Elementary Conundrum

3 x 3 cube problem

Challenge:

Twenty-seven identical cubes have two opposite faces in black with the remaining four faces white; these are used to make a 3x3x3 cube. What is the greatest fraction of surface area that may appear black?

Given Answer:

Imagine a large 3x3x3 cube made out of 27 unit cubes. Of the 27 unit cubes, 1 is completely hidden and cannot be seen, and each of the remaining 26 cubes has one, two, or three of its faces exposed. Three faces are exposed on 8 of the 27 unit cubes (at the 8 corners of the large cube), 12 of the 27 unit cubes have two faces exposed (on the edges of the large cube), and 6 of the 27 cubes have 1 face exposed (in the middle of each face of the large cube). Each of the 26 unit cubes that have one or more faces exposed has at most one black face exposed. A large 3 Å~ 3 Å~ 3 cube made out of 27 unit cubes has a surface area of 54 unit squares. Therefore, the answer to our question is 26/54 = 13/27.

Credit:

NCTM Mathematics Teacher, October 2016, Vol. 110, Issue 3,  Calendar and Solutions

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Adding and Subtracting Integers

Find this and other free lessons at www.enasco.com

• Put an equation on the board, but note that this time they will be subtracting. Subtraction can also be called take away or the opposite of. 
• Set up this example: -8 – (-4) = 
• You will have one pile of 8 red counters and another of 4 red counters. When you subtract, you will turn over the counter pile that is directly after the minus sign because subtract means to do the opposite. Therefore, you will now have 8 red counters and 4 yellow counters. You now have an addition problem and the answer is -4. 
• When subtracting integers, you make two moves: one is to change the subtraction sign to addition and the other is to change the sign of the number directly following the subtraction sign.

Workshop Reviews

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Football Math

In this interactive game from Idaho Public Television, students add and subtract plays on a football field to practice working with negative and positive integers on the number line.
http://idahoptv.pbslearningmedia.org/resource/mket-math-ns-ratnumb/football/

Each set of randomized drives consists of seven to nine plays and ends with a touchdown. Plays are represented with equations. When students place the football at the correct point for each play, the answer to the corresponding equation is automatically filled in. The accompanying activity suggests ways that teachers can integrate the interactive into the classroom setting.