Make sense of quantities
22
How do quantities fit into the problem
23
Abstract and Recontextualize
24
Mathematical Practice:
Construct viable arguments and critique the
reasoning of others.
 Build logical progression of statements to
explore conjectures
 Recognize and use counterexamples
 Justify their conclusions and respond to other’s
ideas using drawings, diagrams, actions
25
5th Walls With Windows
26
Draw a picture
27
Use a table
28
Build on previous knowledge or
calculations
29
Mathematical Practice:
Model with mathematics.




Solve problems in everyday life
Write equation to describe a situation
Solve a design problem
Make assumptions and approximations to
simplify a complicated situation
 Interpret results to see if they make sense in
terms of the situation
30
Singing at the Ballpark- 4th
21 students need to get to the ballpark.
Each car will carry one adult and up to
4 students.
5 ¼ cars please?
31
Interpret results into context to check
for reasonableness
32
Check for Reasonableness
33
Check for Reasonableness
You can’t leave someone behind!!!!
34
Solve a design problem – 5th
35
Real Life Situation
36
Write an equation to model
37
Map relationships between Quantities
38
Mathematical Practice:
Use appropriate tools strategically.
 Use paper pencil, concrete models, ruler,
protractor, calculator, spreadsheet, etc.
 Tools might also include choosing an appropriate
mathematical strategy
39
Strategy - Slope
40
Tools - Calculator
41
Tools Calculator – Strategy Convert to
decimals
42
Mathematical Practice:
Attend to precision.
 Communicate precisely with clear definitions
 State meaning of symbols
 Calculate accurately, efficiently, and use
appropriate level of precision
43
Have a Clear Definition
• Teacher comments on 2-4 tests: Why do the
tasks use the word angle when our textbooks
use “vertex”?
44
Words are tools for thinking
Not having words to use limits the mathematics
we can think about. – Harold Asturias,
Lawrence Hall of Science
Teacher comments: Why did you use the word
“dimensions” on the 5th grade task box of
cubes? Why didn’t you just ask for length,
width, and height?
45
Cognitive Demand
• How many dimensions are there?
46
Dimensions?
47
Dimensions?
48
Diagonal
49
Mathematical Practice:
Look for and make use of structure.
 Discern pattern or structure
 See complicated things, such as algebraic
expressions as single objects or as composed of
several objects
50
5th- Box of Cubes- Structure
51
Making Sense of Structure
Seven most important words to
transform education: How did you
Figure that out?
52
7th Grade Freezing in Fargo
• How many times colder is Wednesday Feb.
25th than Tuesday Feb. 3rd?
Almost 40% of the students in the sample
subtracted.
53
Mathematical Practice:
Look for and express regularity in repeated
reasoning.
 Look for repeated calculations in both general
methods and for short cuts
54
6th Freezing In Fargo
• Which week (Sunday through Saturday)
recorded the average lowest temperature?
A student noticing that all the averages are
divide by 7 days should realize that comparing
totals will yield some comparative results
without needing to divide.
55
Tools For Practices and Standards
• Use MARS Tasks
– Define the meaning of the standards and practices
– Raise expectations for teachers about what
students are capable of accomplishing
– Help teachers anticipate misconceptions so that
they can be surfaced and addressed in class
discussion and re-engagement lessons
56
Resources
• SVMIMAC.org website
• Inside Mathematics.org directly or through
link in NCSM
57
TOOLS BY SUBJECTAlgebra &
FunctionsAlgebraic Properties
& RepresentationsData
AnalysisFunctions &
RelationsGeometry &
MeasurementMathematical
Reasoning & ProofsNumber
OperationsNumber
PropertiesPatterns, Functions
& AlgebraProbabilityStatistics
3rd GradeCore IdeasRecognize and use characteristics,
properties, and relationships of two-dimensional
geometric shapes and apply appropriate techniques to
determine measurements.Choose appropriate units
and tools for particular tasks and use these units and
tools to estimate and measure (length, weight,
temperature, time, and capacity).Identify and compare
attributes of two-dimensional shapes and develop
vocabulary to describe the attributes.Calculate
perimeter and area and be able to distinguish between
the two measures. (Area may be measured by covering
a figure with squares.)Use visualization, spatial
reasoning, and geometric modeling to solve
problems.Recognize geometric ideas and relationships
and apply them to problems.MARS TasksLooking Glass
LandTaskRubricCore Mathematical Ideas and
ChallengesQuestions for Teacher ReflectionDiscussion
of Successful Examples of Student WorkDiscussion of
Student MisconceptionsGraph and Analysis of the
MARS Task DataSummary of Student Understandings
and MisunderstandingsImplications for Instruction
58
59
60
Practices Require Content
• Looking at and Understanding Number System
• Using Place Value Strategies to Make Sense of
and Solve Problems
• Understanding Number Line as a basic
mathematical concept and tool
Butterfly and Moth Collection
How much longer was the longest wingspan
than the shortest?
Research Suggests:
• Number lines help students understand
fractions as a “single number” instead of two
– unique point or location on the line
• Number line concepts and reading fractions
can be introduced through rulers, clocks,
scales
• Number lines help students develop the
ability to generalize about number and
operations
Knowing Fractions
Knowing Fractions
MARS 2012 Grade 4 Task 1
Butterfly and Moth Collection
25%
20%
15%
10%
5%
0%
Series1
0
6%
1
10%
2
13%
3
18%
4
19%
5
8%
6
5%
7
21%
MARS 2012 Grade 5 Task 3
Knowing Fractions
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
Series1
0
2%
1
10%
2
15%
3
16%
4
13%
5
9%
6
9%
7
11%
8
5%
9
10%
MARS 2012 Grade 6 Task 3
Fraction Match
30%
25%
20%
15%
10%
5%
0%
Series1
0
26%
1
14%
2
16%
3
13%
4
10%
5
10%
6
7%
7
5%
Preparing for Geometry
• To do the type of work needed to be
successful in geometry, students need to have
a variety of experiences at earlier grades.
• Ideas build over time.
Purpose of Resources
MARS 2012 Grade 4 Task 2
Looking for Shapes
25%
20%
15%
10%
5%
0%
Series1
0
8%
1
14%
2
22%
3
15%
4
14%
5
14%
6
14%
MARS 2012 Grade 5 Task 2
A Box of Cubes
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Series1
0
14%
1
4%
2
10%
3
4%
4
6%
5
14%
6
9%
7
38%
MARS 2012 Grade 6 Task 5
Unfolding a Box
40%
35%
30%
25%
20%
15%
10%
5%
0%
Series1
0
36%
1
24%
2
12%
3
4%
4
5%
5
4%
6
5%
7
4%
8
4%
9
3%
MARS 2012 Grade 7 Task 1
Similar Figures
25%
20%
15%
10%
5%
0%
Series1
0
22%
1
16%
2
8%
3
17%
4
9%
5
14%
6
10%
7
1%
8
2%
MARS 2012 Course 2 Task 4
The Company Logo
35%
30%
25%
20%
15%
10%
5%
0%
Series1
0
0%
1
5%
2
26%
3
19%
4
32%
5
4%
6
1%
7
1%
8
4%
9
3%
10
4%
MARS 2012 Course 2 Task 5
Trapezoid Blocks
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Series1
0
8%
1
7%
2
8%
3
11%
4
41%
5
3%
6
8%
7
4%
8
5%
9
1%
Developing Algebraic Thinking
Describe T pattern 5.
MARS 2012 Grade 2 Task 5
Misha's Marbles
30%
25%
20%
15%
10%
5%
0%
Series1
0
3%
1
10%
2
3%
3
11%
4
8%
5
25%
6
17%
7
23%
Lattice Fence – 6th
Define the pattern to explore relationships.
MARS 2012 Grade 6 Task 4
Lattice Fence
25%
20%
15%
10%
5%
0%
Series1
0
7%
1
21%
2
14%
3
13%
4
17%
5
10%
6
5%
7
9%
8
1%
9
2%
Aussie Fir Tree - Algebra
MARS 2012 Course 1 Task 5
Aussie Fir Tree
16%
14%
12%
10%
8%
6%
4%
2%
0%
Series1
0
7%
1
12%
2
8%
3
8%
4
7%
5
9%
6
10%
7
15%
8
8%
9
9%
10
7%
Proportional Reasoning
MARS 2012 Grade 7 Task 1
Similar Figures
25%
20%
15%
10%
5%
0%
Series1
0
22%
1
16%
2
8%
3
17%
4
9%
5
14%
6
10%
7
1%
8
2%
Rate Concentrate – 6th
MARS 2012 Grade 6 Task 1
Rate Concentrate
25%
20%
15%
10%
5%
0%
Series1
0
23%
1
15%
2
3%
3
24%
4
15%
5
4%
6
17%
Using the 8th Grade Test
8th Grade - Tool
• Understand the 8th grade Mathematics Common Core
Course and the deep rich mathematical expectations
for students
• Includes rich Algebra Strand but also works to expand
and deepen understanding and facility with other
strands
• Use as Placement Test or Summative Test - Could your
7th graders pass this test? Could your Algebra and
Geometry students meet standard on test?
• Facilitate Course Discussion with Staff and Parents – Is
this the mathematics we want students to be able to
know and do?
Implementing Formative Assessment
Lessons- Forming Professional
Learning Communities
– How do we create classroom culture?
– How do we facilitate “staff” doing mathematics
together to understand purpose of the lesson,
understand the mathematics before giving it to
students, work as a learning community to discuss
techniques, purpose of using student work, types of
comments to put on student papers, how to have a
plenary discussion? What are the important
mathematical ideas to be drawn from students? What
teacher moves help hold all students accountable for
their own learning?
– Making time for lessons?