transitioning to the common core state standards – mathematics pam hutchison [email protected]
TRANSCRIPT
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AGENDA
Fractions Fractions on a Number Line
Naming and Locating Fractions, Whole Numbers and Mixed
Numbers Comparing Equivalent
Assessing Fractions Stoplighting the Standards
Making Math Visible
David spent of his money on a game.
Then he spent of his remaining
money on a book. If he has $20 left,
how much money did he have at first?
Spending Spree
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Fractions
Fraction Concepts
Four children share six brownies so that each child receives a fair share. How many brownies (or parts of brownies) will each child receive?
Fraction Concepts
Six children share four brownies so that each child receives a fair share. What portion of each brownie will each child receive?
Fractions NF.1. Understand a fraction 1/b as the
quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Fraction Concepts
Illustrative Mathematics
The importance of the unit or whole Naming the whole for the fraction Implication for instruction
So what is the definition of a
fraction?
Definition of Fraction:
Start with a unit, 1, and split it into ___ equal pieces.
Each piece represent 1/___ of the unit.
When we name the fraction__/__,
we are talking about ___ of those 1/___ size pieces .
Fraction Concepts
Fractions Task - NC
Mr. Rogers started building a deck on the back of his house. So far, he finished ¼ of the deck. The fraction of the completed deck is below. Draw 2 pictures of what the completed
deck might look like. Use numbers and words to explain how you created your picture.
Fraction Task - NC
Martha is making a scarf for her sister. Each day she knits 1/6 of a scarf. What fraction of the scarf will be
complete after three days? What fraction of the scarf will be
complete after six days? How can you use a number line to
prove that your answers are correct?
Fraction Concepts
What fraction of the rectangle is shaded? How might you draw the rectangle in another way but with the same fraction shaded?
Fractions
on the
Number Line
Fractions
NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number
line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
How many pieces are in the unit?
Are all the pieces equal?
So each piece represents
0 1
●
51
How far (how many pieces) is the point from 0?
We name that point…….
0 1
51
51
●
How many pieces are in the unit?
Are all the pieces equal?
So the denominator is
And each piece represents .
0 1
7
●
71
How far is the point from 0?
So the numerator is
And the name of the point is ……
0
1
71
●
71
Fractions
NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. b. Represent a fraction a/b on a number
line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
How many pieces are in the unit?
Are all the pieces equal?
So each piece represents
0 1
71
●
How far is the point from 0?
How many pieces from 0?
So the name of the point is ….
0 1
74
71
●71
72
73
74
Definition of Fraction:
When we name the point ,
we’re talking about a distance
from 0 of ___ of those ___
pieces.
4 71
74
How many pieces are in the unit?
Are all the pieces equal?
So the denominator isand each piece represents
0 1
51
5
●
How far is the point from 0?
So the numerator is and the fraction represented is
0 1
53
3
53
●52
51
The denominator is So each piece represents
The numerator is And the fraction is
0 1
65
6
61
5
65
●
Academic Vocabulary What is the meaning of denominator? What about numerator?
Definitions should be more than a location – the denominator is the bottom number
They should be what the denominator is – the number of equal parts in one unit
Student TalkStrategy: Rally Coach Partner A: name the point and explain Partner B: verify and “coach” if
needed Tip, Tip, Teach
Switch roles Partner B: name the point and explain Partner A: verify and “coach” if
needed Tip, Tip, Teach
Here is the unit. (SHOW)
The unit is split in ___ equal pieces
Each piece represents
The distance from 0 to the point is ___ of those pieces
The name of the point is .
Explains – Key Phrases
1
Partner Activity 1
Start with a unit, 1, Split it into __ equal pieces.
Each piece represents of the unit
The point is __ of those pieces from 0
So this point represents
0 1
Definition of Fraction: 27
7
2
72
●
71
72
Start with a unit, 1, Split it into __ equal pieces.
Each piece represents of the unit
The pointa is __ of those pieces from 0
So this point represents
0 1
Definition of Fraction: 68
8
6
86
●
81
86
Partner A
5.
6.
7.
Partner Activity 1, cont.
Partner B
5.
6.
7.
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43
32
31
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83
| | | | | | | | |
The denominator is …….
The numerator is ………
Another way to name this point?
0 1 2
33
1
33
| | | | | | | | |
The denominator is ……..
The numerator is ………
Another way to name this point?
0 1 2
36
2
36
| | | | | | | | |
The denominator is ……
The numerator is ………
Another way to name this point?
0 1 2
35
123
35
| | | | | | | | |
The denominator is …..
The numerator is ………
Another way to name this point?
0 1 2
37
213
37
15
| | | | | | | | |
Suppose the line was shaded to 5.
How many parts would be shaded?
So the numerator would be ………
0 1 2
3
30
| | | | | | | | |
Suppose the line was shaded to 10.
How many parts would be shaded?
So the numerator would be ………
0 1 2
3
Rally Coach Partner A goes first
Name the point as a fraction and as a mixed number. Explain your thinking
Partner B: coachSWITCH Partner B goes
Name the point as a fraction and as a mixed number. Explain your thinking
Partner A: coach
Page 93-94
Rally Coach Part 2 Partner B goes first
Locate the point on the number line Rename the point in a 2nd way (fraction
or mixed number) Explain your thinking
Partner A: coachSWITCH ROLES
Rally Coach
Partner B
6.
7.
8.
43
2
415
511
310
611
21
3
Partner A
6.
7.
8.
Connect to traditional
Change to a fraction.
How could you have students develop a procedure for doing this without telling them “multiply the whole number by the denominator, then add the numerator”?
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Connect to traditional
Change to a mixed number.
Again, how could you do this without just telling students to divide?
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Student Thinking
Video Clips 1 – David (5th Grade)● Two clips● First clip – 3 weeks after a conceptual lesson on
mixed numbers and improper fractions● Second clip – 3.5 weeks after a procedural
lesson on mixed numbers and improper fractions
Student Thinking
Video Clips 2 – Background
● Exemplary teacher because of the way she normally engages her students in reasoning mathematically
● Asked to teach a lesson from a state-adopted textbook in which the focus is entirely procedural.
● Lesson was videotaped; then several students were interviewed and videotaped solving problems.
Student Thinking
Video Clips 2 – Background, cont.
● Five weeks later, the teacher taught the content again, only this time approaching it her way, and again we assessed and videotaped children.
Student Thinking
Video Clips 2 – Rachel
● First clip – After the procedural lesson on mixed numbers and improper fractions
● Second clip – 5 weeks later after a conceptual lesson on mixed numbers and improper fractions
Classroom Connections
Looking back at the 2 students we saw interviewed, what are the implications for instruction?
Research
Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first
Initial rote learning of a concept can create interference to later meaningful learning
Discuss at Your Tables How is this different from the way
your book currently teaches fractions?
How does it support all students in deepening their understanding of fractions?
Fractions
NF3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. c. Express whole numbers as fractions,
and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
Compare Fractions
Using
Sense Making
Comparing Fractions
A. B. 98
95
73
75
Comparing Fractions
B. A. 71
41
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91
Common Numerator
Comparing Fractions
A. B. 52
32
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Common Numerator
Benchmark Fractions
| | |0 ½ 1
How can you tell if a fraction is: Close to 0? Close to but less than ½? Close to but more than ½? Close to 1?
Comparing Fractions
B. A.
A. B.
41
32
43
84
84
65
83
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Fractions
NF3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare two fractions with the
same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Equivalent Fractions
Fractions
NF3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. Understand two fractions as
equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
43
Locate on the top number line. 43
●
0 1
43
43
Copy onto the bottom number line. 43
0 1
●
●
43
43
Are the lengths equal?
0 1
●
●
43
Equivalent fractions can be constructed by partitioning equal fractional parts of a whole into the same number of equal parts.
The length of the whole does not change; it has only been partitioned into more equal sized pieces. Since the length being specified has not changed, the fractions that describe that length are equal.
43
0 1
●
●
43
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CaCCSS
Fractions are equivalent (equal) if they are the same size or they name the same point on the number line. (3.NF3a)
86
43
0 1
●
●
43
86
43
Locate on the top number line. 43
●
0 1
43
Then copy onto the bottom number line.
43
Are the lengths equal?
43
0 1
●
●
43
129
129
43
●
●
So
129
43
0 1
Order Matters!
Locate 1st fraction on number line Duplicate on 2nd number line “Are they equal?” Split 2nd number line “Are they equal?” Name point on 2nd number line
So Fraction 1 = Fraction 2
Equivalent Fractions
Partner Activity 3 – Rally Coach
A. Find 3 fractions equivalent to
B. Find 3 fractions equivalent to
21
32
.......30
18
25
15
20
12
15
9
10
6
5
3
.......12
6
10
5
8
4
6
3
4
2
2
1
.......18
6
15
5
12
4
9
3
6
2
3
1
Fraction Families
CCSS-M SMARTER Items
Fraction Items