1 What is Algebra in the Middle School? Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876.

1

What is Algebra in the Middle School?

Dr. Eric Milou

Rowan University

Department of Mathematics

milou@rowan.edu

856-256-4500 x3876

2

Overview

Traditional Algebra

Patterns & Algebra Problems

States’ Test Problems

Algebra I Placement

3

Algebra

If x + y = 28 and xy = 7,

then

€

1x

+1y

=

4

x + y = 28 and xy = 7

1

x+1y=

y+ xxy

=287

=4

5

Find x

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

6

Define “ALGEBRA”

The intensive study of the last three letters of the alphabet.using variables to replace something significant in a problemequations, formulas, numbers, headachesusing letters such as x to represent an unknown number and solving for that letterThe class between prealgebra and geometry

7

Traditional Definitions!

Dolciani Verbs

– Solve

– Simplify

– Factor

– Rationalize

8

We need a BALANCE

Traditional Algebraic Symbolic Manipulation

Patterns & Algebraic Thinking

Contextual problem solving

9

Patterns to Algebra

Swimming Pool Walkway

Given a 2 by 2 pool:

How many 1 by 1

concrete slabs (tiles)

are needed for

the walkway?

10

What if the pool was 3 by 3?

11

Extend to algebra

Pool Size Tiles for walkway

2 by 2 12

3 by 3 16

4 by 4

5 by 5

n by n

20

24

4n + 4

12

Connect the algebra back to the geometry

13

Piles of Tiles:3rd grade algebra

14

Construct a Table

Figure Total Squares

1 4

2 7

3 10

4

5

x

13

16

3x+1

15

Growing Tiles

size # of tiles size # of tiles size # of tiles

0 2 0 -1 0 1

1 5 1 2 1 3

2 8 2 5 3 7

3 11 3 8 5 11

4 14 4 11 7 15

5 17 5 14 9 19

3

3

33

3

3

3

4

4

4

4

3

2

2

2

2

3x+2

3x-1

2x+1

16

Crossing the River

8 adults and 2 children need to cross a river and they have one small boat only available. The boat can hold ONLY:– One adult– One or two children

How many one-way trips does it take for all 8 adults and 2 children to cross?

17

8 adults and 2 children

# of ad. + 2ch.One ways trips1 5

2 9

3 13

4 17

5 21

x 4x+1

18

8 adults and 3 children

# of ad. + 3ch.One ways trips1 7

2 11

3 15

4 19

5 23

x 4x+3

19

a adults and c children

a ad. + # ch. One ways trips2 4a + 1

3 4a + 3

4 4a + 5

5 4a + 7

c 4a + ( )

2c - 3

20

Scratch off game cards

A game card has 5 scratch off spots. Exactly two of them have a million dollar prize. The other three spots are blank.

You can ONLY scratch off two spots.

What is the probability that the two spots you scratch off have the million dollar prize?

€

2

5⋅

1

4=

1

10

21

Change the Number of Spots

Spots Probability3 1/3

4 1/6

5 1/10

6 1/15

7 1/21

x

3

4

5

6

1

1

1

€

1x(x −1)

2

⎛

⎝ ⎜

⎞

⎠ ⎟

22

To Medicate or Not?

Thursday at 9 PM, an athlete is considering taking a cold preparation medicine (16 gr. dose). He will be drug tested before competing on Saturday at 9 AM. The test can detect 1 gram or more or the drug. His body eliminates 1/4 of the drug every four hours. If he takes the 16 gr. dose, will he fail the test?

23

16 gram dose

Number Hours Grams Remaining of Intervals Elapsed in Body

0 0 161 4 122 8 93 12 6.754 16 5.06x 4x

€

163

4

⎛

⎝ ⎜

⎞

⎠ ⎟x

24

Back to 3rd grade

13 x 17 = ?

1 3

x 1 7

1

2

9031

2 2 1

-------

-------

10 7

10

3

1 0 0

3 0

7 0

2 1

221

25

This approach leads to ?

x 7

x

3

x2

3x

7x

21

Algebra: (x + 3) (x + 7) =

26

Texas State Assessment

Herman claimed that the square of a number is always greater than or equal to the number. Which of the following examples disproves Herman’s claim?– A) A comparison of (-1.5)2 with -1.5– B) A comparison of (-0.5)2 with -0.5– C) A comparison of (0.5)2 with 0.5– D) A comparison of (1.5)2 with 1.5

The graph of y = 11x2 + c is a parabola with a vertex at the origin. Which of the following is true about the value of c?– A) c > 0– B) c < 0– C) c = 0– D) c = 11

27

Connecticut State Assessment

Elena plans to rent a motorbike at the beach. She uses a rental service that charges a fixed fee of $25 plus $15 for each hour or part of an hour that she uses the bike. Fill in the table and extend the patterns to show the total charges if she works from 9:15 am to 5:40 pm.

Hours Charges

0 < h ≤ 1

1 < h ≤ 2

2 < h ≤ 3

3 < h ≤ 4

28

Massachusetts State Assessment

The graph below shows the number of milligrams of a medication in the bloodstream from the time it was administered to 300 minutes after administration. Using the information from the graph, which of the following statements is true?

A. The maximum amount of medication in the bloodstream was 12 milligrams.

B. The minimum amount of medication was in the bloodstream 300 minutes after administration.

C. The amount of medication in the bloodstream increased at a faster rate than it decreased.

D. The maximum amount of medication was in the bloodstream 100 minutes after administration.

29

Idaho State Assessment

a. In the space provided, draw in the diagonals for the six-sided polygon.

b. How many diagonals are there in an 8-sided polygon? Show or explain how you found your answer.

c. Explain how you would determine the number of diagonals in a 15-sided polygon.

9

30

Algebra Placement

Districts should not encourage all students to take Algebra I in grade 8; students should be taking Algebra I in grade 8 only if they are highly motivated, have a strong foundation in middle school mathematics, receive high grades in previous courses, intend to study calculus in high school, and only if the Algebra I courses are taught by teachers with mathematics certification.

31

Summary

Algebra Courses:– Traditional Algebraic Symbolic

Manipulation

– Patterns & Algebraic Thinking

– Contextual problem solving

32

Thank You

Dr. Eric Milou

Rowan University

milou@rowan.edu