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Course Profile (for a locally developed course)

Essential Mathematics, Grade 9

Unit 2: Applying Ratio and Rate

Time: 18 hours

Description

Students explore ratio and rate in real-life contexts. They apply proportional reasoning through investigations in real-life contexts to solve problems related to measurement, geometry, and data management. Students build on and extend their understanding of fractions to include ratios, decimals, proportions, and per cent. Manipulative materials, diagrams, charts, and drawings are used to gain a greater understanding of concepts such as scale drawings, unit pricing, and sampling. Students develop the facility to translate between and among equivalent numerical forms choosing the representation to best fit the context of the problem. Opportunities to practise the skills of estimation and judging the reasonableness of an answer will be provided throughout the unit.

Strand(s) and Expectations

Number Sense Strand Specific Expectations: NS 1.01, 06, 07, 08, 09, 10, 11,12, 14, 15,16.

Relationships Strand Specific Expectations: RE1.02, 03, 04, 06.

Measurement Strand Expectations: MG2.01.

Activity Titles

What follows is a suggested sequence, with timing, for teaching Unit 2. To a great extent, these activities are teacher-directed. Working with a partner is often done as parallel work where students provide support and encouragement for one another. The need for remediation and further development of skills can be addressed as they arise within the activities. Proportional thinking is particularly important for student success in the Grade 11 Workplace Course and it is therefore critical that the teaching focus on this key concept.

Activity 1	Defining and Using Ratios and “Many-to-one” Rates	75 minutes
Activity 2	A Neighbourhood’s Recycling Efforts: Reading, Creating, and Interpreting Ratios	90 minutes
Activity 3	About Yourself	120 minutes
Activity 4	Applying Rates	120 minutes
Activity 5	Sampling and Estimating Populations	150 minutes
Activity 6	Only the Shadow Knows	120 minutes
Activity 7	Review and Summative Assessment - A Provincial Park Camping Trip	330 minutes

Time for activity completion and consolidation of unit expectations: 75 minutes

Unit Planning Notes

· This unit incorporates numerous concrete materials that must be organized prior to the activity.

· There are opportunities to modify Activities 1, 3, and 4 to use spreadsheets.

Teaching/Learning Strategies

This unit requires flexibility of timing while at the same time it requires structure so that students are engaged in meaningful tasks. Teachers work diagnostically with students to determine what type of support each student requires. Time has been built into the activities to allow for these opportunities and to further develop skills with a context.

Activity 1: Defining and Using Ratios and “Many-to-One” Rates

Time: 75 minutes

Description

In this introductory activity students read and interpret ratios, compare ratios, and write ratios in three different forms: a:b, (read a to b), ^a/_b, and decimal value: 1, (see Resource 2.1, Sample 1). Students also work with multi-link cubes in order to use ratios to describe the composition of a whole (see Resource 2.1, Sample 2).

Strand(s) and Expectations

Strand(s): Number Sense

Specific Expectations: NS1.06, 08, 09

Planning Notes

· Copies of the two worksheets (Resource 2.1) are required for each student.

· Approximately 50 multi-link cubes are required for each student or pair of students.

Teaching/Learning Strategies

Student Activity:

· Students gather data from their classmates to be used for expressing ratios in the three different forms.

· Students work with multi-link cubes and use ratios to describe the composition of a whole.

Teacher Facilitation:

· The teacher may wish to lead a discussion about the use of ratios in our daily lives. Examples taken directly from a newspaper or a magazine would be appropriate.

· Direct teaching may occur to introduce the three methods for recording a ratio, three- term ratios (e.g., boys : girls : teachers = 16 :10 :1), and equivalent ratios (use simple examples). Emphasis is placed on the relationship between the sum of the terms and the total number of items under consideration.

· The teacher completes worksheet 1 with the class. Students work independently on worksheet 2.

· Worksheet 2 uses multi-link cubes to help students explore the relationship between the parts of a ratio to its whole and the idea of equivalent ratios.

Assessment/Evaluation Techniques

Assess the worksheets for completeness and accuracy. Observe two or three students for initiative and independent work habits using the rubric in Appendix 1. An extension assignment could ask students to listen to the radio or television to record how a ratio is used.

Resource 2.1

Defining and Using Ratios Worksheets

SAMPLE 1

Statement of the Comparison	a:b	a b	decimal : 1 value
The ratio of the number of girls in class to the number of boys	8:10	8 10	0.8:1
The ratio of the number of left-handed students to the number of right-handed students
The ratio of the number of left-handed students to the total number of students
The ratio of the number of students who like chocolate to those who do not
The ratio of the number of students with sisters to those without sisters
The ratio of the number of students with brown eyes to those with blue eyes to those with green eyes			not possible to write in this form
The ratio of the number of students who like broccoli to those who do not

SAMPLE 2

Using Ratio to Describe Composition of the Whole (using multi-link cubes)

As you complete each problem, call the teacher over to check your solution. Ensure that your learning partner and you both agree with the solution.

1. Select 25 cubes. Use them to build anything you like. Write the ratio that describes the number of each colour used.

Select a total of 25 cubes in three colours according to the ratio Colour A:Colour B:colour C =12:7:6. Write the ratio for the colours you selected. Use them to build anything that you like (e.g., blue:green:yellow = 12:7:6).

Select a total of 12 cubes in two colours according to the ratio Colour A:Colour B = 7:5. Write the ratio for the colours you selected. Use them to build anything that you like.

Select a total of 14 cubes in two colours according to the ratio Colour A:Colour B = 2:12. For every 1 cube of Colour A there are ____ cubes of Colour B. This means that 2:12 = ___. This is called an equivalent ratio. (Some discussion may occur here.)

Select a total of 20 cubes in two colours according to the ratio Colour A:Colour B = 12:8. Show that for every 3 cubes of Colour A there are 2 of Colour B. This means that the ratio 12:8 is “equivalent” to the ratio 3:2.

Select a total of 18 cubes in three colours according to the ratio 3:2:1. How many of each colour did you choose? What is the equivalent ratio? 3:2:1 = _________.

Select any number of cubes greater than 20 and less than 45, so that two colours are found in the ratio Colour A:Colour B = 5:3. Record the total number of cubes used. How many of each colour did you choose? What is the equivalent ratio? 5:3 = ___________.

Activity 2: A Neighbourhood’s Recycling Efforts: Reading, Creating and Interpreting Ratios

Time: 90 minutes

Description

Students use diagrams to represent two- and three-term ratios. They also develop the concept of proportionality to solve simple problems.

Strand(s) and Expectations

Strand(s): Number Sense

Specific Expectations: NS1.01, 06, 07, 10, 15.

Planning notes

· Each student requires copies of the worksheets, blank paper, and a ruler.

Teaching/Learning Strategies

Student Activity:

· Students complete worksheets that include writing ratios from diagrams, calculating “many-to-one” ratios, analyzing the data, and using a ratio to draw a diagram.

· Students will complete the worksheets with teacher direction and engage in a discussion of their results and other incidental topics, such as recycling, as they arise.

· Finally, they solve simple problems involving proportionality.

Teacher Facilitation:

· Initiate discussion about the importance of recycling. Related information may be obtained through the geography department or the Internet.

· Related career opportunities could be discussed at this time.

· Review the steps to change a ratio to a “many-to-one” rate in order to make comparisons.

· Lead students through determining the ratios and rates for the first two neighbourhoods. Have students complete the chart while the teacher circulates around the room giving assistance where necessary.

· Direct instruction is required to help students to solve proportion problems. Students should build on their understanding of equivalent fractions to solve these problems.

Assessment/Evaluation Techniques

Collect and assess work for completion and quality of response. Observe a few students for initiative, working independently, or oral communication using the rubric Appendix 1.

Resources

Wiggins, L., J. Musson, G. Hartwig, and G. Barry. Everyday Math 1. Toronto Board of Education, 1980.

Wiggins, L., J. Musson, G. Hartwig, and G. Barry. Everyday Math 2. Toronto Board of Education, 1980.

Resource 2.2: A Neighbourhood’s Recycling Efforts Worksheets

Sample 1

The following is a neighbourhood in which some homes recycle. An R means the household recycles and a blank square means they do not recycle.

Draw a diagram showing the number of homes that do and do not recycle and complete the chart.

Neighbourhood A 20 houses with 12 who recycle.

Neighbourhood B 17 houses with 12 who recycle.

Neighbourhood C 12 houses with 8 who recycle.

Neighbourhood D 21 houses with 7 who recycle.

Neighbourhood E 27 houses with 15 who recycle.

Neighbourhood	Number Who Recycle	Number Who Do Not Recycle	Total Number of Dwellings	Ratio of Recyclers to Non-Recyclers	Decimal:1
Example A	12	8	20	12:8 = ____	1.5 : 1
B
C
D
etc.

2. Which neighbourhood gets the award for being the best recyclers? Explain why you have decided this.

Does this neighbourhood have the most homes?

List the neighbourhoods in order from the one with the greatest rate of recyclers to the one with the worst rate of recyclers.

Draw a possible neighbourhood for each and make up some of your own ratios and draw diagrams for them.

11:9

15:5

3:1

This is an opportunity for students to spend more time developing their skills in representing ratios using diagrams. For example;
salad dressing	oil : vinegar = 3:2

Sample 2

In the neighbourhood you just drew, the ratio of recyclers to non-recyclers was 3:1.

You drew

In a neighbourhood of 20 homes, you drew

If the neighbourhood was larger so that there are 80 homes and the ratio remained the same then;

Therefore, in a neighbourhood of 80 homes we would expect that 60 are recyclers.

This is an opportunity for students to develop their skills in solving problems involving proportions. Problems should extend to questions such as:

2 = ?

7 72

In this case students make judgements to the appropriateness of the answer in relation to the context of the problem.

Activity 3: About Yourself

Time: 120 minutes

Description

Students explore the golden ratio and make connections to the ratios of measures taken from human anatomy. Students graph full height versus navel height and determine the mean line of best fit.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships

Specific Expectations: NS1.06, 08, 09, 16; RE 1.03, 04.

Planning Notes

· Students require rulers, graph paper, calculators, metre sticks, and measuring tapes.

· Chart paper should be posted for students to record their data.

· Attach measuring tapes to the wall or doorjamb in order to accurately measure the heights of students. When measuring the lengths of limbs students should be instructed to measure themselves as accurately as possible. If working with a partner to measure, students must be sure to use measuring strategies that do not involve bodily contact. An example would be in the measurement of the distance from the floor to the navel. If the tape measure is attached to the wall a student is able to find their own navel without uncovering it, place their thumb at the navel, and stretch out the baby finger to the tape measure. This gives the required measurement.

· This activity can also be completed with the use of a spreadsheet. A computer lab should be booked in this case.

· Teachers may wish to provide other information related to the golden ratio or make overhead acetates of different examples where it has been used in art, architecture, or where it occurs in nature.

Teaching/Learning Strategies

Student Activity:

· Students fill in the data on a worksheet (see sample, Resource 2.3) and also place their data on a chart or the blackboard at the front of the class.

· The data could be entered onto a spreadsheet if the technology is available.

· Each student calculates the indicated ratios from the worksheet or using a spreadsheet.

· Students look for ratios that are near to 1.6:1 and a discussion about the golden ratio follows.

· They discover that the ratio of their full height to their navel height is approximately 1.6:1.

· Each student then graphs the full height to navel height ratios for the entire class and discovers that it is a linear relation. They use a grid with labelled scales to draw the mean line of best fit. By choosing any point on the line, students determine the full height to navel height and then convert it to a decimal ratio of 1.6:1.

Teacher facilitation:

· To introduce golden ratios to the class a series of rectangles can be produced on an overhead acetate of which one is in the ratio of approximately 1.6 to 1. Students can be asked to select which rectangle they think is most visual appealing. Statistically most students select the golden rectangle.

· Information about the golden ratio can typically be found in most math textbooks.

· This activity requires great attention to classroom organization. If computers are used, spreadsheets should be set up ahead of time. Attach tape measures to the wall or doorjambs before class. Ample rulers, tape measures, and metre sticks must be available.

· This lesson is best completed in small, teacher-directed portions.

· Supply each student with worksheets and graph paper with labelled axes.

· Chart paper needs to be posted to collect the data for full height and navel height.

· Circulate to provide assistance where needed.

· Teachers can use the results from the "Bounce-ability" activity (Unit 1: Activity 13) to extend this activity.

Assessment/Evaluation Techniques

Evaluate the worksheet and graph for accuracy and completeness. Use Appendix 2 - Observing Students Rubric to observe a few students as they work. If the class has progressed to working well in groups you can also evaluate teamwork skills. The activities are completed together as a class, in small chunks, with teacher direction. You may choose to have a particular part of the exercise, such as measuring, done in an independent fashion and would weight that work more heavily. It is important to know your class and their strengths before deciding which method(s) of assessment is most appropriate.

Resources

Garland, R. “The Golden Proportion Poster”. Addison Wesley Longman, 1990.

Resource 2.3: About Yourself

Sample Worksheet

Comparisons	Ratio Measures	Decimal:1
full height:navel height
full height:double arm span
full height:arm span
knee height:foot length
knee height:hand span
foot length:hand span
fingertip to shoulder:fingertip to elbow
head to waist:head to neck
head to nose:head to chin

Activity 4: Applying Rates

Time: 120 minutes

Description

Students collect data from personal experimentation and use it to form rates. They calculate rates from given information and solve problems involving rates.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement

Specific Expectations: NS1.10, 11, 12, 14, 16; MG2.01.

Planning Notes

· Find a section of hallway or school yard where you can walk and mark off 5- to 10-metre intervals.

· Obtain the use of a stopwatch or watch with a second hand.

· Provide copies of the worksheets (Resource 2.4).

· One student needs a chart drawn on a piece of acetate to fill in the data as it is collected.

· Book the computer lab and prepare for using spreadsheets if you are going to include the use of technology for this activity.

Teaching/Learning Strategies

Student Activity:

· In a small group, student will complete this activity. As one student walks back and forth along a marked section of hallway for one minute, a second student will use a watch and be the timer while the third will keep track of the distance covered by the walker. Another student will record the distance walked in one minute on a piece of acetate.

· When the students return to the class they will place their data on the overhead display for all students to record.

· Students calculate the speed in km/h and answer the questions (see the sample worksheet).

· Students calculate some rates provided on a worksheet.

· Students use rates to solve problems.

Teacher Facilitation:

· Begin with a discussion of “What is a rate?”, “What rates can you name?” (e.g., km/h, $/hour, ¢/Litre).

· In a hallway, empty room, front sidewalk, etc., mark off a straight stretch of walkway in 5- to 10 metre intervals using masking tape.

· Spreadsheets could be used to organize and analyse the data.

· As an additional or alternative activity, teachers could have students measure their heart rates over a 15-second period and determine the number of beats per minute, per hour, per day, and per year. Students can then compare the difference in the number of beats.

Assessment/Evaluation Techniques

A variety of techniques and strategies can be used to assess and evaluate this activity. Parts of the rubrics from Unit 1 - Appendix A and B can be used. The pencil and paper work can be marked for completion and accuracy. Computer use can be assessed if this is the third or fourth time that they have been used. Communication skills can be assessed based on the responses given and the strategies chosen for answering the rate questions at the end of the activity.

Resources

Montesanto, R. and D. Zimmer. Numbers and Patterns. D.C. Heath Canada Ltd., 1995.

Resourse 2.4: Applying Rates Worksheets

Sample 1 (spreadsheet optional)

A	B	C	D
Name	Distance Walked in Metres in 1 Minute	Distance Walked in km in 1 minute (column B ¸ 1000)	Distance Walked in km in 1 Hour (column C x 60)

How do the rates compare to the speed of a car in the city?

Sample 2

Determine the unit rates for each of the following.

480 km driven in 4 hours 480¸4 =___________ rate = __________km/h

3 cans of iced tea for 99¢

3 cans of beans for $1.47 rate =___________$/can

or rate =___________¢/can

etc.

(Once students are comfortable with calculating rates and giving the proper units, proceed with some application problems. Much prompting will be required. It may be necessary to read and explain the questions to the students.)

3. Each student in a club helped with a project. The club was paid $767. The chart shows how long each person worked.

a) What hourly rate of pay did they get?

b) How much did each person earn?

Hourly rate = Total $ paid to club

Total hours worked

Name	Hours Worked	Total Pay
Sue	45	45 x hourly rate =
Emily	20
John	15
Jim	22
Rob	16
Total

The average high school student eats 6 slices of pizza per week.

a) How much pizza would be eaten in one year by one student?

b) How much pizza would be eaten in one year by all the students in your school?

c) If there are 12 slices per pizza, how many pizzas are eaten in each of the above cases?

d) If each 12 slice pizza has 200 g. of sauce on it, how much sauce is used?

e) If a bag of flour costs $11.50 and you get 75 crusts from that amount of flour, how much does the flour cost for each pizza. How much does the flour cost for each of the answers in a) and b)?

What is the mass of garbage produced by one person in one month? (What assumptions are you making?)

How many plastic bags does one person use in one year?

This would be an appropriate time develop skills in the use of unit rates in contexts such as comparison shopping to determine the best buy.

Activity 5: Sampling and Estimating Populations

Time: 150 minutes

Description

The exercises in this activity focus on proportional thinking. Students examine sampling techniques in order to estimate the size of a population. They perform an investigation which simulates a “capture-tag-recapture” method for determining wildlife population size.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships

Specific Expectations: NS1.01, 06, 15; RE1.02, 06.

Planning Notes

· These activities can be completed with pairs of students sharing materials. However, have each student record their work individually.

· Students require copies of worksheets and overhead transparencies of the grids in the worksheets. The teacher also requires transparencies of the worksheets and an overhead projector.

· Each pair of students requires a die and a container with approximately 800-1200 beans, (e.g., white kidney beans). To minimize spillage, students should also be given lid of a box to perform their investigation.

· Materials from the Ministry of Natural Resources (MNR) on wildlife populations, and procedures for tracking endangered species would be helpful in generating interest and questions around these exercises. A guest speaker may be invited to speak with the students.

· Student performance in these activities is assessed using rubrics and therefore it is important to review the criteria with the students at the beginning of the exercises.

Teaching/Learning Strategies

Student Activity:

· The students complete the work on estimating population sizes using simulated aerial photograph worksheets and an overhead transparency (investigation 1, 2 and 3).

· Secondly, students will complete an activity to simulate the capture-tag-recapture method (see investigation 4).

· Students participate in discussions throughout the exercises concerning factors that influence the sampling of populations.

Teacher facilitation:

· Prepare copies of the worksheets and transparencies of the grids to overlay on top of the "aerial photographs".

· Prepare a container holding between 800 and 1200 beans and a box lid.

· Place students into working pairs who will support one another throughout the estimating population activities, though students are required to record their individual work.

Exercise 1

Estimating the Size of a Crowd from an Aerial Photograph

· The teacher leads the class in a discussion about estimating populations and population densities. Questions that could be asked are:

1. How would you count the number of people attending a Canada Day celebration on Parliament Hill or a political rally on Queen Street in front of the Provincial Parliament Buildings?

2. How would biologists count the number of moose in northwestern Ontario, the number of fish in a stream, or the number of Peregrine Falcons in Algonquin Park?

· The teacher encourages discussion about how crowd size is estimated and the factors that make the counting difficult.

· Distribute a copy of an illustration that represents an aerial photograph of a crowd at a rally.

Grid overlay for the diagram

	1	2	3	4	5
A
B
C

Aerial Photograph of a political rally

· The teacher models the procedure for analysing the aerial photographs. Place the grid acetate on top of the photograph.

· A class discussion concerning how many squares to select, how the number of squares influences accuracy, and how to select a square should take place.

· At this point, the teacher can suggest (if students haven't already) the use of a die or pieces of paper numbered 1 through 4 and A through C to help them randomly select locations.

· The teacher may have different students try different number of squares to help illustrate how the number of areas considered influences outcome.

· The teacher and students work together to complete the following computations.

Sample worksheet

1. Complete the following table

Location (code)	Number of people
1A
3C
etc.
Total

2. average number of people per square = total number of people

number of squares counted

3. estimated total = average number of x number of squares in grid

people per square

· The teacher distributes another aerial photograph of caribou on Amethyst Island and students complete the prepared material. As students work to complete the exercise the teacher circulates to provide support.

· The teacher also provides a third aerial photograph of boats in a harbour attending a regatta that could be completed as an individual assignment. Some students will continue to require teacher support.

Exercise 2

Sample worksheet for Caribou on Amethyst Island

Each year a fall caribou hunt is planned for Amethyst Island. It is important to determine how many caribou live on the island before the Ministry of Natural Resources (MNR) issues hunting licenses. The drawing below was taken from the air of the herd on Amethyst Island. Use a technique similar to the one used in the previous exercise, to determine the number of caribou that are on Amethyst Island.

1. Scale 1 square = 1km2 (one km by one km)

total area = ______________________

2. Complete the following chart.

Location (code)	Number of caribou
1A
3C
etc.
Total

3. Average number of caribou per square = total

number of squares counted

4. Estimate the total number of caribou on the island.

Estimated number = average number of x total number

of caribou caribou per square of squares

5. If the MNR has decided that one out of every six caribou can be hunted this year and each hunter can only take one caribou, how many hunting licenses should be issued?

Exercise 3

Sample worksheet for Manitouwadge Lake sailing regatta

Each small boat (no mast) has two people on board.

Each large boat (with mast) has six people on board.

· Students code the grid and choose a reasonable number of squares to count for the sample. They then complete a chart like the one below to determine the population size of the number of people attending the regatta. They explain their reasoning for their choices of grid squares.

Sample Chart

Grid Square Number	Number of small boats	# of people on small boats(2 people/boat)	Number of large boats	# of people on large boats (6 people/boat)	total # of people in the grid square

1. What is the average number of people per grid square?

2. If the Manitouwadge Lake Fishing Association had charged $2.00 per person attending the regatta, how much money could have been earned?

3. If the association had decided to charge $5.00 per small boat and $10 per large boat, how much could have been earned? (teacher direction will be required here)

4. What did you consider when choosing the grid squares for your sample?

Exercise 4

How Many Bass Are There in Lake Nipigon?

· Prior to beginning the experiment, discuss how the MNR sets hunting season regulations for moose and deer or fishing regulations for pickerel, trout, bass, etc. One method used to count animal and fish populations is the capture-tag-recapture method. The teacher explains that they will simulate this method by using a container holding a number of white beans. They are to imagine that each bean is a bass in Lake Nipigon in northwestern Ontario. The student’s job is to estimate the number of bass in Lake Nipigon without actually counting all of them.

· Pass out the container of beans and the student worksheets (a container with a lid would be helpful).

· The teacher will lead the students through steps 1, 2, 3, and 4 and then circulate as the students repeat the procedure four more times. Each student in the working pairs will have two opportunities to draw a sample.

· After the students have completed the investigation, a discussion takes place to consider the results.

Sample Instructions

Your group needs a container with a number of white beans (bass in Lake Nipigon). Work with your partner to perform this experiment.

1. Remove exactly 100 beans from the bucket and mark them with an X using a ballpoint pen or a highlighter. (Make sure it does not smear onto the other beans.)

2. Put the marked beans back into the container, and mix them thoroughly with the unmarked beans.

3. Scoop out a handful of the beans without looking into the bucket, (this usually works out to be about 40-55 beans). Count the number of marked and unmarked beans in the drawn sample. Each of you will record this data onto a chart in your notebook similar to the one below.

Sample	1	2	3	4	5
Number of marked beans
Number of unmarked beans

4. Return the sample to the bucket, and thoroughly mix the beans together again.

5. Repeat steps 3 and 4 at least four more times.

6. Study the data collected and use your results to estimate the total number of beans in your bucket

· Students should be prepared to explain how they made their estimate. Students should be prepared to draw comparisons between what they learned from doing this experiment and how biologists count deer or fish populations.

· Once students have completed the chart a teacher-directed lesson follows that will show students how to use proportions to determine the number of bass in the lake.

Assessment/Evaluation Techniques

· Collect and assess each student’s work on the Manitouwadge Lake Regatta, for accuracy of calculations and completeness. Resource 2.5 can be used to assess student performance.

· To assess student initiative and work habits refer to Appendix 1 - Learning Skills Rubric

Resources

Kelly, B. Impact Math: Data Management and Probability. Ministry of Education and Training, 1998.

Resource 2.5

Rubric for assessing the Lake Manitouwadge Fishing Regatta

	Level 1	Level 2	Level 3	Level 4
Use of appropriate coding technique	- needs much prompting to complete the task	- needs some prompting to complete the task	- needs minimal prompting to complete the task	- independently completes the task
Chooses an appropriate sampling technique (includes low and high density squares)	- has no apparent strategy for choosing an appropriate random sample, needs much teacher direction	- with teacher support, the student chooses and uses an appropriate random sample	- with minimal prompting from the teacher, is able to choose and use an appropriate random sample	- independently chooses and uses an appropriate random sample
Communicates the sampling strategy	- is very vague about the choice of sampling strategy and has no apparent strategy for the choice	- communicates vaguely using incomplete justification and often uses inappropriate terminology	- communicates the strategy used with minimal help and some justification, uses appropriate terminology	- independently and clearly communicates the strategy used and gives a clear justification for the choice using appropriate terminology

Activity 6: Only the Shadow Knows

Time: 120 minutes

Description

Students are introduced to the concept of similarity and how to apply ratios to measure objects in their surroundings.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships, Measurement

Specific Expectations: NS1.06, 08, 15, 16; RE1.03; MG2.01

Planning Notes

· This activity can be done individually, in groups of two or three, or as a class. As students are working outside, safety is an important consideration. The teacher will want to discuss specific safety issues depending on where the students will be taking their measurements. The teacher will also want to place very specific parameters around what the students are supposed to do while outside the classroom.

· Students require metre sticks and/or tape measures. Some discussion about accuracy of measurement will be helpful.

· Worksheets will be required for each student.

· The measurements of objects (e.g., buildings, trees, flagpoles, goalposts, people) need to be taken on a sunny day when distinct shadows are visible. Careful explanation of how the measurements are to be taken is important. For example, if you wish to know the height of the flagpole, measure the shadow from base of the flagpole to the top of its shadow.

· If weather conditions do not permit the measurement of shadows then teachers can construct similar triangles with the aid of clinometers and measure objects within the school.

· An overhead projector is also required for this activity.

Teaching/Learning Strategies

Student Activity:

· Students go outside and carefully measure the lengths of the shadows of three objects. They also measure the length of the shadow of a metre stick placed perpendicular to the ground.

a) Hold a metre stick so zero is touching the ground with the stick perpendicular to the ground.

b) Measure the length of the shadow of the metre stick using a second measurement tool. This provides you with the ratio for calculating the heights (e.g., 1 m:0.65 m).

c) Carefully measure 3 shadows, measuring from the base of the object to the top of the shadow.

d) Record the measurements in the first two columns of a table (see sample, Resource 2.6).

e) Return to the classroom to learn how to do the necessary calculations for the third column.

f) Record the measurements while you are outside and complete the calculations in the third column when you return to the classroom. The teacher will guide you through the first calculation and you will complete the others yourself.

· Students then work with the teacher using the overhead projector to complete the calculations.

a) Project a triangle onto the blackboard and carefully trace it. Move the projector closer to the blackboard and trace the resulting shape (the same as the first one but smaller), then move the projector farther from the blackboard and trace the shape again.

b) Measure the sides of each of the triangles. Use many-to-one ratios to compare the sides to see that the three triangles are proportional to one another. These are called dilatations.

· Each student takes a piece of centimetre square graph paper and draws a right-angled triangle with side lengths 3 cm and 4 cm at the right angle.

· Students draw a right-angled triangle so that its sides are in the ratio 1:2 of the first triangle compared to the second triangle. (Double the lengths of the two sides and draw in the hypotenuse.)

· Measure the hypotenuse of each of the triangles. Take the measurement from the first triangle and divide it by the measurement from the second and see that the answer is also 1:2.

· This process will be repeated two or three times (include a triangle where the image triangle is smaller than the original triangle). Triangles can be presented with some of the measurements marked on them allowing the students to calculate the other dimensions without drawing the scale diagram.

Teacher facilitation:

· Go outside with the students to do the measuring. Make sure the measuring tools are being read correctly. Keep the students in a small, easily supervised area.

· Lead students through the initial calculations each time and then circulate to provide support as they complete the exercise.

· Direct the work with the overhead projector. Show the students how to set up their answers in a suitable manner.

· Provide the graph paper and direct the final activity ending with a worksheet with problems to solve.

Assessment/Evaluation Techniques

Assess the worksheets for accuracy and completion. Use Appendix 1 - Learning Skills Rubric to observe a few students as they work.

Resource 2.6: Only the Shadow Knows

Sample Chart

Object	Shadow Height	Work
Object	Shadow Height	metre stick length = object height metre stick's shadow length object's shadow length
e.g., tree	5.7 m	_1_ = tree height 0.65 5.7 0.65 x tree height = 1.0 x 5.7 tree height = 5.7 ¸ 0.65 =___________m

Activity 7: Review and Summative Assessment - A Provincial Park Camping Trip

Time: 330 minutes

Description

This final activity is comprised of six smaller assignments in which students use ratios and rates to plan a camping trip. The activities involve reading charts of park facilities, calculating distances between parks using an Ontario road map, fuel requirements and costs, travel time, campsite fees, planning a trip, and completing a scale drawing of the campsite. The first five parts of the activity provide the necessary review of the unit to complete the summative assignment. Allow approximately 180 minutes for completion and correction of these assignments. Part six of the assignment could be used as the summative assessment piece for this unit. Allow approximately 150 minutes for its completion. Each year, Ontario Parks publishes a booklet that is full of maps, charts and information about Ontario’s Provincial Parks that are available free of charge (see resources). Other resources may be substituted.

Strand(s) and Expectations

Strand(s): Number Sense, Relationships

Specific Expectations: NS1.06, 07, 08, 09, 10, 12, 15, 16; RE1.06.

Planning Notes

· Obtain Ontario Parks brochures from your local travel information centre or by calling Travel Ontario 1-(800)-ONTARIO or MNR 1-(800)-667-1940 or by visiting a provincial park in your area.

· Obtain Ontario road maps.

· Provide an outline map of Ontario.

· Organize the worksheets with clear instructions and charts to record solutions.

· Provide 0.5 centimetre square graph paper.

· Students should be provided with the assessment rubric for the summative assessment before beginning the activity.

Teaching/Learning Strategies

Student activities and teacher facilitation are given for each part of the assignment.

Part I: Provincial Parks

Student Activity:

· Use the index list in the front of the Ontario Parks booklet to compare the park classifications using ratios.

e.g., Recreational:Historical = 58:1

Natural Environment:Nature Reserve:Wilderness = 40:2:4

Recreational:Total number of parks = 58:107

· Use the chart for your region to obtain information about 7 or 8 parks. Also use the Ontario map in the centre of the booklet and an Ontario road map to locate the cities and towns for the chart.

· Organize your information in a chart:

Sample chart

Name of park
City ( from park address)
Nearest large city(use road map)
Classification(use index page)
Number of sites
Number of sites with electricity
Is there group camping?
Is there swimming?
Are there showers?
Is there a boat launch?
Is there a Nature/ Visitor Centre?

· The teacher will provide some examples to compare park facilities using ratios (e.g., number of sites with electricity to total number of sites for each park, number of park with swimming to number of parks with boat launches).

Teacher facilitation:

· Guide students through the booklet to familiarize them with the layout and coding.

· Provide clear, well-organized worksheets.

· Circulate to provide support.

· Encourage students to make up additional comparisons using ratios.

Part II: Distances Between Parks

Student Activity:

· Students determine the distances between 8-10 pairs of provincial parks using an Ontario road map.

· There are 3 methods for determining the distances:

Method A: Determine the city nearest to the chosen park that is on the distance triangle on the provincial road map. From the distance triangle, read and record the distance in km.

Method B: Carefully follow the highways between two parks and total the distances marked on them.

Method C: Use the scales on an Ontario road map to calculate the distances.

Sample Chart

Parks	Nearest cities	Distance in km
e.g. Bonnechere to Silent Lake	Pembroke to Bancroft
etc.

Teacher Facilitation:

· Initially, it requires much direction to have students use the map index and distance triangle. The Geography teacher may be able to provide assistance or pre-teach this in class.

· Encourage the use of the map in the centre of the booklet to help locate the parks and cities.

· Teachers should discuss with that they are looking for the closest cities to the parks.

Part III: Fuel Expenses

Student Activity:

· Calculate the fuel consumption for your vehicle.

· Each vehicle has a fuel consumption rating in L/100 km.

A sample problem could be:

Your vehicle has a fuel consumption rating of 9.0 L/100 km. (It uses 9.0 L to travel 100 km).

How much fuel is needed to drive from __________________ Provincial Park to

__________________ Provincial Park?

fuel consumption rating = fuel required

100 distance between the 2 provincial parks

9.0 = fuel required

100 distance between the 2 provincial parks

· Calculate the gasoline cost for the given trip by multiplying the fuel needed by the average cost of a litre of gasoline.

e.g., Cost of fuel = fuel required x cost of gas in $/L

Teacher Facilitation:

· Provide worksheets indicating which parks you wish the students to use.

· Provide booklets and road maps.

· Provide a variety of problems to solve (e.g., change the fuel consumption rating, use a variety of parks, or give the amount of fuel used and ask for the distances between parks).

· When calculating the costs, vary the price of the gas. Students tend to have difficulty with the gas prices when using decimals of a cent like 56.9¢/L and may need assistance to write it as $0.569/L.

Part IV: Distance, Speed, and Time

Student Activity:

· Calculate the average time required for each trip.

e.g., time = distance ¸ speed or time = distance

speed

Teacher Facilitation:

· Discuss average speed and time.

· Discuss other reasons that would affect travel time.

· Provide a variety of problems. Give time and speed and ask students to calculate distance or give time and distance and ask students to calculate speed. Provide careful modelling of each type of solution.

Part V: Park Fees

Student Activity:

· Given the number of nights stay (e.g., 4 nights at each park), the number of people, the name of the park, etc., calculate the total fees for the stay.

Teacher Facilitation:

· Provide a variety of problems and guide the students through the pricing key in the Ontario Parks booklet.

· Complete a few examples together and then have the students complete a few on their own.

Part VI: Let's Go Camping

Student Activity:

· Plan a camping trip that will visit four provincial parks. Spend three nights in each park.

· Start and finish at your home town/city.

Sample Chart for Trip Planning

Park	City or town	What I will do there	Distance to the next destination
------	hometown	I will get packed and leave.	140 km
Park A	Town A
Park B	Town B
Park C	Town C
Park D	Town D
-------	hometown		-------

1. What will you do at each park? (Place your answer in the chart above.)

2. Select 4 parks. Mark the parks and the closest city to each on an outline map of Ontario. What order makes the most sense for your route? Connect the parks with a line in that order. Outline maps are available from the Geography department.

3. Determine the distance between each pair of locations. Mark it on the chart like the one above. Use the road map to determine the distances.

4. Determine the travel time between parks and the total travel time if you average 65 km/h.

5. Determine the fuel needed if your fuel consumption rating is 8.2 L/100 km.

6. Determine the cost of fuel used if the average cost is 59¢ / L.

7. What will the cost be for four nights stay at each park?

8. What is the total cost of the fuel?

9. What are the total park fees?

Sample Worksheet for the Calculations

Park	City or Town	Distance from next destination	Travel time	Fuel required	Cost of fuel	Campsite fees
-------	home
Park A




Totals

· The information for the first three columns comes from the sample chart for Trip Planning. The fuel required will have students set up a proportion as in Part III. The campsite fees are calculated from the booklet based on the results of Part V. The total costs are tabulated in the last row of the chart.

· Each student is provided with an outline sketch of a campsite (Resource 2.7). They are given a piece of graph paper with the outline of the tent (a 2x1 rectangle), trailer (a 4x2 rectangle), picnic table (a 2x1 rectangle), two lawn chairs (each one is a 1x1 square), fire pit (a 1x1 square), truck (a 3x2 rectangle) and canoe (a 1x0.5 rectangle). They will enlarge each item by a ratio of side length of 1:2 on the graph paper (use 0.5 centimetre square paper). Cut out the enlarged items and arrange them on the campsite taking into consideration the location of the road, lake, walking areas, safety, etc.

· Students explain their reasoning for the placements orally to the teacher or in writing.

Teacher Facilitation

· The scope of this activity is very large and can be overwhelming to both the teacher and the student. Preparation through the review exercises and presentation of the worksheets are vital for success. The activity should be split into small amounts and this should be reflected in the design and structure of the charts, worksheets, and instructions.

· In order to keep track of exercises, students could be provided with a folder with a checklist on the front of it in which to put each completed piece of work.

· Provide copies of the rubric (Resource 2.8) for assessment.

· Encourage students to use all of their corrected sheets from parts I-V.

· Circulate to provide support and assistance.

· It would be beneficial to have another teacher or a senior student or peer assistant in the room for the classes when the summative assessment is being completed.

· Despite the fact that work is often recorded in charts, students should be encouraged to show their calculations or to explain the steps taken.

· 75 minutes of extra time has been provided for completion of this activity if required.

Assessment/Evaluation Techniques

Due to the scope of the activity, a variety of assessment and evaluation techniques can be employed. The review exercises could be used as formative evaluation. These marks should not count toward the final mark for the students and should be used only as preparation for the summative assessment. For the assignment in Part VI, many of the calculations could be marked for completeness and accuracy.

Resource 2.8: A Provincial Park Camping Trip Rubric could be used to assess the rest of the final assignment. Students must be made aware of the method of assessment and evaluation at the beginning of the activity.

Resources

Ontario Parks (parks guide is available every year free of charge). Obtain copies from your local tourist information bureau, from Travel Ontario 1-(800)-ONTARIO, from the MNR 1-(800)-667-1940 or from your local provincial park.

Resource 2.7

My Campsite at the Provincial Park

Resource 2.8

A Provincial Park Camping Trip Rubric

	Level 1	Level 2	Level 3	Level 4
Learning Skill: Work Habits	- does not refer to notes from review or refuses to work unless given assistance	- rarely refers to the review notes before seeking help	- frequently refers to review notes before seeking help	- always refers to review notes before seeking help
Learning Skill: Work Habits	- does not check solutions before submitting	- rarely checks solutions; will make some corrections if they are pointed out	- frequently checks solutions before submission	- always checks solutions before submission
Knowledge/ Understanding · understands the use of ratios and rates to make comparisons (NS1.06, NS1.07, NS1.08)	- needs assistance to form ratios and rates	- usually forms ratios and rates correctly, sometimes requires assistance	- often forms ratios and rates correctly, occasionally requires assistance	- forms ratios and rates correctly without assistance
Thinking/Inquiry/ ProblemSolving · is able to follow the question being asked · recognizes whether the question uses a ratio or a rate and chooses a method to use in solving the problem (NS1.09,NS1.10, NS1.12)	- follows simple mathematical arguments	- follows mathematical arguments of one or two steps	- follows more complex mathematical arguments and may require assistance	- independently follows more complex mathematical arguments
	- has limited success in choosing the strategy for solving the problems involving costs, distances, etc.	- has some success in recognizing the type of question being asked and choosing an appropriate strategy	- has reasonable success in recognizing the type of question being asked and usually chooses the appropriate strategy with little or no assistance	- has success in recognizing the type of question being asked and chooses the appropriate strategy
Communication · describes orally or in writing what they will do at each park · justifies orally or in writing the decision for placement of items on the campsite (RE1.06)	- gives one word answers, with little variety in choices of activities	- uses phrases or some sentences and has some variety in choice of activities	- with considerable clarity describes the choices in activities	- clearly describes the choices of activities
	- does not base answers on the chart of park services	- bases choices on chart of park services with some accuracy	- bases the choices on the chart of park services with accuracy	- bases the choices on the chart of park services with thought given to time of day and surroundings
	- little thought given to spacing when placing items	- is able to give some justification for placement of items, with thought given to spacing	- justifies placement of items and has given thought to use and spacing	- clearly understands the issues of placement with respect to use and spacing but also to safety

Appendix 1: Learning Skills Rubric

Learning Skills	Needs Improvement	Satisfactory	Good	Excellent
Initiative	- begins task with prompting	- begins task with some prompting	- often begins task promptly	- always begins task promptly
Works independently	- pursues alternate strategies if initial one does not result in a solution with considerable help	- pursues alternate strategies if initial one does not result in a solution, with help	- pursues alternate strategies if initial one does not result in a solution with some prompting	- pursues alternate strategies if initial one does not result in a solution
	- easily distracted	- easily distracted but will usually return to task	- stays on task with little external prompting	- stays focussed without external prompting
Teamwork	- meets the expectation of own role with limited effectiveness	- meets the expectation of own role with moderate effectiveness	- meets the expectation of own role with considerable effectiveness	- meets the expectation of own role with high degree of effectiveness
	- assumes a passive role, allows other group members to regulate activities	- encourages equal participation of all group members some of the time, occasional uses exclusive language or negative tone	- encourages equal participation of all group members most of the time, seldom uses exclusive language or negative tone	- encourages equal participation of all group members at all times, through the use of appropriate language and tone of voice
	- is often the cause of group conflict	- helps deflect group conflict with limited effectiveness	- helps deflect group conflict with considerable effectiveness	- helps deflect group conflict with considerable effectiveness
Work habits	- does not refer to class notes, other resources or refuses to attempt work unless given assistance	- rarely refers to class notes, other resources before seeking help	- frequently refers to class notes, other resources before seeking help	- always refers to class notes, other resources before seeking help
	- does not check solutions, unless prompted, requires considerable help to make corrections	- rarely checks solutions (if available), makes corrections with help	- frequently checks solutions (if available), and makes corrections, sometimes with help	- always checks solutions (if available), and makes corrections without help

Appendix 2: Observing Students Rubric

This rubric provides a variety of areas to assess students at work. Not all categories need to be assesed at one time. Choose only four or five students to observe during any one class.

	Level 1	Level 2	Level 3	Level 4
Contribution to the activity	- needs much encouragement to participate	- participates and needs encouragement to share ideas	- participates and shares ideas	- willingly participates and openly shares ideas and encourages others to do so
Application of the Inquiry Process (collect, organize, analyse)	- accepts with limited understanding arguments in choice of methods for collection, organization and analysis and has difficulty making inferences	- understands and follows arguments in choice of methods for collection, organization and analysis of data and making inferences	- uses arguments in choice of method for collection, organization and analysis of data and making inference with some minor flaws	- uses convincing arguements in choice of method for collection, organization and analysis of data and making inferences
	- collects, organizes data under direction of group	- collect, organizes and analyses data with some help from group	- collects, organizes and analyses data independently	- collects, organizes and analyses data effectively and independently
Communication (oral)	- communicates minimal explanation of the solution with limited or no use of mathematical terminology	- communicates vaguely using incomplete justification and often uses inappropriate mathematical terminology	- communicates frequently using correct and appropriate mathematical terminology and some justification	- communicates clearly and consistently using correct and appropriate mathematical terminology and justification
	- requires assistance to write and organize the solution	- requires minimal assistance while writing	- writes a solution independently	- writes a clear, well organized and thorough solution independently
Communication (other, e.g., poster)	- may have limited ability to clearly express result, requires assistance to complete the product	- may need prompting or extra explanations of expectations for completion of product	- completes product and accurately conveys message through product	- completes product neatly and accurately and independently, conveys message effectively and accurately