Last night I tutored my geometry student for the second time this week. Section 6-6 of the Glencoe text is on -- and don't say I didn't warn you, since I mentioned it earlier --

*trapezoids*. And my student already knew about trapezoids, since I showed them to him during the first tutoring session. Indeed, I didn't given my student another worksheet, since our worksheet from earlier was on trapezoids.

My student had two proofs to complete last night. Naturally, both of them were on

*isosceles*trapezoids, since, as the mathematician Conway pointed out, there's hardly anything to prove about a trapezoid that isn't isosceles.

Given:

*HGK*and

*JKG*are congruent,

*not*parallel to

~~JK~~

Prove:

*GHJK*is an isosceles trapezoid

Glencoe asks the students to write

*flow*proofs for this assignment. I won't post any proof, since this has nothing to do with the current U of Chicago chapter (trigonometry or vectors). But notice out the exclusive definition of trapezoid led necessitated the statement "

*not*parallel to

*" being included in the list of Given statements -- otherwise the figure could be a rectangle. Under the inclusive definition, this statement would not be necessary.*~~JK~~

Given:

*ZYXP*is an isosceles trapezoid.

Prove:

*PWX*is an isosceles triangle.

(The diagram shows that lines

*PZ*and

*XY*intersect at

*W*.)

This is a straightforward proof -- albeit long since Glencoe requires the students to use the definition of congruence to convert between the congruence of segments and the equality of their lengths before one can apply the Segment Addition Postulate.

Also, today I subbed in another math class. This time, it was a sixth grade class. The text that this class is using for Common Core was published by Carnegie Learning. Here is the course layout:

Chapter 1. Factors, Multiples, Primes, and Composites

Chapter 2. Prime Factorization and the Fundamental Theorem of Arithmetic

Chapter 3. Fractions

Chapter 4. Decimals

Chapter 5. Ratios

Chapter 6. Percents

Chapter 7. Introduction to Expressions

Chapter 8. Algebraic Expressions

The class was on the final section of the sixth chapter -- Section 6.5, "Practical Percents Practice." This is just about the right pace for the class to finish Chapter 8 before the SBAC exam. In this section, the students had to deal with the common misconception that a 25% discount followed by a 40% discount equals a 65% discount. I even told them about some bookstores that once had to explain that if a book can be preordered at 40% and members get a 10% discount, the book is only 46% off. The main sixth grade Common Core standard for percents is:

CCSS.MATH.CONTENT.6.RP.A.3.C

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Today I finally post the vector activity that I've been planning this week. But today is supposed to be the review for the Chapter 14 Test. Well, that's no problem -- technically this activity counts as part of the test review.

Even then, I still have problem making this activity organized. I quickly created my worksheets for both Partner A and Partner B, but I didn't have time to scan them before posting them here. Since I can't keep delaying the activity, I will cut and paste what I wanted the worksheet to say right here:

Name
_________________________ Date _______________ Period _____

Partner’s Name
_________________________

**Partner A: Vector Activity**

**Task 1:**What vector did you select? (_____, _____)

Add (2, 1) to this vector.
Answer: (_____,
_____)

Tell this answer to Partner
B. Did Partner B guess your original vector?

**Task 2:**What vector did Partner B tell you? (_____, _____)

Partner B multiplied the
original vector by the scalar 5 to obtain this vector. What is the original
vector? (_____, _____)

Tell your guess to Partner B.
Did you guess correctly?

**Task 3:**What vector did you select? (_____, _____)

Multiply this vector by the
scalar 4. Answer: (_____,
_____)

Add (-2, -2) to this vector.
Answer: (_____,
_____)

Tell this answer to Partner
B. Did Partner B guess your original vector?

**Task 4 (Optional):**Graph the original and final vectors from Task 3.

Name
_________________________ Date _______________ Period _____

Partner’s Name
_________________________

**Partner B: Vector Activity**

**Task 1:**What vector did Partner A tell you? (_____, _____)

Partner A has added (2, 1) to
the original vector to obtain this vector. What is the original vector? (_____, _____)

Tell your guess to Partner A.
Did you guess correctly?

**Task 2:**What vector did you select? (_____, _____)

Multiply this vector by the
scalar 5. Answer: (_____,
_____)

Tell this answer to Partner A.
Did Partner A guess your original vector?

**Task 3:**What vector did Partner A tell you? (_____, _____)

Partner A multiplied the
original vector by 4, then added (-2, -2) to obtain this vector. What is the
original vector? (_____, _____)

Tell your guess to Partner A.
Did you guess correctly?

**Task 4 (Optional):**Graph the original and final vectors from Task 3.

As I've mentioned earlier, this activity is based on the one that the class I subbed for earlier this week were working on, except that this one involves vectors. Also, I pointed out that I designed this activity to run more smoothly than the one in that class, by making sure that Partner A and Partner B have clearly defined roles.

In fact, this one should run even better than my improved 6th period version ran. In that class, I had to tell the Partner A students to cross out the boxes since only the Partner B students need them. But that wasted time, and students on the other side of the classroom were goofing off while I was checking to see that the students on my side were crossing out the boxes. So for my lesson, Partner A and Partner B already have separate worksheets, so there is nothing for anyone to cross out -- so no time is wasted crossing out boxes.

There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.

Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:

For the horizontal component:

Ace through 10 -- valued 1 through 10

Jack -- valued -2

Queen -- valued -1

King -- valued 0

For the vertical component, use the suit:

Clubs -- valued -1

Diamonds -- valued 0

Hearts -- valued 1

Spades -- valued 2

Examples:

Eight of Diamonds -- (8, 0)

Ace of Clubs -- (1, -1)

Jack of Hearts -- (-2, 1)

Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between

*clubs*and

*spades*!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to

*graph*that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.

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