Rules On Exponents

Lammas: Britain's first eco-village

Scattered across an emerald green sweep of Pembrokeshire countryside are a handful of rustic-looking dwellings: some have turf roofs, others smoothly sculpted cob walls; finely wrought hinges adorn massive wooden doors and the comforting smell of wood smoke rises from stove chimneys. Fat hens scratch for grubs and grain and children have the run of the land, climbing trees and making dens.

One resident, Jasmine Dale is currently enjoying a bumper strawberry crop. "We've got so many I don't know what to do with them, the children go foraging in the strawberry plants whenever they want," she says. The vegetable patches are bursting with sweetcorn, spinach, beans and squash. It appears, on the surface, to be an earthly paradise.

Earlier this month, Pembrokeshire County Council issued summonses against some of the pioneers for breaching building regulations. Breaches relate to fire hazards, outside lavatories, the use of ladders for staircases and the uncertain specification of some of the recycled materials the homes incorporate.

The summonses are a huge blow for the residents of Lammas, most of them families with young children.

One resident, Cassie Lishman, mother-of-three, who has built a pink cob reciprocal-framed roundhouse says she is in breach for her cotton ceiling and for the lack of a plastic radon barrier under her handmade earthen floor.

"As for my cob bath, which is heated from underneath with a fire, well the inspectors just didn't know what to make of it." She says there is "no way" she is going to dig up her floor "to put a piece of plastic under it. We're in an untenable situation. There are huge tensions between building regs and low-impact living."

Some materials such as the timber from their woodland and recycled windows do not have standardised ratings recognised by inspectors.

Jasmine's home does not comply because the composting lavatory is not inside the house and there is no mains-connected fire alarm. "How can we have a mains-connected fire alarm if we have no mains?" she asks. "We need to start a debate about an alternative building code for low-impact homes."

It took Lammas pioneer Paul Wimbush four years and a planning appeal to win permission for the village. He and other residents like Simon Dale have built their own homes, with help from volunteers who have come from all over the world to support the project.

Sweeney Math: How I see exponent rules (and log rules)

Exponent rules can be difficult to remember, and memory has never been one of my strong suits. When I was in high school myself learning exponent rules, I would get mixed up just trying to remember them individually, and had to come up with a different way of thinking to condense it into one idea.  What I came up with deals with the levels of complexity of operations: So, you've got your simple functions on the bottom, multiplication and division are a little more complex, and then exponents and roots are more complex. The actual chart above I created after the fact when trying to explain the idea to students later in life.  So, basically when it came to exponent rules all I had to remember was to "go down a level" of complexity. My results with trying to get students to see the same thing I do has been mixed.  I usually end up doing an exploratory learning exercise with exponents , then going through the rules individually and only quickly going through this chart idea on the side.  While it doesn't really connect with every student, when a particular student gets the rules mixed up it can really help because it at least gives them a plan rather than just relying on straight memorization.

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