INTRODUCING VARIABLES AND WORKING TOWARD ALGEBRAMath Smart TLQP Workshop @ Binghamton UniversityDecember 14, 2013

Dr. Rachel BachmanWeber State University

I have more than impression – it amounts to a certainty – that algebra is made repellent by the unwillingness or inability of teachers to explain why we suddenly start using a and b, what exponents mean apart from their handling, and how the paradoxical behavior of + and – came into being. There is no sense of history behind the teaching, so the feeling is given that the whole system dropped down ready-made from the skies, to be used only by the born jugglers. This is what paralyzes – with few exceptions – the infant, the adolescent, or the adult who is not a juggler himself. ~Jacques Barzun, Teacher in America, 1945

THE PROBLEMs = ½ at2 describes the relationship between time and distance fallen for a free falling object, where

s = distance fallen in feeta = acceleration due to gravity (about 9.81 ft/sec2)t = time in seconds

What are the variables represented in this function?

This issue with this problem is that students traditionally thinking that s, a, and t are all variables because they are letters. However, a is not a variable because it is a constant.

THE STRATEGYReal Activity

Extending a Pattern to the General Case

Verbal Communication of Pattern

“Syncopated” Communication of Pattern

Symbolic Communication of Pattern

TODAY’S AGENDA

Clap Your Name The Border Problem Total Wall Area Introducing Algebra Tiles

CLAP YOUR NAMESchaffer, K., Stern, E., & Kim, S. (2001). Math Dance with Dr. Schaffer and Mr. Stern. Santa Cruz, CA: MoveSpeakSpin

Math Dance can be ordered only from mathdance.org. If you have any further questions, you can contact Erik Stern at estern@weber.edu

RACHEL-Start off by clapping your name for the students. Clap your hands for consonants and slap your legs for vowels. See if they can figure out the pattern of clapping and slapping.

-Once you have the clapping/slapping pattern established, have the students practice clapping your name with you.

NOW IT IS YOUR TURN….CLAP YOUR NAME Clap for Consonants Slap for Vowels Y You get to choose clap or slap

Have the students practice clapping their own name. Then have a few students demonstrate. Pick up on any variations they are doing (syncopating, pausing, etc.). Teach the students to repeat their clapping sequence with equal beats.

-Once the students can repeat their sequence, you pick one of their names to clap but this time capitalize their name by clapping/slapping louder on the first letter of their name. See if they can pick up on the change you made. Have them practice capitalizing their name while repeating their pattern.

-Choose two students in the class (with different length names). Have one of the students work with one half of the room to clap his or her name (repeating and capitalizing). Have the other student do the same thing with the other half of the room. Once both groups are well practiced, perform the two names together. Introduce a discussion about what the sequences sounded like together. Ask if the capitalizing was always occurred together.

After how many beats will the two names begin their sequence at the same time?

1.Create a drawing of the two clapping sequences happening at the same time.

2.How many times does each sequence need to be played till they begin at the same time? How many beats is that?

3.With your partner, use your method of drawing to answer for the same questions for your two names.

HOW ABOUT FOR THREE OR FOUR NAMES?

THE BORDER PROBLEMBoaler, J., & Humphreys, C. (2005). Connecting mathematical ideas: Middle school video cases to support teaching and learning. Portsmouth, NH: Heinemann.

Without counting one by one, determine how many squares are in the border of this 10x10 grid.

The chart below shows the six methods for the Border Problem10 x 10 6 x 6 15 x 15 231 x 23110 + 10 + 8 + 8 6 + 6 + 4 + 4 15 + 15 + 13 + 13 231 + 231 + 229 + 229

4(10) – 4 4(6) – 4 4(15) – 4 4(231) – 4

4(8) + 4 4(4) + 4 4(13) + 4 4(229) + 4

4(9) 4(5) 4(14) 4(230)

10 x 10 – 8 x 8 6 x 6 – 4 x 4 15 x 15 – 13 x 13 231 x 231 – 229 x 229

10 + 9 + 9 + 8 6 + 5 + 5 + 4 15 + 14 + 14 + 13 231 + 230 + 230 + 229

Step 1: Figure out the methods for a 10 x 10 gridStep 2: Edit the methods for 6 x 6, 15 x 15, and a 231 x 231 gridStep 3: Pick one of the six methods. Write directions for calculating the number of squares in the border for any square grid using that method. Use only words, no symbols.Step 4: Work on one of the methods with the class to be as specific and precise as possible.Step 5: Edit the instructions by swapping out numerals and symbols, leaving only the words “number of squares on one side of the grid” in the instructions.Step 6: Introduce a symbol to take the place of the words “number of squares on one side of the grid” Step 7: Write all of the methods symbolically

THE BORDER PROBLEM EXTENSION Verify all the strategies are algebraically equivalent. What does the expression mean in reference to the

Border Problem? What does the sentence mean in reference to the

Border Problem? What does the sentence mean in reference to the

Border Problem? What does mean in reference to the Border Problem?

TOTAL WALL AREABachman, R. M. (2013). Building conceptual understanding in a remedial college mathematics classroom: A study of effectiveness. (Order No. 3596992, State University of New York at Binghamton). ProQuest Dissertations and Theses, , 324. Retrieved from http://search.proquest.com/docview/1449822624?accountid=14168. (1449822624).

What is the total wall area in a room 8 feet high, 12 feet wide, and 16 feet long?

Suppose a painter wanted a formula to use to calculate the wall area of any rectangular room. Develop such a formula for the painter.

The most common expression is 2LH + 2WH (finding the area of each kind of wall, doubling each to get the wall just like it, and adding all the wall areas together). Once we come up with one expression, I ask the class to interpret the following expressions with reference to the room…

a. H(2L + 2W) -- perimeter times height…one long wall

b. H(L + W + L + W) -- same as above

c. 2H(L + W) -- the length of half way around the room times height gives half the wall area and then times 2 gives the total wall area

d. -- find average wall length, multiply by four to find the perimeter and then multiply by height to find the wall area.

ILLUSTRATIVE MATHEMATICS PROBLEMSIllustrativemathematics.org

DISTANCE TO SCHOOL - 540

Some of the students at Kahlo Middle School like to ride their bikes to and from school. They always ride unless it rains.

Let d be the distance in miles from a student's home to the school. Write two different expressions that represent how far a student travels by bike in a four week period if there is one rainy day each week

P AND Q - #436

Suppose P and Q are two different animal populations, where Q > P. For which, say what the expressions mean and then decide which expression is larger.

a. AND 2b. AND c. AND 0.50d. ANDe. AND

DELIVERY TRUCKS – #531

A company uses two different-sized trucks to deliver sand. The first truck can transport x cubic yards, and the second y cubic yards. The first truck makes S trips to a job site, while the second makes T trips. What quantities do the following expressions represent in terms of the problem's context?

MIXING CANDIES - #389

A candy shop sells a box of chocolates for $30. It has $29 worth of chocolates plus $1 for the box. The box includes two kinds of candy: caramels and truffles. Lita knows how much the different types of candies cost per pound and how many pounds are in a box. She said, “If x is the number of pounds of caramels included in the box and y is the number of pounds of truffles in the box, then I can write the following equations based on what I know about one of these boxes:

Assuming Lita used the information given and her other knowledge of the candies, use her equations to answer the following:

How many pounds of candy are in the box? What is the price per pound of the caramels? What does the term 12y in the second equation represent? What does 8x+12y+1 in the second equation represent?

IN THE SAME SPIRIT…

1. Player E and Player F are two different baseball players. On average, Player E scores more homeruns per season than Player F. Let e represent the average number of homeruns scored by E each season and f represent the average number of homeruns scored by F each season. a. Write a variable expression that represents the number of additional homeruns

Player E makes on average compared to Player Fin a season.

b. Write a variable expression that represents the ratio of the average number of homeruns made by Player E to the average number of homeruns made by Player F this season.

c. Explain which expression is larger: 𝑒+𝑓 or 2𝑒

d. Interpret the expression 𝑒+𝑓2 in reference to E and F.

e. Write a variable expression that represents the number of total homeruns

made by E and F in their entire careers. Define the variables you use.

INTRODUCING ALGEBRA TILESBachman, R. M. (2013). Building conceptual understanding in a remedial college mathematics classroom: A study of effectiveness. (Order No. 3596992, State University of New York at Binghamton). ProQuest Dissertations and Theses, , 324. Retrieved from http://search.proquest.com/docview/1449822624?accountid=14168. (1449822624).

LET’S REVISIT THE BORDER PROBLEM…

-You can use base 10 blocks to model the border problem strategy of doing four times the side length minus four for the corners. Then transition to a multi-base block other than ten to show the method for a different grid. Finally introduce the algebra tiles to show the method working on any square grid.