kindergarten common core math
DESCRIPTION
The content of this slide deck can be applied to all grade levels. Focuses on the Common Core Standards for Mathematical Practice and (1) unpacking the standards, (2) productive struggle, problem solving, and a deeper understanding of content, (3) prompts, checklists, and rubrics, and (4) problem solving resources. The presentation took place at a district professional development day. 10-13-14TRANSCRIPT
KINDERGARTEN MATH
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PUTTING THE “HOW” BEFORE THE “WHAT”
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There are 25 sheep and 5 dogs in a flock. How old is the Shepherd?
There are 25 sheep and 5 dogs in a flock. How old is the Shepherd?
Three out of four students will give a numerical answer to this problem.
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There are 25 sheep and 5 dogs in aflock. How old is the Shepherd?
25 because a shepherd has the word sheepin it so you have to take away the dogs
and you just get 25. !
I dont haveenough info. I cant
answer this question !
If he started out with 2they reproduced it would take a year(about) for each to be born. And thesame with the dogs. If he started
at the age of about 18
What else is there?
Unpacking the Standards For Practice “I am somewhat familiar with the Practice Standards.”
Productive Struggle, Problem Solving, an Understanding“I am starting to think about teaching with rigor.”
Lesson Resources (prompts, checklists, rubrics)“I have a few resources t0 use across many lessons.”
Everyday Math & Problem Solving Resources “I know where to find problem solving resources.”
Unpacking theStandards For Practice
Contentwhat
Contentwhat
Practicehow
Practicehow
1 2 3 4
5 6 7 8
Make sense of problems & persevere in solving them
Reason abstractly & quantitatively
Construct viable arguments & critique the reasoning of
others
Model with mathematics
Use appropriate tools
strategically
Attend to precision
Look for & make use of structure
Look for & express
regularity in repeated reasoning
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Activity
For each Practice Standard, write one student friendly “I can” statement that
clearly and concisely “summarizes” the standard.
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Leader’s Guide Visit go.solution-tree.com/commoncore to download this page.
Table 2.2: Mathematical Practices—Look-Fors as Classroom Indicators
Mathematical Practice Look-Fors: Classroom Indicators
Mathematical Practice 1: Make sense of problems, and persevere in solving them.
Students:tasks
Teacher: Provides adequate time with formative feedback for students to discuss problem pathways and solutions with peers
Mathematical Practice 2: Reason abstractly and quantitatively.
Students: Are able to contextualize or decontextualize problems
Teacher: Provides access to and uses appropriate representations (manipulative materials, drawings, or online renderings) of problems and asks questions focused on determining student reasoning
Mathematical Practice 3: Construct viable arguments, and critique the reasoning of others.
Students: Understand and use prior learning in constructing arguments
Teacher: Provides opportunities for students to listen to or read the conclusions and arguments of others—as students discuss approaches and solutions to problems, the teacher encourages them to provide arguments for why particular strategies work and to listen and respond to the reasoning of others and asks questions to prompt discussions.
Mathematical Practice 4: Model with mathematics.
Students: Analyze and model relationships mathematically (such as when using an expression or equation)
Teacher: Provides contexts for students to apply the mathematics learned
Mathematical Practice 5: Use appropriate tools strategically.
Students: Have access to and use instructional tools to deepen understanding (for example, manipulative materials, drawings, and technological tools)
Teacher: Provides and demonstrates appropriate tools (like manipulatives)
Mathematical Practice 6: Attend to precision.
Students: Recognize the need for precision in response to a problem and use appropriate mathematics vocabulary
Teacher: Emphasizes the importance of precise communication, including appropriate use of mathematical vocabulary, and emphasizes the importance of accuracy and efficiency in solutions to problems, including use of estimation and mental mathematics, when appropriate
Mathematical Practice 7: Look for and make use of structure.
Students: Are encouraged to look for patterns and structure (for example, when using properties and composing and decomposing numbers) within mathematics
Teacher: Provides time for students to discuss patterns and structures that emerge in a problem’s solution
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
Students: Reason about varied strategies and methods for solving problems and check for the reasonableness of their results
Teacher: Encourages students to look for and discuss regularity in their reasoning
Source: Adapted from Kanold, Briars, & Fennell, 2012.
R E PRO DUCI B LE
Common Core Mathematics in a PLC at WorkTM, Leader’s Guide Visit go.solution-tree.com/commoncore to download this page.
Table 2.2: Mathematical Practices—Look-Fors as Classroom Indicators
Mathematical Practice Look-Fors: Classroom Indicators
Mathematical Practice 1: Make sense of problems, and persevere in solving them.
Students:tasks
Teacher: Provides adequate time with formative feedback for students to discuss problem pathways and solutions with peers
Mathematical Practice 2: Reason abstractly and quantitatively.
Students: Are able to contextualize or decontextualize problems
Teacher: Provides access to and uses appropriate representations (manipulative materials, drawings, or online renderings) of problems and asks questions focused on determining student reasoning
Mathematical Practice 3: Construct viable arguments, and critique the reasoning of others.
Students: Understand and use prior learning in constructing arguments
Teacher: Provides opportunities for students to listen to or read the conclusions and arguments of others—as students discuss approaches and solutions to problems, the teacher encourages them to provide arguments for why particular strategies work and to listen and respond to the reasoning of others and asks questions to prompt discussions.
Mathematical Practice 4: Model with mathematics.
Students: Analyze and model relationships mathematically (such as when using an expression or equation)
Teacher: Provides contexts for students to apply the mathematics learned
Mathematical Practice 5: Use appropriate tools strategically.
Students: Have access to and use instructional tools to deepen understanding (for example, manipulative materials, drawings, and technological tools)
Teacher: Provides and demonstrates appropriate tools (like manipulatives)
Mathematical Practice 6: Attend to precision.
Students: Recognize the need for precision in response to a problem and use appropriate mathematics vocabulary
Teacher: Emphasizes the importance of precise communication, including appropriate use of mathematical vocabulary, and emphasizes the importance of accuracy and efficiency in solutions to problems, including use of estimation and mental mathematics, when appropriate
Mathematical Practice 7: Look for and make use of structure.
Students: Are encouraged to look for patterns and structure (for example, when using properties and composing and decomposing numbers) within mathematics
Teacher: Provides time for students to discuss patterns and structures that emerge in a problem’s solution
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
Students: Reason about varied strategies and methods for solving problems and check for the reasonableness of their results
Teacher: Encourages students to look for and discuss regularity in their reasoning
Source: Adapted from Kanold, Briars, & Fennell, 2012.
“A bad curriculum well taught is invariably a better experience for students than a good curriculum badly taught: pedagogy trumps curriculum. Or more precisely, pedagogy is curriculum, because what
matters is how things are taught, rather than what is taught.” - Wiliam
Khan Academy Does Angry Birds
How does the video represent the PA Core Standards for Mathematical Practice?
Productive Struggle, Problem Solving, an Understanding
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“It is hard to think of allowing - much less planning for - the children in your
classroom to struggle. Not showing them a solution when they are experiencing
difficulty seems almost counterintuitive. If our goal is relational understanding, however, the struggle is part of the
learning, and teaching becomes less about the teacher and more about what the children are doing and thinking.” - VDW
“Teaching mathematics through problem solving generally means that children solve problems to
learn new mathematics, not just to apply mathematics after is has been learned.”
- VDW
Doing Problems vs. Problem Solving
Tatyana’s mother is decorating a cake for Tatyana’s fifth birthday but she only has green and blue candles. If she wants to use exactly 5 candles on the cake, how
many green and blue candles could she use.
“Most, if not all, important mathematics concepts or procedures can best be
taught through problem solving.” - VDW
“I know that8 + 2 = 10 because…”
!
“I don’t know 8 + 2, but 8 + 1 = 9, and
then I can add on 1 more to get the
answer, which is 10.” !
“It would make sense to add 8 and 2
when…”
What is an understanding?
vs.
“An understanding can never be ‘covered’ if it is to be understood.”
- Wiggins & McTighe
!
• Give the answer and ask for the problem.
• Replace a number in a given problem with
a blank or a question mark.
• Offer two situations or examples and
ask for similarities and differences.
• Create a question in which children have
to make choices. - VDW
A variety of strategies you can use to
create open questions include the
following:
Strategy Standard Question Open Question
Replace a number in agiven problem with ablank or a questionmark.
Offer two situations or examples and ask for similarities or differences
Create a question so that children have to make choices.
23 + 68 = _____
Draw a triangle.
What number is10 more than 25?
?3 + 6? = _____
How are these triangles the same? Different?
A number is 10 greater than another number.
What could the number be?
• Measure a length
• Perform a specified or routine procedure
• Evaluate an expression
• Solve a one-step word problem
• Retrieve information from a table or graph
• Recall, identify, or make conversions between and among representations or numbers (fractions, decimals, and percents), or within and between customary and metric measures
• Locate numbers on a number line, or points on a coordinate grid
• Solves linear equations
• Represent math relationships in words, pictures, or symbols
• Compare and contrast figures
• Provide justifications for steps in a solution process
• Extend a pattern
• Retrieve information from a table, graph, or figure and use it solve a problem requiring multiple steps
• Translate between tables, graphs, words and symbolic notation
• Select a procedure according to criteria and perform it
multiple steps and multiple decision points
• Generalize a pattern
• Describe, compare, and contrast solution methods
• Formulate a mathematical model for a complex situation
• Provide mathematical justifications
• Solve a multiple- step problem, supported with a mathematical explanation that justifies the answer
• Formulate an original problem, given a situation
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• Design a mathematical model to inform and solve a practical or abstract situation
NOTE: Level 4 requires applying one approach among many to solve problems. Involves complex restructuring of data, establishing and evaluating criteria to solve problems.
Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.
(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)
Level 1 Recall
Level 2 Skills/Concepts
Level 3 Strategic Thinking
Level 4 Extended Thinking
Examples represent, but do not constitute all Level 1 mathematics performances:
• Recall or recognize a fact, definitions, or term
• Apply a well known algorithm
• Apply a formula
• Determine the area or perimeter of rectangles or triangles given a drawing and labels
• Identify a plane or three dimensional figure
Examples represent, but do not constitute all Level 2 mathematics performances:
• Classify plane and three dimensional figures
• Interpret information from a simple graph
• Use models to represent mathematical concepts
• Solve a routine problem requiring multiple steps, or the application of multiple concepts
• Compare figures or������VWDWHPHQWV
Examples represent, but do not constitute all Level 3 mathematics performances:
• Interpret information from a complex graph
• Explain thinking when more than one response is possible
• Make and/or justify conjectures
• Develop logical arguments for a concept
• Use concepts to solve problems
• Perform procedure with
Examples represent, but do not constitute all Level 4 mathematics performances:
• Relate mathematical concepts to other content areas
• Relate mathematical concepts to real-world applications in new situations
• Apply a mathematical model to illuminate a problem, situation
• ConduFt a project that specifies a problem, identifies solution paths,
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Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.
(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)
Level 1 Recall
Level 2 Skills/Concepts
Level 3 Strategic Thinking
Level 4 Extended Thinking
Examples represent, but do not constitute all Level 1 mathematics performances:
• Recall or recognize a fact, definitions, or term
• Apply a well known algorithm
• Apply a formula
• Determine the area or perimeter of rectangles or triangles given a drawing and labels
• Identify a plane or three dimensional figure
Examples represent, but do not constitute all Level 2 mathematics performances:
• Classify plane and three dimensional figures
• Interpret information from a simple graph
• Use models to represent mathematical concepts
• Solve a routine problem requiring multiple steps, or the application of multiple concepts
• Compare figures or������VWDWHPHQWV
Examples represent, but do not constitute all Level 3 mathematics performances:
• Interpret information from a complex graph
• Explain thinking when more than one response is possible
• Make and/or justify conjectures
• Develop logical arguments for a concept
• Use concepts to solve problems
• Perform procedure with
Examples represent, but do not constitute all Level 4 mathematics performances:
• Relate mathematical concepts to other content areas
• Relate mathematical concepts to real-world applications in new situations
• Apply a mathematical model to illuminate a problem, situation
• ConduFt a project that specifies a problem, identifies solution paths,
������VROYHV�DQG
• Measure a length
• Perform a specified or routine procedure
• Evaluate an expression
• Solve a one-step word problem
• Retrieve information from a table or graph
• Recall, identify, or make conversions between and among representations or numbers (fractions, decimals, and percents), or within and between customary and metric measures
• Locate numbers on a number line, or points on a coordinate grid
• Solves linear equations
• Represent math relationships in words, pictures, or symbols
• Compare and contrast figures
• Provide justifications for steps in a solution process
• Extend a pattern
• Retrieve information from a table, graph, or figure and use it solve a problem requiring multiple steps
• Translate between tables, graphs, words and symbolic notation
• Select a procedure according to criteria and perform it
multiple steps and multiple decision points
• Generalize a pattern
• Describe, compare, and contrast solution methods
• Formulate a mathematical model for a complex situation
• Provide mathematical justifications
• Solve a multiple- step problem, supported with a mathematical explanation that justifies the answer
• Formulate an original problem, given a situation
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• Design a mathematical model to inform and solve a practical or abstract situation
NOTE: Level 4 requires applying one approach among many to solve problems. Involves complex restructuring of data, establishing and evaluating criteria to solve problems.
Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.
(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)
Level 1 Recall
Level 2 Skills/Concepts
Level 3 Strategic Thinking
Level 4 Extended Thinking
Examples represent, but do not constitute all Level 1 mathematics performances:
• Recall or recognize a fact, definitions, or term
• Apply a well known algorithm
• Apply a formula
• Determine the area or perimeter of rectangles or triangles given a drawing and labels
• Identify a plane or three dimensional figure
Examples represent, but do not constitute all Level 2 mathematics performances:
• Classify plane and three dimensional figures
• Interpret information from a simple graph
• Use models to represent mathematical concepts
• Solve a routine problem requiring multiple steps, or the application of multiple concepts
• Compare figures or������VWDWHPHQWV
Examples represent, but do not constitute all Level 3 mathematics performances:
• Interpret information from a complex graph
• Explain thinking when more than one response is possible
• Make and/or justify conjectures
• Develop logical arguments for a concept
• Use concepts to solve problems
• Perform procedure with
Examples represent, but do not constitute all Level 4 mathematics performances:
• Relate mathematical concepts to other content areas
• Relate mathematical concepts to real-world applications in new situations
• Apply a mathematical model to illuminate a problem, situation
• ConduFt a project that specifies a problem, identifies solution paths,
������VROYHV�DQG
Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections-relate ideas within the content area or among content areas-and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs.
Table 1: Math Descriptors - Combined Webb Depth of Knowledge Levels for Mathematics (Webb, 2002), NAEP 2002 Mathematics Levels of Complexity, and Other Descriptors Related to NECAP GLEs.
(M. Petit, Center for Assessment 2003, K. Hess, Center for Assessment, updated 2005)
Level 1 Recall
Level 2 Skills/Concepts
Level 3 Strategic Thinking
Level 4 Extended Thinking
Examples represent, but do not constitute all Level 1 mathematics performances:
• Recall or recognize a fact, definitions, or term
• Apply a well known algorithm
• Apply a formula
• Determine the area or perimeter of rectangles or triangles given a drawing and labels
• Identify a plane or three dimensional figure
Examples represent, but do not constitute all Level 2 mathematics performances:
• Classify plane and three dimensional figures
• Interpret information from a simple graph
• Use models to represent mathematical concepts
• Solve a routine problem requiring multiple steps, or the application of multiple concepts
• Compare figures or������VWDWHPHQWV
Examples represent, but do not constitute all Level 3 mathematics performances:
• Interpret information from a complex graph
• Explain thinking when more than one response is possible
• Make and/or justify conjectures
• Develop logical arguments for a concept
• Use concepts to solve problems
• Perform procedure with
Examples represent, but do not constitute all Level 4 mathematics performances:
• Relate mathematical concepts to other content areas
• Relate mathematical concepts to real-world applications in new situations
• Apply a mathematical model to illuminate a problem, situation
• ConduFt a project that specifies a problem, identifies solution paths,
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Is there any place for drill and practice?!
• Drill is only appropriate when: ! • The desired concepts have been
meaningfully developed • Students have already developed (not
mastered) flexible and useful procedures • Speed and accuracy are [eventually]
needed - VDW
Lesson Resources
Questioning!
Observations do not have to be silent. Probing into student thinking through the use of questions can provide better data and more insights to inform instruction. As you circulate around the classroom to observe and evaluate students’ understanding, your use of questions is one of the most important ways to formatively assess in each lesson phase. Keep the following questions in mind (or on a clipboard, index cards, or a bookmark) as you move about the classroom to prompt and probe students’ thinking:
• What can you tell me about [today’s topic]? • How can you put the problem in your own words? • What did you do that helped you understand the problem? • Was there something in the problem that reminded you of another problem we’ve
done? • Did you find any numbers or information you didn’t need? How did you know that
the information was not important? • How did you decide what to do? • How did you decide whether your answer was right? • Did you try something that didn’t work? How did you figure out it was not going to
work? • Can something you did in this problem help you solve other problems?
NAME: Sharon V.
Estimates fraction quantities
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ERE
YET
ON
TA
RG
ET
AB
OVE
AN
D
BEY
ON
D
CO
MM
ENTS
FRACTIONS
MATHEMATICAL PRACTICES
Understands numerator/denominator
Area models
Set models
Use fractions in real contexts
Make sense of problems and perseveres
Models with mathematics
Uses appropriate tools
Used pattern blocks to show
2/3 and 3/6
Showing greater reasonableness
Stated problem in own words
Reluctant to use abstract models
Names
Lalie
Pete
Sid
Lakeshia
George
Pam
Maria
Topic: !Mental Computation Adding 2-digit numbers
Not There Yet CommentsOn Target Above and Beyond
Difficulty with regrouping
Flexible approaches used
Counts by tens, then adds ones
Beginning to add the group of tens first
Using a posted hundreds chart
3-20
3-18
3-24
3-24
3-20
Can’t do mentally
Has at least one strategy
Uses different methods with
different numbers
3-21
Observation Rubric Making Whole Given Fraction Part
Above and Beyond Clear understanding. Communicates concept in multiple representations. Shows evidence of using idea without prompting.
On Target Understands or is developing well. Uses designated models.
Not There Yet Some confusion or misunderstanding. Only models idea with help.
Fraction whole made from parts in rods and in sets. Explains easily.
Can make whole in either rod or set format (note). Hesitant. Needs prompt to identify unit fraction.
Needs help to do activity. No confidence.
Sally !
Latania !
Greg
John S. Mary
Lavant (rod) !
Julie (rod) !
George (set) !
Maria (set)
Tanisha (rod) !
Lee (rod) !
J.B. (set) !
John H. (set)
“You used the red trapezoid as your whole?” “So, first you recorded your measurements in a table?” “What parts of your drawing relate to the numbers from the story problem?” “Who can share what Ricardo just said, but using your own words?”
Clarify Students’ Ideas
“Why does it make sense to start with that particular number?” “Explain how you know that your answer is correct.” “Can you give an example?” “Do you see a connection between Julio’s idea and Rhonda’s idea?” “What if...?” “Do you agree or disagree with Johanna? Why?”
Emphasize Reasoning
“Who has a question for Vivian?” “Turn to your partner and explain why you agree or disagree with Edwin.” “Talk with Yerin about how your strategy relates to hers.”
Encourage Student-Student Dialogue
Examples of teacher prompts for supporting classroom discussions.
Got it Evidence shows that the student essentially has the target concept or idea.
Not Yet Student shows evidence of major misunderstanding, incorrect concept or procedure, or failure to engage the task.
4 Excellent: Full Accomplishment Strategy and execution meet the content, processes, and qualitative demands of the task. Communication is judged by effectiveness, not length. May have minor errors
3 Proficient: Substantial Accomplishment Could work to full accomplishment with minimal feedback. Errors are minor, so teacher is confident that understanding is adequate to accomplish the objective.
2 Marginal: Partial Accomplishment Part of the task is accomplished but there is lack of evidence of understanding or evidence of not understanding. Direct input of further teaching is required.
1 Unsatisfactory: Little Accomplishment The task is attempted and some mathematical effort is made. There may be fragments of accomplishment but little or no success. N
EED
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PN
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ITE
GO
T IT
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W
Everyday Math & Problem Solving Resources
Activity
Explore the Everyday Math series and the other resources that are available to you.
Select one activity and analyze it by responding to our 3 questions…
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What else is there?
Thank You!
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