Unpacking the Middle School
Mathematics Curriculum
August 16, 2011
2009 Mathematics
Standards of Learning
 Rigor has been increased
 Repetition has been decreased
 Retention and application of content from previous
years required
 Vertical alignment has been improved
2
Enhanced Scope and Sequence
Lesson Plans
• Revised and redeveloped
• New layout
• Provides differentiation strategies for all types of
learners
• Anticipated by Summer 2011
3
Blueprints and Curriculum
 Blueprints are currently available on the VDOE
website to accommodate curriculum development
and instructional planning, but will not become
effective until the 2011-2012 school year
 School divisions should be teaching the new content
from the 2009 SOL in the 2010-2011 school year
since there will be FT items in spring 2011 on the
new content
-4-
New SOL Blueprints
Look for changes in:
 Number of reporting categories
 Number of items in reporting categories
 Asterisks denoting SOL that will be assessed in the
non-calculator section for grades 4 - 7
 SOL that will not be tested
5
6
Grades 6-8 Reporting Categories
Reporting Categories
6 (2001)
6 (2009)
7 (2001)
Number and Number Sense
8
10
7
Computation and Estimation
10
9
7
Measurement and Geometry
12
12
12
Probability and Statistics
Patterns, Functions, and Algebra
Total Questions
7
8
12
50
7 (2009)
16
13
12
19
50
12
50
8 (2001)
7
7
12
8 (2009)
14
14
8
21
50
16
50
2001
15%
16%
24%
2009
33%
26%
19%
22
50
27%
100%
41%
100%
Formula Sheets
 Formula sheets that correspond to the 2009
Standards for grades 6-8 and EOC are currently
available on the VDOE 2011-2012 Ancillary Test
Materials webpage
http://www.doe.virginia.gov/testing/test_administration/a
ncilliary_materials/2011-12/index.shtml
-8-
Click here for documents
Vertical Articulation Documents
9
Vertical Articulation of Content
Why is it important knowledge to have?
• Consistency
• Connections
• Relevance
All these lead to deeper understanding and
long-term retention of content
 The Mathematics Crosswalk Between the 2009 and
2001 Standards (PDF) provides detail on additions,
deletions and changes included in the 2009
Mathematics Standards of Learning.
10
Examine the 5-8 Vertical Articulation
 Identify the similarities and differences between the
grade levels
 What are the key verbs?
 Was there anything that surprised you?
Breaking Down the 6-8 Standards
 List the 5 most important concepts you see in
Grades 6 and 7
 Can you draw a representation of the topics?
11
Number and Number Sense &
Computation and Estimation
Grade 6
12
Number and Number Sense &
Computation and Estimation
Grade 7
13
Number and Number Sense &
Computation and Estimation
Grade 8
14
Measurement and Geometry
Grade 6
15
Measurement and Geometry
Grade 7
16
Measurement and Geometry
Grade 8
17
Probability, Statistics, Patterns,
Functions, and Algebra
Grade 6
18
Probability, Statistics, Patterns,
Functions, and Algebra
Grade 7
19
Probability, Statistics, Patterns,
Functions, and Algebra
Grade 8
20
Today’s Content Focus
1.
2.
3.
4.
5.
6.
Key changes at the middle school level:
Properties of Operations with Real Numbers
Equations and Expressions
Inequalities
Modeling Multiplication and Division of Fractions
Understanding Mean: Fair Share and Balance Point
Modeling Operations with Integers
21
Supporting Implementation of
2009 Standards
• Highlight key curriculum changes.
• Connect the mathematics across grade levels.
• Model instructional strategies.
22
Properties of Operations
23
Properties of Operations: 2001 Standards
7.3 The student will identify and apply the following properties of
operations with real numbers:
a) the commutative and associative properties for addition and
multiplication;
3.20a&b; 4.16b
b) the distributive property;
5.19
c) the additive and multiplicative identity properties;
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
6.19a
6.19c
6.19b
8.1 The student will
a) simplify numerical expressions involving positive exponents,
using rational numbers, order of operations, and properties of
operations with real numbers;
24
Properties of Operations: 2009 Standards
3.20
b) Identify examples of the identity and commutative properties for addition and
multiplication.
4.16b b) Investigate and describe the associative property for addition and multiplication.
5.19
6.19
7.16
8.1a
Investigate and recognize the distributive property of multiplication over addition.
Investigate and recognize
a) the identity properties for addition and multiplication;
b) the multiplicative property of zero; and
c) the inverse property for multiplication.
Apply the following properties of operations with real numbers:
a) the commutative and associative properties for addition and multiplication;
b) the distributive property;
c) the additive and multiplicative identity properties;
d) the additive and multiplicative inverse properties; and
e) the multiplicative property of zero.
a) simplify numerical expressions involving positive exponents, using rational numbers,
order of operations, and properties of operations with real numbers;
8.15c c) identify properties of operations used to solve an equation.
25
Meanings of Multiplication
For 5 x 4 = 20…
Repeated Addition: “4, 8, 12, 16, 20.”
Groups-Of: “Five bags of candy with four pieces of candy in each bag.”
Rectangular Array: “Five rows of desks with four desks in each row.”
Rate: “Dave bought five raffle tickets at $4.00 apiece.” or “Dave walked
four miles per hour for five hours.”
Comparison: “Alice has 4 cookies; Ralph has five times as many.”
Combinations: “Cindy has five different shirts and four different pairs of
pants; how many different shirt/pants outfits can she make?”
Area: “Ricky buys a rectangular rug 5 feet long and 4 feet wide.”
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998, Chapter 5.
26
Multiplication and Area (Grade 3)
Concept of multiplication
Connection to area
2x3
2 groups of 3
2
3
2x3=6
Area is 6 square units
27
Represent Multiplication Using an Area Model (SOL 3.6)
3 x 6 = 18
National Library of Virtual Manipulatives – Rectangle Multiplication
28
Represent Multiplication Using an Area Model (SOL 3.6)
Or does it look like this?
Rotating the rectangle doesn’t change its area.
Commutative
Property:
National Library of Virtual Manipulatives – Rectangle Multiplication
29
Associative Property for Multiplication (SOL 4.16b)
Use your base ten blocks to build a rectangular solid
2cm by 3cm by 4cm
Base: 3cm by 4cm; Height: 2cm
Volume: 2 x (3 x 4) = 24 cm3
Associative Property: The
Base: 2cm by 3cm; Height: 4cm grouping of the factors does
Volume: (2 x 3) x 4 = 24 cm3
not affect the product.
National Library of Virtual Manipulatives – Space Blocks
30
Multiplication and Area (Grade 3 and 4)
Multiplying whole numbers – progression of complexity
8
12
10
23
8 x 10
8 groups of 10
31
Multiplication and Area (Grade 3 and 4)
Multiplying whole numbers
23
20
23
12
3
23  6
10
12
2  20  40
10  3  30
2

“Partial Products”
32
10  20  200
276
Multiplication and Area
(Algebra I)
Connection to Algebra I
x
( x  3)( x  2)
3
2 36
x
2 x  2x
3 x  3x
2
33

This will work for more than
multiplying binomials! (unlike
FOIL). This model is directly
linked to use of algebra tiles.
x xx
2
x  5x  6
2
Multiplication and Area
(Algebra I/Geometry Application)
x
x
original
warehouse
3
The sides of a square
warehouse are increased by
2m and 3m as shown.
The area of the extended
warehouse is 156 m2.
2
What was the side length of
the original warehouse?
New Zealand Level 1 Algebra 1
Asia-Pacific Economic Cooperation – Mathematics Assessment Database
34
Multiplication and Area
(Algebra I/Geometry Application)
30
x
The original warehouse
measured 30m by 50m.
50
original
warehouse
The owner would like to know
the smallest length by which
she would need to extend
each side in order to have a
total area of 2500 m2.
x
New Zealand Level 1 Algebra 1 (modified)
Asia-Pacific Economic Cooperation – Mathematics Assessment Database
35
Strengths of the Area Model of Multiplication
Illustrates the inherent connections between multiplication
and division:
• Factors, divisors, and quotients are represented by the
lengths of the rectangle’s sides.
• Products and dividends are represented by the area of
the rectangle.
Versatile:
• Can be used with whole numbers and decimals (through
hundredths).
• Rotating the rectangle illustrates commutative property.
• Forms the basis for future modeling: distributive
property; factoring with Algebra Tiles; and Completing
the Square to solve quadratic equations.
36
Expressions and Equations
A Look At Expressions and Equations
A manipulative, like algebra tiles,
creates a concrete foundation for the
abstract, symbolic representations
students begin to wrestle with in
middle school.
38
What do these tiles represent?
1 unit
Area = 1 square unit
1 unit
Tile Bin
Unknown length, x units
Area = x square units
1 unit
x units
x units
Area = x2 square units
The red tiles denote negative quantities.
39
Modeling expressions
x+5
Tile Bin
5+x
40
Modeling expressions
x-1
Tile Bin
41
Modeling expressions
x+2
Tile Bin
2x
42
Modeling expressions
x2 + 3x + 2
Tile Bin
43
Simplifying expressions
x2 + x - 2x2 + 2x - 1
Tile Bin
zero pair
Simplified expression
-x2 + 3x - 1
44
Simplifying expressions
2(2x + 3)
Tile Bin
Simplified expression
4x + 6
45
Two methods of illustrating the Distributive
Property:
Example: 2(2x + 3)
46
Solving Equations
How does this concept progress as we move through middle school?
6th grade:
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
7th grade:
7.14 The student will
a)
solve one- and two-step linear equations in one variable; and
b)
solve practical problems requiring the solution of one- and two-step linear equations.
8th grade:
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the
equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
** What does this mean for Course 2 students who go to Algebra 1?
47
Solving Equations
Tile Bin
48
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
x+3=5
Tile Bin
49
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
x+3=5
x+3=5
̵3 ̵3
x+3=5
̵3 ̵3
x=2
x=2
50
Solving Equations
6.18 The student will solve one-step linear equations in one variable involving whole
number coefficients and positive rational solutions.
2x = 8
Tile Bin
51
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
3=x-1
Tile Bin
52
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
2x + 3 = 13
Tile Bin
53
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
2x + 3 = 13
2x + 3 = 13
̵3 ̵3
2x = 10
2
2
2x + 3 = 13
̵3 ̵3
2x = 10
2
2
x=5
x=5
54
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
0 = 4 – 2x
Tile Bin
55
Solving Equations
7.14
The student will solve one- and two-step linear equations in one variable;
and solve practical problems requiring the solution of one- and two-step linear equations.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
0 = 4 – 2x
0 = 4 – 2x
̵4 ̵4
-4 = -2x
2
2
0 = 4 – 2x
̵4 ̵4
-4 = -2x
-2 -2
2=x
-2 = -x
2=x
56
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
3x + 5 – x = 11
Tile Bin
57
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
Condensed Symbolic Representation:
3x + 5 – x = 11
2x + 5 = 11
2x + 5 = 11
-5 -5
2x = 6
2
2
3x + 5 – x = 11
2x + 5 = 11
-5 -5
2x = 6
2
2
x=3
x=3
58
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
x + 2 = 2(2x + 1)
Tile Bin
59
Solving Equations
8.15 The student will
a) solve multistep linear equations in one variable on one and two sides of the equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.
Pictorial Representation:
Symbolic Representation:
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
x + 2 = 4x + 2
-x
-x
Condensed Symbolic Representation:
x + 2 = 2(2x + 1)
x + 2 = 4x + 2
-x
-x
2 = 3x + 2
-2
-2
2 = 3x + 2
-2
-2
0 = 3x
3 3
0 = 3x
3 3
0=x
0=x
60
Inequalities
61
Inequalities
SOL 6.20
The student will graph inequalities on a number line.
SOL 7.15
The student will
a)
solve one-step inequalities in one variable; and
graph solutions to inequalities on the number line.
SOL 8.15
The student will
b)
solve two-step linear inequalities and graph the results
on a number line
62
Inequalities
What does inequality mean in the world of mathematics?
mathematical sentence comparing two unequal expressions
How are they used in everyday life?
to solve a problem or describe a relationship for which there
is more than one solution
63
Equations vs. Inequalities
x=2
x>2
How are they alike?
How are they different?
So, what about x > 2?
64
Equations vs. Inequalities
x=2
x>2
x>2
65
Open or Closed?
x > 16
-5 > y
m > 12
n < 341
-3 < j
and, which way should the ray go?
66
Equations vs. Inequalities
x+2=8
x+2<8
How are they alike?
How are they different?
So, what about x + 2 < 8?
67
Equations vs. Inequalities
x+2=8
x+2<8
How are they alike?
Both statements include the terms: x, 2 and 8
The solution set for both statements involves 6.
How are they different?
The solution set for x + 2 = 8 only includes 6. The solution set
for x + 2 < 8 does includes all real numbers less than 6.
What about x + 2 < 8?
The solution set for this inequality includes 6 and all real
numbers less than 6.
68
Equations vs. Inequalities
x+ 2 = 8
x+ 2 < 8
x+ 2 < 8
69
Inequality Match
Classroom activity: With your tablemates, find as many
matches as possible in the set of cards.
Tidewater Team: Inequality Match Cards
70
X >5
X is greater
than 5
SAMPLE MATCH
71
Modeling Multiplication and Division of
Fractions
72
So what’s new about fractions in
Grades 6-8?
SOL 6.4
The student will demonstrate multiple
representations of multiplication and division
of fractions.
73
Thinking About Multiplication
The
expression…
We read it…
It means…
It looks like…
23
2
1
2

1
3
1
3
74
Thinking About Multiplication
The
expression…
23
2
1
2

1
3
We read it…
It means…
2 times 3
two groups of
three
2 times
1
1
3
2
two groups of
one-third
1
3
times
It looks like…
1
3
one-half group
of one-third
75
Making sense of multiplication of
fractions using paper folding and area
models
 Enhanced Scope and Sequence, 2004, pages 22 -
24
76
The Importance of Context
• Builds meaning for operations
• Develops understanding of and helps illustrate the
relationships among operations
• Allows for a variety of approaches to solving a problem
77
Contexts for Modeling Multiplication of
Fractions
The Andersons had pizza for dinner, and there
was one-half of a pizza left over. Their three boys
each ate one-third of the leftovers for a late night
snack.
How much of the original pizza did each boy get
for snack?
78
1 1
1
 
3 2
6
One-third of one-half of a pizza is equal to one-sixth of a pizza.
Which
meaning of
multiplication
does this
model fit?
79
Another Context for
Multiplication of Fractions
 Mrs. Jones has 24 gold stickers that she
bought to put on perfect test papers. She
1
took 2 of the stickers out of the package,
1
and then she used 3 of that half on the
papers.
 What fraction of the 24 stickers did she
use on the perfect test papers?
80
1 1
1
 
3 2
6
1
One-third of one-half of the 24 stickers is 6 of the 24 stickers.
What
meaning(s) of
multiplication
does this
model fit?
Problems involving discrete items
may be represented with set models.
81
What’s the relationship between multiplying
and dividing?
• Multiplication and division are inverse relations
• One operation undoes the other
• Division by a number yields the same result as
multiplication by its reciprocal (inverse). For
example:
62  6
1
2
82
Meanings of Division
For 20 ÷ 5 = 4…
Divvy Up (Partitive): “Sally has 20 cookies. How many
cookies can she give to each of her five friends, if she gives
each friend the same number of cookies?
- Known number of groups, unknown group size
Measure Out (Quotitive): “Sally has 20 minutes left on
her cell phone plan this month. How many more 5-minute calls
can she make this month?
- Known group size, unknown number of groups
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998.
83
Sometimes, Always, Never?
• When we multiply, the product is larger than the number we
start with.
• When we divide, the quotient is smaller than the number we
start with.
84
“I thought times makes it bigger...”
When moving beyond whole numbers to situations involving
fractions and mixed numbers as factors, divisors, and
dividends, students can easily become confused. Helping
them match problems to everyday situations can help them
better understand what it means to multiply and divide with
fractions. However, repeated addition and array meanings of
multiplication, as well as a divvy up meaning of division, no
longer make as much sense as they did when describing
whole number operations.
Using a Groups-Of interpretation of multiplication and a
Measure Out interpretation of division can help:
Adapted from Baroody, Arthur J., Fostering Children’s Mathematical Power,
LEA Publishing, 1998.
85
“Groups of” and “Measure Out”
1/4 x 8: “I have one-fourth of a box of 8 doughnuts.”
8 x 1/4: “There are eight quarts of soda on the table. How many whole gallons of soda are
there?”
1/2 x 1/3: “The gas tank on my scooter holds 1/3 of a gallon of gas. If I have 1/2 a tank left, what
fraction of a gallon of gas do I have in my tank?”
1¼ x 4: “Red Bull comes in packs of four cans. If I have 1¼ packs of Red Bull, how many cans
do I have?”
3½ x 2½: “If a cross country race course is 2½ miles long, how many miles have I run after 3½
laps?
3/4 ÷ 2: “How much of a 2-hour movie can you watch in 3/4 of an hour?” *This type may be
easier to describe using divvy up.
2 ÷ 3/4: “How many 3/4-of-an-hour videos can you watch in 2 hours?”
3/4 ÷ 1/8: “How many 1/8-sized (of the original pie) pieces of pie can you serve from 3/4 of a
pie?”
2½ ÷ 1/3: “A brownie recipe calls for 1/3 of a cup of oil per batch. How many batches can you
make if you have 2½ cups of oil left?”
86
Thinking About Division
The
expression…
We read it…
It means…
It looks like…
20 ÷ 5
20 
1
2
87
Thinking About Division
The
expression…
20 ÷ 5
We read it…
20 divided
by 5
It means…
It looks like…
20 divided into
groups of 5;
20 divided into 5
equal groups…
How many 5’s are
in 20?
20 
1
2
20 divided
by 1
2
20 divided into
groups of 1 …
2
How many 1 ’s are
2
in 20?
88
88
Thinking About Division
The expression…
1
2

1
3
We read it…
one-half
divided by
one-third
It means…
It looks like…
1
divided into
2
groups of 1 …
3
?
How many 1 ’s are
3
1
in 2 ?
Is the quotient more than one
or less than one? How do you
know?
89
Contexts for
Division of Fractions
The Andersons had half of a pizza left after
1
dinner. Their son’s typical serving size is 3
pizza. How many of these servings will he eat
if he finishes the pizza?
90
1 1
1
 1
2 3
2
1
2
pizza divided into
1
3
1
pizza servings = 1 2 servings
1 serving
1
2
serving
91
Another Context for
Division of Fractions
1
Marcy is baking brownies. Her recipe calls for 3 cup
cocoa for each batch of brownies. Once she gets
1
started, Marcy realizes she only has 2 cup cocoa. If
Marcy uses all of the cocoa, how many batches of
brownies can she bake?
92
1 1
1
 1
2 3
2
1 cup
Three batches (or
Two batches (or
1
2
cup
3
cup)
3
2
3
cup)
1
1 2 batches
One batch (or 13 cup)
0 cups
93
Another Context for
Division of Fractions
1
Mrs. Smith had 2 of a sheet cake left over after
her party. She decides to divide the rest of the
1
cake into portions that equal 3 of the original
cake.
1
How many 3 cake portions can Mrs. Smith make
from her left-over cake?
94
What could it look like?
1
2

1
3
95
What does it look like
numerically?
96
What is the role of common
denominators in dividing fractions?
• Ensures division of the same size units
• Assist with the description of parts of
the whole
97
What about
the traditional algorithm?
• If the traditional “invert and multiply” algorithm is taught,
it is important that students have the opportunity to
consider why it works.
• Representations of a pictorial nature provide a visual for
finding the reciprocal amount in a given situation.
• The common denominator method is a different, valid
algorithm. Again, it is important that students have the
opportunity to consider why it works.
98
What about
the traditional algorithm?
Build understanding:
1
Think about 20 ÷ 2 .
How many one-half’s are in 20?
How many one-half’s are in each of the 20 individual wholes?
Experiences with fraction divisors having a numerator of one
illustrate the fact that within each unit, the divisor can be taken
out the reciprocal number of times.
99
What about
the traditional algorithm?
Later, think about divisors with numerators > 1.
Think about 1 ÷
2
3
.
2
How many times could we take
3
from 1?
1
We can take it out once, and we’d have 3 left. We could only take half
2
of another
3
from the remaining portion. That’s a total of
In each unit, there are
3
2
sets of
2
3
.
100
3
2
.
Multiple Representations
Instructional programs from pre-k through grade 12
should enable all students to –
• Create and use representations to organize,
record and communicate mathematical ideas;
• Select, apply, and translate among
mathematical representations to solve
problems;
• Use representations to model and interpret
physical, social, and mathematical phenomena.
from Principles and Standards for School Mathematics (NCTM, 2000), p. 67.
101
Using multiple representations to
express understanding
Given problem
Contextual situation
Check your solution
Solve numerically
Solve graphically
102
Using multiple
representations
to express
understanding
of division of
fractions
10
Mean:
Fair Share and Balance Point
104
Mean: Fair Share
2009 5.16: The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data;
and
d) describe the range of a set of data as a measure of
variation.
Understanding the Standard: “Mean represents a fair share
concept of the data. Dividing the data constitutes a fair share.
This is done by equally dividing the data points. This should be
demonstrated visually and with manipulatives.”
105
Understanding the Mean
Each person at the table should:
1.
2.
3.
Grab one handful of snap cubes.
Count them and write the number on a sticky note.
Snap the cubes together to form a train.
106
Understanding the Mean
Work together at your table to answer
the following question:
If you redistributed all of the cubes
from your handfuls so that everyone
had the same amount (so that they
were “shared fairly”), how many
cubes would each person receive?
107
Understanding the Mean
What was your answer?
• How did you handle “leftovers”?
• Add up all of the numbers from the
original handfuls and divide the sum by
the number of people at the table.
• Did you get the same result?
• What does your answer represent?
108
Understanding the Mean
Take your sticky note and place it on the wall, so they are
ordered…
•
•
Horizontally: Low to high, left to right; leave one space
if there is a missing number.
Vertically: If your number is already on the wall, place
your sticky note in the next open space above that
number.
109
Understanding the Mean
How did we display our data?
2009 3.17c
110
Understanding the Mean
Looking at our line plot, how can we
describe our data set? How can we
use our line plot to:
- Find the range?
- Find the mode?
- Find the median?
- Find the mean?
111
Mean: Balance Point
2009 6.15: The student will
a) describe mean as balance point; and
b) decide which measure of center is appropriate
for a given purpose.
Understanding the Standard: “Mean can be defined as the point
on a number line where the data distribution is balanced. This
means that the sum of the distances from the mean of all
the points above the mean is equal to the sum of the distances of
all the data points below the mean.”
Essential Knowledge & Skills:
• Identify and draw a number line that demonstrates the
concept of mean as balance point for a set of data.
112
Where is the balance point for this
data set?
X
X
X
X
X X
113
Where is the balance point for this
data set?
X
X
X
X
X
X
114
Where is the balance point for this
data set?
X
X
X
X
X X
115
Where is the balance point for this
data set?
X
X
X
X
X
X
116
Where is the balance point for this
data set?
3 is the
Balance Point
X
X
X
X
X
X
117
Where is the balance point for this
data set?
X
X
X
X
X X
118
Where is the balance point for this
data set?
Move 2 Steps
Move 2 Steps
Move 2 Steps
Move 2 Steps
4 is the Balance Point
119
We can confirm this by calculating:
2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36
36 ÷ 9 = 4
The Mean is the Balance Point
120
Where is the balance point for this
data set?
If we could “zoom in” on the
Move 1 Step
The Balance Point is
between 10 and 11
Move 2 Steps
(closer to 10).
space between 10 and 11, we
could continue this process to
arrive at a decimal value for the
balance point.
Move 2 Steps
Move 1 Step
121
Mean: Balance Point
When demonstrating finding the balance point:
1. CHOOSE YOUR DEMONSTRATION DATA SETS
INTENTIONALLY.
2. Use a line plot to represent the data set.
3. Begin with the extreme data points.
4. Balance the moves, moving one data point from each
side an equal number of steps toward the center.
5. Continue until the data is distributed symmetrically or until
there are only two values left on the line plot.
122
Assessing Higher-Level Thinking
Key Points for 2009 5.16 & 6.15:
Students still need to be able to calculate the mean by
summing up and dividing, but they also need to
understand:
•
•
•
why it’s calculated this way (“fair share”);
how the mean compares to the median and the mode
for describing the center of a data set; and
when each measure of center might be used to
represent a data set.
123
Mean:
Fair Share & Balance Point
“Students need to understand that the mean ‘evens out’ or
‘balances’ a set of data and that the median identifies the
‘middle’ of a data set. They should compare the utility of the
mean and the median as measures of center for different
data sets. …students often fail to apprehend many subtle
aspects of the mean as a measure of center. Thus, the
teacher has an important role in providing experiences that
help students construct a solid understanding of the mean
and its relation to other measures of center.”
- NCTM Principles & Standards for School Mathematics, p. 250
124
Operations with Integers
125
Operations with Integers
2009 7.3a: The student will
a) model addition, subtraction, multiplication and
division of integers; and
b) add, subtract, multiply, and divide integers.
Is this really a “new” SOL?
2001 7.5: The student will formulate rules for and solve
practical problems involving basic operations (addition,
subtraction, multiplication, and division) with integers.
“Model”
126
Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
= -1
=1
What operation does this model?
3 + (-7) = -4
127
Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
=1
= -1
3 • (-4) = -12
What operation does this model?
128
Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
5 5+ -(-17)
17 = =-12
-12
What operation does this model?
129
Assessing Higher-Level Thinking
7.3a: The student will model addition, subtraction,
multiplication, and division of integers.
3 • (-5) = -15
What operation does this model?
130
Another Example of Assessing HigherLevel Thinking
7.5c: The student will describe how changing one
measured attribute of a rectangular prism affects its
volume and surface area.
Describe how the volume of
the rectangular prism
shown (height = 8 in.) would
be affected if the height was
increased by a scale factor
of ½ or 2.
8 in.
3 in.
5 in.
131
Tying it All Together
1.
2.
3.
4.
Improved vertical alignment of content
with increased cognitive demand.
Key conceptual models can be extended
across grade levels.
Refer to the Curriculum Framework.
Pay attention to the changes in the
verbs.
132
Narrowing Achievement Gaps
• Ask high-level questions of all students
• Consistently provide multiple
representations
• Facilitate connections
• Solicit multiple student solutions
• Engage students in the learning process
133
Narrowing Achievement Gaps
• Promote mathematical communication
• Listen carefully to your students’ words
and learn from them
• Provide “immediate” feedback on all work
• Give students challenging but accessible
tasks
134
No Pain, No Gain
• Pertains to learning mathematics as well
• Let kids struggle to make sense of the mathematics
135
VDOE Resources
• Technical Assistance Documents for SOL A.9 and
SOL AII.11
• Mathematics Institutes: Training/instructional
resources for K-Algebra II available through the
Tidewater Team at William and Mary Website
136