Zooming In On Number Lines:connecting whole numbers, decimals, fractions

Cindy Carter

The Rashi SchoolNewton, Massachusetts

NCTM, San Diego, April 2010

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http://thinkmath.edc.org

Children (all children, all programs, all times, in school and out)

are theory builders

Provide environment, smart tools, Provide environment, smart tools, opportunityopportunity Start with what they know Start with what they know SSupport multiple ways of thinking upport multiple ways of thinking (Differentiating (Differentiating

instruction without differentiating students!)instruction without differentiating students!)

Mathematics (in school and out, all times, all topics)

is the study of one discipline

All topics related by underlying mathematicsAll topics related by underlying mathematics Smart tools make that relationship clearSmart tools make that relationship clear

Don’t reinvent the wheel

Better at basic skillsBetter at basic skills Understand underlying mathematics better, tooUnderstand underlying mathematics better, too

Today’s talk – Number Linea smart tool

Whole numbers – early grades Whole numbers – early grades building a strong and useful foundationbuilding a strong and useful foundation

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

Unlike Cuisenaire rods, base 10 blocks, …Unlike Cuisenaire rods, base 10 blocks, …the number line is the number line is not not just a school tool: just a school tool: mathematicians use it, too.mathematicians use it, too.

All the numbers that students encounter All the numbers that students encounter through elementary and middle school live through elementary and middle school live on the number line: counting numbers, zero, on the number line: counting numbers, zero, fractions, decimals, negative numbers.fractions, decimals, negative numbers.

The Number Line is special

The Number Line

Just another way to count?Just another way to count?

Counting and measuring are different!

We use numbers for counting and measuringWe use numbers for counting and measuring Counting starts at 1.Counting starts at 1.

It uses only “counting numbers”It uses only “counting numbers” (no fractions, decimals, negatives)(no fractions, decimals, negatives)

Measuring starts at 0.Measuring starts at 0.It uses fractions, decimals, and negative numbers.It uses fractions, decimals, and negative numbers.

Counters and base-10 blocks support Counters and base-10 blocks support countingcounting, but not measuring., but not measuring.

The number line supports measuring also.The number line supports measuring also.

Number lines represent measuring

The number line is like a rulerThe number line is like a ruler

Numbers are Numbers are addressesaddresses They also name their They also name their distance from zerodistance from zero DistanceDistance is an important meaning of is an important meaning of

numbersnumbers Addition or subtraction? Addition or subtraction? movingmoving Sign (+ or –) indicates directionSign (+ or –) indicates direction

Kindergarten

Video link to comeVideo link to come

Kindergarten

Video link to comeVideo link to come

2 3 4 5 610 11107 8 9

A child stands on the number line at 3A child stands on the number line at 3 Teacher (or another student) writes on the boardTeacher (or another student) writes on the board

+ 3+ 3

– – 22 Where did the child land?Where did the child land?

X XX

One image of + and –

Grade K

First grade

Building Building standard standard notationnotation

2nd grade

Building mental mathBuilding mental math

Shannon’s Grade 2, pairs to 30

Video link to come

Mental Number Linelearning to think mathematically

Mathematics takes place in the mind Mathematics takes place in the mind The right tools let young children develop The right tools let young children develop

that skill!that skill!

–229 +259

30

–1,000,000 +1,000,030

30

Mental Number Linelearning to think mathematically

22ndnd grade not ready for using the standard grade not ready for using the standard algorithm with such large numbersalgorithm with such large numbers

Are ready for mathematical competence and Are ready for mathematical competence and confidenceconfidence

Doing math is exciting!Doing math is exciting! Building foundations for the standard Building foundations for the standard

algorithmalgorithm

Folded Number Lines for bigger numbers

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

Number Line Hotel

How are the numbers in one column alike?How are the numbers in one column alike? How are the numbers in one row alike?How are the numbers in one row alike?

Number Line Hotel

Where are the 40s?Where are the 40s?

Conventional hundreds chartbigger numbers, but…

Number Line

Supports different models of how to do mathSupports different models of how to do math Already talked about subtraction as moving Already talked about subtraction as moving

backwards or down on the number linebackwards or down on the number line Subtraction as distanceSubtraction as distance Example from 4Example from 4thth grade grade

Grade 4, revisiting subtracting

Video link to comeVideo link to come

3rd Grade Subtraction of larger numbers using the Subtraction of larger numbers using the

distance modeldistance model

Thinking about 76 and 47 as addressesThinking about 76 and 47 as addresses Subtraction – finding the distance betweenSubtraction – finding the distance between

76– 47

29

50 70

47 76

7647 50 70

How far is 47 from 76?29

76– 47

3 20 6

Grade 3

47

What “tens” is it between?

40 50

Buidling those 2 skillsWith 2nd graders

40

47

How far from the nearest tens?

40 50

47 50

37

2nd graders get comfortable with tens neighbors and numbers between

Pairs to 10

Grade 2, finding 70 1/2

Video link to comeVideo link to come

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

Find the nearest multiples of 10 between themFind the nearest multiples of 10 between them

So 62 – 37 = 25So 62 – 37 = 25

37 62

Another example of subtraction as distance

4037 6260

3 220

I have saved $37, but I need $62 to buy the …

All numbers live on a number line! Numbers serve as location and as distanceNumbers serve as location and as distance Addition is adding distancesAddition is adding distances Subtraction measures the distance betweenSubtraction measures the distance between Decimals and fractions live there, too.Decimals and fractions live there, too.

Zooming In

300 10 20 40 50 60 70

30 1 2 4 5 6 7

Grade 4

Zooming In More

30 1 2 4 5 6 7

.30 .1 .2 .4 .5 .6 .7 .8 .9 1

Grade 4Decimals are an extension of whole numbers

Zooming In Still More

.30 .1 .2 .4 .5 .6 .7

.13.1 .11 .12 .14 .15 .16 .17 .18 .19 .2

Grade 4

Fractions, too!

30 1 2 4 5 6 7

2 2½ 3

4th graders building the notion ofzooming in on the number linenumbers are preprinted

Grade 4, zooming in on decimals

Video link to comeVideo link to come

Grade 4, zooming in on decimals

Video link to comeVideo link to come

And we zoomed in more…

Video link to comeVideo link to come

Grade 4 of Think Math!

5 7

4.7 7.6

7.64.7 5 7

How far is 4.7 from 7.6?2.9

7.6– 4.7

.3 2 .6

Grades 4 & 5

4.7

What “ones” is it between?

4 5

4

4.7

How far from the nearest “ones”?

4 5

4.7 5

0.7 0.3

Pairs to 1

3 8

2 825

15

82 3 825

15

How far is 2 from 8 ?8

– 2

5

25

15

1525

5 45

35

15

Same story

Find the nearest whole numbersFind the nearest whole numbers Find the distance to eachFind the distance to each Skill practice – Pairs to 1Skill practice – Pairs to 1

20 50

19 58

5819 20 50

How far is 19 from 58?39

58– 19

1 30 8

Addresses on the number line

Zoom in Distances

Tens between

Add it up

2 5

1.9 5.8

How far is 1.9 from 5.8?3.9

5.8– 1.9

.1 3 .8

Addresses on the number line

Ones between

Zoom in Distances Add it up

5.81.9 2 5

4 9

3 957

47

93 4 957

47

How far is 3 from 9 ?9

– 3

5

57

47

4757

5 67

27

47

Addresses on the number line

Ones between

Zoom in Distances Add it up

Adding/Subtracting: moving

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

Adding/Subtracting: moving

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

Adding/Subtracting: moving

What’s 16 + 32? What’s 16 + 32? What’s 1.6 + 3.2?What’s 1.6 + 3.2?

16 46 48

30 2

1.6 4.6 4.8

3 .2

How to start in late elementary (or even middle) school Same path – start when you can Same path – start when you can Build the concepts with whole numbersBuild the concepts with whole numbers Like the video of 4Like the video of 4thth graders working on graders working on

subtraction as distance – move naturally subtraction as distance – move naturally into fractions and decimalsinto fractions and decimals

Zoom in!Zoom in! Skill practice – pairs to 10, pairs to 1 Skill practice – pairs to 10, pairs to 1

(decimals), and pairs to 1 (fractions)(decimals), and pairs to 1 (fractions)

Ideas drawn from Think Math!

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