Download Bakersfield City School District Mathematics UNIT 4 Grade 8

Bakersfield City School District
Mathematics UNIT 4
Grade 8
Algebra
Bakersfield City School District
Equity ● Integrity
Caring ● Collaboration
Personal & Collective Accountability
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Unit 4 Learning Outcomes
Claim 1 Concepts and Procedures
Claim 2 Problem Solving
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
CONCEPTUAL CATEGORY-DOMAIN
CLUSTER
Algebra-Creating Equations.
Creating equations that describe
numbers or relationships. [Linear,
quadratic, and exponential (integer
inputs only); for A.CED.3 linear only]
SBAC Claims
Claim 3 Communicating Reasoning
Claim 4 Modeling and Data Analysis
Mathematical Practices
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
STANDARDS
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on
coordinate axes with labels and scales.
Statistics and ProbabilityInterpreting Categorical and
Quantitative Data. Summarize,
represent, and interpret data on a
single count or measurement
variable.
S.SID.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread
(interquartile range, standard deviation) of two or more different data sets.
Statistics and ProbabilityInterpreting Categorical and
Quantitative Data. Summarize,
represent, and interpret data on two
categorical and quantitative
variables. [Linear focus; discuss
general principle.]
S.SID.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the
context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and
trends in the data.
S.SID.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects
of extreme data points (outliers).
Indicates a modeling standard linking mathematics to everyday life, work, and decision-making (see CA Math Framework)
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Mathematical Performance Task Development
Purpose: At the end of the unit, students should be able to make connections with the learning of concepts and skills, problem solving, modeling, analysis,
communication and reasoning. Students should be able to perform at higher levels of complexity in their thinking and application of the math content.
Task Development
Mathematical tasks should…
• Integrate knowledge and skills across multiple claims and targets
• Measure depth of understanding, research skills, complex analysis
• Take age appropriate development into consideration
•
•
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Engage students in relevant and interesting topics
Have an authentic purpose and connected components
Accessible to all learners
Professional Learning Communities will…
Step 1: Consider the learning targets needed to be mastered throughout the unit.
Step 2: Consider the Depth of Knowledge or level of complexity that students will need to perform.
Performance Task Expectations
(DOK 3)
(DOK 4)
Strategic Thinking/
Extended Thinking/ Reasoning
Complex Reasoning
(DOK 1)
Recall and Reproduction
(DOK 2)
Skills and Concepts/
Basic Reasoning
• Recall of a fact, information or
procedure
• Recall or recognize fact
• Recall or recognize definition
• Recall or recognize term
• Recall and use a simple procedure
• Perform a simple algorithm.
• Follow a set procedure
• Apply a formula
• A one-step, well-defined, and straight
algorithm procedure.
• Perform a clearly defined series of
steps
• Identify
• Recognize
• Use appropriate tools
• Measure
• Students make some decisions as to
how to approach the problem
• Skill/Concept
• Basic Application of a skill or concept
• Classify
• Organize
• Estimate
• Make observations
• Collect and display data
• Compare data
• Imply more than one step
• Visualization Skills
• Probability Skills
• Explain purpose and use of
experimental procedures.
• Carry out experimental procedures
I can identify the coefficient, variable, and
term of an expression/equation.
Given a multi-step equation, I can solve
using inverse operations.
• Requires reasoning, planning using
evidence and a higher level of thinking
• Strategic Thinking
• Freedom to make choices
• Explain your thinking
• Make conjectures
• Cognitive demands are complex and
abstract
• Conjecture, plan, abstract, explain
• Justify
• Draw conclusions from observations
• Cite evidence and develop logical
arguments for concepts
• Explain phenomena in terms of
concepts
• Performance tasks
• Authentic writing
• Project-based assessment
• Complex, reasoning, planning,
developing and thinking
• Cognitive demands of the tasks are
high
• Work is very complex
• Students make connections within
the content area or among content
areas
• Select one approach among
alternatives
• Design and conduct experiments
• Relate findings to concepts and
phenomena
SAMPLES
I can write and solve an equation from a
real world-situation and explain my
thinking.
Given a real world situation, I can make
sense and plan for a situation that requires
multiple steps and high cognitive demands
in my creating and thinking.
To view full list of DOK descriptors, visit http://education.ky.gov/curriculum/docs/documents/cca_dok_support_808_mathematics.pdf
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Collaborative Team Planning Guide
SEGMENT 1: Rational Functions and Equations
What do I want my students to know and be able to do?
Standard
Learning Targets and Application of Mathematical Practices to be determined by Professional Learning Communities.
See last page of unit for relevant verbs and classroom implementation support.
Claim 1:Concepts and Procedures
Claim 2: Problem Solving 3:Communicating Reasoning 4:Modeling and Data Analysis
Learning Targets/Objectives (Skills/Concepts)
Application of Mathematical Practices (Behaviors/Actions)
A.CED.2
Skills/Concepts…Explaining a Procedure
Language Functions and Considerations
Skills/Concept . . .Making Sense/Perseverance
Language Function: Explanation—phrases or sentences to express the
rationale, reasons, causes, or relationships related to one or more
actions, ideas, events, or processes.
Language Function: Problem Solving—define and represent a
problem; determine a solution; words, phrases, or sentences to
express the meaning of a given word, phrase, or expression.
I can simplify a rational expression by first __________________. Then,
I can ______________.
One way of finding _________ is to use ________. Another way is
_______.
I can simplify a rational expression by first factoring the numerator and
denominator. Then, I can eliminate common factors in the numerator
and denominator to reduce to lowest terms.
One way of finding the quotient when dividing polynomials is to use
long division. Another way is factoring.
Curricular Connections:
Chapter 11
Lesson 1 Inverse Variation
Lesson 2 Rational Functions
Lesson 3 Simplifying Rational Expressions
Lesson 4 Multiplying and Dividing Rational Expressions
Lesson 5 Dividing Polynomials
Lesson 6 Adding and Subtracting Rational Expressions
Lesson 7 Mixed Expressions and Complex Fractions
Lesson 8 Rational Equations
Content Specific Vocabulary:
Inverse variation, product rule, rational function, excluded value, asymptote,
rational expression, least common multiple, least common denominator,
mixed expression, complex fraction, rational equation, extraneous solution,
work and rate problems
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How will we know if they have learned it?
Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom.
Grade level teams will develop and implement common formative assessment.
How will we respond when learning has not occurred?
Professional Learning Communities will develop and implement Response to Intervention.
How will we respond when learning has already occurred?
Professional Learning Communities will develop and implement Enrichment.
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Collaborative Team Assessment Planning Guide
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How will we know if they have learned it?
Identify and use inverse variations/graph inverse variations
Identify excluded values/identify and use asymptotes to graph rational functions
Identify values excluded from the domain of a rational expression/simplify rational expressions
Multiply, divide rational expressions
Divide a polynomial by a monomial/binomial
Add and subtract rational expressions with like/unlike denominators
Simplify mixed expressions/simplify complex fractions
Solve rational equations/use rational equations to solve problems
McGraw-Hill
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Collaborative Team Planning Guide
SEGMENT 2: Statistics and Probability
What do I want my students to know and be able to do?
Standard
Learning Targets and Application of Mathematical Practices to be determined by Professional Learning Communities.
See last page of unit for relevant verbs and classroom implementation support.
Claim 1:Concepts and Procedures
Claim 2: Problem Solving 3:Communicating Reasoning 4:Modeling and Data Analysis
Learning Targets/Objectives (Skills/Concepts)
Application of Mathematical Practices (Behaviors/Actions)
S.ID.2
S.ID.3
S.ID.5 (lab 12-7)
Skills/Concept . . . Describing
Language Functions and Considerations
Behaviors/Action…Modeling with Mathematics
Language Function: Describing—words, phrases, or sentences to
express or observe the attributes or properties of an object, action,
event, idea, or solution.
Language Function: Interpretation—Using symbols, phrases, and
When I observe ______, I can see ________.
A ____________ can be used to ______________.
When I observe the two box plots side-by side, I can see an obvious
difference in the medians and a greater interquartile range in the
second box plot.
A scatter plot can be used to display and interpret the data collected
by our class.
Curricular Connections:
Chapter 12
Lesson 1 Samples and Studies
Lesson 2 Statistics and Parameters
Lesson 3 Distributions of Data
Lesson 4 Comparing Sets of Data
Lesson 5 Simulation
Lesson 6 Permutations and Combinations
Lesson 7 Probability of Compound
Lesson 8 Probability Distributions
Content Specific Vocabulary:
sentences to express the intended meaning of information and make
connections between various data displays variety of situations.
population, sample, simple random sample, systematic sample, self-selected
sample, convenience sample, stratified sample, bias, survey, observational study,
experiment, statistical inference, statistic, parameter, mean absolute
deviation(MAD), standard deviation, variance, distribution, negatively skewed
distribution, symmetric distribution, positively skewed distribution, linear
transformation, theoretical probability, experimental probability, relative frequency,
simulation, probability model, permutation, factorial, combination, compound
event, joint probability, independent events, dependent events, mutually exclusive
events, two-way frequency tables, joint frequencies, marginal frequencies, relative
frequency, conditional relative frequency, random variable, discrete random
variable, probability distribution, probability graph, expected value, normal
distribution, normal curve, z-score
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How will we know if they have learned it?
Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom.
Grade level teams will develop and implement common formative assessment.
How will we respond when learning has not occurred?
Professional Learning Communities will develop and implement Response to Intervention.
How will we respond when learning has already occurred?
Professional Learning Communities will develop and implement Enrichment.
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Collaborative Team Assessment Planning Guide
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How will we know if they have learned it?
Classify and analyze samples/studies
Identify sample statistics and population parameters/Analyze data sets using statistics
Describe the shape of a distribution/Use the shapes of distributions to select appropriate statistics
Determine the effect that transformations of data have on measures of central tendency and variation/compare data using measures of central
tendency and variation
Calculate experimental probabilities/design simulations and summarize data from simulations
Use permutations/combinations
Find probabilities of independent and dependent events/find probabilities of mutually exclusive events
Find probabilities by using random variables/find the expected value of a probability distribution
McGraw-Hill
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Phase
Claims
Addressed
MP’s
1-Inquiry
3
All 8
*3,*6
(Whole Class)
Communicating
and Reasoning
Lesson Delivery Model
Instructional Lesson Sequence
Focus- Coherence- Rigor
Step 1: Introduce Essential Question: Provided in Curriculum or adjust as determined by PLC
Step 2: Present a real world math problem that is connected to the essential question
(i.e. K-5 Problem of the day, 6-8 Real World Link) :
• Teacher presents the problem
Step 3: Deconstruct the math problem and state the operations needed to solve the problem(s):
• Teacher and students share thinking
• Teacher asks students what is their understanding of the problem
• Teacher asks students the skills needed to get to the solution
• ___________________________________________________
Engage in Collaborative Conversations and use Language Functions (Develop within PLC and embed within steps):
• A way of thinking about solving this problem is ____________.
• The most important thing to remember in this problem is ___________.
• I believe the question is asking us to __________.
• The learning objective for today is_____________.
• ___________________________________________________
2- Modeling
1
(Teacher Directed)
Skills and
Concepts
All 8
Step 1: State Learning Target(s)/Objective(s):
• I can…
• I can…
• I can…
• ___________________________________________________
Step 2: Introduce and Contextualize Vocabulary (Throughout the remaining phases):
Step 3: Model how and why the math works using Skills and Concepts:
Engage in Collaborative Conversations and use Language Functions (Develop within PLC and embed within steps):
•
•
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In order to _______ you need to _____________.
_______ and _______ are accomplished by____________.
This first step in _____ is to _______, followed by ______.
______________________________________________
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3- Shared Learning
(Student
Interactionpartners, groups)
2&4
All 8
*1,*5,*7,
*8
Problem Solving,
Modeling, Data
Analysis
Step 1: Revisit/ introduce new real world math problem(s) that are connected to the essential question:
Step 2:
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Provide students with opportunities to engage in mathematical discourse:
Solving real-world problems
Describing and illustrating their understanding (speaking and writing)
Justifying and explaining their reasoning in solving problems (speaking and writing)
Asking questions to generate mathematical thinking
_________________________________________________
Engage in Collaborative Conversations and use Language Functions (Develop within PLC and embed within steps):
• Based on _______, I determined that_____________.
• Given that _________we can deduce that ___________.
• I agree/disagree with ___ that ____________.
• Given that…we can deduce that ___________.
• I agree/disagree with ___ that _____________.
•
4- Practice and
Apply
(Student Practice)
1,2,3,4
Skills &
Concepts,
Problem Solving,
Modeling & Data
Analysis,
Communicate
Reasoning
All 8
Provide students time to work individually or in pairs in order to assess mastery of the skills and concepts presented
to them:
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Teacher facilitates learning with specific immediate feedback
Teacher provides differentiation if needed
Teacher provides students with the opportunity to write, justify, and explain their reasoning (math journal)
Students practice and apply skills using curriculum/resources
Students apply problem solving strategies
____________________________________________________
*particular emphasis
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Resources
Grade Level Expectations (Progressions)
th
7 Grade
Geometry
Claim 1
Claim 2
Claim 3
Claim 4
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Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Use properties of operations to generate equivalent expressions
Solve real-life and mathematical problems using numerical and algebraic expressions and equations
Use random sampling to draw inferences about a population
Draw informal comparative inferences about two populations
Investigate chance processes and develop, use, and evaluate probability models
Congruence Experiment with transformations in the plane, Understand congruence in terms of rigid motions, Prove geometric theorems, Make geometric
constructions
• Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations, Prove theorems involving similarity, Define
trigonometric ratios and solve problems involving right triangles, Apply trigonometry to general triangles
• Circles Understand and apply theorems about circles, Find arc lengths and areas of sectors of circles
• Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section, Use coordinates to
prove simple geometric theorems algebraically
• Geometric Measurement and Dimension Explain volume formulas and use them to solve problems, Visualize relationships between two-dimensional and
three-dimensional objects
• Modeling with Geometry Apply geometric concepts in modeling situations
SBAC Claims
Concepts and Procedures
Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and
fluency
Problem Solving
Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of
knowledge and problem solving strategies.
Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze
Communicating Reasoning
Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of
others.
Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve,
Evaluate, Extend, Apply
Modeling and Data Analysis Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve
problems.
Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze
Implementation Support for Mathematical Practices
Classroom environment:
1. Students engage in “mathematical discourse” (collaborative conversations)
2. Students explaining, justifying, reasoning, and critiquing the work of others through oral or written responses
3. Students use problem solving strategies to solve real world and mathematical problems
4. Students persevere and find different solution pathways
5. Students attend to precision by calculating efficiently and accurately
http://bcsd.com/cipd/files/2011/11/Questions-to-Development.pdf
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Conceptual Categories and Domains
Number and Quantity (N) -The Real Number System (RN), Quantities (Q)
Algebra (A) -Seeing Structure in Expressions (SSE), Arithmetic with Polynomials and Rational Expressions (APR), Creating Equations (CED), Reasoning with Equations and
Inequalities (REI)
Functions (F) – Interpreting Functions (IF), Building Functions (BF), Linear, Quadratic, and Exponential Models (LE)
Statistics and Probability (S) –Interpreting Categorical and Quantitative Data (ID)
Fluency
CA Math Framework:
The word “fluent” is used in the standards to mean “reasonably fast and accurate” and the ability to use certain facts and procedures with enough facility that using them
does not slow down or derail the problem solver as he or she works on more complex problems. Procedural fluency requires skill in carrying out procedures flexibly,
accurately, efficiently and appropriately. Developing fluency in each grade can involve a mixture of just knowing some answers, knowing some answers from patterns, and
knowing some answers from the use of strategies.
In the standards for kindergarten through grade six there are individual content standards that set expectations for fluency in computation. Such standards are culminations
of progressions of learning, often spanning several grades, involving conceptual understanding (such as reasoning about quantities, the base-ten system, and properties of
operations), thoughtful practice, and extra support where necessary.
SBAC
Required Fluencies for Grades K-6
Grade Standard
Required Fluency
K
K.OA.5
Add/subtract within 5
1
1.OA.6
Add/subtract within 10
2
2.OA.2
2.NBT.5
Add/subtract within 20 (know
single-digit sums from memory)
Add/subtract within 100
3
3.OA.7
3.NBT.2
Multiply/divide within 100 (know singledigit products from memory)
Add/subtract within 1000
4
4.NBT.4
Add/subtract within 1,000,000
5
5.NBT.5
Multi-digit multiplication
6
6.NS.2,3
Multi-digit division
Multi-digit decimal operations
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