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Leadership Summit
K-12 Mathematics
November 3, 2015Dr. Lynda Luckie
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The Learning System
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Questions to consider…• Where are we in terms of student
achievement and the systems that affect it?
• Where are we along the rubric of the School Keys?
NOT EVIDENT
EXEMPLARY
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The Silent Epidemic:
Perspectives of High School
Dropouts
A Research Report for the Bill and Melinda Gates FoundationMarch 2006
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The Silent Epidemic
• Bill and Melinda Gates Foundation’s research on high school dropouts shows that…
• 45 % of students who drop out did not feel their previous schooling had prepared them for high school.
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The Silent Epidemic• Many felt behind when they
left elementary school.
• 47% said classes weren’t interesting.
• 81% called for more “real-world” learning opportunities.
• Two-thirds said they would have stayed if their schools had demanded more of them.
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The Silent Epidemic
What does this have to do with us as teachers ?
As administrators?
As support personnel?
As district – level leaders?
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So Why Are Kids Having So Much Difficulty with Math?
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The Poverty of “E”s
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The Poverty of “E”s
• The Poverty of Exposure– Poverty because of low-level
questions and classroom work– Rote learning v creativity, grit,
and strenuous mental gymnastics
– A set of destinations and a set of rules to get there v a map and how to read it
– Discrete algorithms for a correct answer v how to attack a new or different kind of equation
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The Poverty of “E”s
• The Poverty of Exposure
• The Poverty of Experience– Traditional problems v rigorous
tasks– Ritual engagement v authentic
engagement– Textbook generated v related to
their world
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The Poverty of “E”s
• The Poverty of Exposure
• The Poverty of Experience
• The Poverty of Expectations– What do you expect them to
learn/produce/achieve?
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Frustrated Math Teachers
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The Research Says…
• The impact of decisions regarding instruction made by individual teachers is far greater than the impact of decisions made at the school level.
» Robert Marzano» What Works in Schools: Translating
Research into Action, 2003
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The Research Says…
• Differences in the effectiveness of individual classroom teachers are the single largest contextual factor affecting the academic growth of students.
» W. Sanders» The School Administrtor
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According to Teacher Keys - GADOE
• It is estimated that only about 3% of the contribution teachers make to student learning is associated with teacher experience, educational level, certification status, and other readily observable characteristics.
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According to Teacher Keys - GADOE
• The remaining 97% of teachers’ effects on student achievement is associated with intangible aspects of teacher quality that defy easy measurement, such as classroom practices.
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Teacher Efficacy
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Teacher Efficacy
Imm
ers
ion in
Conte
nt
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Teacher EfficacyB
est
Practice
/Pedagog
y
Imm
ers
ion in
Conte
nt
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Teacher Efficacy
Com
pelli
ng
Nat
ure
Best
Practice
/Pedagog
y
Imm
ers
ion in
Conte
nt
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It’s Time to Move Our Cheese!
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Best PracticeFocus on Four
1. Good Questioning2. Critical Thinking & Number
Sense3. The Workshop Model for
Differentiated Instruction4. Collaboration
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1. Good Questioning
Best PracticeFocus on Four
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Best Practice…
…it’s all about the questions we ask.
Too often we give our children answers to remember rather than problems to solve, effectively keeping them IN the box.
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What Are Genuine Questions?
–They are questions the teacher asks for which s/he has no way of knowing what the answer will be.
• NOT, “How many angles does a parallelogram have?”
• INSTEAD, “Tell me what you know about a parallelogram.”
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Traditional Question
• What is the name of this shape?
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Application Question
• Tell me everything you know about this shape.
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OR
• If this is a hexagon, draw other kinds of hexagons you know about. Record some shapes that are NOT hexagons.
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OR
• Compare and contrast these two shapes.
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Traditional Question
• What is the name of this shape?
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Application Question
• Tell me everything you know about this shape.
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Or…• If this is a pentagon inscribed in a
circle, draw other pentagons inscribed in circles you know about. Explain how they are different.
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Or…
• Compare and contrast these two figures.
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So…
…what kinds of questions are we
asking?
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Find the Answer
12 ÷ 3
4 x 2
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How Many Ways Can You Represent Each of These?
12 ÷ 3
4 x 2
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Find the Answer
- 16 + (- 8)
(2x + 4) x (3x – 6)
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How Many Ways Can You Represent Each of These?
- 16 + (- 8)
(2x + 4) x (3x – 6)
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Mr. Billingsley’s Trip
A taxi charges: for the first 1.5 miles $2.40
for every additional ¼ mile .10
Mr. Billingsley paid $12.00 for his taxi ride from work to home. How far is Mr. Billingsley’s work place from home?
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Mr. Billingsley’s TripA taxi charges: for the first 1.5 miles $2.40
for every additional ¼ mile .10
Mr. Billingsley paid $12.00 for his taxi ride from work to home. How far is Mr. Billingsley’s work place from home?Renee’s solution:
$12.00 - $2.40 = $9.60$9.60 ÷ $0.10 = 9.69.6 x 1.5 = 14.4Mr. Billingsley’s work place is 14.4 miles from
his home.
There is something wrong with Renee’s solution. Show how you would solve the problem.Explain the error in Renee’s solution.
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What Number Makes Sense?
The following is a textbook question on the topic of numbers and number operations.
Tickets to a concert cost $15 per adult and $8 per child. Mr. Adams bought tickets for 4 adults and 5 children. How much did he spend altogether?
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What Number Makes Sense?
Read the problem. Look at the numbers in the box. Put the numbers in the blanks where you think they fit best. Read the problem again. Do the numbers make sense?
CONCERT TICKETSTickets to a concert cost ______ per adult and ______ per child. Mr. Adams paid _____ for tickets. He bought tickets for _____ adults and ______ children. 4 5 9 $8 $15
$100
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Typical Task
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Typical Task
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Dollar Line Task Function and Pattern
• Think of a situation which could be represented in the graph below.
• Write a full description of the situation (be sure to tell what each axis represents in your situation.)
• What questions could be answered by your completed graph?
From Balanced Assessment
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Typical Textbook Problem
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Make It More Engaging
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Even Better
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Real World?
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ALWAYS ask yourself…
• What mathematics do I want my students to learn by doing this activity/task?
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1. Good Questioning 2. Critical Thinking and Number
Sense
Best PracticeFocus on Four
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Frayer Models
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Conceptual Understanding
• These are…• These are not…
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These are____________________.
These are not________________________.
Which of these are______________?
Explain how you know.
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These are__SQUARES__.
These are not__SQUARES_.
Which of these are_SQUARES__?
Explain how you know.
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These are_numbers that round to 600__.
These are not__numbers that round to 600___.
Which of these are_numbers that round to 600___?
Explain how you know.
597642
563633 576
541 678515
525651
692588
555539
640679
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Three’s a Crowd!
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three-fourths six-eighths two-thirds
centimeter inch millimeter
square rectanglecircle
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What I Know about…
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What’s My Rule?
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WHAT’S MY RULE?Theme: Sports
Yes NoStrike Stick
Split Puck
Pin Hoop
Gutter Goal
Rule: Bowling Terms
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WHAT’S MY RULE?Theme: Geometry
Yes NoTriangle Cube
Rectangle Pyramid
Square Pentagon
Quadrilateral Octagon
Rule: Plane figures with less than 5 sides.
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WHAT’S MY RULE?Theme: _______________
Yes No
Rule: _________________________
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Logic Puzzles
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Number Sense• an understanding that allows
students to approach concepts, ideas, and problems with an intuitive feel for numbers and their relationships
• an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations
Wikipedia
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Another Definition of Number Sense
“Number sense can be described as a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms. No substitute exists for a skillful teacher and an environment that fosters curiosity and exploration at all grade levels.”
From Hilde Howden, “Teaching Number Sense,” Arithmetic Teacher, 36(6). (Feb. 1989), p. 11.
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7 + 6 = ?
How did yousolve this equation?
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38 + 38 = ?How did yousolve this equation?
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395 + 56= ?How did yousolve this equation?
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What Do Kids Say?
6 + 5 = + 4
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What Do Kids Say?
6 + 5 = + 4
Are we teaching them to THINK?
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6 + 5 = +
Even higher level…
Teaching Kids to Think Algebraically
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6 + 5 = + + +
Even higher…
Teaching Kids to Think Algebraically
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6 x 5 = x 3
…with multiplication
Teaching Kids to Think Algebraically
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6 x 5 = x
…with multiplication
Teaching Kids to Think Algebraically
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What Do Kids Say?
.06 + .15 = + 4
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.06 + .15 = +
Higher level
Teaching Kids to Think
Algebraically
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.06 + .15 = + +
Even higher…
Teaching Kids to Think
Algebraically
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.06 + .15 = x 3
…with multiplication
Teaching Kids to Think
Algebraically
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.06 + .15 = x
…with multiplication
Teaching Kids to Think
Algebraically
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.06 + .15 = ? ?
Even higher…
Teaching Kids to Think
Algebraically
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My Personal Favorite
s for Number Sense
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Spotlight on Number Examples
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Critical Thinking/Number Sense
Implications for Teaching
We need to replace the question, “Does the student know it?”
with the question, “How does the student understand it?”
John Van de Walle
Copyright © Allyn and Bacon 2010
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1. Good Questioning 2. Critical Thinking and Number
Sense3. The Workshop Model for
Differentiated Instruction
Best PracticeFocus on Four
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Workshop Model for Math
A Strategic Model for Differentiated
Instruction
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Turn and Talk
• What happens in each setting? Small Groups Whole Group
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• What do the groups look like?
– Heterogeneous or homogeneous?
Nuts and Bolts
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• What do the groups look like?
– Heterogeneous or homogeneous?
Nuts and Bolts
HOMOGENEOUS GROUPS!
WHY??????
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Nuts and Bolts
• How do you determine your small groups?
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Nuts and Bolts
• WITH DATA!
• Tickets out the Door–Weekly assessments
• Formative and Summative Assessments
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How Can You Determine Small Groups?
• Observation of an assigned task• Small group discussion of problem solving
related to the concept to be studied• Written explanation of understanding by
students in their math journals• Paper and pencil pretest• Formative test results (TICKET OUT THE
DOOR)• Performance in earlier work on sequential
math concepts • Checklists and Conferencing
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Sample TOD
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Sample TOD
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Why Tickets out the Door?
• Quick formative assessment
• It’s the DRIVER !• Helps determine your
groups• Does not give students
permission to forget• Wake up call
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Math Workshop Model
Teacher Facilitated
Group
Interactive Practice
At Your Seat
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Math Workshop Logistics
Teacher Facilitated
Interactive Practice
At Your Seat
1st Rotation Group A Group B Group C
2nd Rotation Group C Group A Group B
3rd Rotation Group B Group C Group A
Group A: ________________________________
Group B: ________________________________
Group C: ________________________________
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What Does This Look Like in a 90 Minute
Block?
• Mini Lesson – How long?• Group Rotations – How
long?• Formats for small groups?
Where in the room?
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ALWAYS Ask Yourself…
• What mathematics do I want my students to learn by doing this activity?
• Are there students who may already be proficient in this area?
• Are there students who will need more time?
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Students Need to Know…
• Which group are we in today?
• Where are our meeting areas?
• Do we know what materials we need?
• Do we know our schedule?• Can we work
independently?
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Make It Your Own
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Make It Your Own
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So…how do you keep everyone engaged?
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With QUALITY work!Meaningful math gamesPractice that reinforcesMath JournalsPair/group activities
NOT something they’ve never seen before!!
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Interactive Practice Examples
ALL about PRACTICE
• iPad practice apps• Laptop interactive
activities/programs• Scavenger Hunts• Meaningful math partner
games
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Scavenger Hunts
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The Power of Meaningful Math
Games
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The Power of Meaningful Math
Games
…Meaningful Practice
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Games for Primary
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My Rolls:
1. ________
2. ________
3. ________
4. ________
5. ________
6. ________
Total: ____________
Closest to 100
1. Roll the dice exactly SIX times.2. Decide if your roll will be a “one” or a
“ten.”3. Fill in the grid AND record it in the box.4. Add all six rolls. Closets to 100 wins.
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Closest to 100
• Use a 10 x 10 grid.• Students take turn rolling the dice
exactly SIX times. • When they roll, they decide
whether they want that number of ones or tens. Eg: Student rolls 4…they can get four “ones” or four “tens.” Record on the 10 x 10 grid.
• After six rolls, student closest to 100 without going over wins.
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Domino Drawings
• Students use large Double Six or Double Nine dominoes and can work with a partner.
• Work together to:– Draw domino– Write a number sentence to
show sums of dots.
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Make Ten Concentration
Use two sets of cards
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Fraction Building
1/3one-third
2/4two-
fourths
1/8one-
eighth
1/5one-fifth
Use colored cubes to build and record collections.
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Race to 50
What I Rolled Solution Total
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Race to 50
• Materials– 10 sided dice– +/- die or spinner– Recording Sheet
• Directions– Players take turns rolling two dice
and the +/- die and solve, recording what they rolled and the solution on the recording sheet.
– First player to 50 wins.
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Games for Intermediate Grades
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Pattern Block Fraction Pizza
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Pattern Block Money
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Rectangular Array Game
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Knock Out Three
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Is It True?
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More Is It True?
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Games for MS/HS
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Spin for Expressions
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Spin for Expressions• Materials:
– Plus/minus spinner– A number cube– Expression playing cards
• Pass out all the expression cards FACE DOWN.• The dealer then rolls the number cube to determine
the value of the variable, and then spins the spinner to determine if the value is positive or negative.
• Each player will turn over one card and evaluate his/her expression. The player with the greatest value for that round takes all the cards.
• The player with the most cards at the end of the game is the winner.
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Domino Cards
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Spinning for Polynomials
• Materials– Polynomial Spinner– Dice
• Student one spins spinner 3 times and combines the monomials. Student two does the same.
• Roll dice for value of x, and highest final number wins.
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Spinning for Polynomials
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24 GAME
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Workshop Model Tips…
• Ensure that students understand directions before dismissing them.
• Practice transitions to and from whole group areas.
• Practice moving from station to station.• Set routines for what to do when assignments
are complete (e.g., Anchor Packets).• Establish positive reinforcements for meeting
expectations.
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Workshop Model Tips…
• Set expectations for behavior when working independently or with partners.
• Have back-up seat-work assignments for students who are not on task.
• Establish a signal for redirection and transitions.
• And most of all…
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Biggest Tip…
• Don’t make it harder than it is!!
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At the end of the day…
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So…how do you keep everyone engaged?
With QUALITY work!Meaningful math gamesPractice that reinforcesMath JournalsPair/group activitiesNOT something they’ve never
seen before!!
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The Workshop Model in Action
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This Teacher Said…
“I know more about what my students know and how they think in 8 days than in the entire first semester!”
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Best PracticeFocus on Four
1. Good Questioning2. Critical Thinking & Number
Sense3. The Workshop Model for
Differentiated Instruction4. Collaboration
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• What are you excited about?
• Where does collaboration fit in this model?
• What are your reservations?– How can we deal
positively with them?
Turn and Talk
Remember…it will likely NOT be perfect the first time, or even the second.
You WILL see positive results!