1. Math Institute: By Reda Berry Tristyn Davis Kelly Meeks
Algeblocks *Thinking about Variables *Addition & Subtraction of polynomials *Solving sentences
Math Institute: By
Reda Berry
Tristyn Davis
Kelly Meeks
“Algeblocks gives students a hands-on method of actually discovering concepts. The more ways that a teacher can present an idea, the better chance students have to understand the concept. Algeblocks allows students to form and see representations of variables and numbers. They actually see math concepts at work.” - David Ott, Principal St. Henry High School, Erlanger, KY

2. Lab 1-8: Knowing Algeblocks
One unit X X³ X²
Lab 1-8 Objective To use all the Algeblocks to practice communicating mathematically. Help the students to see that the variable pieces , X and Y, could represent any number. The piece is not the size of any whole number on purpose, so that we think of its length as x or y. Explain why 2^2 is sometimes read as two “squared” and why 2 ^3 is sometimes read as two “cubed.” Compare the two. How are they alike? (same base, 2) How are they different? (the exponentis different) Y X
Y Y³ width x and length Y

3. Lab 6-1: Representing Variable Expressions Example 1: x + 4
Lab 6-1 Objective: To model variable expressions with Algeblocks. Teaching the Lesson. In this lab the various Algeblocks pieces are used to model variable expressions. Ask students to identify each of the pieces. In this example, the coeffficient of x is 1. Use 1 x-piece. − +

4. Lab 6-1: Representing Variable Expressions Example 2: 2x + 4
Example 2: Since the coefficient of x is 2, two x-pieces are needed. Repeated addition can be used to show that: 2x = x + x − +

5. Lab 6-1: Representing Variable Expressions Example 3: -2x + 4
Example 3: The coefficient of -2 means the x pieces are placed on the negative part of the mat. This can be shown by reading -2x as the opposite of 2x. Place 2 x-pieces on the negative side. − +

6. Lab 6-1: Representing Variable Expressions Example 4: -2x - 4
Example 4: The operation sign in front of a term determines whether it is placed on the positive or negative side of the mat. Place 2 x-pieces and 4 unit pieces on the negative side of the mat. − +

7. Lab 6-2: Evaluate Expressions Example: Evaluate 2x – 5 if x=3
Step 2. Sub-
stitute 3 ones
for each x-piece.
Step 3. Remove
blocks that add
to zero.
Step 4.
Answer on
mat. 1
Step 1. Model
expression 2x - 5 + + + +
Lab 6-2: Objective to evaluate a variable expression. Teaching the lesson: Students will first replace variable pieces with ones, modeling substitution. The expression can then be evaluated using the methods practiced in previous labs. In this lab, all the variables are replaced with positive integers. Once this example is evaluated work the same example for x = 4. In step 2, substitute 4 ones for each x-piece. For x=4, 2x-5=3. Emphasize that a variable expression has different values when the variable is replaced with different integers. - - - -

8. Lab 7-1: Combining Like Terms Example: Combine the terms: 3x + 2y – x – 5y or 3x + 2y + -x + -5y
Lab 7-1, Objective: To simplify variable expressions by combining like terms. Teaching the lesson: Step 1. Place the terms on the mat. The operation sign in front of the term determines whether it is placed on the positive or negative section of the mat. This may be confusing for some students. Have those students change the subtraction sign to “plus the opposite” and use the sign on the coefficient to place the blocks on the positive or negative side of the mat. − +
Step 1. Place the terms on the mat.

9. Lab 7-1: Combining Like Terms Example: Combine the terms: 3x + 2y – x – 5y or 3x + 2y + -x + -5y
Lab 7-1, Step 2 Add pieces that are the same. Emphasize that when opposites are combined, the results are zero. − +
Step 2 Add pieces that are the same.

10. Lab 7-1: Combining Like Terms Example: 3x + 2y – x – 5y = 2x – 3y or 3x + 2y + -x + -5y = 2x + -3y
Lab 7 – 1, Step 3. Final answer: 2 x – 3y or 2x + -3y − +
Step 3. Final answer is 2x – 3y or 2x + -3y

11. Lab 10-2: Representing Equations Represent the equation 2x – 3 = -7
Step 1. Place 2x – 3 on the
left side of the equal sign.
Step 2. Place -7 on the right
side of the equal sign. + + − −
Lab 10-2, Objective To connect equations, Algeblocks, and mathematical symbols. Beginning ask students to define an equation and to describe the differences between a phrase and an equation. Teaching the Lesson: The sentences mat is used in this lab for representing equations. The phrases from the two sides of the equation are placed on the two sides of the mat. Equal is thought of as balancing the two sides. − −

12. Lab 10-3 Writing Algebraic Equations
Example 1: A number increased by 4 is -5.
Example 2: Twice a number is -4. + +
“IS” − −
Lab 10-3: Objective to connect equations, Algeblocks, and mathematical symbols. Beginning, As a group write the phrases, Use X to represent the number; a number increased by 4; and twice a number. Teaching the Lesson: The phrases from the Beginning activity are used as parts of equations. When an equation is written in words, the word “is” is frequently used for the equal sign. In this lab students translate situations into algebraic equations. There are different possible Algeblocks representations based on the variable chosen. Students are not expected to find solutions, only to set up a model. However, answers can be found as an extension. − −

13. Lab 10-4: Solving Equations: Undoing Addition Example: Solve x – 4 = -2
Step 1. Represent the equation on the sentence mat. + + − −
Lab 10-4: Objective to solve equations by adding the same number to both sides of the equation. Beginning, ask the students: “Is this equation true if x=-4? 2x + 3 = -5” Teaching the Lesson: Solving an equation means determining the value(s) for the variable that makes the equation true. In the answer, the variable will be the only term on one side of the equal sign. FYI The addition property of equality if a = b then a+c = b+c will be used to solve the equations. The addition of a number to both sides of an equation is thought of as keeping the equation balanced. − − =
X - 4 -2

14. Lab 10-4: Undoing Addition Example: Solve x – 4 = -2
Step 2. To get x by itself, add the opposite of -4 or 4 to each side. + + − −
Lab Step 2. The addition of a number to both sides of an equation is thought of as keeping the equation balanced. − −
X – 4 + 4
-2 +4 =

15. Lab 10-4: continued Solve x – 4 = -2
Step 3. Simplify each side of the Equation. (Don’t forget how to make zeros.) + + − −
Step 3: Simplify each side of the equation. (Remind students how to make zeros.) − −
0 + 2
X + 0 =

16. Lab 10-4: continued Solve x – 4 = -2
Step 4. Read the answer from the mat. X = 2 + + − −
Step 4. Some students will be able to “see” the answer without using Algeblocks. Tell them that the equations will become more difficult and it is necessary to develop a system for solving equations one step at a time. − − X = 2

17. Algeblocks Review Identified the blocks
Thinking about variables and how to represent them
Addition & Subtraction of polynomials
Solving sentences

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19. Resources:
Johnston, Anita M., Algeblocks, South- Western Publishing Co., Cincinnati, Ohio 1994
ISBN
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