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

Problem:
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?

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

Problem:
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.

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

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? 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.

Breaking Up Can Be Sweet

Grade(s):8Strand(s):Patterning & AlgebraProblem: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

nrows ofmpieces?.pdf download link here

Source:Intermediate Math ABQ Package – Summer 2009 (Compiled by Trevor Brown)Submitted by:Mark Ross, Norseman JMSLanor 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):8,7,6Strand(s):Patterning & Algebra, Number SenseProblem: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.

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 facesTotal # of small cubes2x2x23x3x34x4x45x5x56x6x610x10x10nxnxnWhat 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