Grades 35 Session Objectives 2 Understand the application component of rigor called for in the Standards as defined by guiding documents Examine various activities in A Story of ID: 725207
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Slide1
Rigor Breakdown
Part 3: Application
Grades 3-5Slide2
Session Objectives
2
Understand the
application component
of
rigor called
for in the Standards,
as defined by guiding documents
Examine
various activities in
A Story of
Units
that engage students in application; compare and contrast activities
Highlight
Standards for Mathematical Practice in the
application
activities in
A Story of
Units
Recognize the balance
and intensity of
all three components of rigor in
A Story of
Units
Articulate
how the three components of rigor support and relate to each
otherSlide3
Application
3
Early reflection:
What is application?Slide4
ApplicationDefined by the Instructional Shifts
4
“Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so.
Teachers
provide opportunities at all grade levels for students to apply math concepts in “real world” situations.
Teachers
in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content.” Slide5
ApplicationDefined by the Publishers’ Criteria
5
“
The phrase
‘real-world problems’
…is used to establish expectations …for applications and
modeling.”
(page 5)
“Applications take the form of problems to be worked on individually as well as classroom activities centered on application scenarios…” (page 11
)Slide6
AGENDA
6
Application – Word Problems
Application – Real-World Problems
Application
–
Modeling
Application – Word ProblemsSlide7
Application through Word Problems
7
Provide a “real world” situational context, even if the problem itself is not particularly “real-world”
Jason’s allowance is 2 fifths as much as Sarah’s.
Sarah’s
allowance is half of
Max’s allowance. If
Max earns an
allowance
of $20 per week, what is Jason’s
allowance?
Prompt students to consolidate their knowledge of which operation is applicable to a given situational contextSlide8
Application in A Story of Units
8
The Read, Draw, Write (RDW) process for solving word problems is modeled to students by teachers, promoting
perseverance
in reasoning through problems.Slide9
Video Clip – Word Problems
9
Reflections:
How did the teacher model the problem solving process?
What are the impacts and advantages of having students apply mathematical concepts in real world contexts?
How is this different from how you address application in your classroom?
Identify Mathe
matical Practices. Slide10
Video Clip: Library Problem
10
This video clip can be found on
EngageNY.orgSlide11
Video Clip – Word Problems
11
Reflections:
How did the teacher facilitate the RDW problem solving process?
What are the impacts and advantages of this example of problem solving?
How is this different from how your school/district program currently addresses application?
In what ways were Standards
for
Mathe
matical Practice evident? Slide12
More on Application from the Publishers’ Criteria
12
“Materials in grades K-8 include an ample number of single-step and multi-step contextual problems that develop the mathematics of the grade, afford opportunities for practice, and engage students in problem solving.”
(
page
11
)Slide13
Multi-Step Problems in the Standards
13
2
2.OA.1
Two-step, addition and subtraction within 100
3
3.OA.8
3.MD.3
Two-step, involving the four operations
Two-step, how many more and how many less questions using the information presented in scaled bar graphs with several categories
4
4.OA.3
Multi-step, involving whole numbers and the four operations
5
5.MD.1
Multi-step, real world problems that include converting among different-sized standard measurement units within a given measurement systemSlide14
Student Work Sample Grade 2 – Two-Step
Word Problem
14
Jack has $344. Mason has $266 more than Jack. How much do Jack and Mason have altogether?Slide15
Student Work SampleGrade 3 – Two-Step
Word Problem
15
Sam bought 4 bags of candy. Each bag contained 30 pieces of candy. The candy was shared equally among 6 children. How many pieces of candy did each child receive?Slide16
Student Work SampleGrade 3 – Two-Step
Word Problem
16Slide17
Student Work SampleGrade
4 – Multi-Step Word Problem
17
Laney had 1690 tokens. Mia had 380 fewer tokens than Laney. Laney gave some tokens to Mia. In the end, Mia had 3 times as many tokens as Laney. How many tokens did Laney have in the end?Slide18
Student Work SampleGrade
4 – Multi-Step Word Problem
18Slide19
Student Work SampleGrade
5 – Multi-Step Word Problem
19
Mrs. Jones wants to give each of her 18 students a gift for the holidays. She has a budget of $144. She bought 18 identical gifts each costing $6.00. She purchased 18 printed gift boxes for $1 each. She wants to buy ribbon to tie around each box. The ribbon costs $6 per yard. If she uses 9 inches of ribbon for each box, can she afford to buy the ribbon and still keep to her budget of $144? Explain why or why not. Slide20
Student Work SampleGrade
5 – Multi-Step Word Problem
20Slide21
Lesson Engagement – RDW Process
21
Students
may ask themselves these questions to guide them through the problem solving process:
“What do I see?”
“Can I draw something?”
“What can I draw?”
“What can I learn from my drawing?”
After drawing, students write a statement responding to the question. Slide22
RDW
22
Meagan had $1780 and Lisa had $1910. Lisa gave some money to Meagan. In the end Meagan had twice as much money as Lisa. How much money did Lisa give to Meagan?Slide23
Lesson Engagement – RDW Process
23
Reflections:
What surprised you about working through the RDW process yourselves?Slide24
Application – Word ProblemsKey Points
24
Word problems are a form of application.
Word problems give situational contexts to develop students’ schema of which situations call for which operations.
The RDW process encourages visualization, reflection, and perseverance.
Two-step problems begin in 2
nd
grade.
Multi-step problems begin in 4
th
grade.Slide25
AGENDA
25
Application – Word Problems
Application – Real-World Problems
Application –
ModelingSlide26
Real-World Problems in the Standards
26
3
3.MD.7
3.MD.8
Relate area to multiplication and division
Perimeters of polygons
4
4.MD.3
4.MD.7
Area and perimeter formulas for rectangles
Unknown angles on a diagram
5
5.NF.6
5.NF.7
Multiplication of fractions and mixed numbers
Division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions
5
5.MD.1
5.MD.5
Convert among different-sized standard measurement units within a given measurement system
Volume problems
5
5.G.2
Graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Slide27
More on Application from the Publishers’ Criteria
27
“Materials
attend thoroughly to those places in the content standards where expectations for multi-step and real world problems are
explicit.”
(
page
11
)Slide28
Activity – Writing Real-World Problems
28
Choose one standard calling for real-world problems and write your own example of a real-world problem for that standard.Slide29
Application – Real-World ProblemsKey Points
29
Standards for real-world problems are present in grades 3-5
Real-world problems are problems that people are faced with solving in the real world
Find the amount of fence needed to surround a given area
Converting among different sized measurement units
Multiplying fractions involved in doubling or halving a recipeSlide30
AGENDA
30
Application – Word Problems
Application – Real-World Problems
Application – ModelingSlide31
Application - Modeling
31
Early reflection:
What does modeling mean to you?Slide32
Modeling in Pre-Kindergarten
32
Modeling of environment using basic shapes
From Pre-Kindergarten Overview: “[Students] use basic shapes and spatial reasoning to model objects in their environment.”Slide33
Modeling in Kindergarten
33
Modeling real world objects using geometric shapes and figures
K.G.5 Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
Modeling of early addition and subtraction situations using concrete materials
From Kindergarten Overview: “Model simple joining and separating situations with sets of objects.”Slide34
Modeling in Grade 1
34
Modeling all forms of addition and subtraction situations using concrete materials:
From Grade 1 Overview: “Use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations.”
1.
NBT.4
and 1.NBT.6 Slide35
Modeling in Grade 2
35
Place Value Models Applied to Addition and Subtraction
From
Grade 2 Overview: “They solve problems within 1000 by applying their understanding of models for addition and subtraction”
2
.NBT.
7
Add and subtract within 1000, using concrete models or
drawings and
strategies based on place value, properties of operations,
and/or the
relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers
, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Slide36
Modeling in Grades 3-5
36
Equal-sized groups, array and area models for multiplication and division
3.MD.7,
4.NBT.5, 4.NBT.6, 5.NBT.6, 5.NBT.7
Visual
fraction
models
Most standards in the NF domain for grades 3-5
Place
value and number line models extended
to operations with
decimals and
fractions“
They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths.”Slide37
Modeling in Grades 6-12
37
Modeling real life objects with geometric figures
Modeling relationships using equations, inequalities, and sets of equations and inequalities
Modeling relationships between two data sets using functions
Investigating chance processes with probability modelsSlide38
More on Application from the Publishers’ Criteria
38
“Modeling builds slowly across K-8, and applications are relatively simple in early grades.”
(
page
11
)Slide39
Video Clip: Visual Fraction Models
39
This video clip can be found on
EngageNY.orgSlide40
Lesson Engagement – Modeling
40Slide41
Lesson Engagement – Modeling
41
Reflection:
Analyze
the impacts and advantages of
modeling in grades P-5.Slide42
Modeling is a Mathematical Practice
42
4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to
solve problems
arising in everyday life, society, and the workplace. In early grades, this
might be
as simple as writing an addition equation to describe a situation. In middle grades
, a
student might apply proportional reasoning to plan a school event or analyze
a problem
in the community. By high school, a student might use geometry to solve
a design
problem or use a function to describe how one quantity of interest depends on another. (continued…)Slide43
Modeling is a Mathematical Practice
43
(continued…) Mathematically
proficient students who can apply what they know
are comfortable
making assumptions and approximations to simplify a
complicated situation
, realizing that these may need revision later. They are able to
identify important
quantities in a practical situation and map their relationships using
such tools
as diagrams, two-way tables, graphs, flowcharts and formulas. They can
analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.Slide44
Application – ModelingKey Points
44
Modeling is a required at all grade levels.
Modeling builds across grade levels, primarily concrete in grades K-3, transitioning into pictorial in grades 3-5.
Modeling is a Mathematical Practice.Slide45
Application in A Story of Units
45
Interesting problems that connect mathematics to students’ environment, other disciplines, and to the mathematics
itself.
Goal is for students to gain understanding and modeling skills needed to solve problems they have never seen
before.
Concrete materials and pictorial diagrams develop students’ ability to visualize quantitative relationships, reinforcing understanding.Slide46
Application in A Story of Units
46
Time allotted to application varies, but is commonly 10-20 minutes of the lesson.
The placement of an application problem may go before or after the conceptual development:
Placement before can provide important context and structure to understanding a new concept.
Placement after gives usefulness of a just-learned concept.
Application also appears imbedded in conceptual development, worksheets, and homework, providing opportunity to work
both as a class and individually
.Slide47
Key Points
47
Application involves using relevant conceptual understandings and appropriate strategies
even
when not
prompted to do so.
Application
is called for in the Standards in three ways: single and multi-step word problems, real-world problems, modeling.
A
Story of
Units
provides frequent, rich opportunities for students to practice application both as a group and individually.
Application problems
are often also opportunities to nurture the Standards of Mathematical Practice.Slide48
Next Steps
48
How will you incorporate this information about application into instruction?
How
will
you
share
this information about application
with
your
colleagues?Slide49
A Call for Equal Intensity and Balance
49
The Instructional Shifts:
“
Students are
practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in ‘drills’ and make use of those skills through extended
application
of math concepts...”
Slide50
A Call for Equal Intensity and Balance
50
The Publishers’ Criteria:
“To help students meet the expectations of the Standards, educators will need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding, procedural skill and fluency, and applications.” (page 5)
“Materials and tools reflect the balances in the Standards…” (page 9)Slide51
Intensity and Balance in A Story of Units
51Slide52
Intensity and Balance in A Story of Units
52Slide53
Three Components of Rigor
53
Reflection:
How do the three components of rigor support and relate to each other
?Slide54
Three Components of Rigor
54
Publishers’ Criteria:
“The three aspects of rigor are not always separate in materials. (Conceptual understanding needs to underpin fluency work; fluency can be practiced in the context of applications; and applications can build conceptual understanding.)” (page 11)