Marie Curie Math & Science Center
 
CURRICULUM DESIGN
Nancy Freeman, South Orangetown Middle School, Group 4
This unit is aimed for a pre-algebra course at the eighth grade level
 

Commencement content standard from MST (one or more of the seven)-

  • Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

Content standards
 

  • Students use mathematical modeling/ multiple representation to provide a means of presenting, interpreting, communicating and connecting mathematical information and relationships
  • Students use mathematical operations and relationships among them to understand mathematics
  • Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Performance standards

  • Students use concrete materials and diagrams to describe the operation of real world processes
  • Students use operations with integers
  • Students develop methods to solve equations

Content standards or outcomes

  • Students will use manipulatives (cups and color tiles/counters) to demonstrate an understanding of solving equations involving integers, such as: 2a + 3 = 3a - 4 ; 2a + 10 = -2 ; 2(a-3) = -16
  • Students will use manipulatives (color tiles/counters) to show understanding of operations with integers
  • Students will model additive inverses using color tiles/counters to demonstrate their understanding
  • Students will use manipulatives to show an understanding of the distributive property and its use in. solving equations


ACTIVITY I

Students will demonstrate understanding of inverse operations.

Activity:

  • Students work in small groups on activity cards.
  • Begin by writing Newton's Law on the board: " FOR EVERY ACTION THERE IS AN OPPOSITE REACTION".
  • Ask students to brainstorm to make a list:
  • ACTION --- REACTION
  • After class discussion of group results, ask " Does every action have an opposite?".
  • Define INVERSE as OPPOSITE.
  • List arithmetic operations on the board, having students match opposites.
  • Ask " Do all operations have an inverse operation?".
  • Define ADDITIVE WWRSES as PAIRS WITH A SUM= 0
  • ASSESSMENT: List integers on board and have students name each additive inverse.
  • Ask: Do all numbers have an additive inverse?
  • Define MULTIPLICATIVE INVERSE as PAIRS WITH A PRODUCT= 1

ASSESSMENT: List integers on the board and have students name each multiplicative inverse.  Ask: Do all numbers have a multiplicative inverse?



ACTIVITY 2
Students will model balancing equations

Activity

  • Students work in small groups.
  • Demonstrate "balance " by using a two-pan balance scale or the teeter-totter (a meter stick balanced on a pivot).
  • Explain and demonstrate how to play the computer game "KEEP YOUR BALANCE" (Sunburst Communications), using paper clips, lima beans and pushpins on the overhead projector.
  • Small groups play the game on computers, using worksheets (from manual which comes with the software) to show their work.

ASSESSMENT : Students will model balancing an equation by showing steps to solving a puzzle from the game.

ALTERNATE ASSESSMENT (without software): Students work with pins, clips, and lima beans to model steps to a teacher-made puzzle.



ACTIVIY 2

Rules for "KEEP YOUR BALANCE:
Students begin with an empty pan balance.  They may add objects to, or remove objects from the scale only if they do the same thing to both sides (to maintain balance) at same time.  The goal is to get the desired outcome in the least number of steps.  Different objects which are given as balancing each other, may be used as substitutions on one of the scale.  Example: If a bean balances two pins, a bean may be replaced on one with the two pins without ruining the balance



ACTIVITY 3 (2 days)

Students will model solutions to equations x + b =c; b,c are integers

Activity

  • Students work in small groups.
  • Begin by distributing cups and color tiles/counters to each group.
  • Explain that cups, represent the " missing value", called a variable, and the tiles/counters represent integers, called a constant, (RED tiles are positive integers, BLUE tiles negative integers)
  • Demonstrate r r r means +3; b b means -2
  • Demonstrate the equation x + 3 = 4 as ___r r r = r r r r (using one cup and red tiles).
  • Students model the equation x+(-1)= -3 with a cup and blue tiles (___b= b b b).
  • Demonstrate the solution for x + 3 = 4 as:   ___ r r r = r r r r (refer to this as "balancing" by subtracting 3 from each side).  The solution is ___= r    (refer to this as x = 1).
  • Explain the, alternate solution as adding b b b to each side to make 3 zeros on each side, since additive inverses have a zero sum!  Reinforce that zero is the identity for addition.

___r r r + b b b = r r r r + b b b (This helps establish the relationship between subtraction and addition using additive inverses)

ASSESSMENT.  Students model the solution for each equation using cups and tiles, and record method on worksheet:
 

EQUATION SOLUTION( for teacher use)
1) x+2=1  ___r r+b b=r+b b(add b b to each side)
___=b (x= -I)
2) x + -1 = -3 ___b + r = b b b+ r (add r to each side)
___ = -2
3) x+ 1 =-2 ___r + b = b b + b (add b to each.side)
___=b b b (x = -3)

Teachers could use worksheet/text work to reinforce the symbolic solution of  equations.



ACTIVITY 4

Students will I model solutions to equations ax = c where a> 1; c an integer
 

  • Students work in small  groups.
  • Explain "separating" a group into equal parts is division
  • Distribute cups and tiles to groups
  • Demonstrate the expression 3x as ___  ___  ___.
  • Students demonstrate 2x .
  • Demonstrate 3x = 6 as ___ ___ ___ = r r r r r r
  • Demonstrate "separating" as:

___  r r
___  r r
___  r r  The solution is x=2

ASSESSMENT.  Students model the solution,for each equation using cups and tiles, and record method on worksheet:
 

EQUATION  SOLUTION( for teacher use)
1) 2x = -6  1)___     bbb
___ b b b (x=-3)
2) 4x = 8 ___ r r
___ r r
___ r r
___ r r (X=2)

Teachers should use worksheet/text work to reinforce the symbolic solution of these equations.



ACTIVITY 5 (2 + days)

Students will model the distributive property to solve equations: a(x+b) = c; a>O, bc
integers

  • Students work in small groups
  • Distribute cups and tiles to eac group.
  • Remind students that parentheses are grouping symbols.  Explain that a number in front, or in back of ( ) means multiply, and that multiplication is a shortcut for repeated addition. This means that the number outside the tells "how many" groups are being added.
  • Demonstrate that the expression 2(x + 3) means (x + 3) + (x + 3) or 2x + 6. Demonstrate the expression 2(x + 3) as 2 sets of ___ r r r, which equals___ ___r r r r r r

ASSESSMENT 1
Students model 3(x + -2), then write th result in symbolic form
Students model 2(3x + 1), then write the result in symbolic form.

Demonstrate the solution of 2(x + 3) = 10

___ r r r + ___ r r r = r r r r r    r r r r r
___ + ___ r r r r (subtract r r r r r r, or add  b b b b b b on each side)

___=r r (x= 2) (separate, divide, each side in two)

ASSESSMENT 2
Students model solution of each equation and record method on worksheet:

 
 

EQUATION   SOLUTION (for teacher)
1) 2(x + 1) = 8  ___ r + ___ r = r r r r r r r r
___ ___ r r = r r r r r r r r
___ ___ r r r r r r (add bb or subtract r r)
2) 3(x-2)=3 ___ b b + ___ b b + __ b b=r r r
___ ___ ___b b b b b b=r r r
___ ___ ___= r r r r r r r r r )add 6r)
___ = r r r (devide by 3)

Teacher should use worksheet/text work to reinforce symbolic solution of these equations.



ACTIVITY 6 (2 +days)

Students will model the solution of equations: ax +b = cx + d ; a,c >0 b,d integers
 

  • Students work in small groups.
  • Distribute cups and tiles to each group.
  • Demonstrate 2x + I = x + 5 as:

___  ___ r = ___ r r r r r ___ r = r r r r r (subract ___, on each side) ___ = r r r r      (subtract r, or add b on each side)

  • Explain that a "negative cup", representing -x, is upside down, and that x + -x =O.
  • Demonstrate that a negative cup could be added on each side, instead of subtracting.

ASSESSMENT
Students model the solution for:
 

EQUATION  SOLUTION (for teacher)
1) x-3=3x+5  1)       (Remind them that x -3 = x+ -3)
___b b b=___ ___ ___ r r r r r
b b b=___ ___r r r r r (subtract ___)
b b b b b b b b = ___ ___ (add  b b b b b)
b b b b = ___ (divide by 2)
-4=x
2) 2x-1=x+4 2) ___ ___ b = ___ r r r r
___ b = r r r r (subtract ___)
___ = r r r r r (add r)
x=5

Teacher should use worksheet/text work to re- inforce the symbolic solution of these equations.

 

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