Attacking TAKS Objectives Through the Use of Manipulatives ...
Attacking TAKS Objectives Through the Use of Manipulatives and Graphing Calculator Strategies
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University of Houston – Central Campus
EatMath Workshop
April 18, 2009
Focus Activity: Lateral Thinking Exercise
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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
1. How many [pic] are in [pic]?
2. How many [pic] are in [pic] ?
3. How many [pic] are in [pic] ?
4. How many [pic] are in [pic] ?
5. How many [pic] are in [pic] ?
6. How many [pic] are in [pic] ?
Pattern Block Activity 2
1. If [pic] = 1, [pic] = ___ .
2. If [pic] = 1, [pic] = ___ .
3. If [pic] = 1, [pic] = ___ .
4. If [pic] = 1, [pic] = ___ .
Pattern Block Activity 3
1. If [pic] + [pic] = 1, what is [pic]?
2. If [pic] + [pic] = 1, what is [pic]+ [pic]?
3. If [pic] + [pic] = 1, what is [pic] + [pic]?
4. If [pic] + [pic] = 1, what is [pic]?
5. If [pic] - [pic] = 1, what is [pic]+ [pic]?
Summarize the Trapezoids Data and Look for Patterns
Find the pattern.
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|Hexagon Number |Process Column |Trapezoids Used |
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|1 | |2 |
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|2 | | |
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|3 | | |
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|4 | | |
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|5 | | |
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|6 | | |
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|n | | |
Summarize the Rhombus Data and Look for Patterns
Find the pattern.
| | | |
|Hexagon Number |Process Column |Rhombi Used |
| | | |
|1 | |3 |
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|2 | | |
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|3 | | |
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|4 | | |
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|5 | | |
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|6 | | |
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|n | | |
Summarize the Triangles Data and Look for Patterns
Find the pattern.
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|Hexagon Number |Process Column |Triangles Used |
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|1 | |6 |
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|2 | | |
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|3 | | |
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|4 | | |
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|5 | | |
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|6 | | |
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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
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Yellow Red
[pic] [pic]
Green Red
[pic] [pic]
Blue Red
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1. (+3) + (+1) =
2. (-2) + (-1) =
3. (+3) + (-1) =
4. (+4) + (-4) =
After students have seen many examples of addition, have them formulate rules.
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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.
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← 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) =
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1. (+6)/(+2) =
2. (-8)/(+2) =
← A negative divisor will mean “take the opposite of.” (flip-over)
1. (+10)/(-2) =
2. (-12)/(-3) =
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1. X + 2 = 3
2. 2X – 4 = 8
3. 2X + 3 = X – 5
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1. 3(X + 2) =
2. 3(X – 4) =
3. -2(X + 2) =
4. -3(X – 2) =
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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)
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1. 3x + 3
2. 2x – 6
3. x2 + 6x + 8
4. x2 – 5x + 6
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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
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“Teaching with Manipulatives: Middle School Investigations”. Cuisenaire (1994).
's%20Do%20Algebra%20Tiles.ppt
tea.state.tx.us
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Addition of Integers
* Addition can be viewed as “combining”.
* Combining involves the forming and removing of all zero pairs.
* For each of the given examples, use algebra tiles to model the addition.
* Draw pictorial diagrams which show the modeling.
Subtraction of Integers
* Subtraction can be interpreted as “take-away.”
* Subtraction can also be thought of as “adding the opposite.”
* For each of the given examples, use algebra tiles to model the subtraction.
* Draw pictorial diagrams which show the modeling process.
Multiplication of Integers
* Integer multiplication builds on whole number multiplication.
* Use concept that the multiplier serves as the “counter” of sets needed.
* For the given examples, use the algebra tiles to model the multiplication. Identify the multiplier or counter.
* Draw pictorial diagrams which model the multiplication process.
Division of Integers
* Like multiplication, division relies on the concept of a counter.
* Divisor serves as counter since it indicates the number of rows to create.
* For the given examples, use algebra tiles to model the division. Identify the divisor or counter. Draw pictorial diagrams which model the process.
Solving Equations
* Algebra tiles can be used to explain and justify the equation solving process. The development of the equation solving model is based on two ideas.
* Variables can be isolated by using zero pairs.
* Equations are unchanged if equivalent amounts are added to each side of the equation.
Distributive Property
* Use the same concept that was applied with multiplication of integers, think of the first factor as the counter.
* The same rules apply.
3(X+2)
* Three is the counter, so we need three rows of (X+2)
Modeling Polynomials
* Algebra tiles can be used to model expressions.
* Aid in the simplification of expressions.
* Add, subtract, multiply, divide, or factor polynomials.
Factoring Polynomials
* Algebra tiles can be used to factor polynomials. Use tiles and the frame to represent the problem.
* Use the tiles to fill in the array so as to form a rectangle inside the frame.
* Be prepared to use zero pairs to fill in the array.
* Draw a picture.
Dividing Polynomials
* Algebra tiles can be used to divide polynomials.
* Use tiles and frame to represent problem. Dividend should form array inside frame. Divisor will form one of the dimensions (one side) of the frame.
* Be prepared to use zero pairs in the dividend.
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