Transcript From Buttons to Algebra: Learning the Ideas and Language

Zooming In On Number Lines:
connecting whole numbers, decimals, fractions
Cindy Carter
The Rashi School
Newton, Massachusetts
NCTM, San Diego, April 2010
http://thinkmath.edc.org
Created in PowerPoint 2008
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Children
(all children, all programs, all times, in school and out)
are theory builders
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Provide environment, smart tools, opportunity
Start 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 discipline


All topics related by underlying mathematics
Smart tools make that relationship clear
Don’t reinvent the wheel
Better at basic skills
 Understand underlying mathematics better, too
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Today’s talk – Number Line
a smart tool
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Whole numbers – early grades
building a strong and useful foundation
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Decimals & Fractions – look a lot the same
using the number line to take advantage of
knowledge about whole numbers
The Number Line is special
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.
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The Number Line
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Just another way to count?
Counting and measuring are different!
We use numbers for counting and measuring
 Counting starts at 1.
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It uses only “counting numbers” (no fractions, decimals, negatives)
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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.
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Number lines represent measuring
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The number line is like a ruler
Numbers are addresses
 They also name their distance from zero
 Distance is an important meaning of numbers
 Addition or subtraction? moving
 Sign (+ or –) indicates direction
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Kindergarten
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Video link to come
Kindergarten
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Video link to come
One image of + and –
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0
A child stands on the number line at 3
Teacher (or another student) writes on the board
+3
–2
Where did the child land?
1
2
X
3
X
4
5
X
6
7
8
9
10 11
Grade K
First grade
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Building
standard
notation
2nd grade
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Building mental math
Shannon’s Grade 2, pairs to 30
Video link to come
Mental Number Line
learning to think mathematically
–1,000,000
+1,000,030
30
–229
+259
30
Mathematics takes place in the mind
 The right tools let young children develop
that skill!
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Mental Number Line
learning to think mathematically
2nd grade not ready for using the standard
algorithm with such large numbers
 Are ready for mathematical competence and
confidence
 Doing math is exciting!
 Building foundations for the standard
algorithm
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Folded Number Lines for bigger
numbers
Number Line Hotel
folding is practical and …
Start at 47 and go down 10
 What is 10 higher than 23?
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Number Line Hotel
How are the numbers in one column alike?
 How are the numbers in one row alike?
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Number Line Hotel
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Where are the 40s?
Conventional hundreds chart
bigger numbers, but…
Number Line
Supports different models of how to do math
 Already talked about subtraction as moving
backwards or down on the number line
 Subtraction as distance
 Example from 4th grade
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Grade 4, revisiting subtracting
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Video link to come
rd
3
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Grade
Subtraction of larger numbers using the
distance model
76
– 47
29
Thinking about 76 and 47 as addresses
 Subtraction – finding the distance between
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How far is 47 from 76?
47
76
70
50
3
47
76
– 47
29
20
50
6
70
76
Grade 3
What “tens” is it between?
47
40
50
Buidling those 2 skills
With 2nd graders
How far from the nearest tens?
47
40
50
Pairs to 10
7
40
3
47
50
2nd graders get comfortable with tens neighbors and numbers between
Grade 2, finding 70
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Video link to come
1/
2
Another example of subtraction as
distance
I have saved $37, but I need $62 to buy the …
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How can we do 62 – 37?
It means “how far from 37 to 62?”
37
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62
Find the nearest multiples of 10 between them
37
40
3
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So 62 – 37 = 25
60 62
20
2
All numbers live on a number line!
Numbers serve as location and as distance
 Addition is adding distances
 Subtraction measures the distance between
 Decimals and fractions live there, too.
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Zooming In
0
10
20
30
40
50
60
70
0
1
2
3
4
5
6
7
Grade 4
Zooming In More
0
0
1
.1
2
.2
3
.3
.4
4
.5
5
.6
.7
Decimals are an extension of whole numbers
6
.8
7
.9
1
Grade 4
Zooming In Still More
0
.1
.1
.11
.2
.12 .13
.3
.14 .15
.4
.5
.16
.17
.6
.18 .19
.7
.2
Grade 4
Fractions, too!
0
1
2
3
4
2
2½
3
5
6
7
4th graders building the notion of
zooming in on the number line
numbers are preprinted
Grade 4, zooming in on decimals
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Video link to come
Grade 4, zooming in on decimals
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Video link to come
And we zoomed in more…
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Video link to come
Grade 4 of Think Math!
How far is 4.7 from 7.6?
4.7
7.6
7
5
.3
4.7
7.6
– 4.7
2.9
2
5
.6
7
Grades 4 & 5
7.6
What “ones” is it between?
4.7
4
5
How far from the nearest “ones”?
4.7
4
5
Pairs to 1
0.7
4
0.3
4.7
5
2
5
1
5
How far is 2 from 8 ?
1
8 5
2
2 5
2
2 5
5
8
3
3
5
8
–2
1
5
5
3
1
5
2
5
4
5
8
1
8 5
Same story
Find the nearest whole numbers
 Find the distance to each
 Skill practice – Pairs to 1
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How far is 19 from 58?
19
58
58
– 19
39
Addresses on the number line
Tens between
1
19
Zoom in
50
20
30
50
20
Distances
8
Add it up
58
How far is 1.9 from 5.8?
1.9
Addresses on the number line
.1
Zoom in
5
2
Ones between
1.9
5.8
5.8
– 1.9
3.9
3
5
2
Distances
.8
Add it up
5.8
5
7
4
7
How far is 3 from 9 ?
4
9 7
5
3 7
Addresses on
the number line
9
–3
5
4
7
5
7
6
7
9
4
Ones between
2
7
5
3 7
Zoom in
4
7
5
9
4
Distances
Add it up
4
9 7
Adding/Subtracting: moving
What’s 16 + 32? (start: find the 10s row)
 Address: 16
Move: up 3 rows, right 2
 Number Line Hotel
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Adding/Subtracting: moving
What’s 1.6 + 3.2? (start: find the 1s row)
 Address: 1.6
Move: up 3 rows, right .2
 Decimal Number Line Hotel
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Adding/Subtracting: moving
What’s 16 + 32?
 What’s 1.6 + 3.2?
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30
16
2
46
3
1.6
48
.2
4.6
4.8
How to start in late elementary
(or even middle) school
Same path – start when you can
 Build the concepts with whole numbers
 Like the video of 4th graders working on
subtraction as distance – move naturally
into fractions and decimals
 Zoom in!
 Skill practice – pairs to 10, pairs to 1
(decimals), and pairs to 1 (fractions)
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Ideas drawn from Think Math!
Learn more at
Think Math! Information Exchange
http://thinkmath.edc.org/