Attacking TAKS Objectives Through the Use of Manipulatives and Graphing Calculator Strategies
University of Houston – Central Campus
EatMath Workshop
April 18, 2009
Focus Activity: Lateral Thinking Exercise
To answer these questions, you have to let your brain think in different ways than you may be used to.
1. How can you throw a ball as hard as you can and have it come back to you, even if it doesn’t hit anything, there is nothing attached to it, and no one else catches or throws it?
2. Two students are sitting on opposite sides of the same desk. There is nothing in between them but the desk. Why can’t they see each other?
3. There are only two T’s in Timothy Tuttle. True or False?
4. Once a boy was walking down the road, and came to a place where the road divided in two, each separate road forking off in a different direction. A girl was standing at the fork in the road. The boy knew that one road led to Lieville, a town where everyone always lied, and the other led to Trueville, a town where everyone always told the truth. He also knew that the girl came from one of those towns, but he didn’t know which one.
Can you think of a question the boy could ask the girl to find out the way to Trueville?
Mystery Shapes
· You will need a geoboard, rubber bands, and sets of clue cards.
· Read all the clues, then use your geoboard to identify the shape.
Clue Set 1
1. It has four sides.
2. All of its angles are congruent.
3. One side is twice as long as another side.
4. The rubber band does not touch the center peg.
What shape did you find? Can you find another shape to fit these clues?
Clue Set 2
1. It has a right triangle.
2. It has one interior peg.
3. It is isosceles.
4. It uses a corner peg.
What shape did you find? Can you find another shape to fit these clues?
Clue Set 3
1. It has four sides.
2. It has no line of symmetry.
3. Its area is six square units.
4. Just one pair of opposite sides is parallel.
What shape did you find? Can you find another shape to fit these clues?
Clue Set 4
1. It is not convex.
2. It has five sides.
3. It has no interior pegs.
4. It has a right angle.
What shape did you find? Can you find another shape to fit these clues?
Pattern Block Activity 1
- How many are in ?
- How many are in ?
- How many are in ?
- How many are in ?
- How many are in ?
- How many are in ?
Pattern Block Activity 2
- If = 1, =___ .
- If = 1, =___ .
- If = 1, =___ .
- If = 1, =___ .
Pattern Block Activity 3
- If + = 1, what is ?
- If + = 1, what is + ?
- If + = 1, what is + ?
- If + = 1, what is ?
- If - = 1, what is + ?
Summarize the Trapezoids Data and Look for Patterns
Find the pattern.
Hexagon Number / Process Column / Trapezoids Used1 / 2
2
3
4
5
6
n
Summarize the Rhombus Data and Look for Patterns
Find the pattern.
Hexagon Number / Process Column / Rhombi Used1 / 3
2
3
4
5
6
n
Summarize the Triangles Data and Look for Patterns
Find the pattern.
Hexagon Number / Process Column / Triangles Used1 / 6
2
3
4
5
6
n
What relations exist between the rules involving trapezoids, rhombi, and triangles? Is there any reasonable explanation for these relations?
Algebra Tiles
¢ Algebra tiles can be used to model operations involving integers.
¢ Let the small yellow square represent +1 and the small red square (the flip-side) represent -1.
¢ Let the green rectangle represent X and the red rectangle (the flip-side represent –X
¢ Let the blue square represent X2 and the red square (the flip-side represent ---X2
Yellow Red
Green Red
Blue Red
1. (+3) + (+1) =
2. (-2) + (-1) =
3. (+3) + (-1) =
4. (+4) + (-4) =
After students have seen many examples of addition, have them formulate rules.
1. (+5) – (+2) =
2. (-4) – (-3) =
3. (+3) – (-5) =
4. (-4) – (+1) =
5. (+3) – (-3) =
After students have seen many examples, have them formulate rules for integer subtraction.
¢ The counter indicates how many rows to make. It has this meaning if it is positive.
1. (+2)(+3) =
2. (+3)(-4) =
¢ If the counter is negative it will mean “take the opposite of.” (flip-over)
1. (-2)(+3) =
2. (-3)(-1) =
1. (+6)/(+2) =
2. (-8)/(+2) =
¢ A negative divisor will mean “take the opposite of.” (flip-over)
1. (+10)/(-2) =
2. (-12)/(-3) =
1. X + 2 = 3
2. 2X – 4 = 8
3. 2X + 3 = X – 5
1. 3(X + 2) =
2. 3(X – 4) =
3. -2(X + 2) =
4. -3(X – 2) =
1. 2x + 3
2. 4x – 2
3. 2x + 4 + x + 2 =
4. -3x + 1 + x + 3 =
Multiplying Polynomials
1. (x + 2)(x + 3)
2. (x – 1)(x +4)
3. (x + 2)(x – 3)
4. (x – 2)(x – 3)
1. 3x + 3
2. 2x – 6
3. x2 + 6x + 8
4. x2 – 5x + 6
1. x2 + 7x +6
x + 1
2. 2x2 + 5x – 3
x + 3
3. x2 – x – 2
x – 2
“Polynomials are unlike the other “numbers” students learn how to add, subtract, multiply, and divide. They are not “counting” numbers. Giving polynomials a concrete reference (tiles) makes them real.”
Resources Used
“Teaching with Manipulatives: Middle School Investigations”. Cuisenaire (1994).
http://math.rice.edu/~lanius/Lessons/
http://www.delmar.edu/aims/Files/Presentations/David_Let's%20Do%20Algebra%20Tiles.ppt
www.tea.state.tx.us