finding focus for mathematics instruction grades 6-8 march 8, 2010tuscola isd administrative...
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Finding Focus for Finding Focus for Mathematics Instruction Mathematics Instruction
Grades 6-8Grades 6-8
March 8, 2010March 8, 2010 Tuscola ISD Administrative Tuscola ISD Administrative BuildingBuilding
March 16, 2010March 16, 2010 Huron Area Technical CenterHuron Area Technical Center
March 10 and 16, 2010March 10 and 16, 2010 Huron ISD; HATCHuron ISD; HATC
Thumb-Area Student Achievement Model and Huron Intermediate School District
Today’s GoalsToday’s Goals
Become familiar with Mathematics CurriculumBecome familiar with Mathematics Curriculum Deepen understanding of Big Math IdeasDeepen understanding of Big Math Ideas Use strategies for explicit vocabulary Use strategies for explicit vocabulary
instruction instruction Understand instructional implications of Understand instructional implications of
researchresearch Use assessment in a 3-tier process Use assessment in a 3-tier process Support instruction for intervention and Support instruction for intervention and
enrichmentenrichment
Be Thinking AboutBe Thinking About
What is one idea you will try in the What is one idea you will try in the next week?next week?
What are 2-3 items you will share or What are 2-3 items you will share or discuss with your colleagues?discuss with your colleagues?
How will that happen?How will that happen?
What Are the Focal Points?What Are the Focal Points?
What Do the Focal Points Look What Do the Focal Points Look Like?Like?
1.1. Work in grade-level teams.Work in grade-level teams.
2.2. Find the focal points for your grade.Find the focal points for your grade.
3.3. Sort the GLCE topics according to Sort the GLCE topics according to focal point. Make a separate pile for focal point. Make a separate pile for “leftover topics.”“leftover topics.”
Compare your sort to MDE list. Compare your sort to MDE list.
Re-arrange if necessary.Re-arrange if necessary.
How Should the Focal Points How Should the Focal Points Impact Instruction?Impact Instruction?
Based on nationally-recognized topicsBased on nationally-recognized topics Related to GLCEs and MEAP:Related to GLCEs and MEAP:
Core expectationsCore expectations Must be related to a focal pointMust be related to a focal point No more than 20 per gradeNo more than 20 per grade Assessed with two items, all studentsAssessed with two items, all students
Extended core expectationsExtended core expectations Not related to a focal pointNot related to a focal point Assessed with no more than one item (sampled)Assessed with no more than one item (sampled)
DiscussionDiscussion
70-80% of instruction 70-80% of instruction should focus on GLCEs should focus on GLCEs related to focal points. related to focal points.
Use your textbook to think Use your textbook to think about your instruction. about your instruction. What topics should be What topics should be emphasized more? Less?emphasized more? Less?
Useful DocumentsUseful Documents
Math GLCEs Math GLCEs Assessed with NC Assessed with NC DesignationsDesignations www.mi.gov/mathematicswww.mi.gov/mathematics
Mathematics Focal Mathematics Focal Points K-8 Alignment Points K-8 Alignment (11-11-09 from SAM)(11-11-09 from SAM) http://www.hisd.k12.http://www.hisd.k12.
mi.us/SAM/main.htmlmi.us/SAM/main.html
Common Core Standards Common Core Standards InitiativeInitiative
Coalition of states who have all Coalition of states who have all agreed to adopt the same state agreed to adopt the same state standardsstandards
48 states, 2 territories (Puerto Rico 48 states, 2 territories (Puerto Rico and the U.S. Virgin Islands), and the and the U.S. Virgin Islands), and the District of ColumbiaDistrict of Columbia
Who’s missing?Who’s missing? Alaska and TexasAlaska and Texas
Common Core Standards Common Core Standards Initiative DRAFT 1-13-2010Initiative DRAFT 1-13-2010
Match CCSI Match CCSI standards to Focal standards to Focal Points and GLCEsPoints and GLCEs
Make new piles if Make new piles if neededneeded
No GLCE for a CCSI No GLCE for a CCSI standard? Check standard? Check another grade.another grade.
GRADE 6GRADE 6 Current MI Focal PointsCurrent MI Focal Points 1-13-2010 DRAFT CCSI1-13-2010 DRAFT CCSI
Operations Operations on rational on rational numbersnumbers
x/x/ fractions fractions+/- integers+/- integersratios and ratesratios and ratesdecimal and % problemsdecimal and % problems
fractions, decimalsfractions, decimalsratios, percents, unit ratesratios, percents, unit ratesrational numbersrational numbersinteger conceptsinteger conceptsproperties of arithmetic (MI Gr. 7)properties of arithmetic (MI Gr. 7)
Variables, Variables, expressions expressions and and equationsequations
word problems and word problems and contextscontextscombine like termscombine like termsevaluate simple evaluate simple expressionsexpressionslinear functions as linear functions as tables, graphs, equationstables, graphs, equationssolve simple equationssolve simple equations
similar expressionssimilar expressionsevaluate simple expressionsevaluate simple expressionssolve equations (w/ fract)solve equations (w/ fract)word problemsword problems
Geometry Geometry and and measurememeasurementnt
convert measurementsconvert measurementsvolume and surface area volume and surface area of cubes and rectangular of cubes and rectangular prisms, using formulasprisms, using formulas
complex areas (subdivision) complex areas (subdivision) (expanded in MI Gr. 8)(expanded in MI Gr. 8)surface area of cubes, prisms, and surface area of cubes, prisms, and pyramidspyramidsvolume v. surface areavolume v. surface areavolume of a cube; exponents and volume of a cube; exponents and rootsroots
Grade 5 – mean and Grade 5 – mean and modemode
Grade 8 – mean, median, Grade 8 – mean, median, mode, biasmode, bias
collect, plot, and interpret datacollect, plot, and interpret datamean and median and measures mean and median and measures of variationof variation
GRADE 7GRADE 7 Current MI Focal PointsCurrent MI Focal Points 1-13-2010 DRAFT CCSI1-13-2010 DRAFT CCSI
ProportionaliProportionality and ty and similaritysimilarity
rates, ratios, unit rate, scaling, rates, ratios, unit rate, scaling, equivalent fractions, equivalent fractions, proportions proportions directly and inversely directly and inversely proportional; linearproportional; lineartables, graphs, formulas, tables, graphs, formulas, storiesstoriessimilar polygons, esp. trianglessimilar polygons, esp. triangles
proportional relationshipsproportional relationshipssimilar polygons, esp. similar polygons, esp. trianglestrianglesvolume of cylinder (MI Gr. 8)volume of cylinder (MI Gr. 8)transformations (MI Gr. 8)transformations (MI Gr. 8)cross-sections of cones, cross-sections of cones, cylinders, pyramids, prisms cylinders, pyramids, prisms (MI Gr. 8)(MI Gr. 8)
Linear Linear functions functions and and equationsequations
slope, intercept, line (CC Gr. 8)slope, intercept, line (CC Gr. 8)linear functions (CC Gr. 8)linear functions (CC Gr. 8)properties of arithmeticproperties of arithmeticcombine expressions and combine expressions and solve equations solve equations irrational numbers (CC Gr. 8)irrational numbers (CC Gr. 8)compute with rationalscompute with rationalscircle graphs, stem and leaf circle graphs, stem and leaf plots, histograms, box-and plots, histograms, box-and whisker-plotswhisker-plotsscatter plots, line of best fitscatter plots, line of best fit
+,-,x,+,-,x, rational numbers rational numbersdistributive property, incl. distributive property, incl. factorfactorequivalent expressionsequivalent expressionsstory problemsstory problemsabsolute valueabsolute valueapply inversesapply inverseshistogram and bar graph histogram and bar graph from experiments or from experiments or simulations; absolute and simulations; absolute and relative frequenciesrelative frequencies
Grade 8 – four GLCEs for Grade 8 – four GLCEs for probability not related to a probability not related to a focal pointfocal point
experimental and experimental and theoretical probabilities (MI. theoretical probabilities (MI. Gr. 8)Gr. 8)
GRADE 8GRADE 8 Current MI Focal PointsCurrent MI Focal Points 1-13-2010 DRAFT CCSI1-13-2010 DRAFT CCSI
FunctionsFunctions non-linear functionsnon-linear functionsvertical line testvertical line testquadratic functions in factored quadratic functions in factored form, parabola, root, vertex, x-form, parabola, root, vertex, x-interceptsintercepts
linear functionslinear functionsslope-intercept form of a lineslope-intercept form of a linesolve linear equations with solve linear equations with rational number coefficientsrational number coefficientsidentify and compare rational identify and compare rational numbersnumbers
Surface Surface area and area and volumevolume
volume and surface area formulas volume and surface area formulas for cylinders, cones, pyramids, and for cylinders, cones, pyramids, and spheres (CC Gr. 7)spheres (CC Gr. 7)sketch solids (CC Gr. 7)sketch solids (CC Gr. 7)
See Grade 7See Grade 7
Distance Distance and angleand angle
Pythagorean Theorem; distance Pythagorean Theorem; distance formulaformuladefinition of a circle; definition of a circle; circumference and area formulascircumference and area formulascomplex area (subdivision) complex area (subdivision) (expanded from CC Gr. 6)(expanded from CC Gr. 6)dilation from a point; similar dilation from a point; similar figures (CC Gr. 7)figures (CC Gr. 7)reflective and rotational reflective and rotational symmetries; transformations (CC symmetries; transformations (CC Gr. 7)Gr. 7)
Pythagorean Theorem; Pythagorean Theorem; distance formuladistance formulainterior and exterior angles of interior and exterior angles of convex polygonsconvex polygonsangle measures; parallel lines; angle measures; parallel lines; transversal; vertical anglestransversal; vertical anglesgeometric constructionsgeometric constructions
DataData mean, median, mode (CC Gr. 6)mean, median, mode (CC Gr. 6)BiasBias
scatterplots (MI Gr. 7)scatterplots (MI Gr. 7)probability (MI Gr. 7)probability (MI Gr. 7)
Common Core (CC) – Common Core (CC) – When???When???
From Michigan’s Race to the Top application From Michigan’s Race to the Top application (January 19, 2010):(January 19, 2010):
Adopt the CC Readiness Standards (CCRS) and Adopt the CC Readiness Standards (CCRS) and K-12 Standards (CCK-12) by June, 2010 (p. 51)K-12 Standards (CCK-12) by June, 2010 (p. 51)
Roll-out sessions in August, 2010 (dates to be Roll-out sessions in August, 2010 (dates to be announced June, 2010) (p. 58)announced June, 2010) (p. 58)
Revise and update the Michigan Curriculum Revise and update the Michigan Curriculum Framework to be consistent with CCRS, CCK-Framework to be consistent with CCRS, CCK-12 anticipated by May 2011 (p. 59)12 anticipated by May 2011 (p. 59)
What about testing?What about testing?
““Use of the existing assessment system . . . from Use of the existing assessment system . . . from spring 2010 through spring 2014.” (p. 64)spring 2010 through spring 2014.” (p. 64)
““Move to common annual summative assessment Move to common annual summative assessment provided by the common assessment consortium that provided by the common assessment consortium that Michigan joins when those assessments are available Michigan joins when those assessments are available and validated for such use beginning in Fall 2014.” (p. and validated for such use beginning in Fall 2014.” (p. 64)64)
Competitive grants to consortia of ISDs to develop Competitive grants to consortia of ISDs to develop annual summative assessments in subject/grade annual summative assessments in subject/grade combinations not currently tested at the state level combinations not currently tested at the state level beginning in fall 2011 (pp. 64-65):beginning in fall 2011 (pp. 64-65): Develop specifications by spring 2012Develop specifications by spring 2012 Develop review procedures by spring 2014Develop review procedures by spring 2014
Multi-state AssessmentMulti-state Assessment
SMARTER (p. 52)SMARTER (p. 52) Summative Multi-state Assessment Resources for Teachers Summative Multi-state Assessment Resources for Teachers
and Educational Researchersand Educational Researchers MI is a lead stateMI is a lead state Develop summative assessments based on the CCK-12 Develop summative assessments based on the CCK-12
Standards in ELA and mathematicsStandards in ELA and mathematics MOSAIC (p. 52)MOSAIC (p. 52)
Multiple Options for Student Assessment and Instruction Multiple Options for Student Assessment and Instruction ConsortiumConsortium
““Interim benchmark and formative assessments . . . Interim benchmark and formative assessments . . . designed to complement a summative assessment system” designed to complement a summative assessment system” (i.e., SMARTER)(i.e., SMARTER)
Also collaboratively develop a curriculum framework and Also collaboratively develop a curriculum framework and instructional support and integration materialsinstructional support and integration materials
But . . .But . . .
It was announced Friday morning It was announced Friday morning that MI did not get any money for the that MI did not get any money for the first round of grants.first round of grants.
MI will reapply when applications are MI will reapply when applications are accepted in June.accepted in June.
What does it take to pass the What does it take to pass the Math MEAP?Math MEAP?
FALL FALL 20092009
Lowest Raw Lowest Raw Score for Level 2Score for Level 2
Percent to Percent to PassPass
Grade 3Grade 3 15 out of 4515 out of 45 33%33%
Grade 4Grade 4 19 out of 5319 out of 53 36%36%
Grade 5Grade 5 21 out of 5221 out of 52 40%40%
Grade 6Grade 6 19 out of 5219 out of 52 37%37%
Grade 7Grade 7 19 out of 5319 out of 53 36%36%
Grade 8Grade 8 19 out of 5119 out of 51 37%37%
What does this mean?What does this mean?
Percent Proficient is meaninglessPercent Proficient is meaningless Cut scores are low because of the Cut scores are low because of the
composition of the testcomposition of the test
Look at Item Analysis:Look at Item Analysis: Not reliable to item or GLCE levelNot reliable to item or GLCE level Look at groups of items – by Focal Point Look at groups of items – by Focal Point
or Topicor Topic
Looking at your item Looking at your item analysisanalysis
80% or better80% or better 60-79%60-79% 59% or lower59% or lower Prioritize by topic Prioritize by topic
or focal pointor focal point Track multi-year Track multi-year
trendstrends
Analysis of Several Local Analysis of Several Local DistrictsDistricts
Grade 7 MEAP (Grade 6 Grade 7 MEAP (Grade 6 content):content):
Rational number operations:Rational number operations: Multiply and divide fractionsMultiply and divide fractions Integers and rationals: +,-Integers and rationals: +,- Decimal, %, and rational Decimal, %, and rational
numbersnumbers Expressions and equations:Expressions and equations:
Variables, combine like Variables, combine like termsterms
Solve equationsSolve equations Properties of 3D shapesProperties of 3D shapes
Convert in measurement Convert in measurement systemssystems
Grade 8 MEAP (Grade 7 Grade 8 MEAP (Grade 7 content):content):
Proportionality and Proportionality and similaritysimilarity Rates, ratios, & proportionsRates, ratios, & proportions Directly proportional, linearDirectly proportional, linear Similar polygonsSimilar polygons
Functions, linear equationsFunctions, linear equations Represent linear functionsRepresent linear functions Expressions & equationsExpressions & equations Compute with rational Compute with rational
numbersnumbers Represent & interpret dataRepresent & interpret data
Focal Points Grades 6-8Focal Points Grades 6-8
Grade 6 Grade 7 Grade 8
Operations on rational numbers
Proportionality and similarity
Expressions and equations
Linear functions and equations
Non-linear functions
Volume and surface area
Volume and surface area formulas
Pythagorean Theorem and Transformations
Data
Rational Number ProjectRational Number Projecthttp://www.cehd.umn.edu/rationalnumberprhttp://www.cehd.umn.edu/rationalnumberproject/oject/
From the home page, scroll down and choose From the home page, scroll down and choose “publications in chronological order”“publications in chronological order”
Two documents released 2009: Two documents released 2009: RNP1 – “Initial Fraction Ideas” RNP1 – “Initial Fraction Ideas” RNP2 – “Fraction Operations and Initial Decimal Ideas”RNP2 – “Fraction Operations and Initial Decimal Ideas”
Correct NotationCorrect Notation
Examples of multiplicative language:Examples of multiplicative language: If I double the total parts then the shaded If I double the total parts then the shaded
parts double.parts double. If the total number of parts is multiplied by If the total number of parts is multiplied by
3, then the shaded parts are three times 3, then the shaded parts are three times as many too.as many too.
If the number of shaded parts is multiplied If the number of shaded parts is multiplied by 4 then total number of parts is by 4 then total number of parts is multiplied by 4.multiplied by 4.
Visual and Mental Models for Visual and Mental Models for FractionsFractions
FREE:FREE: Fractions Model I from Fractions Model I from
IlluminationsIlluminations Factor Tree from National Factor Tree from National
Library of Virtual ManipulativesLibrary of Virtual Manipulatives Chapter 6 Templates from Chapter 6 Templates from
MMPI (MMPI (www.michiganmathematics.orgwww.michiganmathematics.org))
Available for purchase from Available for purchase from Scholastic:Scholastic:
FASTTMathFASTTMath Fraction NationFraction Nation
Think - WriteThink - Write
Choose an algebra Choose an algebra focal point for your focal point for your grade. What are grade. What are the critical ideas – the critical ideas – the mathematics the mathematics that students must that students must understand – for understand – for that topic?that topic?
Read the Read the introductiointroduction.n.
Share with Share with a a neighbor.neighbor.
How many focal How many focal points are at your points are at your grade level?grade level?
The GLCE topics The GLCE topics are the same as are the same as the the Core and Core and Extended Extended DesignationsDesignations document from document from MDEMDE
National Math National Math Panel Panel Benchmarks are Benchmarks are checkpointscheckpoints
Benchmarks are Benchmarks are often a grade or often a grade or two past where two past where the topic is the topic is typically taughttypically taught
Find this chart Find this chart for each focal for each focal point at your point at your grade.grade.
The columns are The columns are the same as the the same as the 11” x 17” K-8 11” x 17” K-8 Alignment chartAlignment chart
Three Sections for Each Focal Three Sections for Each Focal PointPoint
From the 1-13-2010 DRAFT of the Common Core Standards
Initiative
Explore the Explore the Finding Focus Finding Focus DocumentDocument
There are two places in this document to find the There are two places in this document to find the list of GLCE topics for focal a point. Where are list of GLCE topics for focal a point. Where are those two places?those two places?
Find the “leftover” GLCEs for your grade. Compare Find the “leftover” GLCEs for your grade. Compare the chart to the the chart to the 11” x 17”11” x 17” K-8 Alignment. K-8 Alignment.
Find the fractions focal point for your grade. What Find the fractions focal point for your grade. What number is it?number is it?
How many “Extended Core” expectations are How many “Extended Core” expectations are related to the fraction focal point?related to the fraction focal point?
Choose any focal point at your grade. Compare Choose any focal point at your grade. Compare the GLCEs for the focal point to the DRAFT CCSI the GLCEs for the focal point to the DRAFT CCSI standards.standards.
The Big Ideas are The Big Ideas are NOTNOT
Topics for Topics for instructional instructional planningplanning
GLCEs for GLCEs for assessing studentsassessing students
Big Mathematical Big Mathematical Ideas and UnderstandingsIdeas and Understandings
The Big Ideas AREThe Big Ideas ARE The mathematics The mathematics
YOU should keep in YOU should keep in mind as you plan mind as you plan instructioninstruction
Critical ideas that Critical ideas that are true at all are true at all grade levelsgrade levels
SILENT ReadingSILENT Reading
Find the “Big Mathematical Ideas and Find the “Big Mathematical Ideas and Understandings” for the algebra focal Understandings” for the algebra focal point you brainstormed about earlierpoint you brainstormed about earlier
Read, re-read, highlight, and take Read, re-read, highlight, and take notesnotes
Add to the list you brainstormed of Add to the list you brainstormed of critical ideas for your gradecritical ideas for your grade
BreakBreak
Big IdeasBig Ideas
With your grade level, discussWith your grade level, discuss What stood out to you from the Big What stood out to you from the Big
Ideas?Ideas? What questions do you still have?What questions do you still have? Revise your notes.Revise your notes.
Children’s Difficulties in Children’s Difficulties in Beginning AlgebraBeginning Algebra
Using Variables:Using Variables: Do #1, 3, 4, 6, 7, 9, Do #1, 3, 4, 6, 7, 9,
1212 What What
misconceptions so misconceptions so students have?students have?
Student Misconceptions from Student Misconceptions from “Using Variables”“Using Variables”
l
Students struggle with . . .Students struggle with . . .
The focus of algebraic activity and The focus of algebraic activity and the nature of “answers”the nature of “answers”
The use of notation and convention The use of notation and convention in algebrain algebra
The meaning of letters and variablesThe meaning of letters and variables The kinds of relationships and The kinds of relationships and
methods used in arithmeticmethods used in arithmetic
The Nature of “Answer”The Nature of “Answer”
14-year-old Wendy14-year-old Wendy
16-year-old Marie16-year-old Marie
14-year-old Michael14-year-old Michael
Student Misconceptions from Student Misconceptions from “Using Variables”“Using Variables”
14x
Notation and Convention in Notation and Convention in AlgebraAlgebra
15-year-old Wayne15-year-old Wayne
ImplicationsImplications
““2+3” means “add 3 to 2” but it also means 2+3” means “add 3 to 2” but it also means “the number that is 3 more than 2”“the number that is 3 more than 2”
Students think that 3n means 3+n. Consider Students think that 3n means 3+n. Consider delaying conjoined terms (3delaying conjoined terms (3nn) and use the full ) and use the full product (3 x product (3 x nn).).
The “2 apples plus 5 bananas” approach to The “2 apples plus 5 bananas” approach to 22aa + 5 + 5bb may not be helpful. Students use may not be helpful. Students use this to justify that 2this to justify that 2aa + 5 + 5bb = 7 = 7abab because 2 because 2 apples plus 5 bananas is 7 apples-and-apples plus 5 bananas is 7 apples-and-bananas.bananas.
Letters in Algebra with 15-year-old Letters in Algebra with 15-year-old PeterPeter
And what does the And what does the yy mean, in a question like that mean, in a question like that [[Add 3 to 5yAdd 3 to 5y]? Does it mean anything, does it ]? Does it mean anything, does it stand for anything, or is it just a letter, or what?stand for anything, or is it just a letter, or what?
15-year-old Trevor and the Notion of 15-year-old Trevor and the Notion of “Variable”“Variable”
So What do We Do to Help So What do We Do to Help Students?Students?
Four activities:Four activities: Variables Representing Variables Representing
Counting Numbers Counting Numbers (H30)(H30)
Variable Expressions Variable Expressions and Relationships (H31-and Relationships (H31-H33)H33)
Variables as Pattern Variables as Pattern Generalizers (H34-H37)Generalizers (H34-H37)
Classifying Uses of Classifying Uses of Variables (H-38)Variables (H-38)
At your tables:At your tables: Each person do Each person do
problems from a problems from a different activitydifferent activity
Discuss the pros and Discuss the pros and cons of each and what cons of each and what students would learn students would learn from the activityfrom the activity
Math VocabularyMath Vocabulary
Targeted Vocabulary with Focal Targeted Vocabulary with Focal Points on Page 5Points on Page 5
Suggested Vocabulary at end of Suggested Vocabulary at end of documentdocument
How would you describe each list? How would you describe each list?
How would you use each list?How would you use each list?
Importance of Importance of Explicit Vocabulary InstructionExplicit Vocabulary Instruction
Find “Targeted Vocabulary” on Page 5
VariableVariable
var·i·able var·i·able ver-ē-ə-bəlver-ē-ə-bəl
Merriam-Webster:Merriam-Webster: able or apt to able or apt to vary; subject to variation or changes vary; subject to variation or changes
What does it mean to “vary?” What What does it mean to “vary?” What kinds of things vary?kinds of things vary?
1.1. Write the termWrite the term
2.2. Outline the termOutline the term-- category or category or
synonymsynonym details details
3.3. Add an exampleAdd an examplemath sentences math sentences
and personal and personal connections also connections also workwork
variable: A letter or other symbol that represents a number or other mathematical thing. If it represents a number, then it is also called a numerical variable. In the equation 2x + y = 9, x and y are numerical variables.
variable- symbol
represents a number or other mathematical thing
- numerical variable
variable that represents a number
2x + y = 9
variables
not a variable
variable: A letter or other symbol that represents a number or other mathematical thing. If it represents a number, then it is also called a numerical variable. In the equation 2x + y = 9, x and y are numerical variables.
variable- representation
letter or symbol
of a number or other mathematical thing
- numerical variable
variable that represents a number
2x + y = 9
variables
not a variable
When do variables vary? When do variables vary? When do they not?When do they not?
ProportionalProportional
pro·por·tion·alpro·por·tion·al prə-poprə-po ̇̇r-shə-nəlr-shə-nəl
In what contexts have you heard the In what contexts have you heard the word?word? Comes from: Comes from: propro "for" + "for" + *partio*partio
"division," related to "division," related to parspars (part) (part)
http://www.etymonline.com/http://www.etymonline.com/
proportional: having the same or constant ratio
proportional
Any strong
connection
•Examples and non-examples
•Multiple examples
Alternate Ideas:
- Story problem with matching equation
- Dictionary definition sentence
My weight should be proportional to my height.
y = 3x y = 3x + 2
Two mixes are proportional if they have the same ratio. For example, a bag with 3 blue and 4 red candies is proportional to a bag with 6 blue and 8 red candies.
SlopeSlope
What words relate to slope?What words relate to slope? How would you introduce the term in How would you introduce the term in
a way that activates activate a way that activates activate students’ prior knowledge?students’ prior knowledge?
Explicit Vocabulary Instruction Explicit Vocabulary Instruction – Word Banks– Word Banks
Students see relationships between Students see relationships between wordswords
Students practice using math Students practice using math languagelanguage
Create a word Create a word bank for slopebank for slope
Share at your Share at your tabletable
Textbook AnalysisTextbook Analysis
How does your textbook present new How does your textbook present new terms?terms?
Does the teacher’s guide include any Does the teacher’s guide include any strategies for explicit vocabulary strategies for explicit vocabulary instruction?instruction?
LunchLunch
Finding Focus Finding Focus DocumentDocument– – Instructional Implications Instructional Implications
SectionSectionIn grade-level groupsIn grade-level groups Each person reads the “Instructional Each person reads the “Instructional
Implications” for a different focal pointImplications” for a different focal point Share the highlights with your groupShare the highlights with your group
Something newSomething new Something not newSomething not new Something you question or wonder aboutSomething you question or wonder about
AssessmentAssessment
Think-Pair-Share: Think-Pair-Share: In what ways do you assess your In what ways do you assess your
students’ learning?students’ learning? What is the purpose of each method?What is the purpose of each method?
3.3
Different assessments serve different purposesDifferent assessments serve different purposes
AssessmentAssessment PurposePurpose ExampleExample
CBM Measure CBM Measure (universal screening)(universal screening)
Identify students at riskIdentify students at riskEvaluate core instructionEvaluate core instructionMonitor overall progressMonitor overall progress
Quantity DiscriminationQuantity DiscriminationMixed ComputationMixed Computation
Formative classroom Formative classroom assessmentassessment
Inform instructionInform instructionProvide student feedbackProvide student feedback
Teacher observationTeacher observationExit questionsExit questions
Summative classroom Summative classroom assessmentassessment
Evaluate studentsEvaluate studentsEvaluate core instructionEvaluate core instruction
Quizzes and testsQuizzes and testsProjectsProjects
District assessmentsDistrict assessments Evaluate studentsEvaluate studentsEvaluate programsEvaluate programs
Common “short-cycle” Common “short-cycle” assessmentsassessmentsCommon course examsCommon course exams
State assessmentsState assessments Evaluate studentsEvaluate studentsEvaluate schoolsEvaluate schools
MEAPMEAPMMEMME
Single-skill or multi-Single-skill or multi-skill assessments skill assessments (mastery (mastery measurement)measurement)
Identify strengths and Identify strengths and weaknessesweaknessesGroup students by needGroup students by needMonitor specific progressMonitor specific progress
MMPI diagnostic MMPI diagnostic assessmentsassessmentsDistrict assessmentsDistrict assessmentsClassroom assessmentsClassroom assessments
CBM v. Mastery CBM v. Mastery MeasurementMeasurement
Curriculum-based Curriculum-based measurement (CBM):measurement (CBM):
Measures general progressMeasures general progress Brief, timed “probes”Brief, timed “probes” Selected skills Selected skills
representative of the representative of the whole year’s curriculumwhole year’s curriculum
Less useful for diagnosing Less useful for diagnosing specific needsspecific needs
Mastery measurement:Mastery measurement: Determines specific Determines specific
needs (diagnostic pre-needs (diagnostic pre-assessment)assessment)
Tracks mastery of each Tracks mastery of each skill or standardskill or standard District short-cycle District short-cycle
assessmentsassessments NWEA also an exampleNWEA also an example
Usually not timedUsually not timed Does not assess Does not assess
overall “health”overall “health”
Math Measures Used for Math Measures Used for Universal Screening in SAMUniversal Screening in SAM
MeasureMeasure SourceSource TimeTime
Tests of Early Tests of Early Numeracy (TEN)Numeracy (TEN)Grades K-1; 4 measuresGrades K-1; 4 measures
AIMSwebAIMSweb 1 min. 1 min. each;each;
individually individually administeradministereded
Mixed Mixed ComputationComputation Grades Grades 1-61-6
AIMSwebAIMSweb 2 – 4 min.;2 – 4 min.;
paper/paper/pencilpencil
Algebra MeasuresAlgebra Measures Grade 6 – 8 and Algebra Grade 6 – 8 and Algebra 11
AAIMS – AAIMS – Iowa Iowa State State UniversityUniversity
5 – 7 min.;5 – 7 min.;
paper/paper/pencilpencil
Progress MonitoringProgress Monitoring
Algebra Algebra Basic Skills Basic Skills Grades 6-7Grades 6-7
Solving basic fact Solving basic fact equationsequations
Applying the Applying the distributive distributive propertyproperty
Working with Working with integersintegers
Combining like Combining like termsterms
Applying Applying proportional proportional reasoningreasoning
Algebra FoundationsAlgebra FoundationsGrades 7, 8, and Algebra IGrades 7, 8, and Algebra I
Writing and evaluating variables and Writing and evaluating variables and expressionsexpressions
Computing expressions (integers, Computing expressions (integers, exponents, and order of operations)exponents, and order of operations)
Graphing expressions and linear equationsGraphing expressions and linear equations Solving 1-step equations and simplifying Solving 1-step equations and simplifying
expressionsexpressions Identifying and extending patterns in data Identifying and extending patterns in data
tablestables
Content Analysis (Multiple Content Analysis (Multiple Choice)Choice)
Grade 8 and Algebra IGrade 8 and Algebra I Samples first 2/3 of Algebra I text:Samples first 2/3 of Algebra I text:
Writing equations of a line (slope-Writing equations of a line (slope-intercept and point-slope)intercept and point-slope)
Graphing inequalities on a number lineGraphing inequalities on a number line Simplifying expressionsSimplifying expressions Solving single-variable equationsSolving single-variable equations Evaluating expressionsEvaluating expressions Solving two-variable systemsSolving two-variable systems
Kuta SoftwareKuta Softwarehttp://kutasoftware.comhttp://kutasoftware.com
Worksheet Worksheet generator:generator:
Infinite Algebra 2Infinite Algebra 2 Infinite Algebra 1Infinite Algebra 1 Infinite Pre-AlgebraInfinite Pre-Algebra
Track skill mastery for each studentTrack skill mastery for each student Track class mastery for each skillTrack class mastery for each skill State assessment data State assessment data Demographic information and grades from Demographic information and grades from
an SIS an SIS District-administered tests District-administered tests Daily classroom assessments given by Daily classroom assessments given by
teachersteachers
Supporting InstructionSupporting Instruction
Conceptual Conceptual understanding and understanding and procedural fluencyprocedural fluency
Materials and Materials and resourcesresources
Scheduling optionsScheduling options Educator’s round Educator’s round
tabletable
Conceptual Understanding and Conceptual Understanding and Procedural FluencyProcedural Fluency
Developing computational skill and developing Developing computational skill and developing understanding are often seen as competing understanding are often seen as competing
for attention in school mathematics. But for attention in school mathematics. But pitting skill against understanding creates a pitting skill against understanding creates a false dichotomy. Understanding makes it false dichotomy. Understanding makes it
easier to learn skills, while learning easier to learn skills, while learning procedures can strengthen and develop procedures can strengthen and develop
mathematical understanding.mathematical understanding.
- Kilpatrick and Swafford, 2002,- Kilpatrick and Swafford, 2002, p. 13p. 13
Understanding Before Speed Understanding Before Speed Developing Procedural FluencyDeveloping Procedural Fluency
Guidelines to Achieving Fluency:Guidelines to Achieving Fluency:
1.1. Essential core skills onlyEssential core skills only2.2. Concepts and strategies firstConcepts and strategies first3.3. Distributed practiceDistributed practice4.4. Research-based strategiesResearch-based strategies
• softwaresoftware• increasing ratio reviewincreasing ratio review
5.5. Develop relationships and strategic thinkingDevelop relationships and strategic thinking• 8 = 2 x 48 = 2 x 4
- Sarama and Clements, pp139-140- Sarama and Clements, pp139-140
Skills vs. ProcessesSkills vs. Processes
Both produce some form of productBoth produce some form of product Both require multiple opportunities for Both require multiple opportunities for
review and practice over timereview and practice over time A “process” is more complex A “process” is more complex Step-by-step procedures and Step-by-step procedures and
“processes” may need even more “processes” may need even more distributed practicedistributed practice
PLEASE try to build distributed practice PLEASE try to build distributed practice into you annual planinto you annual plan
Materials and ResourcesMaterials and Resources
Getting the most from Getting the most from your textbook seriesyour textbook series
For remediation, consider resources For remediation, consider resources from a previous gradefrom a previous grade
Ancillary materials contain extension, Ancillary materials contain extension, expansion, or intervention lessons expansion, or intervention lessons useful for remediation or enrichmentuseful for remediation or enrichment
Technology OptionsTechnology Options
Simulations.Simulations. Distance Distance LearningLearning
Web Based Web Based InstructionInstruction
Word Word processing, data processing, data base, base, PowerpointPowerpoint
scanners, MP3scanners, MP3
Interactive A.P.Interactive A.P.
Interactive Interactive within HISD, within HISD, Virtual H.S.Virtual H.S.
PLATO,PLATO,
Study IslandStudy Island
Education 2020Education 2020
Microcassettes- Microcassettes- Mp3 digital Mp3 digital recordersrecorders
Textbook Textbook WebsitesWebsites
Types of Instructional Types of Instructional SoftwareSoftware
TutorialsTutorials Drill and practiceDrill and practice SimulationsSimulations Problem-solving Problem-solving DiscoveryDiscovery GamesGames
8.6
Computer-Aided Instruction Computer-Aided Instruction Research:Research:
Individualized instructionIndividualized instruction Timely feedbackTimely feedback
Programs:Programs: No clear evidenceNo clear evidence Odyssey from Compass LearningOdyssey from Compass Learning
http://www.compasslearningodyssey.com/http://www.compasslearningodyssey.com/ “Solutions” “Solutions” Cognitive Tutor from Carnegie LearningCognitive Tutor from Carnegie Learning
http://www.carnegielearning.com/software_features.cfmhttp://www.carnegielearning.com/software_features.cfm
Illuminations: Circle ToolIlluminations: Circle Tool
NCTM E-ExamplesNCTM E-Examples
www.nctm.orgwww.nctm.org Standards and Standards and Focal Points Focal Points under Principles under Principles and Standards, and Standards, choose “e-choose “e-Examples”Examples”
National Library of Virtual National Library of Virtual ManipulativesManipulatives
www.nlvm.usu.eduwww.nlvm.usu.edu
Algebra Balance Scales – Algebra Balance Scales – NegativesNegativesfrom NLVMfrom NLVM
Project InteractiveProject Interactive
http://www.shodor.org/interactivate/http://www.shodor.org/interactivate/
Surface Area and Volume Surface Area and Volume from Project Interactivefrom Project Interactive
KeyMath3:KeyMath3:An Assessment-Intervention An Assessment-Intervention
SystemSystem Concepts, skills, and problem solving Concepts, skills, and problem solving K-5 content (remediation through Grade 8)K-5 content (remediation through Grade 8)
““Essential Resources” is the intervention Essential Resources” is the intervention componentcomponent Stand-alone or with the assessment componentStand-alone or with the assessment component Scripted lessons for small-group interventionScripted lessons for small-group intervention Lessons 30-40 minutes eachLessons 30-40 minutes each
SchedulingSchedulingOptionsOptions
Block Scheduling (1 ½ hour math sections)Block Scheduling (1 ½ hour math sections) Common grade schedules to allow sharing Common grade schedules to allow sharing
between classroomsbetween classrooms Extra helpers in the classroom (co-Extra helpers in the classroom (co-
teachers, aides, special education, teachers, aides, special education, Chapter)Chapter)
Before and after school, Saturday schoolBefore and after school, Saturday school During “specials” timeDuring “specials” time Lunch buddiesLunch buddies
Educators’ Round Table Educators’ Round Table OptionsOptions
Meet with colleagues Meet with colleagues from your districtfrom your district
Exchange ideas with Exchange ideas with teachers from other teachers from other districtsdistricts
Ask specific questions Ask specific questions of the presentersof the presenters
Explore materials Explore materials Illuminations bindersIlluminations binders
BreakBreak
ReflectionReflection
What ideas did you learn from the What ideas did you learn from the educators’ round table?educators’ round table?
What questions do you still have?What questions do you still have? What will you do next?What will you do next?
““Scaling Away” from Scaling Away” from IlluminationsIlluminations
Prerequisite: Prerequisite: Compute surface area and volume.Compute surface area and volume.
Sources for prerequisite skills:Sources for prerequisite skills: Cubes Applet from IlluminationsCubes Applet from Illuminations Surface Area and Volume – Project Surface Area and Volume – Project
InteractiveInteractive
Scale FactorScale Factor
Scaling AwayScaling Away
1.1. Choose an object Choose an object and describe its and describe its shape.shape.
2.2. Answer Question Answer Question 1 if the scale 1 if the scale factor was 8, then factor was 8, then STOP.STOP.
DiscussionDiscussion
What do you think will happen to the What do you think will happen to the volume when you enlarge a common volume when you enlarge a common object by a scale factor of 8?object by a scale factor of 8?
What do you think will happen to the What do you think will happen to the surface area?surface area?
Examine the Original ObjectExamine the Original Object
Answer Questions 2, 3, and 4. Answer Questions 2, 3, and 4.
Put on your Teacher Hat:Put on your Teacher Hat:Measurement IssuesMeasurement Issues
Surface Area Surface Area OverheadOverhead
Computing with Computing with decimals in metric decimals in metric v. fractions in U.S. v. fractions in U.S. customarycustomary
Which is easier for Which is easier for students – volume students – volume or surface area?or surface area?
Be the Student AgainBe the Student Again
Choose a scale factor Choose a scale factor from 3 to 7from 3 to 7
Find the new Find the new dimensions (Question dimensions (Question 5)5)
Answer Questions 6-7. Answer Questions 6-7. Be sure to calculate Be sure to calculate your answers using your answers using the dimensions you the dimensions you wrote for Question 5!wrote for Question 5!
Compare the RatiosCompare the Ratios
Answer Questions 8-9. Answer Questions 8-9.
You’ve found 3 ratios: You’ve found 3 ratios: the scale factor for the the scale factor for the dimensions, the ratio dimensions, the ratio of the two volumes, of the two volumes, and the ratio of the and the ratio of the two surface areastwo surface areas
Which (if any) of these Which (if any) of these ratios are the same?ratios are the same?
Why Did the Ratios Work this Why Did the Ratios Work this Way?Way?
1.1. Find someone with the same scale Find someone with the same scale factor as you. Did you have the same factor as you. Did you have the same ratios for volume and surface area?ratios for volume and surface area?
2.2. What is the relationship between the What is the relationship between the three ratios? Make a prediction for a three ratios? Make a prediction for a different scale factor between 3 and 7.different scale factor between 3 and 7.
3.3. Find someone with that scale factor Find someone with that scale factor and compare to your prediction.and compare to your prediction.
Algebraic JustificationAlgebraic Justification
For a prism with For a prism with dimensions dimensions ll, , ww, , hh::
Original volume: Original volume: lwhlwh
New volume (scale New volume (scale factor factor nn): ):
((nlnl)()(nwnw)()(nhnh) = n) = n33((lwhlwh))
Teacher Talk:Teacher Talk: Would you show Would you show
this algebra to this algebra to students? When?students? When?
Would you start Would you start with surface area with surface area or volume?or volume?
SummarySummary
When the dimensions of a figure are When the dimensions of a figure are enlarged by a scale factor, the surface enlarged by a scale factor, the surface area and volume also increase.area and volume also increase.
The surface area and volume do not The surface area and volume do not increase by the original scale factor.increase by the original scale factor.
The surface area and volume do not The surface area and volume do not increase by the same factor as each increase by the same factor as each other.other.
Scale Factor of 8Scale Factor of 8
If you had used a scale factor of 8, by If you had used a scale factor of 8, by what factor would the surface area what factor would the surface area have increased? By what factor have increased? By what factor would the volume have increased? would the volume have increased? How do you know? Write your How do you know? Write your answer to Question 11.answer to Question 11.
ClosureClosure
Compare your answer to Question 1 to Compare your answer to Question 1 to your answer to Question 8. Was your your answer to Question 8. Was your hypothesis correct? Why or why not? hypothesis correct? Why or why not?
Explain what you have discovered about Explain what you have discovered about multiplying a side length by a scale factor. multiplying a side length by a scale factor. What happens to the surface area? What What happens to the surface area? What happens to the volume? Write your answer happens to the volume? Write your answer for Question 10.for Question 10.
Side, Length, Volume, and Side, Length, Volume, and Surface Area of Similar Solids (NCTM)Surface Area of Similar Solids (NCTM)
http://standards.nctm.org/document/http://standards.nctm.org/document/eexamples/chap6/6.3/part2.htmeexamples/chap6/6.3/part2.htm
E-example 6.3 from NCTM Principals E-example 6.3 from NCTM Principals and Standardsand Standards
Crossing the Bridge to Formal Crossing the Bridge to Formal Proportional ReasoningProportional Reasoning
Mathematics Teaching in the Middle Mathematics Teaching in the Middle SchoolSchool, April, 2003, April, 2003
Multiplicative Relationships Multiplicative Relationships “Within” and “Between” Ratios“Within” and “Between” Ratios
Scalar methodScalar method uses relationships uses relationships within ratioswithin ratios
Functional methodFunctional method uses relationships uses relationships between ratiosbetween ratios
Levels of Proportional Levels of Proportional ReasoningReasoning
Level 0 – additive strategies and random Level 0 – additive strategies and random methodsmethods
Level 1 – pictures, models, or Level 1 – pictures, models, or manipulativesmanipulatives
Level 2 – models along with numeric Level 2 – models along with numeric calculations; uses multiplication / divisioncalculations; uses multiplication / division
Level 3 – formal strategies such as finding Level 3 – formal strategies such as finding equal ratios or cross-multiplying to solve equal ratios or cross-multiplying to solve proportionsproportions
Are Students Ready to Move Are Students Ready to Move from Level 2 to Level 3?from Level 2 to Level 3?
Informal SolutionInformal Solution
Student used a Student used a picture combined picture combined with calculationswith calculations
Student sees the Student sees the multiplicative multiplicative relationships relationships between ratiosbetween ratios
Formal SolutionFormal Solution
Student set up a Student set up a proportionproportion
Student used Student used multiplicative multiplicative relationships to relationships to find the missing find the missing valuevalue
Students Compare the Informal Students Compare the Informal and Formal Strategiesand Formal Strategies
““The methods look different since the first one uses The methods look different since the first one uses number sentences, but the other uses ratios. They number sentences, but the other uses ratios. They are similar because, in each, you are multiplying are similar because, in each, you are multiplying and dividing the same numbers.”and dividing the same numbers.”
““Both methods use the same lengths. Both use Both methods use the same lengths. Both use multiplication and division. It’s just that in one, you multiplication and division. It’s just that in one, you use a proportion.”use a proportion.”
““They are similar because, in the end, you end up They are similar because, in the end, you end up dividing 7 by 0.2 to get the scale factor.”dividing 7 by 0.2 to get the scale factor.”
““The proportion is more organized. The height is The proportion is more organized. The height is over the length, and you have an arrow going over over the length, and you have an arrow going over to the other height and length. In the other to the other height and length. In the other method, there’s a picture and number sentences all method, there’s a picture and number sentences all over the place.”over the place.”
SAM Math WikiSAM Math Wikiwww.sammath.wikispaces.comwww.sammath.wikispaces.com
ReflectionReflection
What is one idea What is one idea you will use in the you will use in the next week?next week?
What are 2-3 items What are 2-3 items you will share or you will share or discuss with your discuss with your colleagues?colleagues?
How will this How will this happen?happen?
Contact InformationContact Information
Jenni TrusockJenni Trusock989/269-6406989/269-6406 [email protected]@hisd.k12.mi.us
Kristen LegaultKristen Legault 989/269-6404989/269-6404 [email protected]@hisd.k12.mi.us
Craig WalterCraig Walter 989/269-6406989/269-6406 [email protected]@hisd.k12.mi.us
Joanne ZangJoanne Zang 989/269-6406989/269-6406 [email protected]@hisd.k12.mi.us
ReferencesReferences
Auman, Marueen and Valette, Debbie (2009) Auman, Marueen and Valette, Debbie (2009) Step Up to Step Up to Writing in MathWriting in Math. Longmont, CO: Sopris West Educational . Longmont, CO: Sopris West Educational Services.Services.
Booth, Lesley R. (1988). “Children’s Difficulties in Booth, Lesley R. (1988). “Children’s Difficulties in Beginning Algebra.” Beginning Algebra.” The Ideas of Algebra K-12The Ideas of Algebra K-12, 1988 , 1988 yearbook National Council of Teachers of Mathematicsyearbook National Council of Teachers of Mathematics
Chapin, Suzanne H. and Anderson, Nancy Canavan (2003). Chapin, Suzanne H. and Anderson, Nancy Canavan (2003). Crossing the Bridge to Formal Proportional Reasoning.” Crossing the Bridge to Formal Proportional Reasoning.” Mathematics Teaching in the Middle School Mathematics Teaching in the Middle School (8, 8). pp. 420-(8, 8). pp. 420-425.425.
Cramer, Kathleen; Wyberg, Terry; Leavitt, Seth. (2009) Cramer, Kathleen; Wyberg, Terry; Leavitt, Seth. (2009) Rational Number Project: Fraction Operations & Initial Rational Number Project: Fraction Operations & Initial Decimal IdeasDecimal Ideas. National Science Foundation. Pulled from . National Science Foundation. Pulled from http://cehd.umn.edu/rationalnumberproject/http://cehd.umn.edu/rationalnumberproject/ on 1/30/2010. on 1/30/2010.
ReferencesReferences
Kilpatrick, J. and Swafford, J. (2002)Kilpatrick, J. and Swafford, J. (2002) Helping Children Learn Helping Children Learn Mathematics. Mathematics. Washington, DC: National Academy Press. Washington, DC: National Academy Press.
Marzano, Robert J., Pickering, Debra J., and Pollock, Jane E. Marzano, Robert J., Pickering, Debra J., and Pollock, Jane E. (2001) (2001) Classroom Instruction that Works.Classroom Instruction that Works. Alexandria, VA: Alexandria, VA: Association for Curriculum and DevelopmentAssociation for Curriculum and Development
Sarama, Julie and Clements, Douglas H. (2009) Sarama, Julie and Clements, Douglas H. (2009) Early Early Childhood Mathematics and Education Research.Childhood Mathematics and Education Research. New York, New York, NY: Routledge.NY: Routledge.
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