Decatur City 6-8 Math October 31, 2014 JEANNE SIMPSON AMSTI MATH SPECIALIST.

Decatur City

6-8 MathOctober 31, 2014

JEANNE SIMPSON

AMSTI MATH SPECIALIST

Welcome

Name School Classes you teach What do your students struggle to learn?

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He who dares to teach must never

cease to learn.John Cotton Dana

Agenda

Major Work of the GradesStandards of Mathematical PracticeDoing Some Math

Major Work of the Grades

Major Work of Grade 6

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Progressions Documents

K–6 Geometry

6-8 Statistics and Probability

6–7 Ratios and Proportional Relationships

6–8 Expressions and Equations

6-8 Number System

These are the documents currently available. They are working on documents for the other domains (Functions, Geometry 7-8).

http://ime.math.arizona.edu/progressions/

Value of Learning Progressions/Trajectories to Teachers

Know what to expect about students’ preparation.

Manage more readily the range of preparation of students in your class.

Know what teachers in the next grade expect of your students.

Identify clusters of related concepts at grade level.

Provide clarity about the student thinking and discourse to focus on conceptual development.

Engage in rich uses of classroom assessment.

Standards for Mathematical

Practice

Standards for Mathematical Practice

Mathematically proficient students will:

SMP1 - Make sense of problems and persevere in solving them

SMP2 - Reason abstractly and quantitatively

SMP3 - Construct viable arguments and critique the reasoning of others

SMP4 - Model with mathematics

SMP5 - Use appropriate tools strategically

SMP6 - Attend to precision

SMP7 - Look for and make use of structure

SMP8 - Look for and express regularity in repeated reasoning

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Students: (I) Initial (IN) Intermediate (A) Advanced1a Make sense of

problems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns. Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

SMP Instructional Implementation Sequence

1.Think-Pair-Share (1, 3)

2.Showing thinking in classrooms (3, 6)

3.Questioning and wait time (1, 3)

4.Grouping and engaging problems (1, 2, 3, 4, 5, 8)

5.Using questions and prompts with groups (4, 7)

6.Allowing students to struggle (1, 4, 5, 6, 7, 8)

7.Encouraging reasoning (2, 6, 7, 8)

Students: (I) Initial (IN) Intermediate (A) Advanced1a Make sense of

problems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found. Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution. Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns.

Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

Grouping/Grouping/Engaging Engaging ProblemsProblems

Grouping/Engaging Grouping/Engaging ProblemsProblems


Pair-SharePair-ShareShowing Showing ThinkingThinking

Showing Showing ThinkingThinking

Questioning/Wait Questioning/Wait TimeTime

Questioning/Wait Questioning/Wait TimeTimeQuestioning/Wait Questioning/Wait TimeTime

Questions/Questions/Prompts for Prompts for

GroupsGroups

Questions/Prompts for Questions/Prompts for GroupsGroups

Pair-SharePair-Share

Grouping/Engaging ProblemsGrouping/Engaging Problems

Questioning/Wait Questioning/Wait TimeTime




Allowing StruggleAllowing Struggle




Showing Showing ThinkingThinkingEncourage Encourage

Reasoning Reasoning Grouping/Engaging Grouping/Engaging ProblemsProblems


Showing Showing ThinkingThinkingShowing Showing ThinkingThinking

Encourage Encourage Reasoning Reasoning



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Resources

Illustrative Mathematics

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Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards.

http://www.illustrativemathematics.org/

Chocolate Bar Sales

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Foxes and Rabbits Given below is a table that gives the population of foxes and rabbits in a national park over a 12 month period. Note that each value of t corresponds to the beginning of the month.

Who is the Better Batter?

P-23

Below is a table showing the number of hits and the number of times at bat for two Major League Baseball players during two different season:

a.For each season, find the players’ batting averages. Who has better batting average?

b.For the combined 1995 and 1996 seasons, find the players’ batting averages. Who has the better batting average?

c.Are the answers to (a) and (b) consistent? Explain.

Season Derek Jeter David Justice1995 12 hits in 48 at bats 104 hits in 411 at bats1996 183 hits in 582 at bats 45 hits in 140 at bats

Shrinking Catrina read that a woman over the age of 40 can lose approximately 0.06 centimeters of height per year.

a.Catrina’s aunt Nancy is 40 years old and is 5 feet 7 inches tall. Assuming her height decreases at this rate after the age of 40, about how tall will she be at age 65? (Remember that 1 inch = 2.54 centimeters.)

b.Catrina’s 90-year-old grandmother is 5 feet 1 inch tall. Assuming her grandmother’s height had also decreased at this rate, about how tall was she at age 40?

Gotham City Taxis

US Garbage, Version 1

US Garbage, Version 1

Mathematics Assessment Project

Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise:

Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving.

Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents.

Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM.

Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests.

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http://map.mathshell.org/

6th GradeProportional ReasoningLaws of ArithmeticEvaluating Statements About Number OperationsInterpreting Multiplication and DivisionA Measure of SlopeReal-Life EquationsUsing Coordinates to Interpret and Represent DataMean, Median, Mode, and RangeRepresenting Data Using Grouped Frequency Graphs and Box Plots

7th GradeProportion and Non-Proportion

SituationsUsing Positive and Negative

Numbers in ContextPossible Triangle ConstructionsSteps to Solving EquationsApplying Angle TheoremsProbability GamesStatements About Probability

8th GradeApplying Properties of ExponentsEstimating Length Using Scientific

NotationInterpreting Time-Distance GraphsClassifying Solutions to Systems of

EquationsSolving Linear Equations in One

VariableRepresenting and Combining

Transformations

Playing Catch-up 8.EE.5 - Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationship represented in different ways.

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Dan MeyerMath Class Needs a MakeoverThree Act Math TasksBlog

How MAD are You?(Mean Absolute

Deviation) Fist to Five…How much do you know about Mean Absolute Deviation?

◦ 0 = No Knowledge

◦ 5 = Master Knowledge

Create a distribution of nine data points on your number line that would

yield a mean of 5.

Card Sort

Which data set seems to differ the least from the mean?

Which data set seems to differ the most from the mean?

Put all of the data sets in order from “Differs Least” from the mean to “Differs Most” from the mean.

The mean in each set equals 5.

333 32

11 4 6

Find the distance (deviation) of each point from the mean.

Use the absolute value of each distance.

Find the mean of the absolute deviations.

How could we arrange the nine points in our data to decrease the MAD?

How could we arrange the nine points in our data to increase the MAD?

How MAD are you?

Solving Proportions

If two pounds of beans cost $5, how much will 15 pounds of beans cost?

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Solving Proportions

Solve

The traditional method of creating and solving proportions by using cross-multiplication is de-emphasized (in fact it is not mentioned in the CCSS) because it obscures the proportional relationship between quantities in a given problem situation.

Kanold, p. 94

If two pounds of beans cost $5, how much will 15 pounds of beans cost?

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Implications for Instruction

Proportional reasoning is complex and needs to be developed over a long period of time.

The study of ratios and proportions should not be a single unit but a unifying theme throughout the middle school curriculum.

Students need time to explore a variety of multiplicative situations, to coordinate both additive and relative perspectives, to experience unitizing, and to explore informally the nature of ratio in different problem contexts.

Instruction should begin with physical experiments and situations that can be visualized and modeled.

The cross-product rule should be delayed until students understand and are proficient with informal and quantitative methods for solving proportion problems.

1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throws at practice, how many free throws did Omar make?

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1. The ratio of free throws that Omar made to the ones he missed at practice yesterday was 7:3. If he attempted 90 free throw at practice, how many free throws did Omar make?

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made 7

missed 3

Attempted 90


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made 7

missed 3

Attempted 90

90 ÷ 10 = 9

9

9 9 9

9 9 9 9 9 9


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made 7

missed 3

Attempted 90

9 x 7 = 6390 ÷ 10 = 9

Omar made 63 free throws.

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9 9 9

9 9 9 9 9 9

2. At FDR High School, the ratio of seniors who attend college to those who do not is 5:2. If 98 seniors do not attend college, how many do?

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3. At Mesa Park High School, the ratio of students who have driver’s licenses to those who don’t is 8:3. If 144 students have driver’s licenses, how many students are enrolled at Mesa Park High School?

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4. Of the black and blue pens that Mrs. White has in a drawer in her desk, 18 are black. The ratio of black pens to blue pens is 2:3. When Mrs. White removes 3 blue pens, what is the new ratio of black pens to blue pens?

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http://www.risd.k12.nm.us/instruction/mathdrawingbook.cfm

Contact InformationJeanne Simpson

UAHuntsville AMSTI

[email protected]

[email protected]

Decatur City 6-8 Math October 31, 2014 JEANNE SIMPSON AMSTI MATH SPECIALIST.

Documents

mathematical symbols

solution representation

students preparation

gradesmajor work of

previous solution attempts

proficient students

students solutions

challenging problem