Sixth Grade
Algebra
Standard 6-3: The student will demonstrate through the
mathematical processes an understanding of writing,
interpreting, and using mathematical expressions, equations, and inequalities.
The indicators for this standard
are grouped by the following major concepts:
The indicators that support each of those major concepts
and an explanation of the essential learning for each major concept follows.
Patterns, Relationships, and
Functions
Indicator
6-3.1 Analyze numeric and algebraic patterns and
pattern relationships.
The study
of patterns is extensive throughout elementary school. Students begin the process of transitioning
from the concrete to the abstract and symbolic in the 5th grade, and
learn to represent patterns in words, symbols, and algebraic
expressions/equations for the first time.
In 6th grade, students progress to analyzing numeric and
algebraic patterns and representing them in simple expression, equations, and
inequalities. The focus in the 6th
grade should be on patterns that relate to linear functions which have a
constant rate of change. This will lay
the foundation for the study of slope in the 7th and 8th grades.
Teachers should encourage 6th grade students to explain their observations of a pattern in their own words. This verbalization will enable students to begin to write a mathematical rule for a pattern. Students should also be encouraged to represent patterns using tables, graphs, and equations. Discussions should be used to determine which representation makes it easier to describe the pattern, extend the pattern, and/or make predictions with the pattern. Students will need an in-depth experience discussing real world patterns and patterns which provide concrete examples before they can begin to represent them symbolically. Teachers will need to model many examples that involve moving from the concrete to the symbolic.
Representations, Properties, and Proportional Reasoning
Indicators
6-3.3 Represent algebraic relationships with
variables in expressions, simple equations, and simple inequalities.
6-3.4 Use the commutative, associative, and
distributive properties to show that two expressions are equivalent.
In 4th
and 5th grades, students used variables to write a mathematical
expression in symbolic form. This knowledge should be reinforced in the 6th
grade and refined to understand that variables are more than letters or symbols
that represent a number. The focus in 6th
grade is for students to see the "use" of variables for a specific
unknown in representing algebraic relationships. This focus is then extended into writing simple
one-step equations and inequalities that model a mathematical situation. This
standard is the foundational step for algebraic concepts taught in 7th
and 8th grades. Seventh grade will reinforce these concepts and focus
on the study of proportional relationships with graphs, tables, and equations.
Please note
that a more in depth understanding of the concept of inequality is crucial in
the 6th grade. Students have
been using the inequality symbols >, ≥, <, and ≤ since the 2nd
grade in grade appropriate applications.
It is imperative at this level that students' think of an inequality as
much more than "the alligator eats the biggest piece". Sixth grade students must be encouraged to
view inequalities as a way to describe and represent a relationship
between/among quantities.
Please note that there is often a
common misunderstanding in regards to the concept of equivalence in the 6th
grade. Prior to sixth grade, students may
have simply written an answer after the equal sign. Now, students must clearly
understand that the equal sign does not mean perform an operation, it means
that there is a relationship of equivalence between the two expressions on
either side of that equal sign. Also, students should demonstrate a clear
understanding of the concepts of equivalence by using the commutative,
associative, and distributive properties. These properties should be used in
situations that involve all operations with whole numbers, addition and
subtraction of fractions and decimals, and powers of 10 through 106.
Connections
to:
Other 6th
Grade Indicators:
How these
indicators can be connected is expressly described in the above paragraph.
6-2.4 Apply an algorithm to add and subtract
fractions.
6-2.7 Apply strategies and procedures to determine
values of powers of 10, up to 106.
Solve Mathematical Situations
Indicators
6-3.2 Apply order of operations to simplify
whole-number expressions.
6-3.5
Use inverse operations to solve one-step
equations that have whole-number
solutions and variables with whole-number
coefficients.
Please note the above indicators apply to whole-numbers ONLY and
does not include negatives, fractions or decimals.
Sixth grade is the first time students
are introduced to using order of operations to evaluate a numerical
expression. In 4th and 5th
grades, students used variables to write mathematical expressions in symbolic
form and to write an open sentence representing a given mathematical
relationship. However, students did not evaluate expressions. "Evaluating
an expression", “Solve the expression” and “Find the solution to the
expression” each with the same meaning will be new and important phrases for
students to understand. Please note that
evaluating at this grade level is ONLY to be done with WHOLE-NUMBER expressions.
Order of operations is sometimes a
difficult concept for middle school students to understand and this may be due
to the way it is often introduced. It is
highly recommended that the introductory lessons help students discover and
understand why an agreement for the order of operations is necessary and
important. Many students are simply introduced to the concept with the phrase “Please Excuse My Dear Aunt Sally”, often
referred to as PEMDAS. While this is a
helpful mnemonic devise, it can easily lead to some common misconceptions. Many students come to believe that
multplication is always done before division and that addition is always done
before subtraction. By being taught this
mnemonic device, students do not fully understand that the operations of
multiplication and division (or addition and subtraction) are performed in the
order that they appear, from left to right.
After students have been given opportunites to discover why an agreement
for the order of operations is necessary, it is sugggested that order of
operations be introduced using a table format with students being taught that the
higher in the table an operation is, the more important it is and must be done
first. The table format enables students
to see that all grouping symbols must be done first – not just the “P” for
parentheses in the mnemonic. After students have mastered an understanding of
this format, the PEMDAS mnemonic may then be introduced if desired.
Order
of Operations
|
Level 1 |
{ [ ( ) ]
} All Grouping Symbols |
|
Level 2 |
Exponents |
|
Level 3 |
Multiplication
and Division Proceeding from
Left to Right |
|
Level 4 |
Addition and
Subtraction Proceeding from
Left to Right |
Although students
work with informal equations in the early grades, sixth grade is the first year
they are introduced to both inverse operations and their use in solving
one-step equations. As stated earlier,
there is often a common misunderstanding in regards to the concept of
equivalence in the 6th grade.
Prior to 6th grade, students have simply written an answer
after the equal sign and often have the misconception that an equal sign means
“provide an answer after performing an operation” rather than seeing the equal
sign as an indicator of equality. It is
imperative that the 6th grade student understand that the equal sign does not
mean perform an operation, but that there is a relationship of equivalence
between the two terms on either side of that equal sign. Therefore, it is highly recommended that students
be given the opportunity to review and discuss the concept of equality to
identify and address any misconceptions prior to the introduction of solving
one-step equations.
Because
this is the first experience students have had with the additive inverse (the
sum of a number and its opposite is zero), they will need sufficient
opportunities to develop an understanding of and demonstrate the usefulness of
this property. The connection must be
made between the inverse operation (additive identity) and its use to isolate
the variable. Keep in mind that this is
the first time that the concept of solving equations is introduced, therefore
it is extremely important that students be given the opportunity to build on
prior knowledge, use manipulatives to model the equations and the process of solving
one-step equations, and then gradually progress to solving numerically by using
inverse operations. Although this is the
first year that solving one-step equations using inverse operations is
introduced, it is expected that students reach fluency with this concept before
leaving the 6th grade.