Class+Summaries

​We'll post summaries of each class session here. Here is a sample format I'd like you to refer to when writing the summary. The following mathematical ideas were the focus of today's class: • idea 1 • idea 2 • idea 3

The way we developed idea 1 (2, 3, etc) was ...

An important thing to remember about idea 1(2, 3, etc) is ...

Idea 3 (ideas developed late in the class session) is something that we'll revisit in later class periods but we got a start on this idea by discussing ...

STUDY SESSION MONDAY 10AM IN ROOD 3375 (3RD FLOOR ACROSS FROM THE COMPUTER LAB).

 * DAY 26: MONDAY APRIL 19, 2010**

We reviewed that a square is a special rectangle, so a square can be considered a square or a rectangle jlh

Square prisms are different from cubes because their faces aren't all congruent. SG Just to be clear: All cubes ARE square prisms but not all square prisms are cubes.

Oblique 3-D shapes do not have a right angle, that is formed from base 1 to base 2. They look "tilted." BC Careful: not all 3D shapes have two bases. You would need to say rectangular prisms for your case. -Has 2 congruent bases that are regular polygons and are parallel to each other -All other faces are rectangles -only have one base -the specific name is based on the shape of that base. (i.e. if it is a triangle, it is called a tetrahedron).
 * Prism:**
 * Pyramids:**

//Cylinders// have 2 congruent circular bases, parallel bases and a rectangle wrapped around the circular faces. //Surface Area// = a covering in square units of all the faces.

Cone- A 3d shape with a circular base and a vertex opposite the base. JH

A 3-D figure kicked to the side is called oblique. NS RESTATE! As a teacher, you have to use geometrical terms. As a visual, you can think of it as "kicked to its side" but what is the term that describes that? Please ADD!

Truncated 3-D shapes have part of the shape cut off. Usually seen when the top of a cone or pyramid is cut off or the corner of a cube.

Volume is the filling of a space.

- Volume is measured in cubic units. -Cubic units are used because when trying to find the volume we are trying to find how many cubes we could fill the space with -Can be written as cubic units or the unit to the third power HH

-A box's net and actual box can both be used to figure out how many cubes will fit in the constructed box. MB OR
 * You can use the net to actually count the surface area and then go from there to decide how many boxes will fit AD
 * You can use the constructed box by counting finding the ( surface area of one side **)** **of the box and then multiplying that surface area by how many rows or columns there are in the box (how many times will that surface area** **be placed behind one another to create the 3D box shown) AD** should say "area of one face". not really using just the area but a stack of cubes where the area provides the number of the cubes in the stack or column.

We did more activities with nets and found "fold" lines where we could cut out and then fold a net into boxes of all different dimensions. SB

A //prism// has two congruent regular polygon bases that are parallel to each other. JF

FYI: The word //hypotenuse// derives from the [|Greek] ὑποτείνουσα (hypoteinousa), a combination of hypo- ("under") and teinein ("to stretch") [|[1]] The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timeus 54d and by many other ancient authors. ___so there you have it!!___

Day 25: April 14 Big Idea:
 * 3-D Shapes


 * Tetrahedron**- The base of a pyramid is a triangle.
 * Faces**- The sides of a 3-D object
 * Edge**-The coming together of two faces

In class we worked with connecting cubes. We looked at pictures of the top, front and side of a 3-D shape and then tried to recreate the shape with our cubes. We also used the program, building perspective, on our calculators to look at and create 3-D shapes.

Things to think about:
 * What other 3-D shapes can be made when 3-D shapes are put together.

Homework:
 * Pg. 254-257 (all)

**Day 24: April 12**
Administrative stuff: we talked about homework problems first. Dr B gets bummed when we come to class without having homework done. I suppose we should really try and get it done, seeing as how we want to be awesome elementary teachers. Dr. B also showed us a tangram page where we can move the tangram shapes around and make characters or objects or animals. That web page is on the "What's up for this week" page here on the Wiki site.


 * Big ideas**
 * We can find lengths of straight, "tilty", and "curvy" lines.
 * We developed area rules (area is a covering in square units) for rectangles, parallelograms, and triangles. We discussed several ways to find the area for a trapezoid.
 * We found an area rule for the circle: πr^2
 * Began naming 3D shapes using p 236 in Course Pack

Finding lengths
 * with straight sides, we can simply use a grid paper to determine length.
 * with tilty sides, we can build a square off of the side, find the area of the square, then find the side length of the square by using square roots
 * with curvy lines, we can use the circumference rule we developed, C=πd (d is the diameter)

Finding area rules
 * We already had rules for rectangles, parallelograms, and triangles. Trapezoid had a few. 1: drop perpendiculars down from the vertices on the shorter base to the longer base; cut off the two triangles formed, push them together to make one bigger triangle with area (b2-b1)h; remaining piece of trapezoid is a rectangle with area hb1; area of trapezoid is the sum of those two areas: (b2-b1)h + hb1 (b1 is the shorter base). 2: take two congruent trapezoids and arrange them to make one bigger parallelogram; area of parallelogram is (b2 + b1)h; area of trapezoid is half of that. 3: cut the trapezoid into two triangles by drawing in a diagonal. We needed to finish that rule for homework. All 3 rules should be equivalent since we are finding the area of the same trapezoid.
 * For the circle, we completed page 219, making a soccer field cake from a round one. This was cool!!! We cut up a circular cake into 8 identical pieces and laid the pieces next to each other so that something that looked somewhat like a parallelogram was formed. Dr. B cut up her cake pieces again, cutting each in half, so she had 16 pieces to lay next to each other (they flipped each time you laid a piece beside one so the cake would begin to look like a soccer field). We reasoned that if we kept cutting up the pieces into halves, eventually the cake would approach the shape of a rectangle. Regardless, we had this parallelogram now and to find that area, we used bh. But, here's the cool part. The height of the parallelogram was really the radius of the original cake and the base of the parallelogram was half the circumference of the cake or 0.5(2πr) or just πr. So bh is really πr * r or πr^2. Awesome! We found the rule for the area of a circle.

We were now finally finished with investigating two dimensional shapes and we are moving up a dimension to 3D shapes. We had a bucket of 3D shapes that we dumped out on our tables to look at and sort using p 236. We were trying to remember what prisms, pyramids, cylinders, cones, and polyhedra were. We put the letters of the shapes into the boxes we could. We had to focus on what feature of the shapes helped us be sure they were the shape we thought they were. For example, for pyramids, we said they all had a point on them, where all the triangular faces came together. Then Dr. B gave us a set of definitions from a middle school curriculum (the Connected Math Project). We need to compare those definitions with what we thought the shapes were. AHE; Finish from the previous set (listed below) all problems connected to areas of circles (and any that you didn't do yet!) CP p. 212-215 #7, 10-31 CP p. 216-218 #21-27, 29, 37, 38 CP p. 225-231 #15-27, 36-38, 49 CP p. 232 CP p. 233 all (Good problems for the final exam!!!) New stuff: P 51 #9 P 235 shapes, put into a category on p 236 P 244-246 How many cubes? and Drawing silhouettes. Working on your spatial visualization. complete http://connectedmath.msu.edu/CD/Grade8/Painted/PaintedCubes.html After you have completed that task with the painted sides, use the applet and try to complete p 248. Print the pages of the drawings you finally make or draw a 3D sketch. plus finish the "add 3 inches to the string around your waist" problem Projects due Wed April 21 Final course reflections due on Monday!!! Last chance to edit.

Day 23: April 7

We started with talking about what we did last week which was building squares of of legs of triangles, and figuring out their area. We found that the area of the square of of leg 1 + the area of the square of off leg 2 = the area of the square of the hypotenuse. This equation relates to a^2+b^2=c^2

We looked at pg. 190 and found there was two ways too find the answer. 1. Find a^2 and b^2 first so we would look at numbers that added up to 13 then look at their square roots to find our answer. 2. or we could find a and b first so we would look at two numbers and see if you squared them their sum would be 13

Then we wanted to see if their was a relationship between circumference and diameter to do this we used the equation y = mx + b where m= slope and b= y intercept we found that when you look at circumference and diameter that the slope is about 3 or //pie//. And that the y intercept should be zero. so the equation we found is circumference = //pie// x diameter

Lastly we put together some rules for finding area b= base h=hight rectangle- A = b x h parallelogram- A = b x h triangle- A =1/2b x h

__Day 22: April 5

Today we went through a lot of information, it was very productive. We went through some homework problems on Pg 83 2c, pg 174 along with the concave problem from our Midterm. Here are the ideas that we discussed. IDEA 1: Midterm problem-__ (n-2)180 __we worked on what a concave is and what our thoughts were of what would be a good definition for a concave n hexagon. Are all side s not congruent? what make s it concave? Some classmates said it has to deal with either being a regular or irregular shape. But is this what makes it concave? Be clear. Someone should add the definition of concave here that we came up with. another idea was that the sides were congruent so we know that atleast one side is a reflex angle. -Dr. Browning brought back the idea that we discussed with the turtle with the Supplementary angle= adjacent angles whose sum is 180 degrees and Complementary angle= adjacent angles whose sum is 90 degrees. I didn't make clear that the angles need to be adjacent so I'm adding that here. They were drawn that way in the examples but it wasn't explicitly stated. -How does one make a concave hexagon or concave shape in general? s Seems like this should be earlier in the summary. Looking back on Pg 96 in the course book, we look at the hexagons, find a vertex and see how many triangles we can make on the interior of the hexagon by connecting vertices ; we found 4 triangle with 6 sides. there is 180 degrees in the sum of the interior angles of a triangle so 4x180=720 (sum of interior). we found that we can use this for a concave problem. we look back at the problem that Darkriesha was using and we saw that either way, she knew at least 5 angles and trying out all those angles we find that it is a true regular (it is not regular) shape because no matter everything will add up to the sum of 720 degrees.

IDEA 2: Pg 83 problem 2c- how many quads can we make? When we have all side s put together, we can find how many depending on the interior angle s. how much of a change of an angle you make makes a different shape, (you can make big enough angles to make a convex kit e or a concave kit e ). we found out that the answer i s that we can create an infinite collection of 4 angles to have a sum of to have 4 angles equal 360. Some students or parents might have a different thought. they might think that there should be a time to stop counting so we might say Phynite. finite limit to how many we can make since there is a fixed sum of 360º. How can we convince them the collection of 4 angles is actually infinite?

IDEA 3: On pg 175 we were asked to see how many different squares we can make in the 8 or so 10x10 dot-to-dots. we found 8. This lead to Dimensions, Area(Sq units), and side lengths (units). Is there a relationship between a n area and side length? If you know the side length and you square it to find the area so 8x8=64 units squared. s If you know the area, then you can take the square root of that area to find the corresponding side length. - what is a rational number? It picks up your fractions including whole numbers. Extra: More formally, it is any number that can be expressed as a/b (with b≠0) with a and b from the set of integers.

we worked on pg188 in the course packet but didn't finish it so we will on Wednesday. AHE: Dr. Browning did email us our AHE but here it is again: Pg 180- do at least one but if you want to work more on it you can do the others. 190, and then 191 1-7,13-17 and 27. s Delayed this piece until Wed UNLESS you want to try and do them.

checked by: JLK


 * Day 21: March 31**

Today we made two statements in relation to the connection between having three or four sides of differeing lengths and whether or not they could create a closed figure. idea 1 : The Triangle Inequality- I can make a triangle if the sum of the two shorter sides is greater than the length of the longest side -an example of three numbers that would create a triangle are 10,5,7. -an example of three numbers that would not create a triangle are 3,6,15. idea 2: Quadrilaterals- I can make a quadrilateral if the three smaller sides are greater than the longest side. - an working example is, 4,6,7,10 - an example where a quadriliateral would not be created is, 3,6,4,18 For the Quadrilateral it's the sum of the three smaller sides have to be greater than the longest side. ​E.M.

We developed these ideas by using polystrips. 1)We discovered that when we created a triangle using three radom numbers that fell under the requirements of idea one, it was fixed. We could not change the angles or the appearence of the triangle. 2)Counter to the three sided figure, when we had four different side lengths that connected to make a quadrilateral we could move the poly strips around to create an infinite number of different quadrilaterals Next we moved on to using a colored piece of paper. Here were our steps - We first made a square and threw out the left over strip of paper. - We folded the square on a diagonal line to make two large right isocelese triangles. //Diagonal- a line segment that connects opposite vertices//. - We took one of the large triangles and took the tip of the right angle and brought it down to the center of the bottom line of the triangle. - When we tore along this line the out come was a trapezoid and a smallelr triangle. -we discovered that the smaller triangle fit into the larger one 4 times, and yet it had a 1/2 scale factor. -We took the trapezoid and tore it in half to make two right isoceles trapezoids. - One tapezoid we folded to create a square and a right isocelese triangle. (triangle had a 1/3 s.f.compared to origanal triangle) - The other trapezoid we tore to create a right isoceles triangle and a parallelogram.

The outcome of this was tangram peices

Comming up: looking at shapes and putting them into familiar shapes to find the area.

AHE: In our lists' in our calculators, we are to pick a horizontal line and flip our hat over that line HW:p.83 (add 2) c. How many quads can you make? p.81 #1-12 AMAYN p.147 FINISH take home quiz


 * Day 20: March 29**

Idea 1 We started today by working more on orientation. We discussed whether or not orientation was changed when we rotate a shape and slide a shape in a direction. In an in class poll, we asked whether an object's orientation changes when we translate it. The final was 9-7 that the oreintation does not change.

The No people said that the shape had to be rotated for orientation to change.

We took another poll asking if a shape has to rotate, or can it just slide the object to change orientation. The class answered in favor of the fact that an object must be rotated in order to change its orientation. just sliding an object doesn't change its orientation, just its location. - Orientation also can change when an object is reflected for an axis. a reflection over the x-axis will change the orientation of a shape.

Idea 2 We worked on reflection further. We were trying to figure out how to find the degrees of angles when lines of reflection intercept. <1+<4=180 <1(congruent to)<3 m<2=180-<1 <2+<3=180 <2(congruent to)<4 m<2=180-<3 <1+<2=180 180-<1= 180-<3 <3+<4=180 -180 -180 (-1)-<1= -<3(-1) <1=<3 If the lines of reflection intercept, it is rotation. These equations show how measures of the angles of roation relate to each other to form 180 degrees..

Homework questions. We started by working on scale factor. We refreshed our understanding of the definition: -Two geometric shapes are similiar if the corresponding angles in the image are congruent and the shape is preserved, and the correspondingside lengths are scaled to the given factor. -New sized shapes must always be compared to the original shape when you are trying to find the scale factor of the two shapes. - You must label and be consistant when finding scale factor. - If you have a triangle with the sides of 2,2,5 andthe scale factor is 3, your new triangle will have sides of 6,6,15. Each side is multiplied by the same number, which in this case was 3. -All sides need to be factored to the same number for this to work.

More poll questions were asked. 1.Are dialations isometries? This is no, because dialation stretches and smushes a shape, while isometries is a transformation that keeps the shape congruent. 2. Are squares rectangles? Squares are considered a special rectangle and, so a square can be considered a rectangle, but a rectangle is not considered a square. 3. Are congruent shapes similiar? All congruent shapes are very similar, so yes they are. 4. Are similiar shapes congruent? No. Not all similiar shapes are congruent.

AHE - We will work on pg. 166 #33on wednesday. - Finished the exam question handout for class wednesday. - Every one who hasn't found their three number and seen if they can form a trinagle need to do that. - We will get an e-mail about the rest of our homework.__

CHECKED BY: __JLK


 * Day 19: March 24

Idea 1: We formed a definition of what a translation is by continuing the statement "//A translation matches any two points, X and Y, with image points, X' and Y', so that..."// ** These were original ideas we had as we thought through together what a translation does: __
 * Move the same distance and direction from original points as indicated by the translation
 * Different orientation
 * Image is congruent to original image
 * 1 distance is always preserved with transformations (ex. with a rotation the distance from the original point c to the point of rotation, p, is the same as the distance from image point c and the point of rotation, p.

Our definition: The image points have moved, or transformed, the same distance and direction from the original points, as indicated by the translation.

Rubber band Stretcher: Conjecture: If we use three rubber bands and conduct our rubber band stretcher our new image will have three times the perimeter and 9 times the area
 * Idea 2:

When we stretch two rubber bands these properties apply to the new image: **


 * Shapes are preserved
 * Length is twice as long
 * Area is four times as much/perimeter is two times as much as the original
 * All angels are preserved

Dilation-Creates similar figure NOT congruent (the figure will appear to be the same shape with similar ratios but the measurements will not be the same) We're still working on our definition of what similar means. So far, everything we have examined does have different measures given the scale factors we have used (see scale factor in Idea 3 below). Isometries-Creates congruent figure NOT similar (the measurements are exactly the same) Something to think about! Is it true that congruent figures would not be similar? We learned about scale factors. A scale factor is the percentage to which a shape is increased or decreased. **
 * Idea 3:


 * If you have a line segment with a measurement of 1 cm. and you dilate it to 10 cm. you can find the scale factor by taking 1 cm. and dividing it by 10 cm. Your answer will be .1. So this means that your scale factor is 10% (all decimals must be rounded to the nearest //hundredth// on any homework or exams).
 * Once you solve to find the scale factor, you are now able to solve to find all the other side lengths.
 * When you see an equal sign with a dot above it, this indicates that the amounts on either side of the symbol are about the same. The only time it is appropriate to use the equal sign is when BOTH numbers on either side of the equal sign are EXACTLY the same.

Page 153 in the course pack question number 3- we tested this conjecture and worked with our groups to see what we could come up with. **
 * Idea 4:


 * Corresponding angels angles are congruent (this is true because when you copy the image, you are just making the side lengths longer or shorter, and doing so does not affect the angel angle measurement)
 * Corresponding side lengths are scaled to the given factor (scale factor-10% means that the side lengths are either ten percent bigger or ten percent smaller than the original shape) Compare this idea with a scale factor of 75% and 150% that we examined in class. Is there a way to know from the percent itself whether the shape is getting bigger or getting smaller?? If we really wanted a figure 10% bigger, what should be the scale factor?
 * Shapes are preserved (once a rectangle is dilated larger or smaller, it will still be a rectangle, just with larger or smaller side lengths, same for all shapes (triangles, squares, rhombuses, etc.))


 * AHE:**
 * 1) Look up the definition for 'orientation'.
 * 2) Page 155 (all)/154 (identify two shapes that are similar and two shapes that are not similar-remember what similar means)/page 161 (#2,3,5,6,7,9-14,28-33)
 * 3) Make a triangle with the three numbers that you get from the random function on your calculator
 * Select the MATH function on your calculator
 * Scroll over to select the PRB function
 * Scroll down to the randInt ( function on your calculator and hit enter
 * RandInt ( should now be on your screen with a blinking tab. Enter in the digit 1 then a comma then the digit 18 so your screen will look like this ranInt (1,18 and hit enter
 * A random number between 1 and 18 should appear on the right side of your screen once you hit enter
 * Hit enter another two times and using the three random numbers you now have on the right side of your calculator, see if it is possible to use these three number to make a triangle


 * Day 18: March 22**

The following mathematical ideas were the focus of today's class:
 * Congruency
 * Reviewed translations
 * Reviewed rotations
 * Dilations

The way we discussed congruency was by developing an ending to the question “Two shapes are congruent if…” As a class, we concluded that two shapes are congruent if: -All corresponding angle measurements are congruent -Corresponding side lengths are congruent -Angle measurements and side lengths are all to the same scale -If we transform the shape by either translation, reflection, or rotation We also concluded that all isometries (transformations) produce congruent shapes.

The way we reviewed translations was by completing the Course Pack page 125. We used patty paper to complete the “synthetic translation” and discussed how we knew where to move the shape and the distance it had to be moved. We decided that by placing a letter (Q) at the beginning of the guide line and tracing the line, we could move Q along the line until Q reached the end of the guide line. Our shape will then be translated to the appropriate spot.

We reviewed rotation by first doing reflection on page 131 #31. After reflecting shape ABC over the line L to create A’B’C, we reflected A’B’C over line M to create triangle A”B”C”. We then noticed that: -The product of two reflections is a rotation -The angle of rotation is twice the angle between two reflection lines

At the end of class, we began working on dilations. This is when the shape is stretched larger and the ( wouldn't say "mirror" because that does mean congruent) mirror image does not remain congruent with the original shape. By doing the dilation activity on page 139 in the Course Pack, we discovered: -Corresponding angle measures are preserved -The lengths of the line segments in the new figures hat and body doubled -The area of the new figure quadrupled -The perimeter of the new figure doubled We will continue to discuss dilations in the next class. Homework: AHE: Draw a triangle -Have two lines of reflection -Investigate the angle of rotation and compare to the angle between the two reflection lines CP page 132 #33

- Revisited Reflections - Revisited Transformations -Rotations The way we revisted Reflections was by looking at problem B on page 116 in our coursepacks. We started to talk about how we knew how to reflect quadrilateral JKLM at line n. Some of the points brought up were: -the points of the quadrilateral JKLM and the reflected JKLM are the same distance from the reflecting line; line n. -We can use an angle ruler to create a perpendicular line through any of the points. - When making perpendicular lines through any points to the line of reflection it will create a 90 degree angle. We can use an angle ruler, or a protractor and straight edge, to make sure this is exact by lining up the the Initial ray( ray that does not move) on the line of reflection and the Terminating ray (the ray that is the one that turns) with a point on the original polygon. In our example we used point K. With our example if we line up the angle ruler along line n, the line the reflection, and turn the ruler 90 degrees to line up with point K on the quadrilateral JKLM, we can create a line and then measure along that line the distance between point K and line n. We can then place point K' on the other side of the line n using the same distance from line n. Make sure that this point is on the same perpendicular line of point K. We will then have to repeat this process 3 more times for each of the other 3 points in quadrilateral JKLM. Next we worked on pg 114 part D. We came up with a definition completing the sentence in part D. A line reflection in a line //m// matches each point X on a figure to an image point X' so that point X will be the same distance to the line //m// as the prime point X' on a line perpendicular to //m.//
 * Day 17: March 17**

The review of translations we revisited the rules: Horizontal translations by b units; (x,y) --> (x+a, y) Vertical translations by b unites; (x,y) --> (x, y+a) Translations in the direction of line y = x; (x,y) --> (x+a, y+a) Translations keep shapes congruent and do not change them at all. The orientation or direction the shape is facing may change but the size of the shape or the shape itself does not. This is called ISOMETRIES. Iso (same) - metries ( metric or measure). The final group presented Rotations. We used the definition on page 118 in our coursepack to define a rotation. The transformation that turns a figure about a point, matching each point to an image point is called a rotation. The class traced the compass star on page 118 with patty paper and determined that the relationship between any point X, its image, and the center of the compass star is that they are always the same distance from each other, 90 degrees. With keeping this in mind we rotated the flag at the bottom of page 118, in a 60 degree counterclockwise rotation. We did not use patty paper but were able to complete the rotation with an angle ruler remembering to use the 90 degree rule. Page 85 in the supplement worksheets, we came up with rules for rotation. 90 degree rotation about the origin: (x,y) --> ( -y,x) 180 degree rotation about the origin: (x,y) --> (-x,-y) 270 degree rotation about the origin: (x,y) --> (y,-x) 360 degree rotation about the origin: (x,y) --> (x,y) As an example of this we used points (1,4) 90 degree rotation about the origin --> (-1,4) 180 degree rotation about the origin --> (-1,-4) 270 degree rotation about the origin --> (4, -1) 360 degree rotation about the origin --> (1,4) We will review Rotations next class. AHE pg 117 reflect, pg 123 part C finish the definition, Use figure on page 121 to answer the question part C1 on page 119.


 * Day 16: March 15 -- Midterm**

**Day 15: March 10** - Today we broke back into our groups and discussed our findings from Mondays assignment. - we decided what we were going to present to the class from our problems that each group did. -The first group to go was the Reflections group. - we learned when a shape is reflected over a line the original point and its prime are the same distance from the line. -synthetically: we used the patty paper to fold the reflected line and then we used rulers and angled rulers to make the reflection by measuring the distance between points and the center line and reflecting the points the same distance on the other side of the line. -Coordinates: we learned that to do a reflection over the x and y axis and y=x line that we needed to change the coordinates to their opposites or flip the coordinates Original Reflected -The next group to go was the Translation group -Coordinates: In order for it to be a translation you need to use addition and subtraction. You have to do the same operation to every x in a shape to move it horizontal and leave the y alone. To move it vertical you do the same operation to every y in a shape and leave the x alone. to move it along the y=x line do the same operation to both the x and y coordinates. -We will continue with the rest of translations and rotation on wednesday. -DONT FORGET TO STUDY!!!! MIDTERM MONDAY!!!!
 * to reflect over the x axis: (x,y) (x,-y) The negative sign means opposite not negative
 * to reflect over the y axis: (x,y) (-x,y) The negative sign means opposite not negative
 * to reflect over the y=x line: (x,y) (y,x) The y coordinate becomes the x coordinate it doesnt mean that you plot the y coordinate first

-Today we looked at polygons and how we would classify them. (rotation, reflection, both or none.) We used shapes that we cut out of magazines. -Reflection- You need to find a line where you can fold a shape and both sides match up. -Rotation- If the shape can turn and make the same shape. (Turn less then 360 degrees)
 * Day 14: March 8**

- We looked at our responses from page 107 -Regular polygons- All sides and angles are congruent. (We have to say both) -Complete turn- Start and end in the same spot pointing in the same direction. - We found that 360/n n= number of sides Is the simple formula for exterior angle of any regular polygon. -Vertex angle- 180 – 360/n n= number of sides

We looked at page 112+ handouts We were split into 3 groups 3 groups -slides (translations) -flips (reflections) -Turns (rotations)

Homework- pg 89- 5,9,11,13,14,15, 19-27 Pg 126 – 1,3,4,5,6,8,10-13

checked: KA

Day 13: Feb 24
• Using a computer program like Scratch, we can utilize shape properties to draw or build polygons • Scratch can help us solidify geometrical ideas as we put them into practice, instructing the sprite

We were in the lab using Scratch. We were so productive before class, talking about homework, Dr. B was ecstatic! We rock! We worked through the Scratch exercises, thinking about properties of all the polygons we were drawing. We wrote up our ideas to Exercise #5 and handed those in before we left. We know we have to finish all of the Scratch exercises before Monday, when we return. But, since we can download the software for free, no worries. We could finish them at home if we didn't get them done by tomorrow.

No new homework; just need to make sure we have all other homework problems completed, especially "Hannah's rule" we quite haven't finished. **Day 12: February 22nd

Today in class we reviewed and concluded the discussion about the double triangle formula using both of the previous talked about formulas. We haven't concluded it yet! Dr B

While going over the homework we discussed how to convince someone that the measure of the interior angles of a concave quadrilateral is 360 degrees. These were the steps we took. 1. Cut the quadrilateral in half to form two triangles. 2. We have the pervious knowledge that all the interior angles of a triangle equals 180 degrees. 3. Knowing that information about a triangle, since we have two, you add 180 + 180 = 360. 4. The sum of a quadrilateral is 360 degrees.

We also came across exterior angles. While those were talked about briefly in previous classes, we discussed it further to extract a better understanding of exterior angles. Most people believe the exterior angle to be the rest of the angle after finding the interior angle. REFLEX ANGLE. While that may be a vague description of someone's exterior angle, it is not the accepted definition. After extending one of the sides that make up the interior angle, you know that it takes 180 degrees to make up a line. So you take 180 and subtract the measure of the interior angle to get the true exterior angle.

People may mistake exterior angles for reflex angles. WHAT DO YOU MEAN HERE? A reflex angle, as previously stated in class, is any angle larger than 180 degrees and smaller than 360 degrees. An exterior angle is an angle measured by extending one of the sides of the polygon and measuring the angle that is outside of the shape and also on the same side of the line as the original interior angle. -JF

During class we looked into other ** REGULAR ** polygons like hexagons, octagons and decagons. We used our knowledge of interior angles, and our new found definition of exterior angles to help figure out the measure of angles in these shapes without measuring them. Make sure we recognize these were REGULAR polygons!!!

For example: Using a hexagon 1. Start by finding the center point and dividing the hexagon into equal triangles. You will end up making six triangles in the shape. 2. All interior angles around the center point adds up to 360 degrees. Also known as the central angles. So we take that number and divide it by 6, the number of angles the hexagon is divided into. We get 60 degrees. 3. Once you take 60 and apply it to each central angle, you can then find the other angles because you are aware that each triangle inside adds up to 180 and each triangle is an equilateral triangle since all sides of the hexagon are congruent. We can then figure out that all interior angles are 120 degrees 4. Next to find the exterior angle, one of the sides is extended, and when subtracting 180 from 120, we get to 60. Noting that the exterior angle is 60 degrees.

After following these steps with various types of REGULAR polygons we noticed a pattern. That if you have a central angle, you have an exterior angle. In all cases, the central and exterior angles ended up to be the same measure.

For the remainder of the class we used the Scratch program and did the exercises starting on page 102 in our course packet. edited by jlh **

Suggested: Download Scratch program at home and work on exercises. Last day to work on in class is Wednesday Project 1 due Wednesday**
 * AHE: P. 94 #5-7, 10 & 11

Today in class we talked a little bit more about how Hannah's double triangle formula worked, drawing an equilateral triangle using the turtle on ExpDraw, and rotational and reflection symmetry. 1. You can construct an equilateral triangle by only using the commands of foward, back, right, and left. We told it to go so many steps forward, then you would say 120 degrees left or right. When using this program you found out that using the interior angles it won't make a triangle, you have to use the exterior angle. AHE- to try and make different shapes using sprite commands. 2. Also we tried to figure out Hannah's double triangle formula. Why when you add angle C, and the two small angles you get angle D?-AHE 3. We also talked about what a reflex angle is, an angle between 180 degrees and 360 degrees. 4. We also talked about what shapes have rotational symmetry, a pentagon, square, rectangle, ect.
 * Day 11: February 17th**

We also talked about what shapes have reflection symmetry, which the three quadrilaterals above also have reflection symmetry. We made a formual to find the interior angles in a pentagon, 180-360/n=the interior angles, n=number of sides.

-Also in class we talked about positive and negative angles and how they cordinate on the quadrants, going counter clockwise. -Some new definitions were= -Initial ray: starting ray, terminal ray: ending ray -Regular Polygon: polygon that has all side lengths and interior angles congruent. -S.B.

Today in class we went over last classes homework assignments, revisited Hannah's double triangle formula, looked at and changed working definitions of rectangle and square, and went over our reflection, rotation and none posters.
 * Day 10: February 15th**

1. We can construct an equilateral triangle on sketch pad by:
 * 1) Draw a circle
 * 2) Place a point on the circumference of the circle
 * 3) Form another circle around the point on the circumference on the original circle, and stretch it until it intersects with the center point of the original circle.
 * 4) Place a point at the intersection of both circles (only one of the itersections is necessary). When both circles become highlighted you have found the exact intersection point, if only one circle is highlighted you have not found the true intersection.
 * 5) Connect the two center points with a line segment.
 * 6) Connect each of the center points to the point where the two circles intersect with line segments.
 * 7) Those three line segments will form an equilateral triangle.

2. Validation of the equation used on page 49. Hannah's formula: The measure of angle D= The measure of angle C+(180-the measure of angle C)/2. The equation the class found was: 180-(180-the measure of angle C)/2.
 * We will revisit this later. **AHE- Why does this work?**

3. Isosceles triangles and scalene triangles are disjoint. A isosceles triangle must have 2 congruent sides and a scalene triangle cannot have any congruent sides. Therefore it impossible to have a triangle both isosceles and scalene simultaneously.

Working definitions: Rectangle: Is a parallelogram with 2 right angles. Square: A rhombus with 2 right angles or 3 right angles or all congruent angles.

Definitions given in class: Spatial visualization: Being able to visualize or imagine in your mind. Regular polygons: Polygons whose sides are all congruent, central angles are all congruent, vertex angles are all congruent and will have both reflection and rotation symmetry. Irregular polygons: If it's not a regular polygon, it is an irregular polygon.

It's important to remember that a square is a special __type of rectangle.


 * AHE- Can you rotate a hexagon on it's middle lines of reflection and not on the vertex lines of reflection?

AHE- Create instructions for having a turtle "walk" and make an equilateral triangle using only forward, backward, left turn, right turn. Forward and backward can have distances, like 20 steps backwards, and left turn/right turn can have angles, like 50 degrees right.**

HH JLH

Today in class we focused on talking about quadrilaterals and came up with our class definition for our 6 types. We talked about the Venn Diagrams of quadrilaterals, which became our AHE. At the end of class we started to make posters about symmetry using triangles, quadrilaterals and other polygons. (We did not finish this exercise and will be working on it more next class.)
 * Day 9: February 10th**

Our Class Definitions for Quadrilaterals: We spent some time discussing what needs to be included in a definition. What information about a quadrilateral is necessary to include in its definition. -MT

1. Quadrilateral: A four sided polygon. 2. Kite: A quadrilateral with two pairs of congruent adjacent sides.__ 3. Square: A parallelogram with all 4 angles congruent. 4. Rhombus: A parallelogram with four congruent sides.
 * The issue was that if that is the definition of a kite than a square or a rhombus could also be considered a kite and most of the class disagreed with that idea.
 * This became part 2 of our AHE: research the definition of a kite.
 * There was also a disagreement with this definition so we thought it might be okay to say that it is a rhombus with two congruent angles. The reason we said rhombus is because it meets all the criteria of a square, but we added the two congruent angles because not all rhombuses have that characteristic.

5. Rectangle: Parallelogram with two congruent angels. 6. Parallelogram: A quadrilateral that has two sets of parallel sides. 7. Trapezoid: A quadrilateral with ONLY one set of parallel sides.
 * We discussed that you don't need to say four congruent angles because when you have two you always have four, because it is a parallelogram.

After we finished with our definitions we began talking about how we could make a large Venn Diagram using the parallelogram as the outer most part and working in toward the center. The other three that we needed to add were the square, rectangle and rhombus. The problem was not all rhombuses are not rectangles so how could it work? This was our AHE. How and where would you place all four of there in one Venn Diagram? Is it possible? If it is what would the diagram look like? when looking at the definition, we tried to figure out what order we should put the Parallelogram, rhombus, square, and rectangle. We had to figure out what is necessary or sufficient when figuring out the order we need to put these shapes in.

Another part that of our AHE was to finish page 37 in the course pack if you haven't already.

We finished class but beginning to work on symmetry using Triangles, Quadrilaterals and other Polygons. We were putting them into groups of if they had: no symmetry, some symmetry or rotational symmetry. We didn't get very far on this and I am guessing we will pick up with that next class period.

Last thing there is a quiz next Wed. TD

Today we focused on the pictures on the board from last class (that had to do with the game we played- using Venn Diagrams). We talked about the writing assignment, got a new AHE, talked about quadrilaterals, and worked in the computer lab on sketch Pad. We also talked about the reading we had to do a few class periods ago.
 * Day 8: February 8th**

The board work (3 examples of Venn Diagrams): We talked about having to sets of parallel sides in a big circle and then within that circle we had another circle that said all right angles. We discussed: -Parallel sides in a polygon do not need to be the same measure. -Parallel lines don't meet. -The distance between the two parallel lines stay the same. Another Venn Diagram example was all sides are congruent, and contains no congruent sides. These two would have their own circle (they are disjoint). This is because they are opposite from one another and they can't be or fit into the other. -Side note from this would be that a square has all right angles and a rhombus doesn't have to, looks like a diamond. A square can be a rhombus, just at a different angle. Rhombus can be two acute angles and two obtuse angles, depending on the way you turn it. The third example was a big circle with "contains at least one obtuse angle" and a smaller circle inside of that one saying "contains one angle larger than 180 degrees".

Definition for a Quadrilateral: -Polygon with four sides. Questions we need to ask ourselves now: What is a kite, square, rhombus, trapezoid, rectangle, parallelogram. AHE: Use shape cards, and write "some of the quads have..." and put properties in. (Look, what do you see). Then attach the names of a quad to properties and form a definition, like we did for triangles. AHE: Finish Bottom of page 37 (Venn Diagram). AHE: Google Search about angle measures. (Reflex angles).

AHE: Play around with Geoboard...can you really make an equilateral triangle.

HOMEWORK: Pages 50, 51, 52. (# 1, #2, #6)

We worked with the protractor. -What is a circle? A collection of points all the same distance radius away from the center. (Just collection of outside points)-Mathematically. (See circle as a space like a ball)-Childs view point.

We then used sketch pad. We made circles and triangles and figured out how to make equilateral triangles and isosceles triangles. You can have an equilateral triangle when two circles have the same distance. We worked from page 68. BN Focus of today's class: Class today was bascially a review of homework and making sure that everyone knew what angles were as we took a quiz at the end of class to write down what we have learned so far in MATH 1510. We used the term independent reflective learner. -independent learner focuses on YOU, ask questions, talk with each other. -the reflective part is thinking about what you have learned
 * Day 7: February 3rd**

We went over number 5 on our homework from page 49 in our coursepack -a formula that researched was D=(C+((180-C)/2)) this is part of our AHE, we need to find if this formula works for all triangles.

We defined rhombus which is a quadrilateral with all 4 sides congruent A quadrilateral is a polygon that contains 4 sides

We used a compass to make circles. -We went over the fact that a circle is not a 360 degree angle. -A circle is a collection of points that are equal distance away from the point which called the radius or radii (plural)

Before we took the quiz the question to ponder was, can we construct triangles with a compass?

On Monday we will be going to the lab. Focus of today’s class: __-determining how measuring a side is different than measuring an angle -congruency -triangles (definitions, relationships) -quadrilaterals
 * Day 6: February 1st**

We started class by looking at the measuring side/angle definitions on page 87 of the course-pack. We did a quick poll and decided to discuss #8 and #2 because we thought they were good definitions.

- 8 is good because:__
 * it describes how to measure a line segment (using length)
 * it describes where an angle in located in a shape (the space between where 2 line segments come together, which uses one of our angle definitions), and it also tells us what an angle actually is

- 2 is good because:
 * it describes //what// you measure (the angle is located between two sides that meet at a point) this sounds more like "where" vs "what". Reread #8's again.
 * also uses units such as degrees (which can be defined as **small pieces**) pieces of what?
 * we also decided that there is another way to measure angles, using **radians**
 * The only issue that we found was that we were not sure if it was correct to say " we are measuring the degrees." We talked about how degrees is what we are measuring an angle in. Was this the attribute or the units? ** - BC** This is a great question. We need to know the difference between the attribute or characteristic we are trying to measure and the unit we use to measure the attribute. Let's hit this again on Wed.

We also began talking about the difference between //units//, and the idea of //what// we are measuring (the attribute).
 * the question posed was, “what is the difference between saying “length”, and using a specific measure (inches, cm, etc) in a definition
 * length is an attribute we want to measure
 * units=specific measure (degrees, inches, etc)

Congruency.
 * congruency is the mathematical term for saying that two things (sides, angles) are the same
 * when dealing with congruent shapes, the two shapes must match exactly
 * the symbol ≅ is used when comparing two shapes
 * the symbol = is used when comparing two quantities

We then reviewed our triangle definitions from last week and replaced the word "equal" with "congruent" when needed.
 * **a triangle is a polygon with 3 sides**
 * acute triangle is a triangle with 3 acute angles
 * obtuse triangle is a triangle with 1 obtuse angle
 * equilateral triangle is a triangle that has 3 //congruent// sides
 * isosceles triangle is a triangle with only 2 //congruent// sides
 * scalene triangle is a triangle that has no //congruent// sides

We also looked at the chart on page 36 to discuss the relationships between triangle types using Venn Diagrams.
 * acute triangles and obtuse triangles are not related at all, so they would each have their own circle (disjoint) these circles represent sets or collections, in this case, sets of triangles.
 * isosceles triangles and obtuse triangles are related (you can have a triangle that is both obtuse and isosceles), so they would have intersecting circles
 * we used Geoboard on our calculators to draw different triangles and we measured them to determine their side and angle measurements

We ended class by talking about quadrilaterals.
 * we decided that you can create two triangles in every quadrilateral by drawing lines to connect vertices
 * we also determined that the sum of the angles in a quadrilateral is equal to 360 degrees, and we were asked to explain why this is true in our AHE
 * we did start to explain that because we know that a triangle's angles all add up to 180 degrees; so therefore if there is two triangles in every quadrilaterals it is okay to assume that all the angles in a quadrilateral will equal 360 degrees. (180+180=360) -BC

We will pick up next time discussing the Venn Diagrams that we created during the Quadrilateral Property Game that we played last week.

Focus of Today's class:**
 * Day Five: January 27th


 * Triangle definition
 * Different types of triangles and their definitions
 * Venn Diagrams
 * Quadrilaterals

Today we started by creating a solid definition for a triangle we all agreed on. We decided **a triangle is: a polygon that has three sides**. (We kept it short and to the point, deciding not to include facts like they have three interior angles, and the sum of all the angles always equals 180 degrees).

We talked about a lot of different types of triangles and thier definitions, including:
 * obtuse triangle- a triangle that has one obtuse angle (or possibly to break it down: a three sided polygon with one angle that is larger than 90 degrees. We kept both on the board and decided it would depend on who you were teaching/talking with depending on which definition to use).
 * acute triangle- a triangle with three acute angles.
 * right triangle- triangle that has one right angle
 * equilateral triangle- a triangle that has three equal sides
 * isosceles triangle- a triangle with only two equal sides
 * scalene triangle- a triangle with no equal sides
 * We concluded that equilateral triangles are not isosceles triangles.

Next we discussed Venn Diagrams and the several different types there are. Such as:
 * Two completly seperate cirlces, not touching at all (demonstarted by now and laters and tootsie pops, different)
 * two seperate circles not touching each other, but both inside a larger circle (same as above, two types of different candy, but both still inside the larger cirlce of "candy")
 * a small circle inside another larger circle (large "candy" circle with small "red candy" inside it)
 * two circles overlapping in the center, creating three categories (used example of obtuse triangle and scalene triangles. Both could be seperate from each other and could also overlap in middle and be both)

Lastly today we briefly began discussing quadrilaterals. We did not yet come up with a definition, but listed these key points: We ended class by playing "Quadrilateral Property Sort Game" with our small groups. We played the game four times, by picking two descriptions and seeing how and if they could go together or not. Then we would describe them and draw them each in relation together using one of the types of Venn Diagram, with an example of each quadrilateral in each circle.
 * have four sides
 * are polygons
 * have four angles

We will pick up class next week talking more about quadrilaterals and making a solid definition!

Focus of Today's class:
 * Day Four: January 25th**
 * finalizing the angle definition
 * went over how angles are measured
 * started triangles

All triangles have: Some triangles have: - 3 sides - sum of the angles = 180 - 3 angles Some triangles have:__ -equal side lengths - have different angles - 1 obtuse angle and 2 acute angles - have 1 right angle - three acute angles - three different side lengths - two sides have the same lengths 1 different side length
 * 1) Our working definition: An angle is formed by two line segments (lines or rays) at shared endpoint called the vertex. It is the turn represented by the two lines, line segments or rays. It could also be the area between two sides of a shape or lines where they meet or come together.
 * 2) A degree is the common unit of measure for angles. To measure angles we need to find a unit to measure the space between two line segments. A unit could be a degree, a slice or a sliver etc. We measured angles in class today using an angle ruler. Using the angle ruler goes along with our turn definition of an angle. The angle ruler shows the turn between two line segments. We came to the conclusion that the biggest angle we can make is infinite. Some younger students might believe that the biggest angle they can make is 360 degrees and that is because they are attaching the term 360 degrees to a circle. Their protractor only shows angles to 360 degrees it does not show that it can keep going.
 * 3) Triangles were the beginning of our next discussion. We made a list of qualities that all triangles have and qualities that only some triangles have. **We started a working definition of triangles which was, a triangle is a polygon that has three sides, three interior angles whose sum is 180 degrees.**

We have a beginning definition for triangle. Please include that! I already did... but it is bolded now.

Focus of Today's Class:**
 * Day Three: January 20th
 * Angles
 * Measuring Angles
 * Degree

Angles
 * We worked on making a definition of an angle.
 * Working Definition #1: An angle is formed by 2 line segments/rays at the shared endpoint called the vertex.
 * Working Definition #1.5: The area between two sides of a shape or lines where they meet or come together
 * Working Definition #2: The turn between represented by two rays.
 * Characteristic #1: At any vertex there are at least two angles possibe.
 * Characteristic #2: Angles have an interior and exterior.
 * An arc mark is irrelevant, yet is helpful so we know what angle we are talking about/looking at.
 * We also talked about how we use angles in everyday life. ( Examples : Angles of roads, angle of which you look at something, doing a 360 on your skateboard.)
 * Degree**


 * How we measure angles
 * A unit of measurement
 * In class, we discussed how we would measure an angle using the patty paper protractors everyone made for homework, but not using the word "degrees". We came up with some examples and made up our own unit of measurements. Thus, made us understand the use of a degree.
 * We began talking about how a degree came to be the unit of measurement for an angle because if you were to make a triangle type shape ( maybe we can call it a wedge or pie piece?), it would take 360 of those triangles to fill up a complete circle (let's use the word turn instead of circle; why am I suggesting that?) , therefore 360 degrees makes one **complete turn**. And why is it that 360 of these wedges called degrees should fill up the complete turn.

AHE 1. Complete the sentence on the bottom on p 85 and email that to me by Sat evening, 5pm. 2. Refer to the triangle shapes on p182 (you used those in class already). Double check the 2 lists you made in class last week, All triangles have and some triangles have, so that you have all of the characteristics you can see and verify. We will work from these lists on Monday so it is important you all have these lists from which to work. 3. If you get a calculator before Monday, get it to me WELL BEFORE class starts so I can get it ready.

Writing Assignment: Due Wed Jan 27 Number the responses on p 87 from 1 to 13 (notice two in the middle are not separated). Pick one response, tell me which one it is by # and write it out, then explain why you believe it is a good, solid, appropriate thinking. Pick a second response, tell me which one it is by # and write it out then explain why you believe it shows signs of misunderstanding what an angle is and how we measure it. Refer to p 7 for general criteria for writing assignments. Be sure to clearly support your responses with evidence from the response much like we did in class with student # 260 on p 88.

Tentative quiz set for Wed Feb 3. We first off defined what a polygon was. A polygon ("Many Angles") is any angular shape. We then went more in depth. We figured out the difference between line and line segment.
 * Day Two: Jan 13

Line- Goes on forever Line Segment- Piece of a line that has two endpoints

We continued with how a polygon doesn't have any gaps, breaks, or is continuous at all. All segments, or sides, are connected, making it a //closed// shape.

Our working definition of a polygon was: A //Closed// curve of //Straight// sides.

We correctly identified whether or not some shapes were polygons or not, until coming to a particular shape: ** Since the sides of a polygon shouldn't intersect, we changed our working definition of a polygon to: A //Simple Closed// curve of //Straight// sides.

We also started going over the definition of an angle based on different students definition. We didn't finish, but we've defined it a little better. PLACE OUR WORKING DEFINITION HERE.

The ray of the angle has one endpoint, and the other end goes on forever. We determined, using examples in class, some irrelevant facts about angles. 1. Doesn't need an arc-mark to be considered an angle 2. Doesn't need one ray segment to be horizontal. 3. Orientation is irrelevant. NS
 * Make sure you tell your students it's a rectangle or square instead of saying it's a box. A box is a 3-DIMENSIONAL SHAPE.

Day One: Jan 11 ** Big Ideas of Class: Shapes and how they can be identified-by their features Sides- a. The number of sides b. The length of the sides-if they are all the same, all different, or some of them are the same c. Concave sides, the sides cave in d. Convex sides, the sides bulge outwards

Angles-The angles that the sides make where they join

Overall appereance- a. Symmetrical b. Asymmetrical

Congruent shapes- shapes that match or are the same even though they are positioned or oriented in unsimiliar manners

Polygons- Angular shapes- Poly(many) gon(angles)

Triangles- The sum of all angles in a triangle is 180° -Ways to prove: a. measure every single triangle possible with a protractor b. cut out any size triangle from paper, draw an arrow in each corner pointing towards the corner, cut or rip off the corners of the triangle, set aside the body of the original traingle, put all the corners next to each other with the points pointing towards the same spot. All the corners together should make a straight line, a straight line is 180°, this shows that the sum of all the corners/angles of a triangle equal 180° without measuring the angles. We also discovered that all triangles have 3 sides.

edited by jh