Mathematics Common Core State Standards

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MathematicsCommon Core State Standards
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The user has control

• Sometimes a tool is just right for the wrong use.

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Old Boxes

• People are the next step
• If people just swap out the old standards and put
  the new CCSS in the old boxes
• into old systems and procedures
• into the old relationships
• Into old instructional materials formats
• Into old assessment tools,
• Then nothing will change, and perhaps nothing
  will

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Standards are a platform for instructional systems

• This is a new platform for better instructional
  systems and better ways of managing instruction
• Builds on achievements of last 2 decades
• Builds on lessons learned in last 2 decades
• Lessons about time and teachers

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Grain size is a major issue

• Mathematics is simplest at the right grain size.
• Strands are too big, vague e.g. number
• Lessons are too small too many small pieces
  scattered over the floor, what if some are
  missing or broken?
• Units or chapters are about the right size (8-12
  per year)
• Districts
• STOP managing lessons,
• START managing units

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What mathematics do we want students to walk away
with from this chapter?

• Content Focus of professional learning
  communities should be at the chapter level
• When working with standards, focus on clusters.
  Standards are ingredients of clusters. Coherence
  exists at the cluster level across grades
• Each lesson within a chapter or unit has the same
  objectives.the chapter objectives

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Lesson study and chapter planning

• Lesson study may be more than wonderful, it may
  be necessary
• We have to learn more about the way students
  think about specific mathematics in specific well
  designed problems
• We have to learn how to get student thinking out
  into the open where we can engage it

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Social Justice

• Main motive for standards
• Get good curriculum to all students
• Start each unit with the variety of thinking and
  knowledge students bring to it
• Close each unit with on-grade learning in the
  cluster of standards

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Why do students have to do math problems?

• to get answers because Homeland Security needs
  them, pronto
• I had to, why shouldnt they?
• so they will listen in class
• to learn mathematics

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Why give students problems to solve?

• To learn mathematics.
• Answers are part of the process, they are not the
  product.
• The product is the students mathematical
  knowledge and know-how.
• The correctness of answers is also part of the
  process. Yes, an important part.

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Wrong Answers

• Are part of the process, too
• What was the student thinking?
• Was it an error of haste or a stubborn
  misconception?

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Three Responses to a Math Problem

1. Answer getting
2. Making sense of the problem situation
3. Making sense of the mathematics you can learn
   from working on the problem

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Answers are a black holehard to escape the pull

• Answer getting short circuits mathematics, making
  mathematical sense
• Very habituated in US teachers versus Japanese
  teachers
• Devised methods for slowing down, postponing
  answer getting

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Answer getting vs. learning mathematics

• USA
• How can I teach my kids to get the answer to this
  problem?
• Use mathematics they already know. Easy,
  reliable, works with bottom half, good for
  classroom management.
• Japanese
• How can I use this problem to teach the
  mathematics of this unit?

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Butterfly method
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Use butterflies on this TIMSS item

• 1/2 1/3 1/4

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Set up

• Not
• set up a proportion and cross multiply
• But
• Set up an equation and solve
• Prepare for algebra, not just next weeks quiz.

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Foil FOIL

• Use the distributive property
• It works for trinomials and polynomials in
  general
• What is a polynomial?
• Sum of products product of sums
• This IS the distributive property when a is a
  sum

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Canceling

• x5/x2 xx xxx / xx
• x5/x5 xx xxx / xx xxx

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Standards are a peculiar genre

• 1. We write as though students have learned
  approximately 100 of what is in preceding
  standards. This is never even approximately true
  anywhere in the world.
• 2. Variety among students in what they bring to
  each days lesson is the condition of teaching,
  not a breakdown in the system. We need to teach
  accordingly.
• 3. Tools for teachersinstructional and
  assessmentshould help them manage the variety

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Differences among students

• The first response, in the classroom make
  different ways of thinking students bring to the
  lesson visible to all
• Use 3 or 4 different ways of thinking that
  students bring as starting points for paths to
  grade level mathematics target
• All students travel all paths robust, clarifying

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Social Justice

• Main motive for standards
• Get good curriculum to all students
• Start each unit with the variety of thinking and
  knowledge students bring to it
• Close each unit with on-grade learning in the
  cluster of standards

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Mathematical Practices Standards

• Make sense of complex problems and persevere in
  solving them.
• Reason abstractly and quantitatively
• Construct viable arguments and critique the
  reasoning of others.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• Attend to precision
• Look for and make use of structure
• 8. Look for and express regularity in repeated
  reasoning.
• College and Career Readiness Standards for
  Mathematics

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• What key messages to communicate with principals?
• Why elementary science and middle school math?
• How will this work connect to other initiatives
  and efforts?
• How will principal professional learning and
  support be organized?
• What is the plan for REXO support collaboration
  next year?

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Expertise and Character

• Development of expertise from novice to
  apprentice to expert
• Schoolwide enterprise school leadership
• Department wide enterprise department taking
  responsibility
• The Content of their mathematical Character
• Develop character

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Two major design principles, based on evidence

• Focus
• Coherence

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The Importance of Focus

• TIMSS and other international comparisons suggest
  that the U.S. curriculum is a mile wide and an
  inch deep.
• On average, the U.S. curriculum omits only 17
  percent of the TIMSS grade 4 topics compared with
  an average omission rate of 40 percent for the 11
  comparison countries. The United States covers
  all but 2 percent of the TIMSS topics through
  grade 8 compared with a 25 percent non coverage
  rate in the other countries. High-scoring Hong
  Kongs curriculum omits 48 percent of the TIMSS
  items through grade 4, and 18 percent through
  grade 8. Less topic coverage can be associated
  with higher scores on those topics covered
  because students have more time to master the
  content that is taught.
• Ginsburg et al., 2005

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U.S. standards organization

• Grade 1
• Number and Operations
• Measurement and Geometry
• Algebra and Functions
• Statistics and Probability

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U.S. standards organization

• 12
• Number and Operations
• Measurement and Geometry
• Algebra and Functions
• Statistics and Probability

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Draw a line to represent a 30 inch race course.
Show where all four snails are when the first one
crosses the finish line.

• Snail A
• 5 inches in 10 minutes
• Snail B 3 inches in 20 minutes
• Snail C 1 inch in 15 minutes
• Snail D 6 inches in 30 minutes

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Making Sense of Tom, Dick and Harry

• Suppose that it takes Tom and Dick 2 hours to do
  a certain job, it takes Tom and Harry 3 hours to
  do the same job and it takes Dick and Harry 4
  hours to do the same job.
• 1.How long would it take Tom, Dick and Harry to
  do the same job if all 3 men worked together?
• 2.Prepare an explanation on how to make sense of
  this problem. You will explain to elem. Teachers.

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Angel ran 30 laps around the gym in 15 minutes.
(A lap is once around the track.)

• After her run Angel made this calculation
• 30 15 2
• What does the 2 tell you about Angels run?
• How do you know that is what it tells you?

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Assessment

• I. assessment and motivation
• II. what students produce to be assessed

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Angel ran 30 laps around the gym in 15 minutes.
(A lap is once around the track.)

• Her coach made this calculation
• 15 30 0.5
• c. What does the 0.5 tell you about Angels
  run?
• d. How do you know that is what it tells you?

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1. Patti runs a 2.5 miles in 30 minutes.

• Make a table and a double number line for Pattis
  running show at least 5 times.
• How far did Patti run (distance, d) in 5 minutes?
• Write a formula to calculate Pattis distance (d)
  for any time (t)?
• d. How long did it take (time, t) Patti to run 2
  miles?
• e. Write a formula to calculate Pattis time (t)
  for any distance (d)?

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Her dog, Boe, runs twice as fast as Patti.

• Represent his data in a table or double number
  line
•   Write a formula to calculate Boes distance (d)
  for any time (t)?
• c. Write a formula to calculate Boes time (t)
  for any distance (d)?

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How would you calculate the rate, r, in miles, d,
per minute, t?

• Write formulas using r, d, and t that show
• Pattis rate
• Boes rate
• The rate is the speed

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Make up problems

• Here are three quantities 4 pounds, 5, 1.25
  per pound
• Make up a problem that uses two of these
  quantities as givens and has the third as the
  answer.
• Make up another problem switching around the
  givens and the answer.
• Make up a third problem switching givens and
  answer.

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K-5 quantities and number line

• Compare quantities, especially length
• Compare by measuring units
• Add and subtract with ruler
• Diagram of a ruler
• Diagram of a number line
• Arithmetic on the number line based on units
• Representing time, money and other quantities
  with number lines

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Number line

• Ruler and number line concepts that are often
  underdeveloped. In elementary grades
• A number is a pointa location
• and also a length from 0. (like order and
  cardinality)
• Lengths between whole numbers are equal
• The length from 0 to 1 is the unit length
• Any length can be partitioned into any number of
  equal length parts

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Fractions Progression

• Understanding the arithmetic of fractions draws
  upon four prior progressions that informed the
  CCSS
• equal partitioning,
• unitizing,
• number line,
• and operations.

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Units are things you count

• Objects
• Groups of objects
• 1
• 10
• 100
• ¼ unit fractions
• Numbers represented as expressions

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Units add up

• 3 pennies 5 pennies 8 pennies
• 3 ones 5 ones 8 ones
• 3 tens 5 tens 8 tens
• 3 inches 5 inches 8 inches
• 3 ¼ inches 5 ¼ inches 8 ¼ inches
• ¾ 5/4 8/4
• 3(x 1) 5(x1) 8(x1)

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Unitizing links fractions to whole number
arithmetic

• Students expertise in whole number arithmetic is
  the most reliable expertise they have in
  mathematics
• It makes sense to students
• If we can connect difficult topics like fractions
  and algebraic expressions to whole number
  arithmetic, these difficult topics can have a
  solid foundation for students

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Grade 3unit fractions

• The length from 0 to1 can be partitioned into 4
  equal parts. The size of the part is ¼.
• Unit fractions like ¼ are numbers on the number
  line.

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Adding and multiplying Unit Fractions

• Whatever can be counted can be added, and from
  there knowledge and expertise in whole number
  arithmetic can be applied to newly unitized
  objects.
• Grade 4
• ¼ 1/4 ¼ ¾
• Add fractions with like denominators
• 3 x ¼ ¾
• Multiply whole number times a fraction n(a/b)
  (na)/b

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Grade 5

1. Add and subtract fractions with unlike
   denominators using multiplication by n/n to
   generate equivalent fractions and common
   denominators
2. 1/b 1 divided by b fractions can express
   division
3. Multiply and divide fractions

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Fraction Equivalence Grade 3

• Fractions of areas that are the same size, or
  fractions that are the same point (length from 0)
  are equivalent
• recognize simple cases ½ 2/4 4/6 2/3
• Fraction equivalents of whole numbers 3 3/1,
  4/4 1
• Compare fractions with same numerator or
  denominator based on size in visual diagram

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Fraction equivalence Grade 4

• Explain why a fraction a/b na/nb using visual
  models generate equivalent fractions
• Compare fractions with unlike denominators by
  finding common denominators explain on visual
  model based on size in visual diagram

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Fraction equivalence Grade 5

• Use equivalent fractions to add and subtract
  fractions with unlike denominators

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Fraction Item

• 4/5 is closer to 1 than 5/4. Show why this is
  true on a number line.

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Fraction Item

• 4/5 is closer to 1 than 5/4. Show why this is
  true on a number line.

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Students perform calculations and solve problems
involving addition, subtraction,and simple
multiplication and division of fractions and
decimals

• 2.1 Add, subtract, multiply, and divide with
  decimals add with negative integers subtract
  positive integers from negative integers and
  verify the reasonableness of the results.
• 2.2 Demonstrate proficiency with division,
  including division with positive decimals and
  long division with multidigit divisors.

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Students perform calculations and solve problems
involving addition, subtraction, and simple
multiplication and division of fractions and
decimals

• 2.3 Solve simple problems, including ones
  arising in concrete situations, involving the
  addition and subtraction of fractions and mixed
  numbers (like and unlike denominators of 20 or
  less), and express answers in the simplest form.
• 2.4 Understand the concept of multiplication and
  division of fractions.
• 2.5 Compute and perform simple multiplication and
  division of fractions and apply these procedures
  to solving problems.

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Use equivalent fractions as a strategy to add and
subtract fractions.

• 1. Add and subtract fractions with unlike
  denominators (including mixed numbers) by
  replacing given fractions with equivalent
  fractions in such a way as to produce an
  equivalent sum or difference of fractions with
  like denominators. For example, 2/3 5/4 8/12
  15/12 23/12. (In general, a/b c/d (ad
  bc)/bd.)

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Use equivalent fractions as a strategy to add and
subtract fractions.

• 2. Solve word problems involving addition and
  subtraction of fractions referring to the same
  whole, including cases of unlike denominators,
  e.g., by using visual fraction models or
  equations to represent the problem. Use benchmark
  fractions and number sense of fractions to
  estimate mentally and assess the reasonableness
  of answers. For example, recognize an incorrect
  result 2/5 1/2 3/7, by observing that 3/7 lt
  1/2.

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CA CST grade 5 item

• It takes Suzanne 1/6 hour to walk to the
  playground and 1/3 hour to walk from the
  playground to school. How much time does it take
  Suzanne to walk to the playground and then to
  school?
• A 2/9 hour
• B 1/3 hour
• C 1/2 hour
• D 2/3 hour

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Solving equations vs. functions

• .25a - 3 0 f(a) .25a - 3
• a2 5a 0 f(a) a2 5a
• ? f(a) 2a

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CA CST 5th grade

• Yoshi spent 1 and 1/3 hours reading and ¾ hour
  doing chores. How many total hours did Yoshi
  spend on these activities?
• A 1 1/3
• B 1 4/7
• C 2 1/12
• D 2 1/6

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The most important ideas in the CCSS mathematics
that need attention.

1. Properties of operations their role in
   arithmetic and algebra
2. Mental math and algebra vs. algorithms
3. Units and unitizing
4. Operations and the problems they solve
5. Quantities-variables-functions-modeling
6. Number-Operations-Expressions-Equation
7. Modeling
8. Practices

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Progression quantities and measurement to
variables and functions
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K - 5 6
- 8 9 - 12

Equal Partitioning, division
proportional and linear relationships
multiplication
Measurement of quantities, units
Systems of linear equations
Unit Rate
Number line, graphs
slope
ratio
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Representing quantities with expressions
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Mental math

• 72 -29 ?
• In your head.
• Composing and decomposing
• Partial products
• Place value in base 10
• Factor X2 4x 4 in your head

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Operations and the problems they solve

• Tables 1 and 2 on pages 88 and 89

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Properties of Operations

• Also called rules of arithmetic , number
  properties

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From table 2 page 89

• a b ?
• a ? p, and p a ?
• ? b p, and p b ?
• 1.Play with these using whole numbers,
• 2.make up a problem for each.
• 3. substitute (x 1) for b

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Nine properties are the most important
preparation for algebra

• Just nine foundation for arithmetic
• Exact same properties work for whole numbers,
  fractions, negative numbers, rational numbers,
  letters, expressions.
• Same properties in 3rd grade and in calculus
• Not just learning them, but learning to use them

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Using the properties

• To express yourself mathematically (formulate
  mathematical expressions that mean what you want
  them to mean)
• To change the form of an expression so it is
  easier to make sense of it
• To solve problems
• To justify and prove

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Properties are like rules, but also like rights

• You are allowed to use them whenever you want,
  never wrong.
• You are allowed to use them in any order
• Use them with a mathematical purpose

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Properties of addition
Associative property of addition (a b) c a (b c) (2 3) 4 2 (3 4)
Commutative property of addition a b b a 2 3 3 2
Additive identity property of 0 a 0 0 a a 3 0 0 3 3
Existence of additive inverses For every a there exists a so that a (a) (a) a 0. 2 (-2) (-2) 2 0
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Properties of multiplication
Associative property of multiplication (a x b) x c a x (b x c) (2 x 3) x 4 2 x (3 x 4)
Commutative property of multiplication a x b b x a 2 x 3 3 x 2
Multiplicative identity property of 1 a x 1 1 x a a 3 x 1 1 x 3 3
Existence of multiplicative inverses For every a ? 0 there exists 1/a so that a x 1/a 1/a x a 1 2 x 1/2 1/2 x 2 1
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Linking multiplication and addition the ninth
property

• Distributive property of multiplication over
  addition
• a x (b c) (a x b) (a x c)
• a(bc) ab ac

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Find the properties in multiplication table
patterns

• There are many patterns in the multiplication
  table, most of them are consequences of the
  properties of operations
• Find patterns and explain how they come from the
  properties.
• Find the distributive property patterns

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Grade level examples

• 3 packs of soap
• 4 dealing cards
• 5 sharing
• 6 money
• 7 lengths (fractions)
• 8 times larger ()

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K -5 6
8 9 - 12

Quantity and measurement
Ratio and proportional relationships
Operations and algebraic thinking
Functions
Expressions and Equations
Modeling (with Functions)
Modeling Practices
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K -2 3
- 6 7 - 12

Equal Partitioning
Rates, proportional and linear relationships
Unitizing in base 10 and in measurement
Rational number
Fractions
Number line in Quantity and measurement
Properties of Operations
Rational Expressions
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Functions and Solving Equations

• Quantities-variables-functions-modeling
• Number-Operations-Expressions-Equation

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Take the number apart?

• Tina, Emma, and Jen discuss this expression
• 5 1/3 x 6
• Tina I know a way to multiply with a mixed
  number, like 5 1/3 , that is different from the
  one we learned in class. I call my way take the
  number apart. Ill show you.

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Which of the three girls do you think is right?
Justify your answer mathematically.

• First, I multiply the 5 by the 6 and get 30.
• Then I multiply the 1/3 by the 6 and get 2.
  Finally, I add the 30 and the 2, which is 32.
• Tina It works whenever I have to multiply a
  mixed number by a whole number.
• Emma Sorry Tina, but that answer is wrong!
• Jen No, Tinas answer is right for this one
  problem, but take the number apart doesnt work
  for other fraction problems.

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What is an explanation?

• Why you think its true and why you think it
  makes sense.
• Saying distributive property isnt enough, you
  have to show how the distributive property
  applies to the problem.

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Example explanation

• Why does 5 1/3 x 6 (6x5) (6x1/3) ?
• Because
• 5 1/3 5 1/3
• 6(5 1/3)
• 6(5 1/3)
• (6x5) (6x1/3) because a(b c) ab ac

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Mental math

• 72 -29 ?
• In your head.
• Composing and decomposing
• Partial products
• Place value in base 10
• Factor X2 4x 4 in your head

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Locate the difference, p - m, on the number line
p
m
0
1
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For each of the following cases, locate the
quotient p/m on the number line
p
m
0
1
m
0
p
1
p
m
0
1
m
1
p
0
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Misconceptions about misconceptions

• They werent listening when they were told
• They have been getting these kinds of problems
  wrong from day 1
• They forgot
• The other side in the math wars did this to the
  students on purpose

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More misconceptions about the cause of
misconceptions

• In the old days, students didnt make these
  mistakes
• They were taught procedures
• They were taught rich problems
• Not enough practice

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Maybe

• Teachers misconceptions perpetuated to another
  generation (where did the teachers get the
  misconceptions? How far back does this go?)
• Mile wide inch deep curriculum causes haste and
  waste
• Some concepts are hard to learn

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Whatever the Cause

• When students reach your class they are not blank
  slates
• They are full of knowledge
• Their knowledge will be flawed and faulty, half
  baked and immature but to them it is knowledge
• This prior knowledge is an asset and an
  interference to new learning

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Second grade

• When you add or subtract, line the numbers up on
  the right, like this
• 23
• 9
• Not like this
• 23
• 9

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Third Grade

• 3.24 2.1 ?
• If you Line the numbers up on the right like
  you spent all last year learning, you get this
• 3.2 4
• 2.1
• You get the wrong answer doing what you learned
  last year. You dont know why.
• Teach line up decimal point.
• Continue developing place value concepts

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Research on Retention of Learning Shell Center
Swan et al
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Lesson Units for Formative Assessment

• Concept lessonsProficient students expect
  mathematics to make sense
• To reveal and develop students interpretations
  of significant mathematical ideas and how these
  connect to their other knowledge.
• Problem solving lessonsThey take an active
  stance in solving mathematical problems
• To assess and develop students capacity to apply
  their Math flexibly to non-routine, unstructured
  problems, both from pure math and from the real
  world.

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Mathematical Practices Standards

• Make sense of complex problems and persevere in
  solving them.
• Reason abstractly and quantitatively
• Construct viable arguments and critique the
  reasoning of others.
• 4. Model with mathematics.
• 5. Use appropriate tools strategically.
• Attend to precision
• Look for and make use of structure
• 8. Look for and express regularity in repeated
  reasoning.
• College and Career Readiness Standards for
  Mathematics

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Mathematical Content Standards

• Number Quantity
• Algebra
• Functions
• Modeling
• Statistics and Probability
• Geometry

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Concept focused v Problem focused
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Optimization Problems Boomerangs

• Projector Resources

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Evaluating Sample Responses to Discuss

• What do you like about the work?
• How has each student organized the work?
• What mistakes have been made?
• What isn't clear?
• What questions do you want to ask this student?
• In what ways might the work be improved?

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Alexs solution
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Dannys solution
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Jeremiahs solution
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Tanya's solution
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Progressions (http//ime.math.arizona.edu/pro
gressions/) Illustrative Mathematics Project
(http//illustrativemathematics.org). Technical
manual NCTM-AMTE-NCSM-ASSM task force (see
description at http//commoncoretools.wordpre
ss.com NCTM sample tasks for reasoning and
sense-making, http//www.nctm.org/hsfocus Mc
Callum blog, Tools for the Common Core,
http//commoncoretools.wordpress.com Daro video
and slides serpinstitute.org
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Making Sense of Word Problems
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Word Problem from popular textbook

• The upper Angel Falls, the highest waterfall on
  Earth, are 750 m higher than Niagara Falls. If
  each of the falls were 7 m lower, the upper Angel
  Falls would be 16 times as high as Niagara Falls.
  How high is each waterfall?

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Imagine the Waterfalls Draw
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Diagram it
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The Height of Waterfalls
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Heights
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Height or Waterfalls?
750 m.
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Heights we know
750 m.
7 m.
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Heights we know and dont
750 m.
d
d
7 m.
7 m.
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Heights we know and dont
750 m.
d
d
7 m.
7 m.
Angel 750 d 7 Niagara d 7
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Same height referred to in 2 ways
16d 750 d
750 m.
16d
d
d
7 m.
7 m.
Angel 750 d 7 Niagara d 7
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d ?
16d 750 d 15d 750 d 50
750 m.
16d
d
d
7 m.
7 m.
Angel 750 50 7 807 Niagara 50 7 57
Angel 750 d 7 Niagara d 7
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Activate prior knowledge

• What knowledge?
• Have you ever seen a waterfall?
• What does water look like when it falls?

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Take the number apart?

• Tina, Emma, and Jen discuss this expression
• 5 1/3 x 6
• Tina I know a way to multiply with a mixed
  number, like 5 1/3 , that is different from the
  one we learned in class. I call my way take the
  number apart. Ill show you.

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Which of the three girls do you think is right?
Justify your answer mathematically.

• First, I multiply the 5 by the 6 and get 30.
• Then I multiply the 1/3 by the 6 and get 2.
  Finally, I add the 30 and the 2, which is 32.
• Tina It works whenever I have to multiply a
  mixed number by a whole number.
• Emma Sorry Tina, but that answer is wrong!
• Jen No, Tinas answer is right for this one
  problem, but take the number apart doesnt work
  for other fraction problems.

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What is an explanation?

• Why you think its true and why you think it
  makes sense.
• Saying distributive property isnt enough, you
  have to show how the distributive property
  applies to the problem.

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Example explanation

• Why does 5 1/3 x 6 (6x5) (6x1/3) ?
• Because
• 5 1/3 5 1/3
• 6(5 1/3)
• 6(5 1/3)
• (6x5) (6x1/3) because a(b c) ab ac

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In the Description box

• Today I valued
• Id like to spend more time on
• Make sure to put the correct date June 29, 2011