Naked Math Gets a CTE Cover-up
UNIVERSITY OF ALASKA ANCHORAGE
ALASKA DEPARTMENT OF EDUCATION & EARLY
DEVELOPMENT
UAA
&
EED
Partnered in 2011 & 2012
Research by National Research Center for Career & Technical Education
Questions to think about . . .
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Do math and CTE teachers collaborate at your school
or district?
Do math teachers know what math concepts kids
need in CTE courses?
Do CTE teachers use the same math vocabulary and
algorithms that are used in math class?
Let’s Look at Some Trends
3.8 math
credits
3.4 math
credits
3.6 math
credits
1.7 Math
Credits
Source: NAEP Trends in Academic Progress
How Can We Increase Math Achievement?
 One way – not THE ONLY way – to help increase
math achievement
 A model of curriculum integration and pedagogy to
increase CTE students’ math achievement while
maintaining technical skill attainment.
 Students showed significantly higher math
achievement on Terra Nova and Accuplacer
 For complete research results, see NRCCTE
Core Principles of the Model
 Community of practice is critical
 Begin with the CTE curriculum –
NOT the math curriculum
 Math is an essential workplace skill
 Maximize the math in the CTE curriculum
 CTE teachers are teachers of math-in-CTE –
they are not math teachers
What is the Model?
 1 CTE Teacher + 1 Math Teacher = 1 Team
 Each team
 Maps the CTE curriculum
 Identifies embedded math concepts
 Creates math-enhanced lessons
 CTE teacher delivers the lessons
 CTE teacher and math teacher continue to
collaborate before and after each math-enhanced
lesson is delivered
What is a “Math-Enhanced CTE Lesson” ?
 Introduce the CTE lesson
 Assess students’ math awareness
 Work through the math example embedded in CTE
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lesson – using standard math vocabulary
Work through related, contextual
math-in-CTE examples
Work through “naked math” examples
Formative assessment
Summative assessment includes math questions
Sample Curriculum Map – Healthcare
CTE
Course
or Unit
Diseases
CTE Concepts or Applications
Embedded Math Concepts
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Growth
and
Development
Health
Careers
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Work-place safety, body
mechanics and disease
prevention practices
Basic human anatomy and
physiology, growth,
development, wellness and
disease.
Relationship between diseases
/disorders to the environmental
or genetic causes.
Basic human anatomy and
physiology, growth,
development, wellness and
disease.
Potential health science careers
required education, and
opportunities.
Measure and perform
calculations.
U.S health care system and the
interdependence of careers and
professionals.
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Statistics
Whole numbers
Interpreting data
Temperature
Charts and graphs
Percentages
Graphing
Probability
Proportion
Charts and graphs
Estimation
Weights
Percents
Reading interpreting
data
Whole numbers
Statistics
Cost/benefit ratio
Computation
Ratios
Decimals
Conversion
Trends
Charts/ graphs
CTE
Course or
Unit
CTE Concepts or
Applications
Embedded Math
Concepts
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Basic human anatomy and
physiology, growth,
development, wellness and
disease.
Basic anatomy and physiology of
body systems and topographic
terms.
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Quadrants
Planes
Measuring
Ratios
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Basic anatomy and physiology of
the skeletal system.
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Integumentary
System
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Body surface area
Wound Area
Respiratory
System
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Major structures of the
respiratory system
Managing the airway.
Measurements
Angles
Formulas
Positive and
negative numbers
Estimation
Percent
Surface area
Area
Volume
Estimation
Dilation
Body
Structures
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Skeletal
System
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Sample Curriculum Map – Construction
CTE Course
or Unit
CTE
Concepts or
Applications
Construction
Floor systems
Construction
Construction
Scaling/
conversions
Electricity
Embedded Math Concepts
•
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•
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•
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Area perimeter
Measurement
Estimation
Ratio
Whole numbers
Interpret tables
Two dimensional drawings
Scaling
•
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Ratios/ proportions
Fractions & Decimals
Measurement
Factors
Inverse fractions
Two and three dimensional drawings
Point of reference
Linear equations
Quadratic equations
Area
•
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•
•
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Units
Direct variation
Indirect variation
Solve equations
Schematics
Formulas
Percent
Average
CTE Course
or Unit
CTE Concepts or
Applications
Embedded Math Concepts
Construction
Doors and windows
•
•
•
•
•
Ratios
Tolerances
Formulas
Whole numbers & Fractions
Measurement
Construction
Squaring
•
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•
•
Pythagorean theorem
Congruence
Measurement
Whole numbers & Fractions
Construction
Measurement
•
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Linear measurement
Area
Angle measurement
Fractions
Ratios/ proportions
Construction
Walls
•
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•
•
•
Measurement- linear
Area
Whole number operations
Fractions & Decimals
Pythagorean theorem
Construction
Roofing
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Slopes
Trig
Area
Conversion
Fractions
Pythagorean theorem
Linear equations
The Model does NOT . . .
 Force extra math into the CTE program
 Create a mentoring or coaching relationship – the
teachers are partners
 Include developing or re-designing curriculum
 Use “team-teaching”, i.e., math teacher does not
teach in the CTE class
 Above all, it does NOT make the CTE class
into a math class
Statewide Participants in Math-in-CTE
 2010-2011
 Anchorage
 Denali
 Fairbanks
 Ketchikan
 Mat-Su
* 8 Construction Teams
* 4 Health Careers Teams
 2011-2012
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Bering Strait
Craig
Fairbanks
Kenai
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Ketchikan
Mat-Su
Unalaska
Valdez
UAA
* 5 Construction Teams
* 5 Health Careers Teams
* 1 Transportation Team
Alaska Team Reactions
 CTE teachers:
Now I know why my construction students can’t
subtract ¼” from 15” in their heads!
 Now I know the correct math vocabulary for the
3-4-5 stair riser lesson.
 You mean a ratio is not the same as a proportion?
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 Math teachers:
I had no idea there was so much math in the CTE class.
 I see that my students need practice in performing
‘mental math’ for use in real life.
 Now I know why we really do teach this stuff!
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A Sample Math-Enhanced Lesson
SCALING
DRAWINGS
Developed by Dave Oberg and Jen Nelson, Service High School, Anchorage School District, 2010
What would you need to know before
you begin your drawing?
1. Dimensions of the building.
2. Size of the paper you are going
to print the plans on.
What are standard paper sizes?
A: 8.5” X 11”
B: 11” X 17”
C: 18” X 24”
D: 24” X 36”
What is the relationship
between the size of the
building and the size of
the paper?
What units would be used on the drawing?
The building?
How many times
larger is the building
footprint than the
paper it must fit on?
What does the term “scale” mean?
The fraction used to represent the
ratio making the drawing and
building proportional.
So, what is meant by “ratio”?
A ratio is a comparison of two
things expressed as a fraction. In
drafting, the drawing measure is
always given first, followed by the
measurement of the actual
building.
Then, what is a proportion?
A proportion is an equation
showing that 2 ratios are
equivalent.
Let’s say we have a building
whose floor plan footprint is
120’ X 40’. What size paper
would we need to use if our
drawing is to be done at a scale
of ¼” = 1’?
¼” = 1’;
therefore the ratio is 1/4.
Set up a proportion for each dimension:
Inches
Feet
1 = L
4 120
and
1 = W
4 40
Cross multiply to create an equation
4L = 120
4
4
and
4W = 40
4
4
Divide by 4 to solve.
L = 30 inches and W = 10 inches
Therefore, we would need to use D-size (36” X 24”) paper.
Your client wants you to design a warehouse
that is 40’ x 200’. What size paper should be
used, and what scale should be used, for the
blueprints of the building?
Trying the ratio of ¼ first:
1 = W
and
1 = L
4 40
4 200
Cross multiply to create an equation,
4W = 40
and
4L= 200
4
4
4
4
Divide by 4 to solve.
W = 10 inches and
L = 50 inches
The width at this scale is too large for D-size paper.
Trying the ratio of 1/8:
1 = W
and
1 = L
8 40
8 200
Cross multiply to create an equation,
8W = 40
and
8L = 200
8
8
8
8
Divide by 8 to solve.
W = 5 inches and
L = 25 inches
At this scale the size
would fit on D-size
paper, but would not fit
on C-size paper.
Therefore, the drawing
must be at 1/8” = 1’ on
D-size paper.
Each day, the seals at an aquarium
are each fed 1 pound of food for
every 10 pounds of their body
weight. A seal at the aquarium
weighs 280 pounds. How much
food should the seal be fed per
day?
One pound of food per 10 pounds of body weight is equivalent to a
ratio of 1/10. Set up a proportion using food to body weight.
Pounds of food
Body weight of seal
1 = _x_
10 280
Cross Multiply to get the equation:
10x = 280
10
10
Divide by 10 to solve.
x = 28 pounds of food per day
“Naked Math” Problems
In teams of 2-3 students, use a tape measure to find
the dimensions of this classroom (wall to wall).
Determine the appropriate scales to use for each
common paper size (if possible).