Attached below are some helpful documents for construction unit test prep: topic list, TIC TAC TOE questions from game, and a Build-your-own Notes Sheet for points of concurrency.
Today we discovered and practiced applying the last of our content in the construction unit. We are busy acquiring knowledge about and the ability to construct the four points of concurrency in triangles: orthocenter, incenter, circumcenter, and centroid. Incenter and circumcenter are the centers of the inscribed and circumscribed circles respectively. The centroid in the center of gravity or mass of a triangle. Orthocenter and Circumcenter can be outside of the triangle (obtuse) or on the triangle (right triangle). Centroid and incenter are always inside the triangle.
C-12 - The incenter of a triangle is equidistant from the sides of the triangle.
C-13 - The circumcenter of a triangle is equidistant from the vertices of a triangle.
The centroid investigation is attached below.
C-14 ... are concurrent at the centroid
C-15 .... twice...
C-16 - The centroid is the center of gravity or mass of a triangle.
Attach your cardstock in your G.T. for a sketch.
The half-worksheet of practice on constructing points of concurrency is attached below. Get this done.
The homework #16 is p 179:1-5 (no sketches necessary... just state whether circumcenter or incenter based on new conjectures), p 186:1-5 (use C-15-16), p 191:1-2,8-24. For the T/F, correct or sketch for false. For matching, answer only is enough. It is due Tues-Wed (the next block).
Quizzes must be made up by zero hour tomorrow. I am returning the quiz during the next block.
Topics will be available tomorrow as well.
The test is Thurs-Fri, Oct 19-20, about 70 pts.
Today we worked at gaining speed at basic constructions. We did an investigation on points of concurrency of altitudes, angle bisectors, and perpendicular bisectors. The investigation is attached below. If you were not in class, you need to complete the investigation.
HW #14 was checked for 4 points and corrected.
We took Quiz 3.1-4 over synthesis of basic constructions. - 22 pts
If you missed the quiz, please take during advisory tomorrow.
Hw #15 p p 169:1-3, p 159:8, p 181-2: 6-7, 20-24, is due next time we meet.
Quiz 3.1-4 is 22 pts, Oct 11-12, Wed-Thurs. It is constructions only, no T/F, no matching, no Multiple Choice, no writing.
The Unit Test will be Oct 19-20, about 70 pts. It will have multiple choice, algebra, explain why, and 6 constructions.
Block 20 is just below. Scroll down!
Another day in our construction unit: frequently referred to as the "confusing day". Lots of information chasing around in our heads; not enough skills to differentiate the look of different constructions, especially when you are doing multiple things in a triangle.
Today's goal: copy a triangle by copying three segments; construct 90 degree angle, 45, and 135; construct altitudes in triangles; construct parallel lines; construct angle bisectors in triangles; discover that points on the angle bisector are equidistant from the sides of the angle, allowing us to construct a circle that touches both sides of the angle.
We wrote up the definition of altitude and copied sketches from p 154. We wrote up C-7 from p 153. The fill-in-the-blank is "perpendicular". The sketch is above the conjecture. Label: PM is the shortest distance. We also added problems 7 and 9 from p 159 to HW #13. A grade was taken on HW #13 for 4 pts.
Quiz topics were passed out, as well as homework #14 worksheet. Add to your worksheet: p 159:10, p 162: 1 or 2, p 155:10 in some classes. Topics and worksheet are downloadable below.
We did an investigation together to discover C-8: Every point on the angle bisector of an angle is equidistant from the sides of the angle. This was done on patty paper. Can be glued into your geometric truth.
We tried a construction of parallel lines based on looking at several. If you have trouble with this on your homework, google: mathopenref constructions
Goal of this block was to finish "filling" our tool belt with our set of basic construction skills.
We recorded the following terms into geometric truth: segment bisector, perpendicular bisector, median, midsegment.
We also wrote up C-5 and C-6 from pp 148-9. The points on the perpendicular bisector are equidistant from the endpoints of a segment; any point on the perpendicular bisector of a segment is equidistant from its endpoints. Leading us to the circumscribed circle construction that we did on problem 7 of our homework. We then looked at the relationship between problems 7 and 8 on HW #12... what happens if the triangle is obtuse? Right? Connect median and midsegment definitions to actual construction. etc
Then we investigated the shorted distance from a point to a line and figured out how to construct the perpendicular from a point to an infinite line. We practiced in our notes. Then we turned one of these constructions into a triangle, where the shortest distance is an altitude. We constructed 3 altitudes in an acute triangle. Then we figured out how to construct a perpendicular from a point on a line.
Our finale was to construct an angle bisector, a simple construction that we also viewed on mathopenref.
In most classes, we also sketched out how to construct a square by questioning.
A grade was taken on hW #12, and #11 if not already done.
Attached is information about the geometric truth write-ups.
If you need a link to mathopenref constructions, look at your Pre-AP Geometry, Look for Resources. The first link is to the constructions. The third link is to condensed lessons about the current content.
Hw #13 - p 154:2-5, p 158:1-5
Quiz next week on Wed-Thurs, even in 6th period.
This block our topic was "perpendicular bisector".
We started with p 65:22 - a problem from a previous unit that asks us to locate several points in the coordinate plane that would make an isosceles triangle with the segment in the book. In other words: several points that are equidistant from the endpoints of segment RY. Connecting 5 of these points allowed us to locate the perpendicular bisector of RY.
What is a perpendicular bisector? A line that passes through the MIDPOINT of a segment at a RIGHT ANGLE. We discussed the difference between this and just a segment bisector (passes thru midpoint at any angle). See good sketches on p 147. We did the investigations on pp 147-8 to discover that points on the perpendicular bisector are equidistant from the endpoints of a segment, AND just like the warm-up: if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector. This converse was used to discover the actual construction for perpendicular bisector.
After sharing good homework and measuring our answers for accuracy, we looked at medians on p 149 and made up a definition based on the sketch. None of our definitions or conjectures got written up today.
We then made triangles with a straightedge and practiced constructing perpendicular bisectors of the sides, as well as locating a median after this construction (use perpendicular bisector to find the midpoint, connect back to a vertex of the triangle with a ruler).
HW #12: pp 149-50: 1-4, 7-8
We also viewed tests (except 3rd, which got new seats).
Link to mathopenref is in the post below this one.
We are beginning a new unit that should test around Oct 17-18. The big topic is "Constructions". There is much more to the unit that learning to do some things with the compass and straightedge. We are learning to lean on the idea of compass setting as a distance that is not associated with a number and can therefore be relied upon as a form of deductive reasoning. Today we read and re-created Euclid's first proposition: given a line segment, construct an equilateral triangle. Questioning led us to why it works.
The daisy designs were collected for 5 pts. We did a daisy design in our notes to construct a regular hexagon inscribed in a circle. And why it works.
We started our "construction tool belt" today. Our current basic skills are 1) copy a segment, 2) copy an angle, 3) construct an equilateral triangle or a 60 degree angle (same thing).
No new definitions or conjectures today. Homework requires a compass. p 145:1-8.
For problem 7 on the homework, make the triangle fairly big. Can you you "copying a segment" and "copying an angle" to copy a triangle?
Tests will be viewed next time.
For links to help on daisy design, choose Pre-AP Geometry, Look for Resources, first link.
Block 16 was our unit test. The only homework is a daisy design. You should have completed your basic compass design at the end of the test block and all you lack is coloring it.
A day: if you need to use a compass to finish your design, come by during A & E on Friday.
Use p 11 in your textbook for how-to's or creative suggestions. The link mentioned above is very specific about how to do the construction.
The "rubric" - 5 pt grade for constructing at least one daisy with some decoration or elaboration, or a sequence of them (see p 11 for ideas). Use a piece of copy paper. You may use a ruler to add segments if wanted. The daisy must be constructed with a compass. Color. (colored pencils are available during A&E in my room).