# MATH Connections: A Secondary Core Curriculum

**Review Materials:**

Field-test materials for all three grades were reviewed. A
Teacher Resource

Package will be available and was reviewed. The package includes a Teacher

Edition for each grade, which contains student pages, Teacher Commentary,

suggestions about pacing, blackline masters, and solutions to all problems. The

package also contains "Books in Brief," which describe the role present

materials and its relation to past and to future material. Also included in the

package is a set of student assessments. A second set of assessments is also

available. Ancillary materials are under development including booklets with

challenging problems and projects.

**Format/Description: **This is a complete three-year secondary school
mathematics curriculum

intended for all students. The curriculum is integrated; meaning that material
from several areas of

mathematics are included in any given year. For our purposes, we have centered
our description

around six strands: **algebra/number/function, geometry, trigonometry,
probability and statistics,
discrete mathematics and logic and reasoning.** Not all strands appear in each
year. The student

material is content-centered. That is, mathematical topics take center stage in each section usually

followed by concepts applied or connected to various contextual and "real-world" situations. Each

grade level is divided into two books, a. and b, for ease of using; especially with block scheduling. The

a. section is meant to precede the b. section. Each section is divided into chapters and each chapter is

divided into several sub-sections. Each sub-section begins with stated learning objectives for that subsection

and several student activities within explorations followed by a problem set. The activities are

coded with icons indicating either a discussion topic, a writing topic, or an activity that should be done

before proceeding. Some sub-sections contain ideas for longer student projects. The margins of the

student materials contain

**Thinking Tips, About Symbols,**and

**About Words**(notes that detail how

some everyday words have more specific meanings in mathematics). Appendices for each level detail

technology information helping students learn to use a TI-82 (83) Graphing Calculator, use a

spreadsheet, program a TI-82 (83), etc.

The Teacher Commentary includes an overview and general comments on the student materials as well

as suggestions concerning the implementation of the material in the classroom. It contains comments

and answers to the activities and problems contained in each sub-section. Blackline Masters are

included, also.

**Pedagogy:** The curriculum delivery in the classroom
is based on directed student activities where

students "try out" approaches suggested by the materials or apply concepts with
which they are

working. There are follow-up homework problems. The classroom activities may be
directed by the

teacher using the text along with a more conventional teaching approach or
students can read the text

and do activities on their own with the teacher acting as both a guide and
provider of additional material

or explanation. These materials have been implemented with students working
individually as well as

with students working in groups. Sometimes the Teacher Commentary or student
materials suggest

student interaction such as comparing answers to activities or general
discussion. Other times, choices

are left to the teacher.

**Technology:** Graphing calculators are essential for the delivery of the
curriculum. Such calculators

are used in the majority of the material. The TI-82 (83) series is supported by
the student materials. In

addition, spreadsheet capabilities are used in some units. For example,
spreadsheets are used in the

material on linear programming. (The use of Excel is supported in an appendix to
that material.) In

addition, students will use technology to produce scatterplots, fit regression
lines, graph functions,

observe patterns, and do tedious calculations. Some programming on graphing
calculators is included.

The availability of a dynamic geometry program such as the Geometer's Sketchpad
is suggested for

some of the geometry material.

**Assessment:** Assessments are provided for each sub-section of each book.
These assessments

include "quizzes" for each chapter and end-of-section assessments.

**Content Overview: MATH** Connections is an integrated curriculum in the
sense that content

associated with algebra, geometry, and data analysis appears at each level.
(Note that some coordinate

geometry regarding straight lines appears in the first year of the curriculum
and is listed under the

**algebra/number/function** section for Year 1 below.) In addition, this
review identifies several strands

that occur in the curriculum taken as a whole: algebra/number/function,
geometry, trigonometry,

probability and statistics, discrete mathematics and logic and reasoning. Not
all strands appear in each

year. See below for details concerning the appearance of these strands in the
various years. It may be

noted that Year 1 material is heavily concentrated in algebra, Year 2 material
is heavily concentrated in

geometry, and Year 3 contains considerable material in pre-calculus and discrete
mathematics. While

the above emphases suggest a more traditional division of material, it must be
noted that many of the

concepts within these topics are non-traditional. Moreover, "traditional" topics
associated with these

emphases may appear elsewhere in the curriculum. For example, axiomatic proof
involving geometry is

covered in Year 3 despite the emphasis on geometry in Year 2. As a second
example, while the

concept of function may be more traditionally included in a course on
pre-calculus, the topic of functions

appears throughout the entire **MATH** Connections curriculum. In the words
of the developers, **"a
major goal of the curriculum is the development of higher-order thinking
skills."** The

developmental approach most often used is for the students to actively explore a concept in order to

develop experimental evidence and/or recognize a pattern, be assured the pattern is accurate, represent

the pattern by a symbolic formula, and then apply the formula in real-world situations. There is a strong

symbolic component throughout the curriculum. It is the philosophy of the developers that a great deal

of mathematical power comes from being able to " understand formulas and be able to transfer use them

to a variety of contexts." Graphical and tabular representations are also included throughout. Many

times students are asked to think about and discuss or write about what they are doing and why they

are doing it. The problem set reinforces and extends, to some degree, the material in the sub-section.

It usually does not contain traditional simple drill and practice problems. It often does contain additional

material and/or cases not covered in the preceding narrative as well as applications. Our description of

what students completing this curriculum will know and be able to do is given in terms of the strands

mentioned above. Some topics, such as coordinate geometry, could be listed under more than one

strand. However, when topics are closely integrated, the decision has been made to list them only

under one strand in each year. Moreover, the description below does not necessarily describe the

order in which topics are encountered in the curriculum within each year. For the most part, topics

which are mentioned or worked with briefly but not emphasized to any degree are omitted. However, in

some places we say, simply, that students are "exposed" to such topics.

Year 1

**Algebra/number/function:**Using appropriate variables and constants, students will be able to write an

equation to represent one quantity in terms of one or more other quantities. They will be able to

evaluate equations or formulas, including exponential equations, for specified values. Given a table of

values, they will look for any patterns and, if found, write a formula to represent a generalization of the

pattern. They will be able to use the correct order of operations to compute an arithmetic expression or

to simplify an algebraic expression. Students will be able to solve an equation in one variable, by

applying the appropriate laws of algebra, including the commutative and associative laws of addition and

multiplication and the distributive law. They will be able to write an exponential expression to represent

a real-world situation involving exponential growth or decay. Students will be able to construct a

rectangular coordinate system and plot points that represent ordered pairs of coordinates. Given the

coordinates of two points, they will be able to compute the slope of a line. They will recognize that a

vertical line has no slope. They will be able to write a linear equation in the form y=mx+b when given

two points or the slope and y-intercept. Using a graphing calculator, students will investigate the effect

of a change in a or b on the graph of a linear equation y=ax+b. They will also explore the effect of

integral changes in integral exponents on the graph of y=x

^{a}, for a ·1. They will be able to graph a linear

equation in the form y=mx+b (with and without the aid of a calculator). They will compare graphs of

linear equations in terms of their slopes. Given a real-world situation, they will be able to write a linear

equation of the form y=mx+b, draw its graph, and use both in the analysis of the problem. Using a tree

diagram to organize possible outcomes, students will develop an algorithm for an efficient way to play a

number guessing game and apply it in solving real-world problems. They will graph inequalities in one

variable on a number line. Students will utilize graphs and/or tables of values to compare two sets of

data. By plotting points, they will graph two linear equations on the same set of axes, find the point of

intersection, and make comparisons. They will also use the calculator to graph pairs of equations (some

non-linear) and find their point(s) of intersection. They will solve systems of two linear equations of the

form y=mx+b algebraically (substitution method). They will graph equations of the form ax+by=c by

plotting points (including the x- and y-intercepts) and solve systems of these equations graphically.

Students will be introduced to a linear programming problem and its terminology (e.g. constraints, region

of feasible solutions). They will graph a system of linear inequalities and investigate where maximum

profit occurs within the feasible region.

Students will be able to determine whether or not a situation describes a function. They will be able to

identify the domain and range of many functions, use function notation, and evaluate a function at

specified values. They will write a formula for a sequence of numbers and use it to find the n

^{th}term in

the sequence. They will use a spreadsheet to generate a sequence. They will determine whether or not

two functions are equal on the domain of counting numbers. Students will be able to make a table

and/or graph to represent a function, including step functions. They will write a function and create a

simple program for the calculator to convert temperature scales (Celsius to Fahrenheit). They will write

a linear or exponential growth (involving compound interest or population growth) function for real-world

situations and graph it on the coordinate plane. They will also graph a function on the calculator

and use the trace feature to evaluate it at specific values. Given two or more functions, students will be

able to find and evaluate a composite function. They will be able to write a function as the composite of

two or more simpler functions. They will be able to describe some real-world processes as composite

functions. They will examine the laws of commutativity and associativity with respect to the operation of

composition (e.g. f·g·g·f).

Students will be able to compute the absolute value of a number. They will be able to calculate a

number written in exponential form. They will recognize the power of exponents by exploring the classic

doubling problem (1 penny, 2 pennies, 4 pennies, 8 pennies, etc.). They will be able to convert

numbers expressed in scientific notation to standard notation and vice versa. Using the calculator, they

will solve problems involving operations with scientific notation. By investigating patterns in multiplying

integers, they will draw conclusions concerning the multiplication of negative integers.

**Probability and statistics:**Students will recognize the need for a systematic approach to counting and

they will investigate and develop a variety of strategies to solve counting problems in several contexts.

They will be introduced to set notation and terminology (e.g. intersection, union, disjoint, empty set).

They will be able to list the elements of a set, count the number of elements, and identify subsets. They

will realize that ••• and ••• are associative operations. Students will use Venn diagrams (2 and 3

circles, with a known quantity for the intersection) to partition sets into disjoint subsets and determine

the number of elements in these subsets. They will use tree diagrams to find all possible outcomes for an

event. In cases that involve two possible outcomes at each level, they will become aware of the

symmetry of the branches and the need to create only a partial tree to ascertain the total number of

outcomes. They will be able to apply the Fundamental Principle of Counting. Students will utilize these

counting strategies in assigning probabilities to events on the basis of intuition, past experience,

experimentation, or theory. They will recognize that a probability must be a number between 0 and 1,

inclusive. They will be able to identify certain and impossible events, assigning an appropriate

probability to each. Using the definition of probability for events involving equally likely outcomes (i.e.

P(E)=# of acceptable or favorable ways/total # or possible ways), students will be able to find the

probability of an event. They will become familiar with expectation as they compute the expected

frequency of an outcome for a given number of repetitions of an experiment. They will be introduced to

conditional probability (not stated as such) as they find the probability of an event given a known

previous occurrence or condition. They will be able to find the probability of the complement of an

event as well as the intersection or union of two events. They will examine the generalization

P(A·B)=P(A)+P(B)- P(A·B). They will apply their knowledge of probability in finding medical indices

that could serve as predictors of survival in trauma cases. Students will explore simulation as a means of

modeling real-world situations (e.g. waiting times at the post office). They will become familiar with

random number tables and use the calculator to random generator feature to simulate the flipping of a

coin N times.

Students will investigate sets of data that have been gathered by them or another source in terms of

measures of central tendency and spread of data. They will be able to compute the mean, median, and

mode (using a calculator with statistical capabilities, if they wish) and determine which is the most

appropriate measure of the center. They will analyze the effect of a change in one or more data values

(including an extreme value) on the mean, median, and mode. Given a set of data, they will be able to

construct a bar graph, a dot plot (line plot), a stem-and-leaf plot, and a histogram. Students will be able

to find the range and mean absolute deviation for a set of data. They will be able to determine the five

number summary for a set of data, construct a boxplot from these figures, and analyze the spread of the

data. They will be able to compare and contrast sets of data represented by two or more boxplots,

histograms, or back-to-back stem-and-leaf plots. They will use a spreadsheet to calculate the mean,

mean absolute deviation, and the five number summary for a set of data. They will explore and calculate

the variance and standard deviation for a set of data. They will compare and contrast two sets of data,

draw conclusions based on an analysis of measures of central tendency and spread of data, and justify

their conclusions. Students will be able to construct a scatterplot from a table of values and use it in the

analysis of the data to observe any overall trends and estimate data. They will determine when a table or

a scatterplot might be the better representation for a set of data. They will also use linear interpolation

and the least-squares (regression) line to estimate data. In addition, they will use linear models for

extrapolation (prediction). Using a calculator, they will be able to find the least-squares line, use it to

make predictions, and determine whether or not their estimates are reasonable.