| Proportional Reasoning | Algebra & Functions | Other Math Topics |
MMAP Design Units, together with Extensions and Investigations, form
a balanced mathematical “diet” for middle schoolers. While all
of the NCTM Standards are addressed, and students have an opportunity to
develop important skills and concepts in each one, the mathematical content
of MMAP hinges on two central areas: proportional reasoning and algebra/function.
Middle school is the time for important transitions within each of these
areas. Consequently, these math topics are revisited again and again over
a series of MMAP units in a variety of real-world and mathematical contexts.
Extensions allow students access to the established mathematical tools and
notations used in each, as well as the opportunity to make explicit some
of the implicit understanding developed within the main units.
Proportional Reasoning
Middle schoolers must make the transition from additive to multiplicative
reasoning about relationships between quantities. This means they need to
be able to use multiplication, ratios and proportions to express relationships
between quantities and make predictions. A base-line problem in this area:
Given a 3 by 5 photo, what will its length be if it is enlarged so that
the width is 6 inches instead of 3? Many young students will answer that
the length is 8 inches, adding 3 onto 5, not recognizing that 3 inches has
been multiplied by two to enlarge the photo. While this transition to using
multiplication is never fully complete (most adults can be “tricked”
into additive reasoning with the right problem), middle schoolers need to
develop the tools to express and manipulate multiplicative relationships.
Important tools include proportion and percents.
Proportional reasoning is addressed through a range of MMAP units over the
course of sixth through eighth grade. When students use ArchiTech in, for
example, the Dream Home unit, they must grapple with scale, the ratio that
defines proportionality between the real world and the paper or screen world.
After Dream Home, students who do the extension, Problems with Proportions,
map their experiences with ArchiTech onto standard proportional notation
and then develop and verify rules for manipulating the proportion equation.
In contrast, HabiTech units develop proportion from a completely different
slant. Here, ratios represented by decimals and percents represent the growth
rates of populations over time. Students compare and manipulate growth ratios
to align populations with historical data. Other extensions help students
examine directly and indirectly proportional co-variation.
By the end of middle-school, MMAP-using students will have had experience
using proportions, ratio fractions and percents to express the relationship
between quantities and find missing quantities. They will be able to recognize
and describe directly and indirectly proportional co-variation. They will
have used and developed their proportional reasoning skills in mathematical
and non-mathematical contexts.
Algebra and Functions
Middle school is the time for intensive work on functions. Central are the
abilities to track changes in two variables and to describe the change of
one in terms of the others. A variety of representations of functions needs
to be developed in middle school, from tables to algebraic formulas. By
the time students leave middle school, they should be comfortable with some
parts of standard algebraic notation, with a rich understanding of variable
and the ability to use functions to solve problems.
The core MMAP Design Units address the areas of algebra and functions not
only from different real-world contexts but from different mathematical
vantage points. Extensions help students solidify and codify their experiences
in the Design Units, introducing and re-enforcing standard mathematical
notation. Units based on ArchiTech software, such as the Antarctica Project,
provide opportunities to develop concepts of variable in two ways: The ArchiTech
sliders, by which “global variables” such as outside temperature
and insulation values can be manipulated, represent variables that are under
the student’s control; and students build tables comparing two variables
such as outside insulation and cost to build. HabiTech-based units, such
as Wolves and Caribou, allow opportunities for students to represent their
knowledge of the world through algebraic functions and then track the results
of those functions over time. Facility with algebraic notation (and with
full words used for variables), is developed through using the software
and the accompanying math opportunities. Finally, units based on Coding
Toolbox allow students to investigate functions from a more purely mathematical
vantage point. Functions are used to build codes, the real-world context,
and then learning about function properties–the relationship between
domain and range, for example–enhances students’ ability to break
codes. A standard algebraic notation for functions–for example, y=x2+3–is
used in the Coding Toolbox.
MMAP Extensions ease students’ transition from informal, implicit use
of functions to understanding and flexibly using the standard symbolic notations
associated with functions. In From Patterns to Functions, students build
on their familiarity with geometric patterns to simple function tables and
graphs. They compare multiplicative and additive linear functions. Functioning
in the Real World examines co-variation in a variety of real-world settings,
as students learn to recognize how a change in one variable affects another.
Connecting to Algebra introduces standard terms and notation of algebra
as students write equations to represent patterns and situations.
By the end of middle school, MMAP-using students will have had experience
designing investigations of co-variation, using functions to represent real-world
phenomena, and exploring the mathematical properties of functions. They
will be able to use standard algebraic notation as well as verbal description
of functions, tables and graphs to solve problems.
Other Math Topics
MMAP materials cover all of the NCTM’s curriculum standards for middle
school. The chart on page 11 shows that coverage. We are not claiming that
all of our units must be connected to proportional or algebraic reasoning,
but we exploit that connection whenever possible to reinforce these two
broad concept areas for middle schoolers. Investigations allow students
to explore, for example, three dimensional geometry in a somewhat fanciful
context–Mobile Math–as well as develop geometric vocabulary and
explore “nets” of 3-D shapes as pure math objects.
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