Applications of
Mathematics 10 -
Shape and Space (Measurement)
This sub-organizer contains
the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources
PRESCRIBED
LEARNING OUTCOMES
It is expected that students
will :
- demonstrate an understanding
of scale factors, and their interrelationship with the dimensions of similar
shapes and objects
- solve problems involving
triangles, including those found in 3-D and 2-D applications
It is expected that students
will:
- calculate the volume
and surface area of a sphere, using formulas that are provided
- determine the relationships
among linear scale factors, areas, the surface areas and the volumes of similar
figures and objects
- solve problems involving
two right triangles
- extend the concepts
of sine and cosine for angles through to 180°
- apply the sine and cosine
laws, excluding the ambiguous case, to solve problems
- select and apply appropriate
instruments, units of measure (in SI and Imperial systems) and measurement
strategies to find lengths, areas and volumes
- analyse the limitations
of measuring instruments and measurement strategies, using the concepts of
precision and accuracy
- solve problems involving
length, area, volume, time, mass and rates derived from these
- interpret drawings,
and use the information to solve problems
SUGGESTED
INSTRUCTIONAL STRATEGIES
Many worthwhile and practical
problems in mathematics involve calculations of shape and space and are most
easily solved by making and labeling a drawing. Some drawings work best when
reduced to scale, while in other cases an annotated sketch is sufficient. Visualizing
shape and space is a critical component of learning mathematics.
- Bring a collection of
spherical balls related to various sports into the classroom. Have students
find the circumference of each ball using a fabric tape, and, from the circumference,
calculate radius-diameter-area of the greatest circle. Ask them to place the
results in a table and find the relationship between each column and classify
it as linear or non-linear.
- Discuss why smaller
animals require faster metabolisms to maintain a constant body temperature,
and why they require more food as volume decreases faster than surface area
decreases.
- Have students use equipment
such as a parallax viewer (for range) and clinometer (for height) to analyse
the motion of a football thrown on a field, resolving the motion into horizontal
and vertical components. Point out that the horizontal motion vs. time is
linear while the vertical motion vs. time is described using a quadratic function
(i.e., nonlinear).
- Provide students with
pictures of non-right triangles that show either two sides and one angle or
two angles and one side. Have students calculate the missing side or angle.
A practical example might involve using an orienteering map.
- Give instructions to
students (in terms of angles and sides) to move to a certain point in two
steps (two sides of a triangle), using Law of Cosines or Law of Sine to calculate
how to return to the starting point.
- Divide the class into
small groups and supply each group with a measuring instrument (e.g., ruler,
meter stick, measuring tape, 10 cm piece of string, 5 cm x 5 cm square of
paper). Give students a list of objects of various sizes and shapes in the
classroom and have them measure specific aspects of each one. In a class discussion,
ask students to determine which measuring instruments were best for measuring
which object, which measurements were most precise, and which measurements
were most accurate.
- Develop a five-station
carousel where each station has one type of measurement problem (e.g., length,
area, volume, mass, time). Include time dependent questions involving rate.
Design a treasure hunt or rally to send students from station to station to
solve problems.
SUGGESTED
ASSESSMENT STRATEGIES
Being able to measure using
instruments and understanding measurement limitation is fundamental to applying
mathematics. Assessing students’ abilities to solve formula type, or drawing
interpretation type problems should be done in a variety of ways, reflecting
real world methodologies.
Observe
- As students approach
measurement problems check on their ability to use instruments and obtain
results that are both precise and accurate. Particularly where group work
is involved, check for individual students’ abilities to measure and report.
- Note students ability
to translate written problems into sketches, including those whose angles
range from 0° to 180°. Observe students’ ability to use calculators and manipulate
angle representations to solve problems.
Collect
- Check students’ solutions
to problems for clarity and presentation. Students should be able to explain
the strategy they used to solve selected problems.
- Check that students
develop criteria that can be used to review their plans for a parking design.
Criteria might include:
- Will a large car
fit in the designed spaces?
- Can cars easily
get in and out of the spaces?
- Are the identified
dimensions reasonable?
- Is there a sufficient
number of spaces?
Self-/Peer Assessment
- Ask students to comment
or reflect on their ability to visualize relative size and predict the scaling
of 3-D objects. For example, ask them if they feel able to predict surface
areas given dimensions of objects such as spheres.
- Have students create
two or three problems involving sine and cosine laws, including solutions.
Have them exchange problems with a partner. Partners complete, and then comment
on, the construction of the problem and its solution.
- Encourage students to
check each other’s measurements and compare results. Team and group analysis
of others performance is an excellent way of providing alternate assessments.
RECOMMENDED
LEARNING RESOURCES
Print Materials
- Applied Mathematics
10, Western Canadian Edition
Ch.1 (Sections 1.1 - 1.8)
Ch.7 (Sections 7.1 - 7.7)
Projects: Jewelery Design, Candy Boxes, Jet Propulsion, Reach for the Top,
Remote Sensing
Software
Games/Manipulatives
©
2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: November 30, 2000
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