Zooming In On Number Lines: connecting whole numbers, decimals, fractions Cindy Carter The Rashi School Newton, Massachusetts NCTM, San Diego, April 2010 Created in PowerPoint 2008 Some parts of this presentation may not play perfectly on earlier versions of PowerPoint. Fonts may align differently on different machines.

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

Mathematics (in school and out, all times, all topics) is the study of one discipline All topics related by underlying mathematics All topics related by underlying mathematics Smart tools make that relationship clear Smart tools make that relationship clear Don’t reinvent the wheel

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

Today’s talk – Number Line a smart tool Whole numbers – early grades building a strong and useful foundation Whole 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 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. 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. 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? Just another way to count?

Counting and measuring are different! We use numbers for counting and measuring We use numbers for counting and measuring Counting starts at 1. It uses only “counting numbers” (no fractions, decimals, negatives) Counting starts at 1. It uses only “counting numbers” (no fractions, decimals, negatives) Measuring starts at 0. It uses fractions, decimals, and negative numbers. Measuring starts at 0. It uses fractions, decimals, and negative numbers. Counters and base-10 blocks support counting, but not measuring. Counters and base-10 blocks support counting, 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 ruler The number line is like a ruler Numbers are addresses Numbers are addresses They also name their distance from zero They also name their distance from zero Distance is an important meaning of numbers Distance is an important meaning of numbers Addition or subtraction? moving Addition or subtraction? moving Sign (+ or –) indicates direction Sign (+ or –) indicates direction

Kindergarten Video link to come Video link to come

Kindergarten Video link to come Video link to come

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

First grade Building standard notation Building standard notation

2nd grade Building mental math Building mental math

Shannon’s Grade 2, pairs to 30 Video link to come

Mental Number Line learning to think mathematically Mathematics takes place in the mind Mathematics takes place in the mind The right tools let young children develop that skill! The right tools let young children develop that skill! – –1,000,000 +1,000,030 30

Mental Number Line learning to think mathematically 2 nd grade not ready for using the standard algorithm with such large numbers 2 nd grade not ready for using the standard algorithm with such large numbers Are ready for mathematical competence and confidence Are ready for mathematical competence and confidence Doing math is exciting! Doing math is exciting! Building foundations for the standard algorithm Building foundations for the standard algorithm

Folded Number Lines for bigger numbers

Number Line Hotel folding is practical and … Start at 47 and go down 10 Start 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 chart bigger numbers, but…

Number Line Supports different models of how to do math Supports different models of how to do math Already talked about subtraction as moving backwards or down on the number line Already talked about subtraction as moving backwards or down on the number line Subtraction as distance Subtraction as distance Example from 4 th grade Example from 4 th grade

Grade 4, revisiting subtracting Video link to come Video link to come

3 rd Grade Subtraction of larger numbers using the distance model Subtraction of larger numbers using the distance model Thinking about 76 and 47 as addresses Thinking about 76 and 47 as addresses Subtraction – finding the distance between Subtraction – finding the distance between 76 – 47 29

How far is 47 from 76? – Grade 3

47 What “tens” is it between? 4050 Buidling those 2 skills With 2 nd graders

40 47 How far from the nearest tens? nd graders get comfortable with tens neighbors and numbers between Pairs to 10

Grade 2, finding 70 1 / 2 Video link to come Video 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 them Find the nearest multiples of 10 between them So 62 – 37 = 25 So 62 – 37 = Another example of subtraction as distance I have saved $37, but I need $62 to buy the …

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

Zooming In Grade 4

Zooming In More Grade 4 Decimals are an extension of whole numbers

Zooming In Still More Grade 4

Fractions, too! ½ 3 4 th graders building the notion of zooming in on the number line numbers are preprinted

Grade 4, zooming in on decimals Video link to come Video link to come

Grade 4, zooming in on decimals Video link to come Video link to come

And we zoomed in more… Video link to come Video link to come

Grade 4 of Think Math!

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

4.7 What “ones” is it between? 45

4 4.7 How far from the nearest “ones”? Pairs to 1

How far is 2 from 8 ? 8 –

Same story Find the nearest whole numbers Find the nearest whole numbers Find the distance to each Find the distance to each Skill practice – Pairs to 1 Skill practice – Pairs to 1

How far is 19 from 58? – Addresses on the number line Zoom inDistances Tens between Add it up

How far is 1.9 from 5.8? – Addresses on the number line Ones between Zoom inDistancesAdd it up

How far is 3 from 9 ? 9 – Addresses on the number line Ones between Zoom inDistancesAdd it up

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

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

Adding/Subtracting: moving What’s ? What’s ? What’s ? What’s ?

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 numbers Build the concepts with whole numbers Like the video of 4 th graders working on subtraction as distance – move naturally into fractions and decimals Like the video of 4 th graders working on subtraction as distance – move naturally into fractions and decimals Zoom in! Zoom in! Skill practice – pairs to 10, pairs to 1 (decimals), and pairs to 1 (fractions) Skill practice – pairs to 10, pairs to 1 (decimals), and pairs to 1 (fractions)

Ideas drawn from Think Math! Learn more at Think Math! Information Exchange