teaching measurement concepts

Teaching MeasurementGrades 3 to 6Carlo MagnoDe La Salle University, Manila

K to 12 Mathematics Content Areas•Numbers and Number Sense •Measurement•Geometry•Patterns and Algebra•Probability and Statistics

Question:•Why is measurement important in the

math curriculum?

Measurement Outcomes:•Choosing units•Measuring•Estimating•Time•Using relationships

Mathematics: A curriculum profile for Australian Schools (1994)

Measurement in K to 12•Use of numbers and measures to

describe, understand and compare mathematical and concrete objects.

•Focuses on attributes such as length, mass and weight, capacity, time, money and temperature.

•Applications involving perimeter, area, surface area, volume, and angle measure.

Grade 4 Competencies•Describes and illustrates the perimeter of a

given figure. •Finds the perimeter of triangles, squares,

rectangles, parallelograms and trapezoids. •Solves word problems involving perimeter of

squares and rectangles, triangles, parallelograms and trapezoids.

•Estimates the area of an irregular plane figure made up of squares and rectangles using non-standard units.

Grade 4 Competencies•Derives inductively the formulas for the

area of squares and rectangles •Finds the area of a figure made up of

squares and rectangles using cm2 and m2. •Estimates the area of triangles,

parallelograms and trapezoids using non-standard units.

•Derives inductively the formulas for the area of triangles, parallelograms and trapezoids.

Grade 4 Competencies•Finds the area of triangles, parallelograms

and trapezoids using cm2 and m2. •Solves word problems involving the area of

a figure made up of squares and rectangles. •Solves word problems involving the area of

a triangle, parallelogram and trapezoid. •Visualizes and builds rectangular prisms

using unit cubes. •Derives inductively the formula for the

volume of rectangular prisms.

Grade 4 Competencies•Finds the volume of a rectangular prism

using cubic units. •Solves word problems involving the

volume of a rectangular prism.

Grade 5 Competencies•Represents and describes the

circumference of a circle. •Uses a model to estimate the circumference

of a circle. •Derives a formula for finding the

circumference of a circle. •Finds the circumference of the given circle

using the formula/s derived. •Solves problems involving circumference.

Grade 5 Competencies• Estimates and uses appropriate units of measure for

area. • Converts sq cm to sq m and vice versa. • Names the appropriate unit of measure used for

measuring area for accuracy. • Estimates and uses appropriate units of measure for

volume. • Converts one cubic unit of measure to a larger or

smaller unit. • Names the appropriate unit of measure used for

measuring the volume of a cube and a rectangular prism for accuracy.

Grade 5 Competencies• Represents and describes the area of a circle. • Uses a model to find the area of a circle. • Derives a formula for finding the area of a circle. • Finds the area of a circle using the formula/s

derived. • Solves problems involving area of circle using

appropriate formulas and procedures. • Describes the volume of cube and a rectangular

prism. • Derives a formula for finding the volume of cube

and a rectangular prism.

Grade 5 Competencies•Solves problems involving volume of a cube

and rectangular prism using appropriate formulas and procedures.

•Describes and estimates the temperature inside and outside of the classroom.

•Identifies the parts of a thermometer. •Reads a thermometer. •Measures temperature using the degree

Celsius. •Solves problems involving temperature.

Grade 6 Competencies• Finds the area of composite figures formed by

any two or more of the following: triangle, square, rectangle, circle and semi-circle.

• Solves word problem involving area of composite figures formed by any two or more of the following: triangle, square, rectangle, circle and semi-circle.

• Identifies the faces of a geometric solid. • Visualizes and describes surface area and name

the unit of measure used for measuring the surface area of solids.

Grade 6 Competencies• Derives a formula for finding the surface area of

cubes, prisms and cylinders. • Finds the surface area of cubes, prisms and

cylinders. • Solves word problems involving measurement of

surface area. • Describes the meaning of the volume of a solid. • Determines the relationship between the volume of a

rectangular prism and of a pyramid and between a cylinder and a cone.

• Obtains formulas for finding the volumes of cylinders, pyramids and cones.

Grade 6 Competencies• Finds the volume of a cylinder, pyramid or a

cone. • Solves problems involving volumes of solids. • Reads and interprets electric and water meter

readings. • Solves problems involving electric and water

consumptions. • Estimates the duration of time in seconds and

minutes. • Measures time using a 12-hour and a 24-hour

clock.

Grade 6 Competencies•Converts measures of time from a 12-hour

to a 24-hour clock and vice versa. •Calculates time in the different world time

zones. •Calculates speed, distance and time. •Solves problems involving average rate

and speed.

Premise in Measurement•Measurement activities are based on

everyday life experiences:▫More or less▫Tall or short▫Faster, slower▫Softer, lighter

Prerequisite skills•Comparison skills•Estimation skills

•Development of measurement understanding:▫Natural (body): The length of my bed is 10

handspans.▫Informal (comparison): My big sister

measured my bed and said its 10 handspans.

▫Formal (standard units): My bed is 2 meters long.

•Regardless whether students use formal or non formal measurement they should make decision about:▫Which quantities should be measured at

hand▫Which units to use▫The measuring tool appropriate

K to 12 Mathematics•The framework is supported by the

following underlying learning principles and theories: ▫Experiential and Situated Learning ▫Reflective Learning ▫Constructivism ▫Cooperative Learning▫Discovery and Inquiry-based Learning

K to 12 Mathematics•Experiential learning as advocated by

David Kolb is learning that occurs by making sense of direct everyday experiences.

•Experiential learning theory defines learning as "the process whereby knowledge is created through the transformation of experience. “

•Knowledge results from the combination of grasping and transforming experience" (Kolb, 1984, p. 41).

K to 12 Mathematics•Situated learning, theorized by Lave

and Wenger, is learning in the same context on which concepts and theories are applied.

K to 12 Mathematics•Reflective learning refers to learning

that is facilitated by reflective thinking. • It is not enough that learners encounter

real-life situations. •Deeper learning occurs when learners are

able to think about their experiences and process these allowing them the opportunity to make sense and meaning of their experiences.

K to 12 Mathematics•Constructivism is the theory that argues

that knowledge is constructed when the learner is able to draw ideas from his own experiences and connects them to new ideas that are encountered.

K to 12 Mathematics•Cooperative Learning puts premium on

active learning achieved by working with fellow learners as they all engage in a shared task.

K to 12 Mathematics•The mathematics curriculum allows for

students to learn by asking relevant questions and discovering new ideas.

•Discovery and Inquiry-based learning (Bruner, 1961) support the idea that students learn when they make use of personal experiences to discover facts, relationships and concepts.

Experiential and Situated Learning• Ask students to record the time it takes them to

travel from home to school in minutes for one week.• Tabulate the results• Ask students to make interpretation

▫Which days took the longest time? Why is this so?▫Which days took the shortest time? Why is this so?

• Open google maps and see how many kilometers is the distance from your house to school.

• What is the average time required to get to school?• How many minutes per kilometer does it take you to

travel?

Experiential and Situated Learning• Watch your favorite cartoons. • Count all the words being said in the cartoons.• How long is the cartoon in minutes?• If the time of the cartoon is doubled, can you

predict the number of words that can be mentioned?

• If you talk in the telephone for a 10 minutes, what is the estimated number if words you can use?

• If you need to tell your friend an important message with 50 words, how much time will it require you to tell him/her?

Experiential and Situated Learning•Measure how much volume can your

bathroom pail contain water.•Determine the time it takes you to fill the pail.• If you double the size of your pail, how much

time will you fill it?•Take the time in consuming the one pail of

water when taking a bath. How many pails do you consume?

•Given the pails of water you consume, estimate how long you take a bath.

Ask for 2 volunteers•Participants will provide their own

example of an activity for experiential and situational learning.

Reflective learning•Use google maps and estimate the

kilometers from your house to where your father is working.

•Ask for the total amount spent when commuting.

•Ask for the total amount spent when driving your won car.

•Show your data to your father and recommend which mode of transportation is better.

Reflective learning•Get the weight of each of your family

members.•Determine their ages and check who is

underweight and overweight.•Given your data make recommendations

on the following:▫Money spent on food▫Menu for the week▫Exercise activities for the family

Reflective thinking•Keep a journal and take note of all your

travels via plane.•Jot down in the journal the time you spent in

the plane and the places you went.•Summarize the following:

▫What places took the longest plane ride? Why?▫What places took the shorted plane ride?

Why?▫What is the relationship between time and

distance?

Reflective thinking• Get the square meter of your land area.• How much does your land cost?• Get the square meter of other land areas.• How much do they cost?• Get the square meter of residential lands in the

city. How much do they cost.• Compare the cost of equal land areas in the city

and in the province? • Is there a difference? Why is this so? • What do you need to do if you want to live in the

city?

Ask for 2 volunteers•Participants will provide their own

example of an activity for reflective learning.

Constructivism• Students will identify a problem in their

community in the following areas:▫Waste management▫Overcrowding▫Increased air temperature

• Collect data to serve as evidence to these problems by:▫Measure the weight of garbage produced by each

household each day for 4 weeks▫Count the number of people living in each household

and the floor area of their house. Report the ratio.▫Get the temperature each day and tabulate it.

• Provide recommendations given the severity of the problem (reflective learning)

Constructivism• What type of body pains did you experience as

the most painful? Why did you had such pain?• List them down.• For each pain indicate how painful it is using a

scale from 0 (no pain) to 10 (very painful)• Ask your classmates to rate the list of pain you

have.• Get the average of the rating for each pain.• Report the standard deviation. • Given the SD are you all having a similar feeling

for each pain? (reflective learning)

Constructivism•Measure the temperature for the months of

April and May.•Go to the malls and ask some sales person

which types of clothes are bought and the quantity.

•Tabulate the number of purchases for each type of clothes for each day and the temperature?

•What does the temperature got to do with the type of clothes sold?

Constructivism•Which model of cell phone do you have?•What do you think is the best cell phone model?•List down different cell phone models and

count the number of people in your community who has it.

•For each model of cell phone, ask the owners how long does their battery last. Ask also what is the total minutes they used to call and text.

•Which characteristic of cell phone do you recommend that can save more energy?

Ask for 2 volunteers•Participants will provide their own

example of an activity for constructivist learning.

Cooperative learning•Form a group and each one will be

assigned to a place to take the air temperature for 7 days.

•Compare the temperature for each person.

•Why is there variation in the temperature?

•Report the findings.

Cooperative learning•Students form three groups and are

assigned to measure the floor area of the classroom.

•One group will only use a one inch paper clip.

•One group will use an 8 inches pencil.•One group will use a 15 inches long stick.•Which group do you think will measure

the floor area the fastest? Why?

Cooperative learning•Students will work together to build an

improvised anemometer and a rain gauge.•When the rain comes they record the

speed of the wind.•They measure the amount of water

collected in the rain gauge.•Students make a report and present to

class the relationship between wind speed and amount of rainfall.

Ask for 2 volunteers•Participants will provide their own

example of an activity for cooperative learning.

Discovery•Students will ask their parents at home

the different tools they use to measure length of objects.

•The students will bring this material and demonstrate to their classmates how the tools are used.

Discovery •Self-study on the procedure to convert oC

to oF.•Show how it is done in class

Ask for 2 volunteers•Participants will provide their own

example of an activity for cooperative learning.

Insights