Download Algebraic Relationships

ALGEBRAIC RELATIONSHIPS
Kindergarten
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize and
extend
patterns
EXPECTATION
*Recognize or
repeat sequences of
sounds or shapes
EXAMPLE
Activity:


The teacher will demonstrate a pattern (e.g., triangle, triangle, square) of
shapes using pattern blocks. The students will then continue the pattern
independently using pattern blocks.
The teacher will demonstrate a sound pattern (e.g., clap, clap, stomp).
Students will then join the teacher, following the same sound pattern.
TEACHER NOTES:
Other ways for students to recognize and extend patterns are through repetitive
songs, rhythmic chants, and predictive poems that are based on repeating and
growing patterns.
B
Create and
analyze
patterns
*Create and
continue patterns
Activity:
Have students use pattern blocks to create their own patterns, which must be
repeated three times to be evident as a pattern.
ALGEBRAIC RELATIONSHIPS—Kindergarten
1
Activity:
Each month,different shapes to represent holidays or special days can be used for
students to form patterns.
Examples:

August—school house, book


September—colored leaves


October—bat, spider, owl


November—turkey, pumpkin,
pilgrim
December—evergreen, star,
deer



January—snowflake, snowman,
bell
February—heart, Lincoln,
Washington
March—shamrock-turned in
various positions
April—rabbit, egg, raindrop,
umbrella
May—dog, cat, flower
Activity note: Depending on the ability level of each student, you may add or delete
shapes. You may create a monthly book of patterns for each child to bind and take
home at the end of the school year.
ALGEBRAIC RELATIONSHIPS—Kindergarten
2
CONCEPT
C
EXPECTATION
EXAMPLE
Classify objects
and
representations
ALGEBRAIC RELATIONSHIPS—Kindergarten
3
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model situations
that involve whole
numbers, using
pictures, objects or
symbols
EXAMPLE
Activity:
“I want to know if we have enough snacks for everyone who is here today. We have
18 cookies. Will that be enough for everyone to get one cookie? When you have
figured out if we have enough or not, be sure you can show me how you know.”
Activity:
Give students pictures of dogs in which to draw the number of spots you specify.
For example, “Draw five spots on the first dog. Draw eight spots on the second dog.”
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols. 2
2
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Kindergarten
4
CONCEPT
EXPECTATION
EXAMPLE
Activity:
There are two dogs and one bird. The dogs have four legs, and the birds have two
legs. How many legs are there altogether?
Activity note: Several students may draw pictures and tallies to solve the problem,
while some may actually add the numbers.
TEACHER NOTES:
“Students should learn to make models to represent and solve problems.”1
1
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Kindergarten
5
ALGEBRAIC RELATIONSHIPS
Grade 1
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize and
extend
patterns
EXPECTATION
*Extend patterns of
sound, shape,
motion or a simple
numeric pattern
EXAMPLE
Activity:
The teacher will demonstrate a pattern (e.g., triangle, triangle, square, rhombus) of
shapes using pattern blocks. The students will continue the pattern independently
using pattern blocks.
Activity:
The teacher will demonstrate a sound pattern (e.g., clap, clap, stomp). Students will
then join the teacher, following the same sound pattern.
Activity:
The teacher will introduce skip counting using even and odd numbers. Students
should complete the next three numbers for each numeric pattern. For example, if
the teacher says, “2, 4, 6,...,” students will complete by saying, “8, 10, 12.”
Problem:
Draw the arrows in the last three boxes.
ALGEBRAIC RELATIONSHIPS – Grade 1
6
Answer:
TEACHER NOTES:
Although skip counting is a great way for students to identify and extend patterns,
there are other ways, too, in which they can recognize and extend patterns, e.g.,
through repetitive songs, rhythmic chants, and predictive poems that are based on
repeating (123, 123, 123,…) or growing patterns (1, 4, 7, 10,... ).
ALGEBRAIC RELATIONSHIPS – Grade 1
7
B
CONCEPT
EXPECTATION
Create and
analyze
patterns
*Describe how
simple repeating
patterns are
generated
EXAMPLE
Activity:
Have students communicate what happens in a repeating pattern. Complete the
next three shapes in the pattern below.
What shape follows the triangle? How do you know?
Problem:
Describe the pattern below, and tell what color the next box will be.
Answer:
The pattern is white, shaded, shaded. The next box will be white.
DEFINITION:
repeating patterns—patterns that are cyclical in nature, with each cycle repeating elements in the same order.
Example: ABCABCABC.1
1
National Council of Teachers of Mathematics (2001). Navigating through geometry in prekindergarten–grade 2 (p.7). Reston, VA
ALGEBRAIC RELATIONSHIPS – Grade 1
8
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Write the next two numbers in the pattern.
1, 2, 4, 1, 2, 4, 1, 2, 4, 1, 2, ___, ___
Answer:
4, 1
TEACHER NOTES:
Students should transfer their knowledge of repeating patterns to all patterns
(numbers, words, colors, designs, etc.).
ALGEBRAIC RELATIONSHIPS – Grade 1
9
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
*using addition or
subtraction,
represent a
mathematical
situation as an
expression or
number sentence
EXAMPLE
Problem:
There were nine children playing on the swings. Then eight more children came to
play on the swings. How many children were playing on the swings? Write a number
sentence to help you solve the problem.
Answer:
9 + 8 = 17
or




9
+




8
=

 



 
17
TEACHER NOTES:
Throughout first grade, students should be encouraged to begin to use
mathematical symbols to represent mathematical situations.
“Through classroom discussions of different representations during the pre-K–2
years, students should develop an increased ability to use the symbols as a means
of recording their thinking. In the earliest years, teachers may provide scaffolding for
students by modeling for them until they have the ability to record their ideas.
Original representations remain important throughout the students’ mathematical
study and should be encouraged. Symbolic representations and manipulation
should be embedded in instructional experiences as another vehicle for
understanding and making sense of mathematics.
ALGEBRAIC RELATIONSHIPS – Grade 1
10
DEFINITION:
expression—a mathematical phrase that represents a number through the combination of operation symbols, numbers, and/or
symbols. Examples: 2 + 3; 5 – 4.2
CONCEPT
EXPECTATION
EXAMPLE
Equality is an important algebraic concept that students must encounter and begin
to understand in the lower grades. A common explanation of the equals sign given
by students is that ‘the answer is coming,’ but they need to recognize that the
equals sign indicates a relationship—that quantities on each side are equivalent, for
example, 10 = 4 + 6 or 4 + 6 = 5 + 5.”3
2
3
Math at hand: A mathematics handbook (p.523). (1999). Wilmington, MA: Great Source Education Group
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.94). Reston, VA
ALGEBRAIC RELATIONSHIPS – Grade 1
11
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
*Model situations
that involve addition
and subtraction of
whole numbers,
using pictures,
objects or symbols
EXAMPLE
Problem:
John has one dog and one duck. How many legs do the two animals have together?
Show how you know your answer is correct.
Answer:
4+2=6
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols.4
4
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS – Grade 1
12
CONCEPT
EXPECTATION
EXAMPLE
Problem:
There are six chairs. How many legs are there altogether?
Answer:
Some students will draw all six chairs and count the legs, while others will make six
sets of tally marks. Still others will do mental math, visualizing the chairs and
counting.
TEACHER NOTES:
“Students should learn to make models to represent and solve problems.”5
5
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS – Grade 1
13
ALGEBRAIC RELATIONSHIPS
Grade 2
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize and
extend
patterns
EXPECTATION
*Describe and
extend simple
numeric patterns
and change from
one representation
to another
EXAMPLE
Problem:
Look at the three patterns below and tell how they are alike and different.
1.
2.
A
B
B
A
B
B
A
B
B
A
B
B
3.
Clap
Snap
Snap
Clap
Snap
Snap
Clap
Snap
Snap
Clap
Snap
Snap
Answer:
They are alike because they are all ABBABBABBABB forms. They are different
because one is made of shapes, one is made of letters, and one is made of sounds.
ALGEBRAIC RELATIONSHIPS—Grade 2
14
CONCEPT
EXPECTATION
EXAMPLE
TEACHER NOTES:
Teachers should help students develop the ability to form generalizations by asking
students questions such as “How could you describe this pattern?”, “How can it be
repeated or extended?” or “How are these patterns alike?” For example, students
should recognize that the color pattern “blue, blue, red, blue, blue, red, blue, blue,
red” is the same in form as “clap, clap, step, clap, clap, step, clap, clap, step.” This
recognition lays the foundation for the idea that two very different situations can
have the same mathematical features and are the same in some important ways.
Knowing that each pattern could be described as having the form AABAABAAB is
an early introduction to the power of algebra for students.1 Teachers can explore
functions with students by pairing counting numbers with a repeating pattern, as in
the figure below.
1
2
3
4
5
6
7
?
8
Help students develop effective questioning strategies to form connections about
patterns. How could this pattern be extended? How is this pattern similar to other
patterns? Help students make generalizations about various patterns. Example:
Red, red, green in a paper chain has the same repeating pattern as clap, clap, slap
in a listening pattern. Both can be identified as an AAB pattern. Transfer this
knowledge of patterns to poetry.
1
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.91–92). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 2
15
CONCEPT
B
Create and
analyze
patterns
EXPECTATION
*Describe how
simple growing
patterns are
generated
EXAMPLE
Problem:
How many squares would be in position 5?
?
1
2
3
4
5
Answer:
Position 5 would have six squares. The number of squares is always one more than the
position.
Problem:
What is the next number in the sequence?
3, 6, 9, ____
Answer:
12
DEFINITIONS:
growing patterns—patterns that show an arithmetic change between pairs of elements in the pattern. For example, growing
patterns may show numbers in decreasing order or buildings in decreasing size. Example: 3, 5, 8, 12,…2
2
Greenes, C., Cavanagh, M., Dacey, L., Findell, C. & Small, M. (2001). Navigating through algebra in prekindergarten–grade 2 (p.4). Reston, VA:
National Council of Teachers of Mathematics
ALGEBRAIC RELATIONSHIPS—Grade 2
16
CONCEPT
EXPECTATION
EXAMPLE
Problem:
1.
Insert the number of feet the ducks in each row have.
1 duck = 2 feet
2.
ALGEBRAIC RELATIONSHIPS—Grade 2
2 ducks =
feet
3 ducks =
feet
4 ducks =
feet
5 ducks =
feet
How many ducks are there in all? How many feet do they have in all?
17
___ ducks = ___ feet
CONCEPT
EXPECTATION
EXAMPLE
Answer:
1.
2 ducks = 4 feet; 3 ducks = 6 feet; 4 ducks = 8 feet; 5 ducks = 10 feet
2.
Total: 15 ducks = 30 feet
Problem:
Find the pattern in the following T-chart to fill in the missing numbers. Write about
the pattern.
Number of ducks
Number of feet
1
2
2
4
3
4
5
ALGEBRAIC RELATIONSHIPS—Grade 2
18
Answer:
Number of ducks
Number of feet
1
2
2
4
3
6
4
8
5
10
The pattern: add two more feet than the duck before.
ALGEBRAIC RELATIONSHIPS—Grade 2
19
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
EXAMPLE
*
Problem:
*using addition or
subtraction,
represent a
mathematical
situation as an
expression or a
number sentence
Sue had 16 pieces of candy. After she gave Ellen some candy, she had 12 pieces
left. Use numbers and symbols to write a problem about Sue’s candy.
Answer:
12 = 16 – ___
or
16 - ____ = 12
or
ALGEBRAIC RELATIONSHIPS—Grade 2
20
16
-
Candy
given to
Ellen
=
-
_____
=
12
DEFINITIONS:
expression—a mathematical phrase that represents a number through the combination of operation symbols, numbers, and/or
symbols. Examples: 2 + 5; 4 – 2.4
4
Math at hand: A mathematics handbook (p.523). (1999). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 2
21
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Bob had 12 goldfish. For his birthday, he got 5 more. How many goldfish does he
have now?
Answer:
+
12
+
=
5
=
17
12 + 5 = 17
Problem:
Sam was playing marbles with his friends. He started with 16 marbles. At the end of
the game, he had 48 marbles. How many marbles did Sam win? Write a number
sentence to help you solve the problem.
Answer:
48 – 16 = 32
Problem:
Bo and his friends went on rides at the amusement park. There were 14 riders on
the first ride, 18 riders on the second ride, 22 riders on the third ride, and 26 riders
on the fourth ride. If this pattern continues, how many riders will be on the 10th ride?
ALGEBRAIC RELATIONSHIPS—Grade 2
22
CONCEPT
EXPECTATION
EXAMPLE
Answer:
50 riders
or
1
14
2
18
3
22
4
26
5
30
6
34
7
38
8
42
9
46
10
50
TEACHER NOTES:
Throughout second grade, students should be encouraged to begin to use
mathematical symbols to represent mathematical situations.
“Through classroom discussions of different representations during the pre-K–2
years, students should develop an increased ability to use the symbols as a means
of recording their thinking. In the earliest years, teachers may provide scaffolding for
students by writing for them until they have the ability to record their ideas. Original
representations remain important throughout the students’ mathematical study and
should be encouraged. Symbolic representations and manipulation should be
embedded in instructional experiences as another vehicle for understanding and
making sense of mathematics. Equality is an important algebraic concept that
students must encounter and begin to understand in the lower grades. A common
explanation of the equals sign given by students is that ‘the answer is coming,’ but
they need to recognize that the equals sign indicates a relationship—that quantities
on each side are equivalent, for example, 10 = 4 + 6 or 4 + 6 = 5 + 5.”5
5
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.159–160). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 2
23
CONCEPT
B
Describe and
use
mathematical
manipulation
EXPECTATION
EXAMPLE
*solve problems
with whole numbers
using the
commutative and
associative
properties of
addition
Activity:
Roll a die several times. Have the students record the numbers rolled. Ask students
to find the sum of the numbers rolled. Ask if the sum will be different if the numbers
are added in a different order.
Activity:
Give students problems such as the following:
2 + 3 = ____
3 + 2 = ____ 23 + 24 = ____ 24 + 23 = ____
18 + 17 = ____ 17 + 18 = ____
5 + 3 = ____
3 + 5 = ____
Ask the students if they notice any similarities in the answers to the problems.
Discuss the commutative property of addition.
DEFINITION:
commutative property of addition: the sum stays the same when the order of the addends is changed. Example: 6 + 4 = 10; 4
+ 6 = 10; 6
of multiplication: Example: 3 x 2 = 6; 2 x 3 = 6
ALGEBRAIC RELATIONSHIPS—Grade 2
24
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Fill in the table below using five counters (manipulatives) and a handout with two
shapes. Place the counters on the shapes to find the missing numbers.
Number of counters
on shape 1
Number of counters
on shape 2
0
5
1
3
3
2
1
5
Answer:
ALGEBRAIC RELATIONSHIPS—Grade 2
Number of counters
on shape 1
Number of counters
on shape 2
0
5
1
4
2
3
3
2
4
1
5
0
25
CONCEPT
EXPECTATION
EXAMPLE
TEACHER NOTES:
Opportunities to explore the commutative property of addition will often
arise when students are allowed to use their own strategies for adding several twodigit numbers. When they have three or more numbers, they will also use the
associative property of addition.
Although it is not necessary to introduce specific vocabulary, teachers should be
aware of algebraic properties used by students at this age.
DEFINITIONS:
associative property of addition —The sum stays the same when the grouping of three or more addends is changed.
Example: (7 + 4) + 3 = 7 + (4 + 3).7
commutative property of addition —The sum stays the same when the order of the addends is changed.
Example: 6 + 4 = 4 + 6.8
7
Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group
at hand: A mathematics handbook (p.519). (1999). Wilmington, MA: Great Source Education Group
8 Math
ALGEBRAIC RELATIONSHIPS—Grade 2
26
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
EXAMPLE
*Model situations
that involve addition
and subtraction of
whole numbers,
using pictures,
objects or symbols
Problem:
Rex had 12 pieces of candy. During the day, he ate some of the candy. By the end
of the day, he had 4 pieces of candy left. How many pieces of candy did he eat that
day? Show how you know your answer is correct.
Answer:
12 – 4 = 8
-
12
-
=
4
=
8
Problem:
There were eight bears eating at the zoo. Then nine more bears came to eat. How
many bears were eating at the zoo? Show how you know your answer is correct.
Answer:
ALGEBRAIC RELATIONSHIPS—Grade 2
27
8 + 9 = 17
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols.10
CONCEPT
EXPECTATION
EXAMPLE
Problem:
John is playing with his toys. Each action figure has two legs, and each animal has
four legs. Altogether, there are 18 legs. How many action figures and animals does
John have?
Answer:
Students may represent the situation in different ways.






10
some may draw a picture representing the number of legs; or
others may represent the situation using symbols, making a first guess, then
adjusting the number of action figures and animals so that the sum of the legs
is 18; or
2 + 4 + 4 + 4 + 4; or
2 + 2 + 2 + 4 + 4 + 4; or
2 + 2 + 2 + 2 + 2 + 4 + 4; or
2+2+2+2+2+2+2+4
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 2
28
TEACHER NOTES:
“Students should learn to make models to represent and solve problems.”9
9
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 2
29
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
*Describe
qualitative
change, such as
students growing
taller
EXAMPLE
Problem:
Two students weighed their dogs. Bill’s dog weighed 23 pounds, and Becky’s dog weighed
25 pounds. Without using numbers, write a sentence to compare the weight of Bill and
Becky’s dogs.
Answer:
Bill’s dog weighs more than Becky’s dog; or, Becky’s dog weighs less than Bill’s
dog.
DEFINITION:
qualitative change—a change (in the quality of something) that can be described by words such as taller, shorter, darker,
lighter, warmer, etc.11
11
Greenes, C., Cavanagh, M., Dacey, L., Findell, C. & Small, M. (2001). Navigating through algebra in prekindergarten–grade 2 (p.4). Reston,
VA: National Council of Teachers of Mathematics
ALGEBRAIC RELATIONSHIPS—Grade 2
30
CONCEPT
EXPECTATION
EXAMPLE
TEACHER NOTES:
During a discussion of data representations, ask students to describe the data
without using numbers. This should bring out students’ use of comparative words,
such as lighter, thinner, etc.
“From a very early age, children recognize examples of change in their environment
and describe change in qualitative terms, such as getting taller, colder, darker, or
heavier. By measuring and comparing quantities, such as when keeping track of
variations in temperature or growth of a classroom plant or pet, children also learn to
describe change quantitatively.“12
“In prekindergarten through grade 2, students can, at first, describe qualitative
change (‘I grew taller over the summer’) and then quantitative change (‘I grew two
inches in the last year’).”13
12
Greenes, C., Cavanagh, M., Dacey, L., Findell, C. & Small, M. (2001). Navigating through algebra in prekindergarten–grade 2 (p.4). Reston,
VA: National Council of Teachers of Mathematics
13 National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.40). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 2
31
ALGEBRAIC RELATIONSHIPS
Grade 3
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize
and extend
patterns
EXPECTATION
Extend geometric
(shapes) and
numeric patterns to
find the next term
EXAMPLE
Problem:
Draw the next figure in the pattern below:
           ____
Answer:

Problem:
What number would come next in the following pattern?
ALGEBRAIC RELATIONSHIPS—Grade 3
32
2, 2, 3, 3, 4, 4, ____
Answer:
5
TEACHER NOTES:
Although algebra is not a commonly heard word in the 3–5 classroom, the
mathematical investigations and conversations of students in these grades
ALGEBRAIC RELATIONSHIPS—Grade 3
33
CONCEPT
EXPECTATION
EXAMPLE
frequently include elements of algebraic reasoning. These experiences not only
provide rich contexts for advancing mathematical understanding, they also serve as
an important precursor to the more formalized study of algebra in the middle and
secondary grades. Algebraic ideas should emerge and be investigated in grades 3–
5 as students
 identify or build numerical and geometric patterns
 describe patterns verbally and represent them with tables or symbols
 look for and apply relationships between varying quantities to make
predictions
 make and explain generalizations that seem to always work in a particular
situation
 use graphs to describe patterns and make predictions
 explore number properties
 use invented notation, standard symbols, and variables to express a
pattern, generalization, or situation”1
1
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.159–160). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 3
34
CONCEPT
B
Create and
analyze
patterns
EXPECTATION
Represent patterns
using words, tables
or graphs
EXAMPLE
Problem:
Lori has discovered a new game at the amusement park. If she puts in 1 token, she receives
three candies; if she puts in 2 tokens, she receives six candies; if she puts in 3 tokens, she
receives nine candies. Complete the table below to determine how many candies she
would receive for each token amount, then make a bar graph to show the information on
the table.
Tokens
Candies
1
3
2
6
3
9
4
5
ALGEBRAIC RELATIONSHIPS—Grade 3
35
CONCEPT
EXPECTATION
EXAMPLE
Answer:
For 4 tokens she would get twelve candies, and for 5 tokens, she would get fifteen
candies.
Candy Received Per Token
15
Candies out
12
9
6
3
0
1
ALGEBRAIC RELATIONSHIPS—Grade 3
2
3
Tokens in
4
5
36
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Find the missing numbers in the table below that shows the number of horses and
riders.
Horses
1
2
3
Riders
2
4
6
4
5
Answer:
8, 10
ALGEBRAIC RELATIONSHIPS—Grade 3
37
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
Using all operation,
represent a
mathematical
situation as an
expression or
number sentence
EXAMPLE
Problem:
Write a number sentence for “what number added to five equals eight?”.
Answer: : ___ + 5= 8; 5 + ___ = 8
Problem:
Write twenty-one plus two as an expression.
Answer:
21 + 2
DEFINITIONS:
expression—a mathematical phrase that represents a number through the combination of operation symbols, numbers, and/or
symbols. Examples: 2 x 60; 3 + .2
number sentence—equations for comparisons. Examples: 3 + 4 = 7; 8 – 2 = 6; 7 > 6.3
CONCEPT
2
3
EXPECTATION
EXAMPLE
Math at hand: A mathematics handbook (p.523). (1999). Wilmington, MA: Great Source Education Group
Cavanagh, M. (2002). Math to learn (p.457). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 3
38
Problem:
Use numbers to write twelve minus three equals nine.
Answer:
12 – 3 = 9
Problem:
Twelve students were playing soccer. At the end of the game, nine players were left. Write
a number sentence using the soccer players.
Answer:
12 – 9 = 3
TEACHER NOTES:
Expressions and number sentences can both contain a combination of variables,
numbers, and operation symbols. An expression represents a mathematical
relationship. A number sentence or equation contains an equal sign indicating that
the amount on one side of the equal sign has the same value as the amount on the
other side.
CONCEPT
EXPECTATION
ALGEBRAIC RELATIONSHIPS—Grade 3
EXAMPLE
39
B
Describe and
use
mathematical
manipulation
Use the
commutative,
distributive, and
associative
properties for basic
facts of whole
numbers
Problem:
Which of the following number sentences gives the same answer as 3 + 5?
A. 3  5
B. 5  3
C. 5 + 3
D. 5 – 3
Answer: C
Problem:
Fill in the blank to show the commutative property of addition.
8 + 4 = ____ + 8
Answer: 4
Problem:
Which of the following number sentences is equivalent to 6x(3+2)?
A. 18+2
B. (6x3)+(6x2)
C. (6x3)x(6x2)
D.
(6+3)+(6+2)
Answer: B
ALGEBRAIC RELATIONSHIPS—Grade 3
40
TEACHER NOTES:
The use of the commutative property of addition will often arise when students are
given the opportunity to use their own strategies for adding several numbers. Once
they have more than two numbers, they are also using the associative property of
addition (sum stays the same even if the order of two or more addends is changed).
At this level, teachers should use the terms “commutative” and “identity” properties.
It is not expected that all students will use these terms.
Problem:
Which of the following is another way to write (5 + 6) x 12?
1. 5  6  12
2. (5  12) + (6  12)
3. 5 + 6 + 12
Answer:
2
ALGEBRAIC RELATIONSHIPS—Grade 3
41
Problem:
Use the grouping property to write 14  (3  5) another way.
Answer:
(14  3)  5
DEFINITIONS:
commutative property of addition—the sum stays the same when the order of the addends is changed.
Example: 6 + 4 = 4 + 6.4
distributive property – when one of the facors of a product is written as a sum, multiplying each addend before adding does not change the
product. Example: 3x(5+4) = (3x5)+(3x4). 5
associative property of addition – the sum stays the same when the grouping of the addends is c hanged. Example: (5+4)+6 = 5+(4+6)6
4
Math at hand: A mathematics handbook (p.519). (1999). Wilmington, MA: Great Source Education Group
at hand: A mathematics handbook (p. 522). (1999). Wilmington, MA: Great Source Education Group
6Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group
5Math
ALGEBRAIC RELATIONSHIPS—Grade 3
42
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model problem
situations, including
multiplication with
objects or drawings
EXAMPLE
Problem:
Model the sets below with counters, then write a multiplication sentence for each.
1. 5 groups of 2
2. 3 groups of 6
Answer:
5  2 = 10
1.     
2.   
3  6 = 18
Problem:
Write a multiplication sentence for the following picture:
ALGEBRAIC RELATIONSHIPS—Grade 3
43
Answer:
3  6 = 18
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols.5
5
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 3
44
CONCEPT
EXPECTATION
EXAMPLE
TEACHER NOTES:
Students in grades 3–5 can model a variety of situations, including geometric
patterns, real-world situations, and scientific experiments. Sometimes they will use
their model to predict the next element in a pattern. At other times, they may make a
general statement about how one variable is related to another variable: for
example, if a sandwich costs $3, you can figure out how much any number of
sandwiches costs by multiplying that number by 3. In modeling situations that
involve real-world data, students need to know that their predictions may not always
match the observed outcomes for a variety of different reasons. Students should
also begin to understand that different models for the same situation can give the
same results.6
6
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.162–163). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 3
45
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
Describe
quantitative
change, such as
students growing
two inches in a year
EXAMPLE
Problem:
Bob’s dog weighed 8 pounds when he got him last year. The dog now weighs 25
pounds. Write a sentence that describes how much weight the dog gained since last
year.
Answer:
Bob’s dog has gained 17 pounds since last year.
DEFINITION:
quantitative change—a change relating to number or quantity; elements can be counted or measured.7
7
Eather, J. (n.d.). A Maths Dictionary For Kids. Retrieved June 5, 2004, from http://www.amathsdictionaryforkids.com
ALGEBRAIC RELATIONSHIPS—Grade 3
46
ALGEBRAIC RELATIONSHIPS
Grade 4
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize and
extend
patterns
EXPECTATION
Describe geometric
and numeric
patterns
EXAMPLE
Problem:
Based on the information in the graphic below, how many circles would it take to make a
30-day-old bug? Explain how you got your answer.
one-day-old bug
two-day-old bug
three-day-old bug
Answer:
60 circles
It takes two circles for each day’s age, so for a thirty-day-old bug, it would
be 30  2 = 60 circles.
Days
1 2 3 4 5
6
7
8
9
10 11 12 13 14 15 16
Circles 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Days
ALGEBRAIC RELATIONSHIPS—Grade 4
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Circles 34
36
38
40
42
44
46
48
50
52
54
56
58
60
47
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Describe the pattern below, and write the number that will go next.
1, 2, 4, 8, 16, 32, 64, ____
Answer:
To get the next number in this pattern, you double or multiply the one before by 2. The
next number here will be 128.
TEACHER NOTES:
Although algebra is not a commonly heard word in the 3–5 classroom, the
mathematical investigations and conversations of students in these grades
frequently include elements of algebraic reasoning. These experiences not only
provide rich contexts for advancing mathematical understanding, they also serve as
an important precursor to the more formalized study of algebra in the middle and
secondary grades. “Algebraic ideas should emerge and be investigated in grades
3–5 as students
 identify or build numerical and geometric patterns
 describe patterns verbally and represent them with tables or symbols
 look for and apply relationships between varying quantities to make
ALGEBRAIC RELATIONSHIPS—Grade 4
48




1
predictions
make and explain generalizations that seem to always work in a particular
situation
use graphs to describe patterns and make predictions
explore number properties
use invented notation, standard symbols, and variables to express a
pattern, generalization, or situation”1
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.159–160). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 4
49
CONCEPT
B
Create and
analyze
patterns
EXPECTATION
Analyze patterns
using words, tables
and graphs
EXAMPLE
Problem:
At a new after-school program, 4 students attended on Monday, 8 students on
Tuesday, and 12 students on Wednesday. If this pattern continues, how many
students would attend the after school program on Thursday? Explain how you got
your answer.
Answer:
16 students. You add 4 students to the previous number to continue the pattern.
ALGEBRAIC RELATIONSHIPS—Grade 4
Monday
Tuesday
Wednesday
Thursday
4
8
12
16
50
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Using the graph below, determine how many ferry boats would be needed for 48
people. Explain how you got your answer.
Answer:
6 ferries would be needed for 48 people. It takes 1 ferry for every 8 people, and 48
divided by 8 equals 6.
ALGEBRAIC RELATIONSHIPS—Grade 4
51
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Explain how the patterns below are alike and different.
Answer:
They are all repeating patterns with two shapes.
In the first two, different shapes are used to make the pattern.
In the third one, different colors of the same shape are used to make the pattern.
ALGEBRAIC RELATIONSHIPS—Grade 4
52
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Complete the table below to show the pattern. Draw a graph to represent the data in the
table.
Cars
1
2
3
4
Number of Riders
4
8
Cars
1
2
3
4
Number of Riders
4
8
12
16
Answer:
Numerous student solutions are possible for the graphs. Example:
ALGEBRAIC RELATIONSHIPS—Grade 4
53
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
Using all operations,
epresent a
mathematical
situation as an
expression or
number sentence
EXAMPLE
Problem:
Five people want to share 20 pieces of candy. Write an expression that shows how to share
the candy equally.
Answer:
20 ÷ 5
Problem:
Allan has enough dog treats to give each of his dogs three treats. Write an expression to
show how many dog treats he has.
Answer:
___ x 3 or 3 x ___ or d x 3 or 3 x d where ___ or d represents the number of dogs
ALGEBRAIC RELATIONSHIPS—Grade 4
54
Problem:
Write a number sentence for twelve times three minus two, then find the
answer.
Answer:
12  3 – 2 = 34
Problem:
Write a number sentence for five plus six minus three equals eight.
Answer:5 + 6 – 3 = 8
DEFINITIONS:
expression—a mathematical phrase that represents a number through the combination of operation symbols, numbers, and/or
symbols. Examples: 23 x 67; 33 – .2
number sentence—an equation or comparison. Examples: 3 + 4 = 7, 8 – 2 = 6, 7 > 6.3
2
3
Cavanagh, M. (2000). Math to know (p.450). Wilmington, MA: Great Source Education Group
Cavanagh, M. (2002). Math to learn (p.457). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 4
55
CONCEPT
EXPECTATION
EXAMPLE
TEACHER NOTES:
Expressions and number sentences can both contain a combination of variables,
numbers, and operation symbols. An expression represents a mathematical
relationship. A number sentence or equation contains an equal sign indicating that
the amount on one side of the equal sign has the same value as the amount on the
other side.
ALGEBRAIC RELATIONSHIPS—Grade 4
56
CONCEPT
B
Describe and
use
mathematical
manipulation
EXPECTATION
EXAMPLE
Problem:
Use the
commutative,
distributive and
associative
properties of
addition and
multiplication for
multidigit numbers
Joann and Steve have been discussing how to arrange the 24 chairs for the school
play. Joann says they can have three rows of eight. Steve said they can have eight
rows of three. Who is correct? Write number sentences or draw a picture to explain
your thinking.
Answer:
Steve’s Work
Joann’s Work
OR
8 x 3 = 24
ALGEBRAIC RELATIONSHIPS—Grade 4
and 3 x 8, so both are correct.
57
DEFINITIONS:
commutative property of multiplication—the product stays the same when the order of the factors is changed. Example:
8  5 = 5  8.4
distributive – when one of the factors of a product is written as a sum, multiplying each addend does not change the product. Example:
3x(5+4)=(3x5)+(3x4).5
associative property of multiplication – the product stays the same when the grouping of the factors is changed. Example: (3x4)x7 =
3x(4x7).6
4
Math at hand: A mathematics handbook (p.519). (1999). Wilmington, MA: Great Source Education Group
at hand: A mathematics handbook (p. 522). (1999). Wilmington, MA: Great Source Education Group
6Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group.
5Math
ALGEBRAIC RELATIONSHIPS—Grade 4
58
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Complete the multiplication sentence below, then use the commutative property to
write a different multiplication sentence.
12  8 = ____
Answer:
96
8  12 = 96
Problem:
Solve and then use the commutative property of multiplication to write a different
multiplication sentence for 5  3.
Answer:
5  3 = 15 3  5 = 15
Problem:
ALGEBRAIC RELATIONSHIPS—Grade 4
59
The number sentence below is an example of what property?
3x(3+7) = (3x3)+(3x7)
A. Commutative property
B. Distributative property
C. Associative property of Addition
D. Associative property of Multiplication
Answer: B
Problem:
Which number sentence is an example of the associative property of multiplication?
A. 2x(5x6)=2 x 30
B. (2x5)x6 = 10 x 6
C. 2x(5x6) = (2x5)x6
D. (2x5)x6 = (6x5)x2
Answer: C
Problem:
The examples below show how Allen used the distributive property to solve
ALGEBRAIC RELATIONSHIPS—Grade 4
60
15  27.
15  (20 + 7)
15  20 + 15  7
300 + 150 = 405
How would Allen solve the following problem using the distributive property?
12  35
Answer:
12  (30 + 5)
12  30 + 12  5
360 + 60 = 420
Problem:
ALGEBRAIC RELATIONSHIPS—Grade 4
61
Does changing the grouping in the following expressions affect the sum?
Explain why or why not.
(5 + 9) + 12 5 + (9 + 12)
Answer:
No.
Addition is associative and either expression equals 26.
Problem:
Does (7 + 8) x 3 equal the same as 7 + (8 x 3)?
Show your work to support your answer.
Answer:
No.
You have to do what’s in parentheses in each problem first.
In the first problem, you add 7 + 8 to get 15, and 15 times 3 is 45.
In the second problem, you multiply 8  3 first to get 24, and add 7 for a total of
31.
ALGEBRAIC RELATIONSHIPS—Grade 4
62
45 is not equal to 31.
Problem:
Show why the following equation is true:
5 (3 + 4) = (5  3) + (5  4)
Answer:
5(7) = (15 + 20)
35 = 35
ALGEBRAIC RELATIONSHIPS—Grade 4
63
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model problem
situations, using
representations
such as graphs,
tables or number
sentences
EXAMPLE
Problem:
1. The cost for admission to a fair is $4 per person. Make a table or graph that shows how
much it would cost for one to six people to attend the fair.
2. Write an expression to show how to find the cost of any number of tickets.
Answer:
1.
Number of people
1
2
3
4
5
6
Cost
$4
$8
$12
$16
$20
$24
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols. 5
5
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 4
64
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
Describe
mathematical
relationships in
terms of constant
rates of change.
EXAMPLE
Problem:
In the table below, how did adding one more student each time change the time
spent on homework?
Number of students
5
6
7
8
Total time spent on
homework
40
minutes
48
minutes
56
minutes
64
minutes
Answer:
Adding one more student increased the time spent on homework by eight minutes
each time.
TEACHER NOTES:
Students should have opportunities to study situations representing different
patterns of change—change that occurs at a constant rate, such as someone
walking at a constant speed, and change that occurs at an increasing or decreasing
rate, as in a growing plant.7
7
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 4
65
ALGEBRAIC RELATIONSHIPS
Grade 5
BIG IDEA (1): Understand patterns, relations and functions
CONCEPT
A
Recognize and
extend
patterns
EXPECTATION
Make and describe
generalizations
about geometric and
numeric patterns
EXAMPLE
Problem:
Find the next number in the following sequence, and explain how you know it’s the next
number.
1, 4, 10, 22, ____
Answer:
The next number is 46.
You take each number, double it, and add 2 to get the next number.
So, 22  2 = 44 plus 2 = 46.
DEFINITIONS:
generalizations—reasoning about the structure of a pattern or rule.1
1
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.159). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 5
66
CONCEPT
EXPECTATION
EXAMPLE
Problem:
John’s electronic piggy bank gives him a reading of how much money he puts in. Today he
notices something strange when he puts money in. When he puts in 10 cents, the bank
reports 23 cents; when he puts in 25 cents, the bank reports 53 cents; and when he puts in
32 cents, the bank reports 67 cents. Complete the Money In/Report Out table below to
help you find out how much money the piggy bank would report if John deposits 45 cents.
Explain how you found your answer.
Money In
Report Out
10 cents
23 cents
25 cents
53 cents
32 cents
67 cents
45 cents
Answer:
ALGEBRAIC RELATIONSHIPS—Grade 5
67
Money In
Report Out
10 cents
23 cents
25 cents
53 cents
32 cents
67 cents
45 cents
93 cents
The report shows double the amount of money put in plus 3 cents.
ALGEBRAIC RELATIONSHIPS—Grade 5
68
CONCEPT
EXPECTATION
EXAMPLE
Problem:
By looking at the number of sides, John says the next shape in the following pattern
could be a square. Do you think John is correct? Explain your answer.
Answer:
Yes, it could be a square. The pattern is a shape that is not four-sided followed by a
shape that is four-sided. Since the last shape in the pattern is not 4-sided, the next
shape must have four sides, so a square could be the next shape.
TEACHER NOTES:
Students in grades 3–5 should investigate numerical and geometric patterns and
express them mathematically in words or symbols. They should analyze structures
of patterns and how they grow and change. In addition, they should organize this
information systematically and use their analysis to develop generalizations about
the mathematical relationships in the pattern.2
2
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.159–160). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 5
69
CONCEPT
B
Create and
analyze
patterns
EXPECTATION
Represent and
analyze patterns
using words, tables
and graphs
EXAMPLE
Problem:
Make a table to show a pattern in which you start with the number 1, then add 4 to
get the next number.
Answer:
1
5
9
13
17
21
25
Problem:
What is the 4th number out in the Number In/ Number Out table below?
Explain how you got your answer.
Number In
Number Out
1
1
2
4
3
9
4
Answer:
The 4th number out is 16. I saw that each number in multiplied by itself gave the
number out, so I took 4 x 4 to get 16;
or, I saw that the numbers out were 1 plus 3, 4 plus 5, so I added 9 plus 7 to get 16
as the 4th number out.
ALGEBRAIC RELATIONSHIPS—Grade 5
70
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Frame 1
Frame 2
Frame 3
Frame 4
Use the pattern above to complete the table below.
Explain how you got your answer.
Frame
1
2
3
4
5
6
20
Number of Shapes
2
6
12
20
Frame
1
2
3
4
5
6
20
Number of Shapes
2
6
12
20
30
42
420
Answer:
I found the missing numbers by multiplying the frame number by itself and then
adding the frame number or (n x n) + n.
ALGEBRAIC RELATIONSHIPS—Grade 5
71
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Describe the pattern below in words.
Frame 1
Frame 2
Frame 3
Frame 4
Answer:


ALGEBRAIC RELATIONSHIPS—Grade 5
The pattern is adding the next odd number of squares each time—
o 1 + 3 gives the second frame of 4, and
o 4 + 5 gives the third frame of 9, and
o 9 + 7 gives the fourth frame of 16 squares
or, the pattern is the frame number multiplied by itself—
o 1 x 1 = 1, and
o 2 x 2 = 4, and
o 3 x 3 = 9, and
o 4 x 4 = 16.
72
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
Using all operations,
represent a
mathematical
situation as an
expression or
number sentence
using a letter or
symbol
EXAMPLE
Problem:
Seven buses each brought an equal number of students to school. Write an
expression to show how many students rode each bus. Let s = the total number of
students.
Answer:
s÷7
Problem:
Which of the following expressions represents three more than twice a number?
A.
B.
C.
D.
3n + 2
5n
2n + 3
3n – 2
Answer:
C
DEFINITIONS:
expression—a mathematical phrase that represents a number through the combination of operation symbols, numbers, and/or
symbols. Examples: 23  67; 3a; x + y.3
number sentence—an equation or comparison. Examples: 3 + 4 = 7, 8 – 2 = 6, 7 > 6.4
3
Cavanagh, M. (2000). Math to know (p.450). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 5
73
CONCEPT
B
Describe and
use
mathematical
manipulation
EXPECTATION
EXAMPLE
*use the
commutative,
distributive and
associative
properties for
fractions and
decimals
DEFINITIONS:
associative property of addition—The sum stays the same when the grouping of the addends is changed. Example:
(5 + 4) + 6 = 5 + (4 + 6).5
associative property of multiplication—The product stays the same when the grouping of the factors is changed.
Example: (3  4)  7 = 3  (4  7).6
distributive property—When one of the factors of a product is written as a sum, multiplying each addend before adding does
not change the product. Example: 3  (5 + 4) = (3  5) + (3  4).7
4
Cavanagh, M. (2002). Math to learn (p.457). Wilmington, MA: Great Source Education Group
Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group
6 Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group
7 Math at hand: A mathematics handbook (p.522). (1999). Wilmington, MA: Great Source Education Group
5
ALGEBRAIC RELATIONSHIPS—Grade 5
74
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model problem
situations and draw
conclusions, using
representations
such as graphs,
tables or number
sentence
EXAMPLE
Problem:
Use the table and shapes below and your pattern blocks to decide what the
perimeter of the 12th shape will be.
Shape Number
Perimeter
1
5
2
8
3
11
12
1
2
3
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols. 8
8
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 5
75
CONCEPT
EXPECTATION
EXAMPLE
Answer:
The perimeter of the 12th shape will be 38.




Some students will draw the shapes and count the sides to determine the
perimeter; or
others will use the pattern blocks to create the shapes and count the sides;
or
others may look at the table and add 3 to the previous perimeter to get the
perimeter for the next number; or
some may see the general pattern that the perimeter for any number can be
determined by multiplying the number by 3 and adding 2.
TEACHER NOTES:
Students in grades 3–5 can model a variety of situations, including geometric
patterns, real-world situations, and scientific experiments. Sometimes they will use
their model to predict the next element in a pattern. At other times, they may make a
general statement about how one variable is related to another variable: for
example, if a sandwich costs $3, you can figure out how much any number of
sandwiches costs by multiplying that number by 3. In modeling situations that
involve real-world data, students need to know that their predictions may not always
match the observed outcomes for a variety of different reasons. Students should
also begin to understand that different models for the same situation can give the
same results.9
9
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.162–163). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 5
76
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
Identify, model and
describe situations
with constant or
varying rates of
change
EXAMPLE
Problem:
The table below shows cell growth over several days. Is the rate of cell growth a constant or
varying rate of change? Explain your thinking.
Change in
Number of Cells
Time (days)
Number of Cells
1
1
2
2
1
3
8
7
4
16
9
Answer:
It is a varying rate of change because the change in the number of cells varies from day to
day.
TEACHER NOTES:
ALGEBRAIC RELATIONSHIPS—Grade 5
77
Students should have opportunities to study situations representing different
patterns of change—change that occurs at a constant rate, such as someone
walking at a constant speed, and change that occurs at an increasing or decreasing
rate, as in a growing plant.10
10
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.163). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 5
78
ALGEBRAIC RELATIONSHIPS
Grade 6
BIG IDEA (1): Understand patterns, relations and functions
B
CONCEPT
EXPECTATION
Create and
analyze
patterns
Represent and
describe patterns
with tables, graphs,
pictures, symbolic
rules or words
EXAMPLE
Problem:
Complete the table below, and write a symbolic rule for the pattern.
Term number
1
2
3
Term
3
5
7
4
5
6
DEFINITIONS:
Rule (for a pattern): a general statement written in numbers or words that describes how to determine any term in a pattern.
Rules or generalizations for patterns may include both recursive and explicit notation. In the recursive form of pattern
generalization, the rules focuses on the rate of change from one element to the next. Example: Next = Now + 2; Next = Now x 4.
In the explicit form of pattern generalization, the formula or rule is related to the order of the terms in the sequence and focuses on
the relationship between the independent variable (the number representing a term in the sequence) and the dependent variable
(the number in the sequence). For example: 5t – 3; t – x; (t+1)x5; Words may also be used to write a rule in recursive or explicit
notation. Example: take the previous number and add two to get the next number; to find the total for any day multiply the day
times five and subtract three.
symbolic rules—rules that use variables and numbers to describe a pattern or express a relationship.1
term number—location of a term in a sequence or pattern. For example, in the sequence 1, 3, 5, 7..., 5 is the third term number
terms—numbers, variables, products, or quotients in an expression. For example, in 6x2 + 5x + 3, there are three terms—6x2, 5x,
and 3.2
1
Navigating through algebra in grades 6–8 (p.3). (2001). Reston, VA: National Council of Teachers of Mathematics
ALGEBRAIC RELATIONSHIPS—Grade 6
79
CONCEPT
EXPECTATION
EXAMPLE
Answer:
Term number
1
2
3
4
5
6
Term
3
5
7
9
11
13




The 4th term is 9,
5th term is 11,
6th term is 13.
The symbolic rule is 2n + 1, n+ term number
Problem:
A video rental store charges a $5 membership fee and $2 to rent a video. Make a table to
show the total cost for renting one, two, three, four, and five videos. Write the symbolic
rule for finding the cost for any number of videos.
Answer:
Number
of videos
Term
1
2
3
4
5
$7
$9
$11
$13
$15
The symbolic rule is 5 + 2n; n represents the number of videos rented.
2
Kaplan, A. (1998). Math on call (p.204). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 6
80
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Use a table, graph, picture, or symbolic rule to represent the following sequence:
1, 5, 9, 13, 17.…
Answer:
 Table
ALGEBRAIC RELATIONSHIPS—Grade 6
Term Number
Term Value
1
1
2
5
3
9
4
13
5
17
81
CONCEPT
EXPECTATION
EXAMPLE
 or graph
 or picture

ALGEBRAIC RELATIONSHIPS—Grade 6
or symbolic rule:
4n – 3
82
CONCEPT
C
Classify
objects and
representations
EXPECTATION
*Compare various
forms of
representations
to identify a pattern
EXAMPLE
Problem:
A sequence can be formed by arranging toothpicks to make rectangles as shown:
F
r
a
m
e
1
F
r
a
m
e
2F
r
a
m
e
3
Make a table, and record the first six term numbers and the number of toothpicks
needed for each term number. Write a general rule describing the relationship
between the term number (n) and the number of toothpicks. Use the general rule to
find the number of toothpicks needed for the 500th rectangle.
Answer:
Frame (term) number
1
2
3
4
5
6
Number of toothpicks
4
7
10
13
16
19
The general rule for describing the relationship between the term number and the number
of toothpicks is 3n + 1. The 500th rectangle would require (500  3) + 1, or 1501 toothpicks.
DEFINITION:
representations—physical objects, drawings, charts, graphs, and symbols that help students communicate their thinking.3
3
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.280). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 6
83
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Find the 625th term in the sequence represented below. Explain how the total is
related to the term number.
Term number
1
2
3
4
5
Total
3
5
7
9
11
625
Answer:
The 625th term is (2 x 625) + 1, or 1,251.
The total is one more than twice the term number, i.e., 2n + 1.
ALGEBRAIC RELATIONSHIPS—Grade 6
84
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Describe how the following graphics can represent the same pattern.
Input
Output
3
7
5
11
8
17
10
21
(10,
21)
(3.7)
Answer: (Answers may vary.)
The input in the table represents the x term of the graph, and the output represents the y
term of the graph.
ALGEBRAIC RELATIONSHIPS—Grade 6
85
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Match the sequence with the appropriate description or rule.
Sequences
1.
2.
3.
10, 12, 14, 16, 18
10, 20, 40, 60, 80
10,
10, 8, 6, 4, 2
Rules
A.
Double the previous
term, first term = 10
B.
Add 2 to the previous
term, first term = 10
C.
Subtract 2 from the
previous term, first term
= 10
Answer:
1. Rule B
2. Rule A
3. Rule C
ALGEBRAIC RELATIONSHIPS—Grade 6
86
D
CONCEPT
EXPECTATION
Identify and
compare
functions
*Identify functions
as linear or
nonlinear from a
table or graph
EXAMPLE
Problem:
Suppose you put $100 in the bank and your money doubles every seven years as
shown below:
Years
Amount in bank
0
7
14
21
28
$100
$200
$400
$800
$1,600
The data of your savings is shown in the graph below. Discuss whether the rate of change is constant from one point to the
next and whether the graph is linear or non-linear and why.
DEFINITIONS:
Functions—relations in which every value of x has a unique value of y; a mathematical rule between two sets which assigns to
each number of the first, exactly one member of the second.4
linear (function) equation— a relationship between two variables that can be expressed as an equation and drawn as a
straight line in a coordinate grid. 5
nonlinear (function) equation— a relationship between two variables x and y is described as nonlinear if it is not of the form
y = ax + b. The graphs will not be straight lines; the equation will not be of the first degree. For example, y = e x and y = x2 are
nonlinear relationships.6
4
Kaplan, A. (1998). Math on call (p.583). Wilmington, MA: Great Source Education Group & Parker, S. (Ed.) (1997). McGraw-Hill dictionary of
mathematics (p.94). NY, NY: McGraw-Hill
5 Berry, J. (2003). Schaum’s A-Z mathematics (p.132). London: McGraw-Hill
6 Berry, J. (2003). Schaum’s A-Z mathematics (p.154). London: McGraw-Hill
ALGEBRAIC RELATIONSHIPS—Grade 6
87
CONCEPT
EXPECTATION
EXAMPLE
Answer:
The rate of change is not constant. The first seven years, the amount increases
$100; the second seven years, it increases $200; and the third seven years, the
amount increases $400. Since the rate of change, or slope, is not constant, the
graph is non-linear.
ALGEBRAIC RELATIONSHIPS—Grade 6
88
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Given the following table of numbers, explain whether or not the pattern is linear.
ALGEBRAIC RELATIONSHIPS—Grade 6
Input
Output
1
5
2
7
3
9
4
11
89
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Identify the non-linear graph below and explain why it is a non-linear graph.
A.
ALGEBRAIC RELATIONSHIPS—Grade 6
90
CONCEPT
EXPECTATION
EXAMPLE
B.
ALGEBRAIC RELATIONSHIPS—Grade 6
91
CONCEPT
EXPECTATION
EXAMPLE
C.
Answer:
Graph B is non-linear because the rate of change is not constant, and the graph is
not a straight line.
ALGEBRAIC RELATIONSHIPS—Grade 6
92
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
EXPECTATION
Represent
mathematical
situations
EXAMPLE
Problem:
Which picture below represents the expression 2h + 3s?
Use symbolic
algebra to
represent unknown
quantities in
expressions or
equations and solve
one-step equations
Explain why.
A.
♥♥♥
B.
♥♥
C.
♥♥
D.
♥
Answer:
B
h is for heart, and s is for star, and the expression is 2 hearts (h) and 3 stars (s), or
2h + 3s.
ALGEBRAIC RELATIONSHIPS—Grade 6
93
CONCEPT
EXPECTATION
EXAMPLE
Problem:
In the table shown below,
1. complete the next three terms;
2. explain how the terms are related to the term numbers (n);
3. write a general rule where: term = ____;
4. show how you use your rule to find the 30th term.
Term number
1
2
3
4
5
6
7
Term
12
24
36
48
Term number
1
2
3
4
5
6
7
Term
12
24
36
48
60
72
84
Answer:
1.
2. The term is the term number times 12.
3. Term = term number  12, or n  12.
4. The 30th term is 30  12, or 360.
ALGEBRAIC RELATIONSHIPS—Grade 6
94
CONCEPT
EXPECTATION
EXAMPLE
Problem:
In the table shown below:
1. complete the missing terms
2. write a symbolic rule to find any missing term
Term number
1
2
3
4
Term
8
15
22
29
Term number
1
2
3
4
Term
8
15
22
29
5
25
200
5
25
200
36
176
1,401
Answer:
1.
2. The symbolic rule to find any missing term is 7n + 1.
ALGEBRAIC RELATIONSHIPS—Grade 6
95
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Use a variable to write an expression that determines the number of dots, no matter
what the term number is.






1
2
3
4
2
4
6
8




Answer:
The term is equal to the term number times two, or n  2, (n = 1, 2, 3, 4....).
Problem:
Use a variable to write an expression for determining any number (term) in the
sequence.
3, 7, 11, 15…
ALGEBRAIC RELATIONSHIPS—Grade 6
Answer:
4n – 1, n= term number
Answer:
4n – 1
96
CONCEPT
B
Describe and
use
mathematical
manipulation
EXPECTATION
EXAMPLE
Problem:
*use the
commutative,
distributive and
associative
properties to
generate equivalent
forms for simple
algebraic
expressions
Does changing the grouping in the following expression affect the product?
Explain why or why not.
(5  b)  7
5  (b  7)
Answer:
No.
Multiplication is associative and either expression equals 35b.
DEFINITIONS:
associative property
of addition—The sum stays the same when the grouping of the addends is changed. Example: (22 + 13) + 12 = 22 + (13 + 12).
of multiplication—The product stays the same when the grouping of the factors is changed. Example: (8  7)  13 = 8  (7  13).8
distributive property—When one of the factors of a product is written as a sum, multiplying each addend before adding does
not change the product. Example: 7  (11 + 13) = (7  11) + (7  13).9
8
9
Math at hand: A mathematics handbook (p.517). (1999). Wilmington, MA: Great Source Education Group
Math at hand: A mathematics handbook (p.522). (1999). Wilmington, MA: Great Source Education Group
ALGEBRAIC RELATIONSHIPS—Grade 6
97
CONCEPT
EXPECTATION
ALGEBRAIC RELATIONSHIPS—Grade 6
EXAMPLE
98
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Tell what property is illustrated in each equation below.
1.
a+b+c=c+b+a
2. (a + b) + c = c + (a + b)
3. (a + b) + c = a + (b + c)
4.
CONCEPT
a(b + c) = ab + ac
EXPECTATION
EXAMPLE
Answer
1. commutative
2. commutative
3. associative
4. distributive
Problem:
Which of the following are equivalent to n + 9?
ALGEBRAIC RELATIONSHIPS—Grade 6
99
1.
9+n
2.
9n
3.
3+n+6
4.
n–9
A. 1 only
B. 2 only
C. 1 and 3 onl
D. 1, 2, and 3 only
Answer:
E
ALGEBRAIC RELATIONSHIPS—Grade 6
100
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model and solve
problems, using
multiple
representations
such as tables,
expressions and
one-step equations
EXAMPLE
Problem:
At the end of February, Ben began to save for a $240 mountain bike. At the time, he
had $113 in savings. His savings increased to $138 in March and $163 in April.
Complete the table below to determine when Ben will have enough if he continues
to save at the same rate:
Month
Feb
Mar
April
Amount saved
$113
$138
$163
Draw a coordinate graph reflecting the information in the table. Describe any
patterns you find in the table or graph.
Answer:
Month
Feb
Mar
April
May
June
July
Aug
Amount saved
$113
$138
$163
$188
$213
$238
$263
DEFINITION:
Model—to represent a mathematical situation with manipulatives (objects), pictures, numbers, or symbols.10
10
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (p.95). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 6
101
CONCEPT
EXPECTATION
EXAMPLE
Ben will have enough money for his bike in August. The pattern is that $25 is added
on each month.
ALGEBRAIC RELATIONSHIPS—Grade 6
102
CONCEPT
EXPECTATION
EXAMPLE
Problem:
‘Talk a Lot’ phone company charges $15 a month for service and $3 for every hour
of long-distance calls. Mary talks long distance for two hours one month, and Tom
talks long distance for four hours the same month. Using an equation or table,
determine the charge that month for both Mary’s phone bill and Tom’s phone bill.
Describe or explain how to determine the price for any number of hours.
Answer:
Each person’s charges can be determined by using $15 + $3n. The following
equations show the cost for each.
Mary—$15 + $3(2) = $21
Tom—$15 + $3(4) = $27
The cost for any number of hours (n) can be found by using $15 + $3n, where n is
the number of hours talked.
Initial
Hour 1
Hour 2
Mary
$15
$18
$21
Tom
$15
$18
$21
Hour 3
Hour 4
$24
$27
TEACHER NOTES:
Students in grades 3–5 can model a variety of situations, including geometric
patterns, real-world situations, and scientific experiments. Sometimes they will use
their model to predict the next element in a pattern. At other times, they may make a
general statement about how one variable is related to another variable.
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103
CONCEPT
EXPECTATION
EXAMPLE
For example: If a sandwich costs $3, you can figure out how much any number of
sandwiches costs by multiplying that number by 3.
In modeling situations that involve real-world data, students need to know that their
predictions may not always match the observed outcomes for a variety of different
reasons. Students should also begin to understand that different models for the
same situation can give the same results.11
11
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics (pp.162–163). Reston, VA
ALGEBRAIC RELATIONSHIPS—Grade 6
104
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
EXPECTATION
Analyze
change
EXAMPLE
Problem:
*construct and
analyze
representations to
compare situations
with constant or
varying rates of
change
Consider a rectangle with a fixed area of 12 square units.
Make a table showing the widths for all possible whole-number lengths of the rectangle up
to a length of 12.
Answer:
Length
1
2
3
4
5
6
7
8
9
10
11
12
Width
12
6
4
3
2.4
2
1.7
1.5
1.3
1.2
1.1
1
Problem:
Would a graph of the relationship between L (length) and W (width) be a straight line?
Why or why not?
Answer:
The graph would not be a straight line.
As the length increases by a constant rate of 1, the width decreases, but not at a constant
rate.
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105
CONCEPT
EXPECTATION
EXAMPLE
Problem:
If you rent 10 videos, would it be cheaper to pay a $15 membership fee and $2 to
rent a video, or no membership fee and $3 to rent a video?
Answer:
15 + 2(10) = 35
3(10) = 30
The second method is cheaper.
Problem:
For what number of videos would the price be the same?
Show how you got your answer.
Answer:
15 videos
15 + 2(15) = 45 and 3(15) = 45
or
ALGEBRAIC RELATIONSHIPS—Grade 6
106
some students may draw a table to show their answer. Example:
$15 membership fee and $2 to rent a video
Number of videos
Cost for videos
0
1
5
10
15
20
$15
$17
$25
$35
$45
$55
No membership fee and $3 to rent a video
ALGEBRAIC RELATIONSHIPS—Grade 6
Number of videos
0
1
5
10
15
20
Cost for videos
$0
$3
$15
$30
$45
$60
107
CONCEPT
EXPECTATION
EXAMPLE
Problem:
In the following pattern:
1. find the perimeters of the next four sets of quadrilaterals, and explain how you
got your answer
2. find the perimeter of the 100th quadrilateral
2
1
1
1
ALGEBRAIC RELATIONSHIPS—Grade 6
Quadrilateral 1
Quadrilateral 2
Quadrilateral 3
Perimeter = 5
Perimeter = 8
Perimeter = 11
Quadrilateral
1
2
3
Perimeter
5
8
11
4
5
6
7
108
Answer:
1. The perimeter increases by three each time.
Quadrilateral
1
2
3
4
5
6
7
Perimeter
5
8
11
14
17
20
23
2. 3x + 2 will give the perimeter of any quadrilateral. So, 3(100) + 2 = 302.
ALGEBRAIC RELATIONSHIPS—Grade 6
109
ALGEBRAIC RELATIONSHIPS
Grade 7
BIG IDEA (1): Understand patterns, relations, and functions
B
CONCEPT
EXPECTATION
Create and
analyze
patterns
Analyze patterns
represented
graphically or
numerically using
words or symbolic
rules, including
recursive notation
EXAMPLE
Problem:
Which equation represents the pattern in the following graph?
A. y= 2x
B. y= 4x
C. y= 2x + 1
D. y= 2x + 5
Answer:
D. y = 2x + 5
DEFINITIONS:
graphically— the plot of points in the plane which constitute the graph of a given real function or a pictorial diagram depicting the
ALGEBRAIC RELATIONSHIPS—Grade 7
110
interdependence of variables. 4
numerically-- pertaining to numbers. 5
Rule (for a pattern): a general statement written in numbers or words that describes how to determine any term in a pattern.
Rules or generalizations for patterns may include both recursive and explicit notation. In the recursive form of pattern
generalization, the rules focuses on the rate of change from one element to the next. Example: Next = Now + 2; Next = Now x 4.
In the explicit form of pattern generalization, the formula or rule is related to the order of the terms in the sequence and focuses on
the relationship between the independent variable (the number representing a term in the sequence) and the dependent variable
(the number in the sequence). For example: 5t – 3; t – x; (t+1)x5; Words may also be used to write a rule in recursive or explicit
notation. Example: take the previous number and add two to get the next number; to find the total for any day multiply the day
times five and subtract three.
symbolic rules-- rules that use variables and numbers to describe a pattern or express a relationship.6
recursive notation-- a process that is inherently repetitive, with the result of each repetition usually depending upon those of the
previous repetition.7 Added Note – Recursive notation requires the previous term (or information to begin) to generate the next
term. This type of function follows a pattern that “reoccurs”. (Juan’s equation on the second problem is an example of recursive
notation).
4
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 102). New York, NY: McGraw-Hill, Inc.
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 170). New York, NY: McGraw-Hill, Inc.
6 Navigating Through Algebra in grades 6-8 (p.3) (2001). Reston, VA: National Council of Teachers of Mathematics.
7 McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 210). New York, NY: McGraw-Hill, Inc.
5
ALGEBRAIC RELATIONSHIPS—Grade 7
111
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Three of your friends are discussing their solution to a problem from last night’s
homework. The question asked you to find an equation that would represent the
following pattern:
Term
Sequence
Number
1
2
3
4
5
4
12
20
28
36
Sammy thought the equation would be: n = 8t –4 if t=term and n=sequence number.
Jenny found the equation to be: n = 8t + 4 if t=term and n=sequence number
Juan believed the equation should be: Next = Now + 8 with the first Now being 4.
Which student(s) is (are) correct? Justify your decision.
Answer:
Sammy and Jenny both used explicit notation. Sammy is correct if t= the term
number and t begins at 1. Jenny can be correct if her t begins at 0, but if not noted
this is assumed not correct. Juan used recursive notation. Juan would be correct
ALGEBRAIC RELATIONSHIPS—Grade 7
112
because he is showing a constant change of 8.
Teacher’s Note: In the recursive form of pattern generalization, students focus on the rate
of change from one element to the next. (Juan would continue adding 8 to determine the
next number in the sequence.) The explicit form of pattern generalization involves
developing a rule or formula that highlights the relationship between the independent
variable (the number representing the position of the term in the sequence) and the
dependent variable (the number in the sequence). (Sammy multiplied the term by 8 and
added four to determine the sequence number.)8
8
Navigating Through Algebra in grades 6-8 (p.7-8) (2001). Reston, VA: National Council of Teachers of Mathematics.
ALGEBRAIC RELATIONSHIPS—Grade 7
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
1. Give a brief description of a situation that could be represented by the
following graph. Include a time, a distance, and a rate.
2. Use numbers and symbols to represent the pattern.
please regenerate-copyrighted material
Answer:
1. Answers will vary. An example would be, “Susan traveled a constant rate of
10 miles per hour. After two hours she traveled 20 miles.”
2. D = 10 t, where D = distance and t = time.
ALGEBRAIC RELATIONSHIPS—Grade 7
114
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Term
Number
Sequence
Number
1
2
3
4
5
3
9
27
?
?
For the table shown:
1. Describe in words how to determine the sequence number from the term
number.
2. Write in symbols the general rule if T = the term number and S = the
sequence number.
Answer:
1. You would multiply 3 by itself as many times as the term number says to.
Or, using recursive notation, it would be Next= Now times 3 with the first
Now being 3.
2. S = 3T
ALGEBRAIC RELATIONSHIPS—Grade 7
115
BIG IDEA (1): Understand patterns, relations, and functions
CONCEPT
C
Classify
objects and
representations
EXPECTATION
Compare and
contrast various
forms of
representations of
patterns
EXAMPLE
Problem:
Benjamin is traveling 55 miles per hour.
Equation: d = 55t, where d=distance and t=tim
Table:
Time (hrs)
Distance (mi)
1
2
3
4
5
6
7
8
55
110
165
220
275
330
385
440
Graph:
Compare and contrast how you would use the equation, the table, and the graph to
determine how far Benjamin traveled in 4 ½ hours.
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116
DEFINITIONS:
representations-- physical objects, drawings, charts, graphs, and symbols.9
Answer:
The equation would answer the question by substituting the time with 4.5 or
d = 55(4.5) which is 247.5 miles. Using the table, 4.5 is halfway between 4 and 5 so
the distance is between 220 and 275, or 247.5 miles. Therefore, the equation and
the table both would give the same answer. The graph is a little more difficult
because I would need to draw a vertical line at the midpoint of 4 and 5 to the
graphed line then follow it horizontally to the y axis. This would give an approximate
value of 250 miles, which is not as exact as the equation or the table.
9
Principles and Standards for School Mathematics (p.280). Reston, VA: Author. National Council of Teachers of Mathematics. (2000).
ALGEBRAIC RELATIONSHIPS—Grade 7
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
Randy bought a compact disc player and needed CD’s. While shopping, he found
that Bob’s Department store sells CD’s for $15.00 each, while Joe’s CD Club sells
them for only $10 each, but has a $25 membership fee. He decided to compare the
costs for CD’s from the two stores using a table, a graph, and an equation.
Table
Bob’s
1
15
2
30
3
45
4
60
5
75
6
90
7
105
8
120
9
135
10
150
Joe’s
35
45
55
65
75
85
95
105
115
125
# of CD’s
ALGEBRAIC RELATIONSHIPS—Grade 7
118
Graph
Equation
Bob’s: C=15n, where C is cost and n is number of CD’s
Joe’s: C=10n+25, where C is cost and n is number of CD’s
1. Use the table, the graph, and the equation to determine where Randy should
buy his CD’s. Explain your reasoning.
2. Which representation- the description, the table, the graph, or the
equation- helps Randy the most in making the decision of where to
buy CD’s? Explain your thinking.
ALGEBRAIC RELATIONSHIPS—Grade 7
119
Answer:
1. If Randy buys less than five CD’s, Bob’s would be the best buy. However, at
more than 5 CD’s, Joe’s Club is the better buy. If he needed exactly five CD’s,
Randy could buy from either Bob’s or Joe’s. Even though the CD’s are
cheaper at Joe’s, Randy would end up paying more for the CD’s because of
the membership fee unless he buys more than 5.
2. I think the table helps Randy the most in making his decision. I looked at
which had the smaller cost for each number of CD’s he might buy. From 1-4
CD’s, Bob’s costs are smaller. If Randy is buying 6 or more CD’s, Joe’s costs
are lower. It would cost $75 for 5 CD’s at both stores. With a table, it’s
easier to just look at the numbers. With graphs, it’s harder to read the exact
number.
Or
I think the graph helps Randy the most in making his decision. Bob’s line is
lower until 5 CD’s, then they switch and Joe’s line is lower. The lines meet at
5 CD’s, so at 5 CD’s the prices are the same. It’s really easy to see with a
picture.
Or
I think the equation helps Randy the most in making his decision. If Randy
knew exactly how many CD’s he wanted to buy he could just use the
equation for each store and find out which would be the better store to buy
the CD’s from without making a table or graph.
Teachers Note: Any representation would be helpful in making Randy’s decision based on personal preference. The
differences in the costs are shown numerically in the table. The graph shows pictorially the differences in the costs. The
equation would probably be the least helpful unless Randy knew exactly how many CD’s he wanted to buy.
ALGEBRAIC RELATIONSHIPS—Grade 7
120
CONCEPT
EXPECTATION
EXAMPLE
Problem:
1. Compare and contrast the table and graph.
2. Which representation more clearly shows a linear pattern?
3. How would you find the rate of change using the table or the graph?
(10,21)
21
15
9
(3,7)
3
2
ALGEBRAIC RELATIONSHIPS—Grade 7
Input
Output
3
7
5
11
8
17
10
21
6
10
121
Answer:
1. The table and graph both represent the same pattern or equation (2n +1).
The difference is that the table shows the pattern using numbers while the
graph shows the pattern using a picture. Another difference is that in the
table the input does not increase by the same amount each time, while in the
graph it does.
2. It is easier to tell that the pattern is linear, or has a constant rate of change,
by the straight line on the graph. In the table it is not as obvious- particularly
since the input does not increase by 1 each time.
3. The rate of change is 2. The rate of change between (3,7) and (10,21) is
14/7 or 2.
Or
By filling in the missing inputs on the table, the output increases by 2
time, therefore the rate of change is 2.
ALGEBRAIC RELATIONSHIPS—Grade 7
Input
Output
3
7
4
9
5
11
6
13
7
15
8
17
9
19
10
21
each
122
BIG IDEA (1): Understand patterns, relations, and functions
CONCEPT
D
Identify and
compare
functions
EXPECTATION
Identify functions as
linear or nonlinear
from tables, graphs
or equations
EXAMPLE
Problem:
For the table shown, determine whether the function is linear or non-linear and
explain why.
Term
number
1
2
3
4
2
4
8
16
5
6
Sequence
Number
Answer:
Nonlinear. The term number increases by 1 (a constant amount), but the sequence
number does not increase by the same amount each time.
DEFINITIONS:
functions-- a mathematical rule between two sets which assigns to each number of the first, exactly one member of the second.
10
10
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 94). New York, NY: McGraw-Hill, Inc.
ALGEBRAIC RELATIONSHIPS—Grade 7
123
linear-- a relationship between two variables that can be expressed as an equation and drawn as a straight line.11
linear equation-- an equation in two variables whose graph in a coordinate plane is a straight line. It is a first degree polynomial
equation.12
nonlinear-- the relationship between two variables x and y is described as nonlinear if it is not of the form
y = ax + b. For example, y = ex and y = x2 are nonlinear relationships.13
Schraum’s A-Z Mathematics (2003). Berry, J, p.132, London, England, McGraw-Hill.
Algebra to Go: A Mathematics Handbook (2000). Great Source Education Group Staff. (p. 493) , Wilmington, MA: Great Source Education
Group, Inc.
13
Schraum’s A-Z Mathematics (2003). Berry, J, p.154, London, England, McGraw-Hill.
11
12
ALGEBRAIC RELATIONSHIPS—Grade 7
124
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Given the following equations, identify which are linear and how you know.
A. Next = Now + 6
B. p  t 2  t C. h  7  v
D. y  x  32
Answer:
A.
B.
C.
D.
ALGEBRAIC RELATIONSHIPS—Grade 7
Linear. Addition of constant – degree of 1.
Nonlinear. The variable is squared-- not a degree of 1.
Linear. Subtraction – degree of 1.
Linear. Addition – degree of 1.
125
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Identify the function in the graph as linear or nonlinear. Explain your reasoning.
Answer:
The graph is nonlinear. The rate of change is not constant; therefore, the graph
does not make a straight line.
ALGEBRAIC RELATIONSHIPS—Grade 7
126
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
use symbolic
algebra to represent
and solve problems
that involve linear
relationships
EXAMPLE
Problem:
The cost for a ferry for different numbers of customers is given by the following
table:
Customers
Ferry Cost ($)
1
2
3
4
5
$3.50
$7.00
$10.50
$14.00
$17.50
Write an equation relating ferry cost, f, and the number of customers, n.
Answer:
f=3.50n
ALGEBRAIC RELATIONSHIPS—Grade 7
127
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Kaziah is four years less than twice Eli’s age. Write an equation to match the
following situation and identify the variables.
Answer:
K = 2E – 4, where K= Kaziah’s age and E= Eli’s age
ALGEBRAIC RELATIONSHIPS—Grade 7
128
CONCEPT
EXPECTATION
EXAMPLE
Problem:
1. Describe the following pattern using symbols and explain your reasoning.
2. Then use your equation to determine how many smiley faces would be in the
10th term.
Term Number
1
2
3
4
5
Term
Answer:
1. The total for a term would be 2s + 1, where s= the number of smiley faces.
There are 2 rows of smiley faces. One row has the same number of smiley
faces as the term number and the other has 1 more than the term number, or
s for one row and s + 1 for the other row.
2. The 10th term number in the pattern would have 21 smiley faces.
ALGEBRAIC RELATIONSHIPS—Grade 7
129
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Which of the following equations represents the pattern?
a)
b)
c)
d)
e)
x
y
0
-3
1
-1
2
1
3
3
y = 3x + 2
y = 2x + 3
y = x-3
y = 2x – 3
y = 3x – 2
Answer:
d) y = 2x -3
ALGEBRAIC RELATIONSHIPS—Grade 7
130
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Use the equation 4x + 9 = y to find the value of y when x is equal to –1. Provide the
work that shows how you arrived at your answer.
Answer:
4x + 9 = y
4(-1) + 9 = y
-4 + 9 = y
5=y
ALGEBRAIC RELATIONSHIPS—Grade 7
131
BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
B
Describe
and use
mathematical
manipulation
EXPECTATION
use properties to
generate
equivalent forms
for simple algebraic
expressions that
include positive
rationals and
intergers
EXAMPLE
Problem:
Which expression has the same value as 3 ( 43 ) ? Explain why.
A. 12 * 12 * 12
B. 3 * 4* 4 * 4 C. 3*3*3*4*4*4
Answer:
B. has the same value as 3 ( 43 ), a and c are both equal to 3343.
ALGEBRAIC RELATIONSHIPS—Grade 7
132
CONCEPT
EXPECTATION
EXAMPLE
Problem:
For each equation, determine whether the statement is true or false. Explain your
reasoning.
1.
2.
3.
4.
n+3=3+n
(n – 5) – 3 = n – (5 – 3)
12 ÷ n = n ÷ 12
5–n=n–5
Answer:
1. True- For example, 2 + 3 =5, and 3+ 2=5. (Addition is commutative.)
2. False- For example, (2 – 5) – 3 = -6, and 2 – (5 – 3)= 0. (Subtraction is not
associative.)
3. False- For example, 12 ÷ 2 = 6, and 2 ÷ 12 = 1/6. (Division is not
commutative.)
4. False- For example, 5 – 2 = 3, and 2 – 5= -3. (Subtraction is not
commutative.)
ALGEBRAIC RELATIONSHIPS—Grade 7
133
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Which of the following is equal to 5(x-2)?
A. x - 10
B.
C.
D.
E.
5x-2
5x - 7
5x - 10
5x - 52
Answer:
D. 5x - 10
ALGEBRAIC RELATIONSHIPS—Grade 7
134
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Generate at least 3 different expressions that are equivalent to 5(x + 4).
Sample Answers:
5x + 20, (x + 4) + (x + 4) +(x + 4) +(x + 4) +(x + 4), x + x + x + x + x + 20,
x + x + x + x + x + 4+4+4+4+4, etc.
ALGEBRAIC RELATIONSHIPS—Grade 7
135
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Are the following two expressions equivalent? Explain why or why not.
2 (10 x n)
10 (2 x n)
Answer:
Yes, they are equivalent. Since multiplication is both commutative and
associative, they both equal 20n when simplified.
2 (10 x n) = (10 x n) 2 = 10 (n x 2) = 10 (2 x n)
commutative associative commutative
or
Yes, they are equivalent. If a number is used in place of n, for example, 3,
both expressions simplify to be the same number.
2 (10 x n)
10 (2 x n)
2 (10 x 3)
10 (2 x 3)
2 (30)
10 (6)
60
60
ALGEBRAIC RELATIONSHIPS—Grade 7
136
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Write an equivalent expression for the following.
2(x +3) + 4x – 2
Answer:
6x + 4, 2x +6 + 4x –2, etc. (There are various answers that could be
accepted.)
ALGEBRAIC RELATIONSHIPS—Grade 7
137
BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical
models
EXPECTATION
Model and solve
problems, using
multiple
representations
such as graphs,
tables, expressions,
and linear equations
EXAMPLE
Problem:
Simone’s family is planning their summer vacation. The website for the area they
plan to visit lists several attractions, including bike riding. The website lists two
companies that offer bike rentals. EZ Rider charges $4 plus $3 per hour that you
have the bike rented. Big Wheels Keep Turning charges $12 plus $1 per hour the
bikes are rented.
Which company should Simone recommend to her family? She should justify her
recommendation using a table and/or a graph.
DEFINITIONS:
Model-- a representation of a given situation that can be used to describe the present situation or predict some aspect of the
situation in the future. A mathematical model is a representation in the form of a mathematical quantity such as a number, a
vector, a formula, an inequality, a graph, a table of values, etc.14
14
Schraum’s A-Z Mathematics (2003). Berry, J, p.146, London, England, McGraw-Hill
ALGEBRAIC RELATIONSHIPS—Grade 7
138
Answer:
Table
EZ Rider
1
$7
2
10
3
13
4
16
5
19
6
22
7
25
8
28
Big Wheels
13
14
15
16
17
18
19
20
# of hours rented
Graph
If the family only wants to bike for a short time (1-3 hrs), Simone should suggest
EZ Rider. If the family plans on biking for a long time (5 or more hours), Simone
should suggest Big Wheels. If Simone’s family wants to bike for 4 hours, they
can rent from either company.
ALGEBRAIC RELATIONSHIPS—Grade 7
139
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Jimmy has $75 in his bank account right now. He is planning on saving $25
each week. His sister, Jenny, has $400 in her account right now, but has to
take out $40 each week to pay for her car. In how many weeks will both
Jimmy and Jenny have the same amount of money in their accounts? Use
equations, tables, or graphs to represent the information given and to solve
the problem.
Answer:
Equation
75 + 25x = 400 – 40 x
x = 5 weeks
Table
Jimmy
0
75
1
100
2
125
3
150
4
175
5
200
Jenny
400
360
320
280
240
200
# of weeks
ALGEBRAIC RELATIONSHIPS—Grade 7
140
Graph
Jimmy and Jenny will have the same amount of money in their accounts in 5
weeks.
ALGEBRAIC RELATIONSHIPS—Grade 7
141
BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
Compare situations
with constant or
varying rates of
change
EXAMPLE
Problem:
Sam is participating in a 10-mile race. The race has checkpoints every 2 miles.
Sam’s times at each checkpoint are recorded in the table below.
Distance (miles)
1
2
2
4
3
6
4
8
5
10
Time (Minutes)
15
25
40
45
57
Checkpoint
1. Between which two checkpoints was Sam going the fastest?
2. Between which two checkpoints was Sam going the slowest?
Answer:
1. Her fastest time was between Checkpoints 3 and 4, when her speed was 2
miles per 5 minutes, or .4 miles per minute.
2. Her slowest time was between Checkpoints 2 and 3 when the speed is 2 mile
per 15 minutes or .13 mile per minute.
Or
1. Her fastest time was between Checkpoints 3 and 4, when it took Sam 5
minutes for 2 miles, or 2.5 minutes for each mile.
2. Her slowest time was between Checkpoints 2 and 3 when it took her 15
minutes to go 2 miles, or 7.5 minutes for each mile.
ALGEBRAIC RELATIONSHIPS—Grade 7
142
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Term 1
Term 2
Term 3
Term 4
1. Complete the table for the total number of tiles used and the distance around
the tiles.
Number of Tiles
1
1
2
3
3
6
Distance around Tiles
4
8
12
Term
4
10
5
6
7
8
2. Will the distance around the tiles always be greater than the number of tiles?
Justify your answer using graphs or tables.
ALGEBRAIC RELATIONSHIPS—Grade 7
143
CONCEPT
EXPECTATION
EXAMPLE
1.
Number of Tiles
1
1
2
3
3
6
4
10
5
15
6
21
7
28
8
36
Distance around Tiles
4
8
12
16
20
24
28
32
Term
2.
40
35
30
25
ts
i
n20
U
Dista nce aroun d Tiles
Number of Tiles
15
10
5
0
0
1
2
3
4
5
6
7
8
9
Term
Before the 7th term, the distance around the tiles is larger than the number of tiles.
On the 7th term, the distance around the tiles and the number of tiles are the same.
After that the distance around the tiles is larger than the number of tiles. Looking at
the pattern, the number of tiles increases by 1 more than the last time, while the
distance around the tiles always increases by 4.
ALGEBRAIC RELATIONSHIPS—Grade 7
144
CONCEPT
EXPECTATION
EXAMPLE
Problem:
The table below shows the charges for renting and racing a go-cart.
Number of Laps
0
1
2
3
4
5
Price ($)
5
8
11
14
17
20
Which graph
BEST represents these prices? Explain your reasoning.
15
Answer:
D. The graph needs to intersect the y-axis at 5 and go through point (5, 20).
15
http://www.cde.ca.gov/ta/tg/sr/documents/rtqgr7math.pdf (California Department of Education)
ALGEBRAIC RELATIONSHIPS—Grade 7
145
ALGEBRAIC RELATIONSHIPS
Grade 8
BIG IDEA (1): Understand patterns, relations, and functions
B
CONCEPT
EXPECTATION
Create and
analyze
patterns
Generalize patterns
represented
graphically or
numerically with
words or symbolic
rules, using
explicit notation
EXAMPLE
Problem:
Given the following pattern, use words or an expression to describe the
pattern.
1, 9, 25, 49, 81, …
Answer:
This pattern takes each odd number, starting with 1, and squares it. The
expression would be (2n-1)2. In recursive notation, the pattern is (the square
root of Now + 2)2, with the first Now being one.
DEFINITIONS:
graphically-- the plot of points in the plane which constitute the graph of a given real function or a pictorial diagram depicting the
interdependence of variables. 16
numerically-- pertaining to numbers. 17
Rule (for a pattern): a general statement written in numbers or words that describes how to determine any term in a pattern.
16
17
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 102). New York, NY: McGraw-Hill, Inc.
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 170). New York, NY: McGraw-Hill, Inc.
ALGEBRAIC RELATIONSHIPS—Grade 8
146
Rules or generalizations for patterns may include both recursive and explicit notation. In the recursive form of pattern
generalization, the rules focuses on the rate of change from one element to the next. Example: Next = Now + 2; Next = Now x 4.
In the explicit form of pattern generalization, the formula or rule is related to the order of the terms in the sequence and focuses on
the relationship between the independent variable (the number representing a term in the sequence) and the dependent variable
(the number in the sequence). For example: 5t – 3; t – x; (t+1)x5; Words may also be used to write a rule in recursive or explicit
notation. Example: take the previous number and add two to get the next number; to find the total for any day multiply the day
times five and subtract three.
symbolic rules-- rules that use variables and numbers to describe a pattern or express a relationship.18
recursive notation-- a process that is inherently repetitive, with the result of each repetition usually depending upon those of the
previous repetition.19 Added Note – Recursive notation requires the previous term (or information to begin) to generate the next
term. This type of function follows a pattern that “reoccurs”.
Problem:
Given the following pattern:
3
4
6
18
19
5
8
10
Navigating Through Algebra in grades 6-8 (p.3) (2001). Reston, VA: National Council of Teachers of Mathematics.
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 210). New York, NY: McGraw-Hill, Inc.
ALGEBRAIC RELATIONSHIPS—Grade 8
147
1. Complete the table expressing width and length.
Width
1
3
2
4
3
5
Length
6
8
10
Term #
4
5
6
7
8
2. Write an expression in simplest form to determine the width of any rectangle
that fits this pattern.
3. Write an expression in simplest form to determine the length of any rectangle
that fits this pattern.
Answer:
1.
Width
1
3
2
4
3
5
4
6
5
7
6
8
7
9
8
10
Length
6
8
10
12
14
16
18
20
Term #
2. Width = n + 2, where n is the term of the rectangle. Using recursive notation,
the width would be Now plus one with the first Now being 3.
3. Length = 2n + 4, where n is the term of the rectangle. (This is the width
doubled or 2(n + 2).) Using recursive notation, the length would be Now plus
2 with the first Now being 6.
ALGEBRAIC RELATIONSHIPS—Grade 8
148
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Explain the following pattern. Then use your explanation to find the next three
numbers in the sequence.
1,1,2,3,5,8,…
Answer:
Each number is the sum of the two numbers before it. For example,
1 + 1= 2, then 1 + 2=3, then 2 + 3= 5, etc. The next three numbers in the sequence
would be 13 (5 + 8), 21 (8 + 13), and 34 (13 + 21). This is known as the Fibbonaci
sequence.
ALGEBRAIC RELATIONSHIPS—Grade 8
149
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Write an equation for each of the patterns below.
1.
x
1
2
3
4
5
5
11
17
23
29
1
2
3
4
5
1
-1
-3
-5
-7
1
2
3
4
5
4.5
5
5.5
6
6.5
y
2.
x
y
3.
x
y
Answers:
1. y = 6x – 1
ALGEBRAIC RELATIONSHIPS—Grade 8
2. y = -2x + 3
3. y = 1/2x + 4
150
CONCEPT
EXPECTATION
EXAMPLE
Problem:
The following table shows the cost for a one-day pass to the Amusement Park over
the last 6 years. If the trend in the cost of passes continues, what do you predict
would be the cost of a one-day pass in the year 2010? Explain your answer.
One-Day Pass Prices to the Amusement Park
Year
Price ($)
2000
2001
2002
2003
2004
2005
18.00
19.25
20.50
21.75
23.00
24.25
Answer:
I think the price will be around $30.50 in 2010. The increase in cost was
$1.25 each year. If it continues at that rate, it will go up $1.25 in 2006, 2007,
2008, 2009, 2010, or 5 x 1.25, which is $6.25. The price in 2005 was $24.25
+ $6.25 = $30.50.
ALGEBRAIC RELATIONSHIPS—Grade 8
151
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Use an equation to describe the pattern in the following graph.
Answer:
y = 2x + 5
ALGEBRAIC RELATIONSHIPS—Grade 8
152
BIG IDEA (1): Understand patterns, relations, and functions
CONCEPT
C
Classify
objects and
representations
EXPECTATION
Compare and
contrast various
forms of
representations of
patterns
EXAMPLE
Problem:
The daily concession stand profit per day at the County Fair can be
represented by the following:
Equation: Profit = $2.50v - $500, where v represents the number of visitors.
Table:
Number of Visitors
0
200
400
600
-500
0
500
1000
800
Profit($)
Graph:
1. How much profit would be expected for 450 visitors?
2. How much profit would be expected for 800 visitors?
3. Which helped most in getting your answers- the equation, the table or the
graph? Why?
ALGEBRAIC RELATIONSHIPS—Grade 8
153
DEFINITIONS: representations-- physical objects, drawings, charts, graphs, and symbols.20
Answer:
1. The expected profit for 450 visitors would be $625.
2. The expected profit for 800 visitors would be $1500.
3. To find the expected profit for 450 visitors, the equation would be the most
helpful, since it is easy to substitute for the variable and find the profit.
(2.50)(450) – 500=625
Or
To find the expected profit for 450 visitors, the table would be the most helpful. If
the profit increases $500 for each 200 visitors, that would be an increase of $250
for each 100 visitors, or $125 for each 50 visitors. I added 500 plus 125 and got
$625.
Or
To find the expected profit for 450 visitors, the graph would be the most helpful,
since you can find where 450 visitors is on the x axis and follow your finger up and
over. The profit would be $625.
-------To find the expected profit for 800 visitors, the equation would be the most helpful,
since it is easy to substitute for the variable and find the profit.
(2.50)(800) – 500=1500
20
Principles and Standards for School Mathematics (p.280). Reston, VA: Author. National Council of Teachers of Mathematics. (2000).
ALGEBRAIC RELATIONSHIPS—Grade 8
154
Or
To find the expected profit for 800 visitors, the table would be the most helpful. If
the profit increases $500 for each 200 visitors, I added 1000 and 500, which is
$1500.
Or
To find the expected profit for 800 visitors, the graph would be the most helpful,
since you can continue the graph for 800 visitors and follow your finger up and over.
The profit would be $1500.
Teachers Note: Any representation could be helpful based on personal preference, as long
as a good explanation is given. Either the equation or the table would be most helpful. It is
easy to see the pattern extension in the chart. The equation is easy to fill in to find the profit.
The graph may be more difficult since 800 visitors are not shown on the graph.
ALGEBRAIC RELATIONSHIPS—Grade 8
155
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Janet has $140 and she adds $50 to her savings at the end of each month. She
asks three of her friends when will she have $1000. Their work is shown below.
Ann:
Y = 140+ 50 X
1000 = 140 + 50 X
1000 - 140 = 140 - 140+ 50 X
860/50 = 50 X/50
17.2 = X
Mark:
Months
0
1
2
4
8
12
16
18
140
190
240
340
540
740
940
1040
Profit($)
Emily: Y = 140+ 50 X
ALGEBRAIC RELATIONSHIPS—Grade 8
156
Explain how each method is alike and how each will give a correct solution.
Answer:
The methods are alike because they are all looking at the equation
y =140 +50x. Each will give the solution by finding the month where y is first equal
to or greater than 1000. This occurs at the end of 18 months. The equation could
be used to find the correct solution, since X = 17.2 when you solve the equation, the
next month (18th month) Janet would have $1000. If the table is extended for an
amount of at least $1000, the answer would also be 18 months. Janet could also
see that it would take 18 months to save $1000 by looking at the graph for the
month where the line is above 1000.
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BIG IDEA (1): Understand patterns, relations, and functions
CONCEPT
D
Identify and
compare
functions
EXPECTATION
Identify fundtions as
linear or nonlear
from tables, graphs
or equations
EXAMPLE
Problem:
Given the equation y = 3x + 2, answer the following questions.
1.
2.
3.
4.
Would the line be straight or curved? How do you know?
Would the slope be positive or negative? How do you know?
What is the slope of the line? How do you know?
What is the y intercept? How do you know?
Answer:
1. It would be a straight line. There are no exponents in the equation.
2. The slope would be positive. The coefficient of x is positive.
3. The slope would go up three for every one you move over. The coefficient is
3.
4. The line would pass through the y axis at (0,2). The constant is 2.
DEFINITIONS:
linear equation-- an equation in two variables whose graph in a coordinate plane is a straight line. It is a first degree polynomial
equation.21
linear function-- a relationship between two variables that can be expressed as an equation and drawn as a straight line.22
21
Algebra to Go: A Mathematics Handbook (2000). Great Source Education Group Staff. (p. 493) , Wilmington, MA: Great Source Education
Group, Inc.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
Given the following 3 representations (of the same linear function), describe how
each representation shows the slope of this line.
Representation 1: y = 2x + 2
Representation 2:
Representation 3:
x
y
1
4
2
6
3
8
4
10
Answer:
In the equation y = 2x + 2, the coefficient of the x term is the slope, therefore the
slope is 2. On the graph, while x changes from 1 to 2, a difference of 1, the y value
increases by 2, so the slope is 2/1 or 2. For the table, each time x increases by 1, y
increases by 2 so the slope is 2/1 or 2.
22
Schraum’s A-Z Mathematics (2003). Berry, J, p.132, London, England, McGraw-Hill.
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BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
A
Represent
mathematical
situations
EXPECTATION
Use symbolic
algebra to represent
and solve problems
that involve linear
relationships,
including recursive
relationships
EXAMPLE
Problem:
The following table represents the cost of renting videos from Video Palace
where a membership fee is charged.
# Videos rented
0
1
2
3
4
5
10
12
14
16
18
20
Cost ($)
Use the table to answer the following questions:
1. What does the $10 represent for zero videos rented?
2. Use an equation to describe the cost for renting n videos.
3. Find the cost of renting 50 videos.
Answer:
1. The $10 represents the membership fee paid upfront.
2. Cost = 2n + 10, where n = the number of videos rented.
3. Cost of 50 videos = 2(50) + 10 = $110
DEFINITIONS:
symbolic rules-- rules that use variables and numbers to describe a pattern or express a relationship.23
recursive notation-- a process that is inherently repetitive, with the result of each repetition usually depending upon those of the
previous repetition. Added Note - Recursive notation requires the previous term (or information to begin) to generate the next
term. This type of function follows a pattern that “reoccurs”. 24
23
Navigating Through Algebra in grades 6-8 (p.3) (2001). Reston, VA: National Council of Teachers of Mathematics.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
Sam earns the same amount each week babysitting. Each week she saves all but
$10 of her earnings. If she has saved $160 after 8 weeks, how much does she earn
per week? Set up an equation to model the situation (where
x = the amount earned in a week). Then use your equation to solve for x.
Answer:
8(x – 10)= 160
8x –80 =160
8x –80 +80 = 160 + 80
8x /8 = 240/8
x = $30 per week
x =30
Problem:
The weight of three cats is less than the weight of 2 dogs. Write an inequality to
match the situation. Let c= the weight of the cats and d=the weight of the dogs.
Answer:
24
McGraw-Hill Dictionary of Mathematics (1997), Parker, S. editor. (p. 210). New York, NY: McGraw-Hill, Inc..
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3c < 2d, where c = weight of the cats and d = weight of the dogs
Problem:
While Ellen is shopping she sees the following sign.
Shirts on sale! $8 each.
(Sales Tax included!)
Ellen cannot spend over $35. Which inequality represents the situation?
A. 8x > 35
B. 8x < 35
C. 8x > 35
D. 8x < 35
Answer: B. 8x < 35
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
You need to order a satellite system for your new home. Super Satellite charges a
$50 installation fee and $65 a month for your satellite.
1. Write an equation to calculate the cost of having satellite.
2. Determine the cost of satellite per year.
Answer:
1. C=65m +50, where C=cost and m=number of months.
2. The cost for 1 year would be 65(12) + 50 or $830.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
As Roger drove by Premier Missouri Bank today, the temperature on the sign read
35 degrees Celsius. Is this an accurate temperature for July? Provide work that
shows how you arrived at your answer.
Formula: F = 9/5C + 32, where F=Degree Fahrenheit and
C= Degree Celsius
Answer:
Yes, this is probably an accurate temperature for July. 35 degrees Celsius is
the same as 95 degrees Fahrenheit if you use the formula.
F=9/5(35) + 32
F= 95
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
In preparing for their spring dance, the Student Council purchased 5 tropical trees to
create a backdrop for the DJ. The total bill for the trees was $251.25,
including $16.99 sales tax. The Student Council Sponsor wants to know the cost of
one tropical tree.
To find the cost of one tree, solve the equation 5t + 16.99 =251.24, where t
represents the number of tropical trees. Provide the work that shows how you
arrived at your answer.
Answer:
5t + 16.99 = 251.24
5t + 16.99 –16.99 = 251.24 – 16.99
5t = 234.25
5t/5 = 234.25/5
t = 46.85
The cost of each tree is $46.85.
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BIG IDEA (2): Represent and analyze mathematical situations and structures using algebraic symbols
CONCEPT
B
Describe
and use
mathematical
manipulation
EXPECTATION
Use properties to
generate equivalent
forms for simple
algebraic
expressions that
include all rationals
EXAMPLE
Problem:
A rectangle has a length that is three less than twice the width.
1. Using the variable w to denote the width, write an equation for the
length in terms of w.
2. Then write two different expressions that represent how to find the
perimeter of this particular rectangle.
3. Simplify both expressions to show they are equivalent.
Answer:
1. Equation: Length = 2w-3
2. Perimeter:
Add up all sides
w + (2w-3) + w + (2w-3)
3.
= w + 2w + 2w + w – 3 – 3 = 6w - 6
2 times the width and 2 times the length
2w + 2(2w-3)
= 2w + 4w – 6
= 6w - 6
Add the width and length and then double it
2( w + 2w-3)
ALGEBRAIC RELATIONSHIPS—Grade 8
= 2w + 4w – 6
= 6w - 6
166
CONCEPT
EXPECTATION
EXAMPLE
Problem:
Write an equivalent expression for 3n4.
Answer:
3 x n x n x n x n, n4 + n4 + n4, 3 x n2 x n2, etc.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
Write the following expression in simplest form. Show work to support your
answer.
4x + 3y – 6(x + 5) + - 2y
Answer:
4x + 3y – 6(x + 5) + - 2y
4x + 3y – 6x – 30 – 2y
4x – 6x + 3y –2y – 30
-2x + y - 30
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BIG IDEA (3): Use mathematical models to represent and understand quantitative relationships
CONCEPT
A
Use
mathematical models
EXPECTATION
Model and solve
problems, using
multiple
representations such
as graphs, tables,
and linear equations
EXAMPLE
Problem:
The school choir is selling boxes of greeting cards to raise money for a trip.
The table represents the amount of profit they receive for selling various
numbers of boxes. Use either an equation or a graph to determine how
many boxes they need to sell in order to make $300 profit.
Boxes of Cards
10
20
30
40
50
-10
10
30
50
70
Profit($)
DEFINITIONS:
model-- a representation of a given situation that can be used to describe the present situation or predict some aspect of the
situation in the future. A mathematical model is a representation in the form of a mathematical quantity such as a number, a
vector, a formula, an inequality, a graph, a table of values, etc.25
25
Schraum’s A-Z Mathematics (2003). Berry, J, p.146, London, England, McGraw-Hill
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Answer:
Equation:
Let b = the number of boxes sold.
Profit = 2b – 30
If 300 = 2b – 30, then b = 165 boxes
Graph:
They need to sell 165 boxes.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
It costs $3 to park at Buoy beach if you buy a special sticker. A sticker costs $50
and can be used all summer. It cost $8 to park without a sticker.
1. Write 2 equations to model your cost for parking at the beach “n” times. One
equation if you have a sticker (y1) and one if you do not have a sticker ( y 2).
2. Determine whether or not you should buy a sticker. You may use graphs,
tables, or the equations.
Answer:
1. y1 = 50 + 3x
y2 = 8x
where x is the number of times you park
2. Equation:
50 + 3x = 8x x = 10 If you park 10 times, the cost will be the same either way. If
you park fewer than 10 times it is cheaper to not purchase the special sticker.
However, if you park more than 10 times it is cheaper to purchase the special
sticker.
Table:
# of times
parking
Sticker
No Sticker
ALGEBRAIC RELATIONSHIPS—Grade 8
1
2
3
4
5
6
7
8
9
10
11
53
8
56
16
59
24
62
32
65
40
68
48
71
56
74
64
77
72
80
80
83
88
171
If you park 10 times, the cost will be the same either way. If you park fewer than 10
times it is cheaper to not purchase the special sticker. However, if you park more
than 10 times it is cheaper to purchase the special sticker.
Graph
If you park 10 times, the cost will be the same either way. If you park fewer than 10
times it is cheaper to not purchase the special sticker. However, if you park more
than 10 times it is cheaper to purchase the special sticker.
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BIG IDEA (4): Analyze change in various contexts
CONCEPT
A
Analyze
change
EXPECTATION
Analyze the nature
of changes
(including slope and
intercepts) in
quantities in linear
relationships
EXAMPLE
Problem:
Lynda borrowed $ 175 from her parents and is paying them back $10 per week. Her
sister Maria borrowed $ 200 and pays back $15 per week.
1. Write equations to model each girl’s pay.
2. Use the equations to determine who would pay back their parents first.
Answer:
1. Equations:
L= 175 – 10w and M= 200 – 15w, where L= the amount Lynda owes, M= the amount
Maria owes, and w = # of weeks.
2. To find who would pay their parents back first:
I used 0 as the amount owed and solved for the variable. Using the equation, I
found it will take Lynda 18 weeks to pay back the $175–
0 = 175 – 10(17.5). It will take Maria 14 weeks to pay back her parents, since 0 =
200-15(13.3). Therefore, Maria will pay back her parents first.
Or
I noticed the slope for Maria is –15 and the slope for Lynda is only –10, which means
Maria is paying back her parents at a much faster rate. The y-intercept for Maria is
only slightly larger (she owed only $25 more than Lynda). Therefore, since the slope
for Maria’s equation is decreasing at a faster rate, she would pay her parents back
in less time.
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CONCEPT
EXPECTATION
EXAMPLE
Problem:
Carla, Drey, and Robert decide to get jobs after school at different restaurants
delivering pizza to earn extra spending money. The following graph shows the pay
plan for each.
1. Who gets paid the least for each additional pizza delivered? How can you tell?
2. How much does each boy get paid for each added pizza delivered?
3. For what number of pizzas will Drey and Carla be paid equal amounts for the
evening?
4. What do the y-intercepts or values where the lines touch the y-axis mean?
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5. When is Carla’s pay plan the best?
6. Whose pay plan is best when 12 or more pizzas are delivered on any given
night? How can you tell on the graph?
Answers:
1. Carla gets paid the least for each additional pizza. The graph of her earnings
has the smallest slope (the line increases at the smallest rate).
2. Carla--$1 per pizza
Drey --$2 per pizza
Robert--$3 per pizza
3. Drey and Carla will be paid equal amounts when they both sell 7 pizzas.
4. The y-intercepts tell you how much each person gets paid before they sells any
pizzas.
5. Carla’s pay plan is the best if only four or less pizzas are sold.
6. Robert’s pay plan is the best when 12 or more pizzas are delivered. This is
because the slope of his line increases at the highest rate.
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