• Learning situation/issues

    This section provides the basic background for this module and its use in middle school mathematics settings.  For additional particular insights about how this module has been used in various settings, please see the history of use section.

    Grade level

    The primary intended audience for this module is middle-grades students of various abilities.  The items are particularly focused on having students use number operations and then reflect on how different operations relate to each other.  This reflective work develops a sound understanding of operations and abstraction of operations as the basis for algebraic topics.  As with many curriculum materials, this module may be used in its entirety or in part with many different learners.


    Mathematical topics

    This module addresses several topics of middle school mathematics.  It is particularly related to number and algebra with considerable attention to geometry and measurement.  The topics are described in the following sections in terms of Pennsylvania Mathematics Standards and Curriculum Connections for at least one school district.


    Relevant Pennsylvania Mathematics Standards

    Standard Number


    Related Items in Module


    Algebra and Functions



    Discover, describe and generalize patterns, including linear

    Variety items involving tessellations, circles, and hexagons


    Create and interpret expressions, equations or inequalities that model problem situations.

    Variety items and several Free-response items


    Select and use a strategy to solve an equation or inequality, explain the solution and check the solution for accuracy.

    Variety items: numerous items involving whole numbers, fractions, decimals, etc.


    Solve and graph equations and inequalities using scientific and graphing calculators

    Variety items involving population increase and/or decrease


    Represent relationships with table or graphs in the coordinate plan and verbal or symbolic rules

    Cell phone plan problems and number of towers needed


    Number, Number Systems and Number Relationships



    Simplify numerical expressions involving exponents, scientific notation and using order of operations.

    Numerous Variety items, Free-response items and parts of the Open-ended projects


    Distinguish between and order rational and irrational numbers.

    Variety items involving populations


    Apply ratio and proportion to mathematical problem situations involving distance, rate, time and similar triangles.

    Variety items involving such as Tower Highs and Lows


    Computation and Estimation



    Complete calculations by applying the order of operations.

    Numerous Variety items and Free-response items


    Add, subtract, multiply and divide different kind and forms of rational numbers including integers, decimal fractions, percents and proper and improper fractions.

    Numerous Variety items and Free-response items.  The conversions scale on the map is fractional thus resulting in the application of fractions and decimals with most operations.


    Determine the appropriateness of overestimating or underestimating in computation.

    Variety items involving population totals


    Identify the difference between exact value and approximation and determine which is appropriate for a given situation.

    Variety items involving population


    Measurement and Estimation



    Develop formulas and procedures for determining measurements (e.g., area, volume, distance).

    Variety items involving tower placement, cell phone usage and coverage


    Measure angles in degrees and determine relations of angles.

    Free-response items: Reaching Out and Service Area


    Estimate, use and describe measures of distance, area, volume, and angles.

    Variety items and Free-response items throughout the module


    Describe how a change in linear dimension of an object affects its area.

    Variety items and Free-response items pertaining to the placement of cell towers and coverage areas


    Use scale measurements to interpret maps or drawings.

    Numerous Variety items and Free-response items that utilize the Dauphin County map


    Mathematical Reasoning and Connections



    Combine numeric relationships to arrive at a conclusion.

    Addressed throughout the entire module


    Mathematical Problem Solving and Communication



    Verify and interpret results using precise mathematical language, notation and representations, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagram.

    Free-response items and the explanations required for Variety items


    Justify strategies and defend approaches used and conclusions reached.

    Free-response and Variety items questions


    Determine pertinent information in problem situations and whether any further information is needed for solution.

    Addressed throughout the entire module


    Statistics and Data Analysis



    Fit a line to the scatter plot of two quantities and describe any correlation of the variables.

    Free-response items and Variety items involving population trends





    Analyze geometric patterns (e.g., tessellations, sequences of shapes) and develop descriptions of the patterns.

    Free-response items involving hexagonal tessellations for coverage areas of cell phones


    Analyze objects to determine whether they illustrate tessellations, symmetry, congruence, similarity and scale.

    Free-response items involving tessellations





    Compute measures of sides and angles using proportions, the Pythagorean Theorem and right triangle relationships.

    Pythagorean Theorem can be used to obtain solutions for several Free-response and Variety items


    Concepts of Calculus



    Describe the concept of unit rate, ratio and slope in the context of rate of change

    Demonstrated in items involving population trends of Dauphin County

    Curriculum connections

    This module particularly fits with the study of measurement, linear relationships, and area in Grades 6 through 8.  There are also considerable opportunities for students to work with large numbers, fractions, and ratios.

    Number and operation

    Several items in this module require students to compare or operate on whole numbers (e.g., F1, V1) or rational numbers (e.g., F2).  Some of the items  provide opportunities for students to think about number operations beyond the actual computations (e.g., V2) and percentages (e.g., V6).

    Algebra, pattern and function

    There are related opportunities for students to write expressions to represent numerical patterns (e.g., V16) and to explore slope (e.g., F4, V18) and intercept (V17) in linear settings.


    Students have opportunities to use scale (e.g., F2, F3) as well as dimensional analysis (e.g., V3) as they think through what they are multiplying or dividing and how the units of measure appear in the calculations and the answers.


    There is a large number of items that involve estimating and computing areas of rectangles (e.g., V4, V11) and of circles (e.g., F6).  There is also some attention to the Pythagorean Theorem (e.g., V10), to similar triangles (e.g., V22) and tessellations (e.g., F7).

    Data analysis

    The analysis of population and business data for Dauphin County underlies several items (e.g., V27, V29, V30).