Patterning+-+Algebra+(7-8)

Breaking Up Can Be Sweet **Grade(s):** 8 **Strand(s):** Patterning & Algebra

I have just bought myself a block of chocolate. It is a normal rectangular-shaped block with 5 rows, and 4 pieces of chocolate to a row, making 20 pieces of chocolate ready for the eating. I want to eat all of it right now, but I want to savour each piece. What is the least number of clean snaps necessary to break the block of chocolate into the 20 individual pieces? What about for a block with //n// rows of //m// pieces?
 * Problem:**

.pdf download link here


 * Source:** Intermediate Math ABQ Package – Summer 2009 (Compiled by Trevor Brown)
 * Submitted by:** Mark Ross, Norseman JMS

**Lanor Exemplars for Grade 7 Patterning:**

Diagnostic, Assessment For & Of Learning: The Diagnostic was taken from the Grade 7 Ministry Exemplars, the Mid-Point and Final Assessment were taken from Susan O'Connell's, The Math Process Series, Reasoning and Proving. Example Pieces of Work

Painted Cubes **Grade(s):** 8,7,6 **Strand(s):** Patterning & Algebra, Number Sense

A toy company wants to create and sell a game that uses solid colored cubes of varying sizes. They use smaller cube-a-links to create (solid) cubes that are 2x2x2, 3x3x3, 4x4x4 and so on. Once the larger cubes are created, they are dipped (fully) into red paint, so that the outsides are painted entirely red. The company wants to keep track of the paint they use, but they must first know how many cube-a-link faces are painted. Can you see the patterns in the table? Can you generalize these patterns (create equations predicting how many cubes with 3, 2,1,0 and total) for a large cube nxnxn? How are those equations (or columns) related to what you see?
 * Problem:**
 * **Size of large cube** || **# of small cubes with 3 red faces** || **# of small cubes with 2 red faces** || **# of small cubes with 1 red face** || **# of small cubes with 0 red faces** || **Total # of small cubes** ||
 * **2x2x2** ||  ||   ||   ||   ||   ||
 * **3x3x3** ||  ||   ||   ||   ||   ||
 * **4x4x4** ||  ||   ||   ||   ||   ||
 * **5x5x5** ||  ||   ||   ||   ||   ||
 * **6x6x6** ||  ||   ||   ||   ||   ||
 * **10x10x10** ||  ||   ||   ||   ||   ||
 * **nxnxn** ||  ||   ||   ||   ||   ||
 * **10x10x10** ||  ||   ||   ||   ||   ||
 * **nxnxn** ||  ||   ||   ||   ||   ||
 * What to look for:** Generalized patterns can be observed by completing the missing values in each column. (Each column generates a separate equation). The nice part about this problem is that it affords an opportunity to relate the equations to visual patterns.

.pdf download link here


 * Source:** Intermediate Math ABQ Package – Summer 2009 (Compiled by Trevor Brown)
 * Submitted by:** Mark Ross, Norseman JMS