Day 3: Solving Equations
Goals




I.
Identify a developmental sequence for solving equations.
Math talk about expressions and equations.
It is all about equivalence and how we communicate this with students.
Discuss properties of equations.
Review Interview for Generalizations
What questions helped most to get your students to explain the explicit rule for different patterns?
II. Solve
Solving Equations - Page |2
III. Solve?
IV. Expressions vs. Equations
directions
problem
What do you write?
Solving Equations - Page |3
V. CGI Algebra Video
How does a student show relational understanding of the equal sign?
another problem
a)
knowledge of equal sign
Solving the Equation
(4th grader, 2 minutes, 42 seconds)
b) Sharing Strategies for Number Sentences
(4th grader, 1 minute, 51 seconds)
c)
Solving the Equation
(4th grader, 39 seconds)
d) Solving the Equation
(4th grader, 51 seconds)
e)
Solving the Equation
(2nd and 3rd grade class, 24 minutes, 13
seconds)
VI. Benchmarks in student thinking about the equal sign (from kick-off day)
Description
Children are asked to be specific about what they think the
equal sign means
Children accept as true some number sentence that is not of
2
the form a + b = c
Children recognize that the equal sign represents a relation
3 between two equal numbers, particularly through
calculation (relational understanding of the equal sign)
Children are able to compare the mathematical expressions
4 using relational thinking without carrying out the
calculations (relational understanding across the equal sign)
Note: These benchmarks are a guide, not a firm sequence
1
sophistication
Solving Equations - Page |4
VII. Methods for solving linear equations of the form
a) Traditional approach
methods
i)
use of number facts (“solve
mentally”)
ii)
generate and evaluate (“guess
and check” or “trial and error
substitution”)
iii) undoing (or working
backwards)
iv) cover-up
v)
transposing (change side change sign)
vi) equivalent equations
(performing the same
operation on both sides)
examples
not-so-good for
equality
benchmark
Solving Equations - Page |5
b) Functions approach (solve: 3x – 4 = x + 6)
i)
methods
table using graphing calculator
(similar to guess and check)
ii)
graphing
VIII. Minnesota standards sort
example
not so good for
Solving Equations - Page |6
IX. Solving equations using number facts – picking examples for grades
Solve mentally.
X. Solving equations using undoing – picking examples for grades
Solve by undoing.
XI. Solving equations using cover-up – picking examples for grades
Solve.
Solving Equations - Page |7
XII. Baseline Assessment
XIII. Equivalence
Expressions
Equations
Directions:
Directions:
Directions:
Check (proof or verification)
Can I add …
Solving Equations - Page |8
Equivalent expressions are …
Equivalent equations are …
XIV. Metaphors for equations
a)
Balance (same weight on both sides)
balance
equation
reason for equivalence and reason
using metaphor
Solving Equations - Page |9
S o l v i n g E q u a t i o n s - P a g e | 10
Problems with balance metaphor
a)
Negatives
b)
Subtraction
initial set-up (Hands-On Equations)
S o l v i n g E q u a t i o n s - P a g e | 11
Adding “convenient zeros” or “neutral pairs”
b) Numbers (same value on both sides)
i.
Open Number sentences (CGI)
ii.
Algebra Tiles (McDougal Littell) or Bags of Gold (CMP)
x
=
x
x
Algebra Tiles – McDougal Littell (Course 2)
CMP – Moving Straight Ahead (7th grade)
iii.
Equation Mat (CPM – Algebra Connections)
+
+
x
x
x
-
= +1
x
x
-
= -1
+
+
+
+
-
-
-
-
S o l v i n g E q u a t i o n s - P a g e | 12
XV. Properties of equations –
description
Addition
Property of
Equality
Words
Adding the same number to each
side of an equation produces an
equivalent equation
Algebra
from McDougal Littell Math Course 2
explain using balance
metaphor
explain using number
metaphor