Equations with Letter Variables
Strand: Patterns
and Relations (Variables and Equations)
Outcomes: 3 and 4
Strand: Shape and Space
(Measurement)
Outcome: 3
Step 1: Identify Outcomes to Address
Guiding Questions
- What do I want my students to learn?
- What can my students currently understand and do?
- What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?
See Sequence of Outcomes from the Program of Studies
Big Ideas
Patterns are used to develop mathematical
concepts and are found in everyday
contexts. The various representations
of patterns, including symbols and
variables, provide valuable tools
in making generalizations of mathematical
relationships. Some characteristics
of patterns include the following
(adapted from Van de Walle and Lovin
2006, pp. 265, 268).
- "A pattern must involve
some repetition or regularity" (Small
2009, p. 3).
- Patterns using concrete and pictorial
representations can be translated
into patterns using numbers to
represent the quantity in each
step of the pattern. The steps
in a pattern are often translated
as the sequence of items in the
pattern.
- Pattern rules are used to generalize
relationships in patterns. These
rules can be recursive and functional.
- A recursive relationship describes
how a pattern changes from one
step to another step. It describes
the evolution of the pattern by
stating the first element in the
pattern together with an expression
that explains what you do to the
previous number in the pattern
to get the next one.
- A functional relationship is
a rule that determines the number
of elements in a step by using
the number of the step; i.e., a
rule that explains what you do
to the step number to get the value
of the pattern for that step. In
other words, for every number input,
there is only one output using
the function rule.
- Variables are used to describe
generalized relationships in the
form of an expression or an equation
(formula).
- "Functional relationships
can be expressed in real contexts,
graphs, algebraic equations, tables
and words. Each representation
for a given function is simply
a different way of expressing the
same idea. Each representation
provides a different view of the
function. The value of a particular
representation depends on its purpose" (Van
de Walle and Lovin 2006, p. 284).
Algebraic reasoning is directly
related to patterns because this
reasoning focuses on making generalizations
based on mathematical experiences
and recording these generalizations
using symbols or variables (Van de
Walle and Lovin 2006, p. 281).
"A variable is a symbol that
can stand for any one of a set of
numbers or objects" (Van de
Walle and Lovin 2006, p. 274). Variables
are used in different ways to generalize
concepts as mathematical literacy
is developed. They can be used (Cathcart,
Pothier and Vance 1994, p. 368):
- in equations as unknown numbers;
e.g., 4 + x = 6
- to describe mathematical properties;
e.g., a + b = b + a
- to describe functions; e.g.,
Input (n): 1, 2, 3, 4...;
and
Output
(3n): 3, 6, 9, 12 …
- in formulas to show relationships;
e.g., A = L × W.
By investigating patterns, students:
- solve problems
- develop understandings of important
mathematical concepts and relationships
- investigate the relationships
among quantities (variables) in
a pattern
- generalize patterns using words
and variables
- extend and connect patterns
- construct understandings of functions.
(National Council of Teachers of
Mathematics 1991, p. 1.)
As students analyze the structure
of patterns and organize the information
systematically, they "use their
analysis to develop generalizations
about the mathematical relationships
in the patterns" (National Council
of Teachers of Mathematics 2000,
p. 159).
Students use patterns to develop
understanding of measurement concepts,
including perimeter of polygons,
area of rectangles and volume of
right rectangular prisms. They generalize
the patterns using words and variables
that can be written as a formula, "a
special algebraic equation that shows
a relationship between two or more
different quantities" (Small
2009, p. 9).
Formulas for finding the perimeter
and area of 2-D shapes and volume
of 3-D objects provide a method of
measuring by using only measures
of length (Van de Walle and Lovin
2006, p. 230).
In using any type of measurement
such as length, area or volume, it
is important to discuss the similarities
in developing understanding of the
different measures; e.g., first identify
the attribute to be measured, then
choose an appropriate unit and finally
compare that unit to the object being
measured (National Council of Teachers
of Mathematics 2000, p. 171). It
is important to understand the attribute
(perimeter, area or volume) before
measuring.
Definitions
Perimeter
"Perimeter is the distance around a polygon.
Perimeter is the sum of the lengths of
all of the sides of
a polygon"
(http://learnalberta.ca/content/memg/index.html).
The standard units for measuring
perimeter are linear units such as "mm," "cm," "m" and "km."
Area
Area is "a measure of the space
inside a region or how much it takes
to cover a region" (Van de Walle
and Lovin 2006, p. 234).
The standard
units for measuring area are square
units such as "mm2," "cm2," "m2" and "km2."
To calculate the area of any rectangle,
multiply the length by the width.
Volume
"Volume is the amount of space that an object takes up" (Van de Walle
and Lovin 2006, p. 244).
The standard units for measuring
volume are cubic units such as "mm3," "cm3," "m3" and "km3."
To calculate the volume of any right
prism, multiply the area of the base
by the height of the prism. Another
way to calculate the volume of a
right rectangular prism is to multiply
the length by the width by the height.
Key ideas in understanding the attribute
of area are described below. Many
of these ideas also apply to perimeter
and volume.
- Conservation—an object
retains its size when the orientation
is changed or it is rearranged
by subdividing it in any way.
- Iteration—the repetitive
use of identical non-standard or
standard units of area to entirely
cover the surface of the region.
- Tiling—the units used to
measure the area of a region must
not overlap and must completely
cover the region, leaving no gaps.
- Additivity—add the measures
of the area for each part of a
region to obtain the measure of
the entire region.
- Proportionality—there is
an inverse relationship between
the size of the unit used to measure
area and the number of units needed
to measure the area of a given
region; i.e., the smaller the unit,
the more you need to measure the
area of a given region.
- Congruence—comparison of
the area of two regions can be
done by superimposing one region
on the other region, subdividing
and rearranging, as necessary.
- Transitivity—when direct
comparison of two areas is not
possible, a third item is used
that allows comparison; e.g., to
compare the area of two windows,
find the area of one window using
non-standard or standard units
and compare that measure with the
area of the other window, especially
if A = B and B = C,
then A = C; similarly
for inequalities.
- Standardization—using standard
units for measuring area such as "cm2" and "m2" facilitates
communication of measures globally.
- Unit/unit-attribute relations—units
used for measuring area must relate
to area; e.g., "cm2" must
be used to measure area and not "cm" or "ml."
(Alberta Education 2006, Research
section pp. 2–4.)
Grade 6
= b, a ≠ 0