Ms. Bearse's Site

Here’s a triangle.  A B and C are internal angles.  D E and F are external angles.  Here is a nice visual proof of some of the theory we’re working on.

The external angles, have a sum of 360 degrees.  Here’s proof.

When we put them all together, they match up perfectly, completing the full circle, which is 360 degrees.

Also, when we put the interior angles together, they make a straight line.  They are supplementary.  They add up to 180 degrees.

Here is an expression: 1x^2+6x-3

We are going to write it in vertex form by completing the square.  To make a square, we take the x term and split it in half.  Half of the x goes vertically and half horizontally.

We notice that we could complete the square with 9 little red squares.  Since we don’t have any, we need to add some zero pairs.

With 9 additional blue and red squares added, we can complete the square.

We now have a square: (x+3)^2 and also 12 blue square tiles.  The expression can now be expressed as (x+3)^2-12.  

We have some sprouts!  We now will start measuring their growth each day and then we can figure out the growth rate.

We have been practicing representing expressions with algebra tiles.  We are getting pretty good at it.  Now that weare solving equations, these representations are becoming useful yet again.  We put a popsicle stick as the equal sign, and have to do the same thing on each side, always.  We can add blue or red tiles to make zeros, or divide up both sides into groups to simplify equations.  
This is 2x=8. To simplify it, we can divide each side in two groups and we can see that 1x=4.

Here we have variables on both sides of the equal sign.  Our goals are as follows

1. Get all variables on one side
2. Eliminate the constant that is on the same side as the variable
3. Eliminate the coefficient beside the variable.

So this equation is 5x=3x-2

We can simplify by either removing 3 red rectangles (3x) from each side, or by placing 3 blue rectangles (-3x) on each side.

This will leave us with 2x=-2 and we split each side up into 2 groups to see that x=-1

Using algebra tiles helps us see what we are doing, and helps us understand.

Lots of assessments happening in room 208 these days.  We’re using our resources, and working hard to communicate what we understand.

The next test is Friday.  Here are the things you’ll need to know.

We are working on surface area and volume formulae for various solids.  Today we looked at spheres and pyramids.

To calculate the volume of a sphere, we used displacement.  We submerged a tennis ball in a displacement tank.  The water displaced overflowed into a juice concentrate container that has been cut so that the height and diameter are equal.  The juice concentrate container was selected as it has the same diameter as the tennis ball.  We can substitute 2r in the place of h in the formula for the volume of a cylinder.

Since the displaced water (the volume of the sphere) is 2/3 the volume of the cylinder, through a bit of calculating we can derive the formula for the volume of a sphere.

We also looked at the area of a sphere.  We used an orange to help us.

The diameter of the orange is close to 8cm.

We then peeled the orange and put the peels into a rectangle.  We know how to calculate the area of a rectangle.

The next step is to make a relationship between the area calculated, and the diameter squared.  We have to compare square units to square units.

The factor we found was always around 3.  This value should be pi.  Since the sphere is not exact, and neither is the rectangle, we don’t expect our pi approximation to be exact either.

We then used some algebra and exponents to derive the formula for the surface of the sphere.

Our pyramid exploration was open ended.  The restrictions were that they needed a square base, and the height needed to be exactly 8 cm.

In order to make the pyramid actually 8cm tall we need to calculate how tall each triangle face should be.  We use the pythagorean theorem to do this.

After a tough quiz today we spent some time working on some puzzles.