Effective Teaching Algebra Feb 2013

Report
“Teachers are thus free to
provide students with whatever
tools and knowledge their
professional judgment and
experience identify as most
helpful for meeting the goals set
out in the Standards.”
~ Introduction to the CCSS
HS Algebra
February 2013
Outcomes
 Align
the regional/district Algebra course
to the PARCC framework
 Create tape diagrams and double
number lines to solve application
problems
 Explain the information we have, need
and will make do with
PARCC Resources

Progressions


Illustrative Mathematics


http://commoncoretools.wordpress.com/
Quality Review Rubrics


http://illustrativemathematics.org/
Common Core Tools


http://ime.math.arizona.edu/progressions/
http://www.achieve.org/files/TriState-MathematicsQuality-RubricFINAL-May2012.pdf
Achieve the Core

http://www.achievethecore.org/
PARCC Components
 Key
Advances from the Previous Grade
 Discussion of Mathematical Practices in
Relation to Course Content
 Fluency Recommendations
 Pathway Summary Tables
 Assessment Limits Tables
Math Practice Meditation 
 Imagine
your best students… consider
how they showed each of these
qualities…
1. Perseverance
2.
Reason abstractly and quant.
3.
Construct and critique
4.
Model
5.
Use tools strategically
6.
Precision
7.
Use structure
8.
Find and express repeated
reasoning
Look For’s in a CCLS Lesson
 Fluency
Task (~10 mins)
 Modeling


Concept Building
Application
 Debrief


Pair Sharing
Exit Ticket (Daily formative assessment)
Module Sources
PK-5
Common Core Inc
6-12
CCI
EduTron
Construct
Scope and
Sequence PK-8
Grade Level
Map
Modules
Lessons
Assessment
Mid Module
Assessment
New
Module
End of
Module
Assessment
Content Gap Instruction

Multiplication

Division Strategies

Algebraic Understanding

Fractions

Number Sense & Place Value
Quiz 1
 What
teaching materials will likely be
available?
 Is
State Ed is making and providing all of
the math materials teachers need?
 What
are important documents to help
build our HS curriculum?
Math Modules
PK-5
6-12
Common Core
Inc
CCI with Support
from EdutTron
Fluency
Application
Conceptual
Fluency
Application
Conceptual
Fluency
Required Fluencies
Grade
Required Fluency
K
Add/Subtract within 5
1
+/- within 10
2
Add/Subtract within 20
Add/Subtract within 100 (paper and pencil)
3
Multiply/divide within 100
Add/Subtract within 1000
4
Add/Subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division
Multi-digit decimal operations
7
Solve px+q=r, p(x+q)=r
8
Solve simple 2x2 systems by inspection
Application
Multiplication Facts
Conceptual
Fluency
X
1
2
3
4
5
6
7
8
9
1
1x1
1x2
1x3
1x4
1x5
1x6
1x7
1x8
1x9
2
2x1
2x2
2x3
2x4
2x5
2x6
2x7
2x8
2x9
3
3x1
3x2
3x3
3x4
3x5
3x6
3x7
3x8
3x9
4
4x1
4x2
4x3
4 x 4
4x5
4x6
4x7
4x8
4x9
5
5x1
5x2
5x3
5x4
5x5
5x6
5x7
5x8
5x9
6
6x1
6x2
6x3
6x4
6x5
6x6
6x7
6x8
6x9
7
7x1
7x2
7x3
7x4
7x5
7x6
7x7
7x8
7x9
8
8x1
8x2
8x3
8x4
8x5
8x6
8x7
8x8
8x9
9
9x1
9x2
9x3
9x4
9x5
9x6
9x7
9x8
9x9
Commutative
Property
Identity
Property
Doubles
Squares
Fives
Challenge
(benchmark) (benchmark)
Gene Jordan’s work but I got the Idea from Gina King’s article:www.nctm.org teaching children mathematics • King, Fluency with
Basic Addition, September 2011 p. 83
Application
Addition Facts
Conceptual
Fluency
Gina King’s article:www.nctm.org teaching children mathematics • King, Fluency with Basic Addition, September 2011 p. 83
Application
Conceptual
Fluency Example
 Finger
Counting
 1,2,3, sit on 10
 High 5
Fluency
Application
Conceptual
Fluency
Fluency
Fast
Frequent
Fun
Application
Conceptual
Fluency
Math Sprints
Ready,
Set,
Go!
Sprint A
Review
Sprint A
Math
Moves
Sprint B
Review
Sprint B
Cool
Down
Conceptual Modeling
Application
Conceptual
Fluency
Conceptual
 Concrete
Pictorial  Abstract
 Moving both ways

Draw a picture of 4+4+4
 Show
your thinking
 Explain, defend and critique the
reasoning of others
Application
Conceptual
Fluency
Concrete Model  Equation
X+3=5
Application
Conceptual
Fluency
Tape Diagram Problems
Tape diagrams are best used to model
ratios when the two quantities have the
same units.
Tape Diagrams: Q1

1. David and Jason have marbles in a
ratio of 2:3. Together, they have a total of
35 marbles. How many marbles does
each boy have?
Tape Diagrams : Q2
 2.
The ratio of boys to girls in the class is
5:7. There are 36 children in the class. How
many more girls than boys are there in the
class?
Tape Diagrams Q3:
Comparing
3
items
Lisa, Megan and Mary were paid $120
for babysitting in a ratio of 2: 3: 5. How
much less did Lisa make than Mary?
Tape Diagrams Q4: Different
Ratios
The ratio of Patrick’s M & M’s to Evan’s is 2: 1
and the ratio of Evan’s M & M’s to Michael’s
is 4: 5. Find the ratio of Patrick’s M & M’s to
Michael’s.
Tape Diagrams Q5: Changing
Ratios
The ratio of Abby’s money to Daniel’s is 2: 9.
Daniel has $45. If Daniel gives Abby $15,
what will be the new ratio of Abby’s money
to Daniel’s?
Double Number Line
Double number line diagrams are best used when the quantities
have different units. Double number line diagrams can help make
visible that there are many, even infinitely many, pairs of numbers in
the same ratio—including those with rational number entries. As in
tables, unit rates (R) appear in the pair (R, 1).
Double Number Line:
Finding average rate

It took Megan 2 hours to complete 3 pages of
math homework. Assuming she works at a
constant rate, if she works for 8 hours, how many
pages of math homework will she complete? What
is the average rate at which she works?
Identify properties of the RDW
modeling technique for
application problems
 Read
(2x)
 Draw a model
 Write an equation or number sentence
 Write and answer statement



Unit
Object
Context
Use RDW to solve Problem
Modeling Challenge
2
boxes of salt and a box of sugar cost
$6.60. A box of salt is $1.20 less than a box
of sugar. What is the cost of a box of
sugar?
Salt
Salt
$6.60
3 parts = $6.60- $1.20
Sugar
$1.20+$1.80= $3.00
$1.20
3 parts = $5.40
1 part = $5.40 ÷ 3
= $1.80
Challenging Problems

The students in Mr. Hill’s class played games at recess. Mika Said: “Four




6 boys played soccer
4 girls played soccer
2 boys jumped rope
8 girls jumped rope
Chaska Said: “For
every girl that
played soccer, two
girls jumped rope.”
more girls jumped
rope than played
soccer.”
Mr Hill Said: “Mika compared girls by looking at the difference and
Chaska compared the girls using a ratio”

1) Compare the number of boys who played soccer and jumped
rope using the difference. Write your answer as a sentence as Mika
did.

2) Compare the number of boys who played soccer and jumped
rope using a ratio. Write your answer as a sentence as Chaska did.

3) Compare the number of girls who played soccer to the number of
boys who played soccer using a ratio. Write your answer as a
sentence as Chaska did.
Challenging Problems
 Compare
3
4
these fractions:
and
3+1
4+1
Which one is bigger than the other? Why?
Application
Application
Conceptual
Fluency
Application
 The
beginning of the year is characterized
by establishing routines that encourage
hard, intelligent work through guided
practice rather than exploration.
 Slower and deeper

Use the Read-Draw-Write (RDW) steps
Application
Conceptual
Fluency
Application
Wrap up
Thanks for coming!
Links
 www.btboces2.org/mathpd
 http://www.parcconline.org/samples/mat
hematics/grade-6-slider-ruler
 http://www.parcconline.org/samples/mat
hematics/grade-7-mathematics
 www.Engageny.org
High School Functions
A--‐REI.4. Solve quadratic equations in one variable.
Seeing Structure in a Quadratic Equation
High School Illustrative
Sample Item
43
A--‐REI.4. Solve quadratic equations in one variable.
Seeing Structure in a Quadratic Equation
High School Illustrative
Sample Item
44
A--‐SSE, Seeing Structure in Expressions
Aligns to the Standards
and Reflects Good
Practice
High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation
Task Type I: Tasks assessing concepts, skills and procedures
Alignment: Most Relevant Content Standard(s)
A-REI.4. Solve quadratic equations in one variable.
a) Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from
this form.
b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the
square, the quadratic formula, and factoring, as appropriate to the initial form of the equation.
Recognize when the quadratic formula gives complex solutions and write them as a  bi for real
numbers a and b.
Alignment: Most Relevant Mathematical Practice(s)
Students taking a brute-force approach to this task will need considerable symbolic fluency to obtain
the solutions. In this sense, the task rewards looking for and making use of structure (MP.7).
45
Aligns to the Standards
and Reflects Good
Practice
High School Illustrative Item Key Features and Assessment Advances
The given equation is quadratic equation with two solutions. The task does not clue the
student that the equation is quadratic or that it has two solutions; students must recognize
the nature of the equation from its structure. Notice that the terms 6x – 4 and 3x – 2 differ
only by an overall factor of two. So the given equation has the structure
2 = 2
where Q is 3x – 2. The equation Q2 - 2Q is easily solved by factoring as Q(Q-2) = 0, hence Q =
0 or Q = 2. Remembering that Q is 3x – 2, we have
3 − 2 = 0 or 3 − 2 = 2.
These two equations yield the solutions 23 and 43.
Unlike traditional multiple-choice tests, the technology in this task prevents guessing and
working backwards. The format somewhat resembles the Japanese University Entrance
Examinations format (see innovations in ITN Appendix F). A further enhancement is that the
item format does not immediately indicate the number of solutions.
46
Gr. 7 Ratios and Proportions:
Equivalent Ratios and Fractions
Gr. 7 Ratios and Proportions:
Equivalent Ratios and Fractions
Gr. 7 Ratios and Proportions:
Equivalent Ratios and Fractions
Gr. 7 Ratios and Proportions:
Equivalent Ratios and Fractions
“Teachers are thus free to
provide students with whatever
tools and knowledge their
professional judgment and
experience identify as most
helpful for meeting the goals set
out in the Standards.”
~ Introduction to the CCSS
Math Sprints
Fluency in a minute

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