Friday, 3 May 2013

On the Concept of Balancing Equations

So often in the early grades kids become accustomed to seeing problems written out as below:
3 + 4 = ________

When the answer blank appears in different places in the equation, such as on the left-hand side, it can help, as can the creative use of manipulatives to represent the symmetry of equations. However, the connection between the equal sign and the demand for an answer may continue to confuse some students. Some students learn to think of the = sign as meaning "insert answer here" rather than as the fulcrum of the equation.

What do I mean by this?

Consider the term "balancing an equation".
If you envision an equation as a balance scale, you can put the = sign at the centre, or fulcrum of the scale. In this way, the equation is balanced when both sides are equal to each other. There is a symmetry in the weight on each side.

This can be used to demonstrate the mathematical meaning of the equal sign in a hands-on concrete way.

For students who have difficulty with the concept, consider having them use the balance with weight manipulatives. x might be the name of the 1 gram weights, y the two gram weights etc. Let them play around.

What if they put 2x on one side and y on the other? When the balance is level, the sides are equal.
What if you put 2y on one side and x on the other? When they balance is tilted one way or the other, the sides are unequal. Instead of an equation, you have an inequality ≠.

You can take this a step further if your scale is the kind that has an arrow on the fulcrum. Label the point of balance with an equal sign =, and the space on either side with an inequality sign ≠.

Encourage students to write the equations as they work with the balance to solve problems and also to make predictions of what expressions will be equations and which will be inequalities, as well as determine what is needed to turn an inequality into a balanced equation.

More of my math activities can be found here.

Sunday, 28 April 2013

Canoe Trip Boredom?

People have asked me in the past how I keep my kids from getting bored on canoe trips. I thought it an odd question, but since those friends are not canoe trippers, I thought little more of it. Then a few weeks ago I read an account from a canoe guru about kids and boredom in the boat. Looking a little further, I found that others too had had this issue.

Having canoed with our kids regularly since their toddlerhoods, I find the whole topic to be rather puzzling, since my own kids never showed any signs of boredom on a canoe trip.

So, with my curiosity tweaked, I have gone through journals, photos etc. to remember just what we did that made the trips not only tolerable, but often even exciting for the kids. It turns out that most of these the kids figured out for themselves. Some of these are posted on the Lemonade website's family camping pages, and some of these I will list here:

  • we stick close to shore where we can observe interesting features including cliff faces, rocky outcrops, interesting trees, wildlife, tracks in the mud, etc. This provides shelter from strong breezes as well as some occasional shade, and easier access to land for emergency bathroom breaks
  • cloud glazing, from sharing "cloud pictures" to predicting the weather
  • water gazing, sometimes using underwater viewer (found on the camping pages on the website)
  • singing--when the kids were young (and often still now that they're teens), we sing on the longer stretches and crossings during our trip
  • rock dropping--fill a bucket with rocks or twigs and drop them one at a time to watch the ripple patterns and also show the speed of travel etc. This is surprisingly amusing for 2-5 year olds
  • float a boat--bring along a small toy boat and tie it to the canoe and watch it float
  • bring an extra map so kids can begin to learn map reading skills; be sure to point out obvious features etc. regularly
  • small paddles or even a whittled dead branch can let young paddlers try their hand at paddling
  • the alphabet game that you can play in a car (finding letters on signs etc.) can be played by kids learning the names of different trees, plants, wildlife, lake and river names, etc. To make it easier, allow letters from any part of the name, not just the first letter
  • becoming the trip photographer

At the site, the kids always found lots to do:

  • pouring water over rocky areas and building stick and rock dams to alter the flow
  • exploring (when younger, we had them wear their PFD with attached whistle to cushion from bumps, provide emergency signal if needed, easy sighting and emergency flotation in case of a fall into water)
  • helping out (gathering firewood, pumping water, putting up tents, finding a good spot to hang the food, etc.)
  • making obstacle courses for each other
  • photography (they really like macro pictures in particular)
  • star gazing
  • hide and seek (with each other, or hiding natural objects)
  • imagination games, such as when a log hanging out over a small bay became their pirate ship etc.
  • swimming
  • wildlife tracking
  • using a pocket microscope to explore the tiny treasures they find

Things we found valuable to bring along on trips:

  • at least one kid-proof bucket and shovel
  • a toy boat, or a knife to whittle one
  • binoculars
  • swim mask or goggles (snorkels were also popular with the kids)
  • a pocket microscope (a later discovery we wish we'd thought of earlier!)

We never paddled for more than 2 hours continuously, always stopping for lunch and a needed leg-stretch part way through. When paddling with the very young, for us anyhow, distance was never the primary goal. We made a point to move at a slower pace so the kids would have time to make and share their discoveries. Doing so gave us a whole new perspective on the wilderness we love so much, and (I hope) helped to foster a love of nature in our kids.

More family camping tips can be found here: