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Pathways from the Past II

Pathways from the Past is a pair of teacher’s manuals and two sets of activity sheets for helping students master various topics in arithmetic, pre-algebra, and algebra.  Building on ideas from their award-winning book, Math through the Ages, Drs. Berlinghoff and Gouvêa have crafted teacher-friendly tools for helping students understand basic topics that often cause trouble.  Their approach implements the spirit of the NCTM document, Reasoning and Sense Making and the “Standards for Mathematical Practice” of the Common Core (CCSSM).  The second set is

Pathways from the Past

II: Using History to Teach Algebra

This is a 72-page manual and a set of reproducible masters for 18 two-page activity sheets that you can use with your students to strengthen their ability to think clearly and reason effectively as they learn algebra.


First Thoughts for Teachers

1. Writing Algebra: Using Algebraic Symbols

Sheet 1-1: Symbols of Arithmetic

Sheet 1-2: Algebra in Italy, 1200 - 1550

Sheet 1-3: Germany and France, 1450 - 1600

Sheet 1-4: Letters for Numbers

2. Linear Thinking: Ratio, Proportion, and Slope

Sheet 2-1: The Rule of Three Direct
Sheet 2-2: The Rule of Three Inverse

Sheet 2-3: False Position
Sheet 2-4: Double False Position

3. A Square and Things: Quadratic Equations

Sheet 3-1: Completing a Square

Sheet 3-2: Algebra Comes of Age

Sheet 3-3: Using Zero

Sheet 3-4: A Method That Always Works

Sheet 3-5: Quadratics in Earlier Times

4. Cubics and Imaginaries: Third-Degree Equations

Sheet 4-1: Boxes, Lines, & Angles

Sheet 4-2: From Shapes to Numbers

Sheet 4-3: The Depressed Cubic

Sheet 4-4: The General Cubic

Sheet 4-5: Impossible, Imaginary, Useful


teacher’s manual  
& blackline masters

for  18 two-page
activity sheets



The Manual

Each of the four major chapters is organized like this:

The Content

The four chapters in this set progress from deciphering basic algebraic symbols to tracing how the pursuit of solutions to cubic equations led to major changes in the way we think about numbers.  There’s something for every level of algebra instruction, from the basic ideas of proportion and linearity to the equation-solving that led to the complex number system.  Connections to European and early American history provide a fertile ground for interdisciplinary student activities.

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Made by Serif

“This is something that I can see fitting into various grade levels.” (A high school algebra teacher)

Teach Algebra thru History

To see some sample activity sheets, click on this bar.

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