Transcript CMC-S - CPM Student Guidebook

TURN
ALGEBRA EXERCISES
into
COMMON CORE PRACTICE
TASKS
CORE PRACTICES
 Make Sense and Persevere.
 Reason Abstractly.
 Construct Viable Arguments.
 Model with Mathematics.
 Look for and Use Structure.
 Look for and Express Regularity.
PROBLEM MAKING STRATEGIES
 Rewrite Equivalent Expressions;
Justify.
 Create and Reconcile Algebraic
Representations.
 Decide on a Strategy for Solving
Solve a Really Hard Problem
 Identify/Explain Errors.
 Work Backwards
 Have Students Create the Problem
 Give an Example of………
 Reverse Your Class Routine
 Play a Game
Rewriting Equivalent Expressions
and Equations:
Rewrite each of the following
expressions in at least two
different ways. Verify using
tables, graphs, or algebra that
each new expression is equivalent
to the original.
a. (x - 3)2 + 7
b.
x +x-6
x-6
c.
2x - x -1
2
x -1
2
2
Ramon’s group was trying to rewrite
3(x + 4) + 2 in simpler form.
3x
They came up with four different
results:
x+6
,
x
x + 14
,
x
3x + 14
,
3x
15
Which are equivalent? Use graphs,
tables, and algebra to explain.
Create and Reconcile
Algebraic Representations
xy
y
y
Create and Reconcile Algebraic
Representations
Work with a partner and use the
one xy piece and one y piece to
form a figure. Record the area
and perimeter for your figure.
How many different results are
there for the perimeter? For the
area?
Using exactly these three pieces build an
arrangement for each of the following
perimeters.
a. 4y + 4
b. 2x + 4y
c. 2x + 6y
Decide on a Strategy to Solve:
With your group, solve
8(x – 5) = 64 for x in at least two
different ways. Explain how you
found x in each case and be
prepared to share your
explanations with the class.
For each equation below, with your
group decide whether it would be
best to rewrite, look inside, or undo.
Then solve the equation, showing
your work and writing down the
name of the approach you used.
a. 40x 2 -120x -120 = 40
b.
7x - 20 = 6
c. x - x + 2 = x
4
3
6
d.
2x - 9
=3
7
e. 18 - 5(x - 6) = 38
(x - 7) = 9
2
a. Solve this equation using all three
approaches studied in this lesson.
b. Did you get the same solution using
all three strategies? If not, why not?
c. Discuss with your group which
method is most efficient. What did
you decide? Why?
a.
b.
c.
x
=8
2
1
x = 27
3
5x 2
=
8
5
x=
d.
2
3
e.
4x =
1
4
9
2
Think before solving!
a.Read the equation and list the
operations on the variable.
b.List the ‘undo’ operations.
c.Write the steps of your solution.
Solve a Really Hard Problem
Now that you have the skills
necessary to solve many interesting
equations and inequalities, work
with your team to solve the
equation below.
( x + 5 - 6) + 7 = 23
2
Identifying & Correcting Errors:
4
3
Solution is complete and is essentially
correct.
Solution would be correct except for a
small error or solution is not complete.
2
Solution is partially correct. There is a
major error or omission.
1
Solution attempted.
Work Backwards: Have Students
Create the Problems
Work with a partner, and the simple
equation x = –4. When you and your
partner have agreed on a complicated
equation, trade your equation with your
team-partners and see if they solve it
and get –4.
Now start with 5 + i.
Work Backwards: Have Students
Create the Problems
Write two linear equations whose
graphs
a. Intersect at (-3, 1).
b. Are parallel.
c. Are the same.
Work Backwards: Give an
Example of……….
 Instead of a set of review problems try
a summary. Start with a brainstorming
session on the big ideas of the chapter.
As a class reach agreement on 2-5 ideas.
 Assignment: Find N problems that to
represent each big idea.
Write each problem and show its
solution.
x2 + x - 6
Work Backwards: Start with the
Problems
You have seen that graphs of equations
in three variables can lead to inconclusive
results. What other strategies can you
use to find the intersection? Consider the
following problem and discuss this with
your group.
2x + 3y + 3z = 6
6x - 3y + 4 = 12
2x - 3y + 2z = 6
a. Elisa noticed she could combine the
first two and get and equation with
only x and y.
b. Can you combine a different pair and
get only x and y?
c. If you solve for x and y you can get z.
Play a Game
Matching representations:
•Linear equations: graph, table,
situation, equation.
•Parabolas: Graph, standard,
form, vertex form, and factored
form of the equation.
•Equivalent expressions: (2 to 4)
Expressions with exponents
Rational expressions
A taxi charges $1 to
begin with, and $3 for
each mile traveled. y
is the amount
charged for traveling
x miles.
TABLE
y = 3x + 1
GRAPH
y = (x + 4) -1
y = (x+ 3)(x+5)
y = x +8x+15
GRAPH
2
2
GAME FORMS:
Group formation in class.
Concentration.
Old Maid.
Go Fish or Authors.
PROBLEM MAKING STRATEGIES
 Identify/Explain Errors.
 Rewrite Equivalent Expressions;
Justify.
 Create and Reconcile Algebraic
Representations.
 Decide on a Strategy for Solving
 Solve a Really Hard Problem
 Work Backwards
 Have students create the problem
 Give an Example of………
 Reverse Your Class Routine
 Play a Game
Enjoy making algebra exercises
more problematic!
 Judy Kysh,
San Francisco State University and
CPM Educational Program
[email protected]