Minecraft is an easy place for math practice. Kids see counting by 3s (when making paper), 4s (making planks), and 6s (when making slabs). They divide stacks of up to 64 items in two regularily. There is plenty of math.

Can Minecraft be used for exploring exponents and negative exponents? My eight year old and I wanted to look at that. I proposed that the relationship between wood logs and sticks is 4² because one log makes four planks and one plank makes four sticks, but my son corrected me. It takes two planks together to make four sticks. So then we have 8 sticks instead of 16.

Positive Exponents and Gold Ingots

A gold block in minecraft is 9 ingots, which in turn are each 9 nuggets. So in that case a gold block is 81 nuggets, or 9². If you wanted 9 gold blocks you would need 9³ nuggets. Why though, would you want 9 gold blocks. You only use eight gold blocks for an enchanted apple. Quick calculations: 9 x 9 x 8 = 648 or 9³ – 81 = 648. We wanted a visual of what this looks like so we went through with screen-shots.

Seeing the gold nuggets laid out in groups of 64 with 8 remainder my son pointed out that the number must be a multiple of 4. We then checked whether it could be written as an exponent of 4, and we found (by repeatedly dividing) that it could not.

Economics of Stairbuilding

Now my eight year old and I both have our own pet peeves with regards to minecraft math. His pet peeve is that stairs require what he considers an unfair amount of wood. Four stairs can be created with 6 wood planks. He considers this a waste, particularily as stairs are not all that more useful than blocks, since you can jump up blocks without all that much more difficulty than walking up stairs. Or you can walk up slabs rather than jumping, so he considers it more efficient (resource wise) to place wooden slabs. We calculated it all out. To go the same height 4 block height you could use 8 slabs to make an 8 block long staircase, or you could use 4 stairs for a 4 block long staircase. The 8 slabs would take 6 planks but leave you with four leftover which could be stacked together to make the equivalent of two planks again (thus using the equivalent of 4 planks, though you have to have 6 at the time you make it, and the new ‘planks’ are still just piles of slabs and can’t be used for crafting other things the same way planks can) . The 4 stairs would take 4 planks with none leftover. No exponents, but fun math and calculations all the same.

Negative Exponents and Crafting Paper

My pet peeve with minecraft math is with regards to making paper. In the game you can convert sugarcane into paper. Three sugar cane, spread out in a line, equals to three paper. But say you have 64 sugar cane you want to convert? You can click to place them in the crafting table grid one at a time, or you can use your other mouse button to drop 1/2 of whatever you are holding in the mouse. It is easy to divide a stack between two squares but not to divide them evenly between three. The whole process seems rather inefficient to me. I’d love to have a way of taking a stack of objects, clicking on any number of empty inventory or crafting squares, and then having the stack of objects divided between the selected squares (with remainders, if necessary – my son points out that when we divide uneven piles the remainder is always left in the square furthers to the left).

Not being able to divide a stack into three even spaces, I find myself using a different method, one that allows me to explore negative exponents again with my children. I take my stack of 64 and divide it between two. I then have 32 and 32. I take half of one of those and move it into the third spot. Now I have 32, 16, and 16. I could click to empty as many as I can into paper, till I have 16 left in the square that used to have 32. I have in effect cleared out 3/4 of the original stack of sugar cane (making it into 48 pieces of paper)  and I have 16 left to make into paper. This could be expressed as 1/4 the stack or four to the power of -1.

Now I divide the 16 remaining canes in half (8, 8) and one of those in half again (8, 4, 4). Click to remove 12 pieces of paper (4 x 3) and I have 4 left in one box. This is one quarter of one quarter the original stack or four to the power of -2.

I divide that in half, and then one of those in half again… I have 2, 1, 1. I remove three sheets of paper and have 1 sugarcane left over for next time! Now I have one quarter of one quarter of one quarter, or four to the power of -3, or 1/64th.

I wanted the kids to get a bit more hands-on experience than clicking on minecraft buttons, so I brought out the math tiles and we modeled 4³.

Next we took those same tiles and declared them to be pieces of minecraft sugarcane. We arranged them into two equal groups, and then split the two equal groups the same way we would split sugarcane, and removed 3/4ths. We repeated everything I described doing above with sugarcane with the tiles.