Handout Digging Down

Math Refresher: Digging Down to the Root of the Cube Robyn Breiman M.Ed. robynbreiman@gmail.com This upper elementary teacher refresher workshop reviews using sensorial experiences as a foundation for three-­‐dimensional analysis of cubed products. The workshop will revisit the Montessori math sequence beginning with re-­‐introducing the cube as a geometric and mathematical concept, and all the lessons through extracting cube roots from very large products. What follows is a sequence of experiences – some of which might require more than one lesson/presentation; and some which might be appropriate for “guided discovery” by the student. Previous Experiences with Cubes 3-­‐6 Experiences with Cubes Sensorial Pink Tower Binomial and Trinomial Cubes Introduction to the Decimal System – Thousand Cubes 6-­‐9 Multiplication/Memorization Bead Cabinet Chains for Squares and Cubes of Numbers 1-­‐10 Decanomial Sequence – Bead Bar Memorization Work Horizontal Layout Vertical Layout Diagonal Layout – Transforming the Decanomial Square into the “Tower of Jewels” Upper Elementary Cubing to Cube Root Cubing  What is a cube?  Building the cube from the cube of one number to the cube of the next number (from a cube to the successive cube)  Sensorially  Mathematically  Building from a cube to a non-­‐successive cube  Sensorially  Mathematically • Building an algebraic binomial cube – starting from the square (a+b)3 to (a+b)3 • Building an algebraic trinomial cube – starting from the square (a+b+c)3 to (a+b+c)3  The Story of the Three Kings (Moving from the Algebraic Trinomial Cube to the Hierarchical Trinomial Cube)  Cubing Trinomials (hierarchically) Cube Roots 1.
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What is cube root? Introducing the radical sign Can any number have a cube root? Creating and Using the Table of Cubes Finding Cube Root using the Table of Cubes 2. The Cube Root of a Binomial  1st passage – sensorial with cubing material  2st passage – sensorial with steps written out – using cubing material  3rd passage – using the binomial pieces of the hierarchical trinomial cube 3 The Cube Root of a Trinomial – using the hierarchical trinomial cube 4 The Cube Root of a Trinomial – abstractly – using the algebraic trinomial cube My hope is that the refresher will encourage upper elementary teachers to present the beautiful, self-­‐
correcting materials that are the culmination of the math operations sequence in a Montessori elementary program. The sequence of lessons supports understanding of the relationship between math and geometry -­‐ aligning arithmetic computation with geometric comprehension; another example of curriculum integration so rooted in a strong Montessori program. Feel free to contact me if you’d like a pdf. copy of the presentation. Robyn Breiman February 25, 2015