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Pathways from the Past I

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 ideas that often cause trouble.  Their approach implements the spirit of NCTM’s document, Reasoning and Sense Making, and the “Standards for Mathematical Practice” of the Common Core State Standards for Mathematics (CCSSM).  The first set is

Pathways from the Past

I: Using History to Teach
Numbers, Numerals, and Arithmetic

This is a 64-page manual and a set of reproducible masters for 19 two-page activity sheets.  Its topics are basic to the curriculum and assessment guidelines of most states and are in the Numbers and Operations parts of the NCTM Focal Points for grades 5 - 8.


First Thoughts for Teachers

1. Writing Whole Numbers: Place Value

Sheet 1-1: Egyptian Hieroglyphics

Sheet 1-2: Babylonian Numerals

Sheet 1-3: Mayan Numerals

Sheet 1-4: Roman Numerals

Sheet 1-5: Hindu-Arabic Numerals

2. Zero Is Not Nothing: Properties of Zero

Sheet 2-1: Using a Placeholder

Sheet 2-2: The Number Zero

Sheet 2-3: Zero in Equations

3. Broken Numbers: Fraction Arithmetic

Sheet 3-1: Unit Fractions

Sheet 3-2: Place Value Fractions

Sheet 3-3: Name and Count

Sheet 3-4: Working with Parts

Sheet 3-5: Decimals

Sheet 3-6: Percent

4. Less than Nothing?: Negative Numbers

Sheet 4-1: What Are Negative Numbers?

Sheet 4-2: Adding & Subtracting Negative Numbers

Sheet 4-3: Multiplying & Dividing Negative Numbers

Sheet 4-4: Fitting In

Sheet 4-5: Powers and (Sometimes) Roots

“I loved the materials. I will be able to use these throughout the year in my class.” (7th grade teacher)

teacher’s manual  
& blackline masters

for 19 two-page
activity sheets



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

The Manual

Each of the four major chapters is organized like this:

The Content

Chapter 1 fits well with the “ancient civilizations” part of any social studies curriculum.  Chapters 2, 3, and 4 explore how the mathematics of non-Western cultures can provide alternative ways to understand some often troublesome ideas about fractions and negative numbers.  The activity sheets for all four chapters encourage students to practice essential skills by presenting them in novel contexts.

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“I am very excited to use these [materials] in the classroom.  The info correlates with the standards I teach and are the areas my students struggle with.” (8th grade teacher)

Teach Math thru History

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