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Fractions with
a comprehensive K-5 curriculum from
© E. Paul Goldenberg 2008, revised July 2011This presentation may be shown for professional development purposes only, and may not be sold, distributed, or altered.
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
Summary
Fractions are just numbers
Named like other numbers Named like other numbers (though written differently)(though written differently)
Live on the number line Live on the number line (have a home; name a distance)(have a home; name a distance)
Not “parts” of numbers or “parts” of thingsNot “parts” of numbers or “parts” of things “half “half ofof,” like “six ,” like “six ofof,” is not the same as ½ or 6,” is not the same as ½ or 6
Arithmetic is like other numbers Arithmetic is like other numbers (+, –, ×, ÷)(+, –, ×, ÷)(Manipulating the (Manipulating the symbol symbol is different; the meaning and effect and model of the arithmetic is the same.)is different; the meaning and effect and model of the arithmetic is the same.)
Not picturesNot picturesPicturesPictures (like pictures of any numbers)(like pictures of any numbers) are just pictures are just pictures
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
Counting nths
11stst grade experiment grade experiment (not in TM!)(not in TM!): : two fifths plus two fifthstwo fifths plus two fifths
““four fifths minus one third” four fifths minus one third” makes no more sense thanmakes no more sense than “four cows minus one chicken”“four cows minus one chicken”
But what But what isis a “fifth”? a “fifth”?
Two twentiethsSix twentieths
Linguistics
Named like other numbers Named like other numbers (but written differently)(but written differently)
One twentieth
If it takes twenty of them to make 1 dollar, then each is a “twentieth” of a dollar.
If seven of them make 1 dollar, then each is a “seventh” of a dollar.
Linguistics
Named like other numbers Named like other numbers (but written differently)(but written differently)
Fractions are just numbers
Named like other numbers Named like other numbers (though written differently)(though written differently)
Live on the number line Live on the number line (have a home; name a distance)(have a home; name a distance)
Not “parts” of numbers or “parts” of thingsNot “parts” of numbers or “parts” of things “half “half ofof,” like “six ,” like “six ofof,” is not the same as ½ or 6,” is not the same as ½ or 6
Arithmetic is like other numbers Arithmetic is like other numbers (+, –, ×, ÷)(+, –, ×, ÷)
Not picturesNot picturesPicturesPictures (like pictures of any numbers)(like pictures of any numbers) are just pictures are just pictures
Zooming in with a strong glass
300 10 20 40 50 60 70
30 1 2 4 5 6 7
Ten “spaces” (nine new numbers, labeled 1 to 9)
300 10 20 40 50 60 70
Zooming in with a strong glass
2320 21 22 24 25 26 27 28 29 30
Ten “spaces” (nine new numbers, labeled 1 to 9)
Zooming in with a weak glass
300 10 20 40 50 60 70
20 25 30
Zooming in with a weak glass
30 1 2 4 5 6 7
2 2½ 3
Zooming in with a weak glass
30 1 2 4 5 6 7
0 ½ 1
Zooming in with a weak glass
30 1 2 4 5 6 7
4 4½ 5
Zooming in with a weak glass
1½0 ½ 1 2 2½ 3 3½
2½ 2¾ 3
mor
e
Zooming in with a stronger glass
30 1 2 4 5 6 7
0 115
25
35
45
55
Zooming in with a stronger glass
30 1 2 4 5 6 7
3 415
25
35
45
55
3
3
3 3 3
155
… 165
175
… … …
Zooming in with a stronger glass
30 1 2 4 5 6 7
3 416
26
36
46
563 33 3 3
263
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
Cutting the unit interval into 6 parts
30 1 2 4 5 6 7
0 1
Cutting the unit interval
Cutting the unit interval into 6 parts
30 1 2 4 5 6 7
0 116
26
36
46
56
6
Whispering
What did I whisper?What did I whisper?
5 2 1
2
2 1
10 5
Fractions are just numbers
Named like other numbers Named like other numbers (but written differently)(but written differently)
Live on the number line Live on the number line (have a home; name a distance)(have a home; name a distance)
Not “parts” of numbers or “parts” of thingsNot “parts” of numbers or “parts” of things “half “half ofof,” like “six ,” like “six ofof,” is not the same as ½ or 6,” is not the same as ½ or 6
Arithmetic is like other numbers Arithmetic is like other numbers (+, –, ×, ÷)(+, –, ×, ÷)
PicturesPictures (like pictures of any numbers)(like pictures of any numbers) are just pictures are just pictures
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
Frac < 1? Frac = 1? Frac > 1?
12
66
96
1100
88
21
224
9
126
Frac < ½? Frac = ½? Frac > ½?
13
36
59
1100
231
2
49
46
4
9
12
Closer to
1Closer to
0Half way between
But how close to 0 and 1?
43
What “tens” is it between?
40 50
40
43
How far from the nearest tens?
40 50
43 50
3 7
50 70
43 71
7143 50 70
How far is 43 from 71?28
71– 43
7 20 1
263
What “ones” is it between?
3 4
3 416
26
36
46
563 33 3 3
263
How far from the nearest ones?
3 4
26
46
How far is 3 from 7 ?26
16
3 4 7
263 1
67
46
163
26
167
–3
563
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
One reason fractions are written the way they are
÷
3
4
16
12
484
5th grade
Fraction machines (multiplying fractions)
Stretching and shrinking, in Stretching and shrinking, in either ordereither order
Fraction machines
÷
5
5
16
16
80
Stretching and shrinking, in Stretching and shrinking, in either ordereither order
If the stretch factor and If the stretch factor and shrink factor are the same, shrink factor are the same, of courseof course “nothing happens”! “nothing happens”!
Fraction machines
÷
6
3
15
30
5
Stretching and shrinking, in Stretching and shrinking, in either ordereither order
If the stretch factor and If the stretch factor and shrink factor are the same, shrink factor are the same, of courseof course “nothing happens”! “nothing happens”!
More stretch than shrink?More stretch than shrink?
Fraction machines
Stretching and shrinking, in Stretching and shrinking, in either ordereither order
If the stretch factor and If the stretch factor and shrink factor are the same, shrink factor are the same, of courseof course “nothing happens”! “nothing happens”!
More stretch than shrink?More stretch than shrink? Twice as much shrinking…Twice as much shrinking…
÷
5
10
16
8
80
Multiplying fractions: based on
(of whole numbers)(of whole numbers)
understanding multiplication
Multiplying whole numbers
3 4 = 4 3
Representing 22 × 17
22
17
Representing the algorithm
20
10
2
7
Representing the algorithm
20
10
2
7
200
140
20
14
2217
200140
20
x
14374
One way of recording
the process
Still true when we use fractions
6
8
1/4
2/3
48
4
2
?
6 ¼8 ⅔
4842
x
?
Recording the process
1 × 1
Multiply ×
17 × 2817
28
23
14
5th grade
23
13
14
24
34
1
1
0
= 212
Still true when we use fractions
6
8
1/4
2/3
48
4
2
?
6 ¼8 ⅔
4842
x
?
Recording the process
⅙ ⅙54 ⅙
What do students need to know?
Fractions are just numbersFractions are just numbers Spoken language: Spoken language: What “three and two sixths” What “three and two sixths” meansmeans
Written language: Written language: How to How to write write “three and two sixths”“three and two sixths”
Their size Their size && their distance from other their distance from other numbersnumbers
Arithmetic with fractionsArithmetic with fractions
Introduction
Fractions are just numbers
Named like other numbers Named like other numbers (though written differently)(though written differently)
Live on the number line Live on the number line (have a home; name a distance)(have a home; name a distance)
Not “parts” of numbers or “parts” of thingsNot “parts” of numbers or “parts” of things “half “half ofof,” like “six ,” like “six ofof,” is not the same as ½ or 6,” is not the same as ½ or 6
Arithmetic is like other numbers Arithmetic is like other numbers (+, –, ×, ÷)(+, –, ×, ÷)
Not picturesNot picturesPicturesPictures (like pictures of any numbers)(like pictures of any numbers) are just pictures are just pictures
Fractions are just numbers
Named like other numbers Named like other numbers (though written differently)(though written differently)
Live on the number line Live on the number line (have a home; name a distance)(have a home; name a distance)
Not “parts” of numbers or “parts” of thingsNot “parts” of numbers or “parts” of things “half “half ofof,” like “six ,” like “six ofof,” is not the same as ½ or 6,” is not the same as ½ or 6
Arithmetic is like other numbers Arithmetic is like other numbers (+, –, ×, ÷)(+, –, ×, ÷)
Not picturesNot picturesPicturesPictures (like pictures of any numbers(like pictures of any numbers)) are just pictures are just pictures
