This is, in essence, the shopkeepers way of making change. Start at the cost and count up in convenient units. Theres nothing wrong with borrowing, but its hard to do in ones head. This method tends to be lots easier.

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