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.
|
