Course Profile Foundations of Mathematics, Grade 10, Applied, Public

Unit 1: Proportional Reasoning

Time: 18 hours

Unit Description

Students practise the five steps in the inquiry/problem solving process: explore, hypothesize, model/formulate, manipulate/transform, infer/conclude, in a variety of contexts that can all be modelled by a proportional problem. Through exploration and generation of examples, students gain an increased depth of understanding of percent, ratio, and rate problems. Forming and testing hypotheses about the type, size and units of results, improves students’ intuition in situations involving proportions. Techniques for efficient creation of models for proportions are taught and practised. Manipulation of the equations resulting from substitution into a proportion model reinforces algebraic skills. At the infer/conclude stage, students are encouraged to communicate their findings, to reflect on the reasonableness of these findings, and to consciously reinforce or modify their hypotheses.

As with the study of any mathematical model, students should be exposed to examples that illustrate when it should and should not be used. Applications of proportionality in this unit will include interest calculations, currency conversions, direct variation problems, and technical drawings. The applications of partial variation problems and distortion of figures resulting from irregular scales will be examined as an example of inconsistent representations of proportionality.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Overall Expectations: PRV.01.

Specific Expectations: PR1.01, PR1.02, PR1.03.

Activity Titles (Time + Sequence)

What follows is a suggested sequence and timing for teaching Unit 1. Mathematical concepts developed through the activities are noted in [square brackets]. Once the stage has been set through an activity, appropriate follow-up skills or homework are identified. The timing shown for an activity includes enough time for the activity itself, extracting the mathematics from the activity, plus development of the suggested follow-up skills. *255 extra minutes have been allotted for practice of essential skills, review, and quizzes. No time has been allotted to skills identified as prior knowledge. If time is needed for review or re-teaching of these skills, it could be taken from the * asterisked time. The teacher might plan for use of these 330 minutes before beginning the unit, or save them for needs as they develop.

*	Time for review, practice, taking and correcting performance tasks and quizzes	255 minutes
Activity 1.1	Getting to Know Each Other [Find ratios and percents using data gathered from the class and school records] Follow-up homework: informal communication assessment piece	75 minutes
Activity 1.2	Go Fish! [Simulate the tagging and releasing of fish; make a prediction about a population from a sample. Introduce the ‘mathematics on the job’ focus of the course. Emphasize ‘exploring’ actions] Follow-up homework: practice of skills and communication about the processes used.	75 minutes
Activity 1.3	Aerial Photography [Use a random sample and ratios to approximate large numbers] Follow-up performance task: an Inquiry	75 minutes
Activity 1.4	A Distorted View [Create proportional and distorted images; set up and solve proportions without a graph.] Follow-up homework: Optimal Viewing and Fish Story	150 minutes
Activity 1.5	Design Problem [Preserve proportions while creating a scale drawing.] Follow-ups: sharing of designs, journal entry	75 minutes
Activity 1.6	Master Mixer [Set up and solve proportions without a graph; introduce ‘What’s right and what’s wrong?’ scenarios which require confirmation or correction] Follow-up homework: Students create and solve similar problems.	75 minutes
Activity 1.7	Trip to Florida [Use ratios, rates, and unit conversions] Follow-up homework: currency exchanges by hand and using a spreadsheet	75 minutes
Activity 1.8	Pet Vet [Calculate drug doses; identify what’s right and what’s wrong in scenarios. No new expectations are addressed, so these problems could be used for review or for gathering assessment data]	75 minutes
Activity 1.9	Catering [Summative Assessment]	150 minutes

Unit Planning Notes

· This unit is placed first since it sets the stage for Unit 2: Trigonometry and requires very little prior knowledge from Grade 9. Success will be likely with good work habits.

· Teachers may choose to do some brief diagnostic assessment of student's knowledge of equations and lines while Unit 1 is underway. Small bits of review and practice could be required of students, as warm-ups or homework, before Unit 3 is begun.

Materials needed for the activities in this unit include:

· a large number of small items of two colours (e.g., toothpicks, candies)

· an aerial photograph that contains a very large number of people, or items which are close to each other in size. Care should be taken that the photograph was not taken at an angle that introduces the challenge of perspective.

Teaching/Learning Strategies

· Students will often be working in pairs or small groups, but growing independence is also a goal.

· Activities that are suggested as teaching tools could be used as assessment tools, and vice versa, since assessment activities should be learning activities.

Assessment and Evaluation

A variety of assessment tools and strategies are suggested for this unit. Since this is recommended as the first unit in the course, it is suggested that the teacher give the students sufficient formative feedback before a formal assessment, to encourage a positive outlook.

Activity 1.1: Getting to Know Each Other

Time: 75 minutes

Description

Using data collected from a class survey, students calculate ratios and percents to predict data for the entire school population.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

PR1.01 – solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations);

PR1.03 – distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning).

Planning Notes

· The teacher will need to know the size of the school population and have small bags of coloured candies, (one for each pair of students), and protractors.

· The teacher may want to investigate actual statistics on the distribution of colours for the candy being used. Some web sites with this type of information are listed under resources.

Prior Knowledge Required

Students should demonstrate facility in operations with percent in constructing circle graphs.

Teaching/Learning Strategies

Teacher Facilitation: Review the concepts of ratio and proportion using pictures (e.g., shaded: unshaded squares/sections in rectangular/circular figures). Collect some data about the class to create class ratios and percents (e.g., boys to girls; jeans to other attire; running shoes to other shoes).

Student Activity

Students calculate the ratios of the given figures (i.e., shaded to unshaded squares), convert to a percent and construct circle graphs. Students calculate the ratios and percents of the class data.

Teacher Facilitation: It would be useful to have students work in pairs for the Pop Shop activity below. Suggest that students compare the number of each flavour to the whole. Students will need to know the total number of students attending the school. There should be some discussion around the ideas of sampling and using small samples make predictions about larger populations.

Student Activity: Pop Shop

The school cafeteria has challenged your class to predict the quantities of pop it should have available to sell and which flavours of pop they should have available.

1. Determine the number of students in your class that are likely to buy pop for lunch.

2. Determine the flavour and number of pops they would choose to buy, if it was available.

3. Using the information gathered from your class and the information you have on the size of the school population:

a) Calculate the total number of pops that would be purchased in the school in 1 day, 1 week, and 1 month.

b) Calculate the quantities of different flavours of pop the cafeteria should have in stock for 1 day, 1 week, and 1 month.

4. Construct two circle graphs, one that shows the percentage of the class that would buy pop and not, and one that displays the different pop preferences. How would the circle graphs for the entire school be different for 1 day? For a week? For a month? Explain your reasoning.

5. Do you think that the information collected from your class is a large enough sample to make predictions for the whole school? Explain why or why not. How could you improve the accuracy of your predictions?

Teacher Facilitation: Take up the students' work by having pairs share. Ensure that the idea that the circle graphs for the entire school will not change from those constructed for the class because the percentages remain constant. This fact will be important in the next student activity.

The next activity will be started in class and completed as homework. Students will be given a small sampling of coloured candies. The teacher could leave this activity open-ended, as posed below in the fan mail, or could guide the class’s work by:

· gathering the numbers for each colour from each student

· ensuring that all pairs see that totals for the entire class should be found to account for variability from sample to sample

· helping the class to predict the ratios that the company uses for brown, yellow, red, green, orange and blue. The teacher may wish to provide actual figures for the particular candy. For example, actual figures for M&M’s® are: 30% Brown, 20% Yellow, 20% Red, 10% Green, 10% Orange, and 10% Blue.

· helping the class set up models for finding the numbers of each different colour of candy that could be expected in small and big bags of candy. These models could involve proportions, or the multiplication of the number of candies in a bag by the appropriate percentages.

The letter below could be adjusted to include the name of the candy used in class.

Student Activity: Fan Mail

A young fan [of this candy] wrote the following note to the company.

To Whom It May Concern:

I’m a huge fan of Red candy and I always save them for last. Today I was greatly saddened to find that I had only1 red candy in my bag after eating the other colours. Last week I’m sure I had at least 6 in the same size bag! What’s going on? Please explain this to me. Has your company cut back on red candies? I hope to hear from you soon.

Your fan,

Julie

It is your job to create the response to this piece of fan mail. Discuss with your partner what information you should send back to this young fan. You will have to provide some specific information that will help explain “what’s going on”. You may wish to include:

· references to percentages for each colour

· how you got those percentages

· a graph that will give the young fan a visual representation of the percentages for each colour

· any other details that you think will help the fan understand

Use class time to gather the data you need for your response. Your homework is to create the response letter to the young fan.

Teacher Facilitation: A discussion of predicting and sampling should follow. Ask students why their predictions may not match the actual figures and what could they do to improve the accuracy of their predictions. Teachers may suggest students who have access to the Internet look up the web site and use the statistics to generate additional questions (e.g., Will the ratio of Reds change if it is Christmas?)

Assessment/Evaluation Techniques

Assessment of the Fan Mail response need only be formative at this early point in the course. The focus of the assessment should be on communication, specifically:

· students’ ability to integrate narrative and mathematical forms of communication;

· the degree of clarity in explanations and justifications.

Accommodations

Teachers may have ESL/ESD students work with a partner who shares the same linguistic background. Some students may benefit from oral reading of the fan letter and use of a scribe for their response.

Resources

M&M’s® Industrial Candy and Magic
http://www.m-ms.com/factory/history/faq1.html

Serra, Michael. Discovering Geometry – An Inductive Approach. California: Key Curriculum Press, 1993. ISBN 1-55953-200-9

Activity 1.2: Go Fish!

Time: 75 minutes

Description

A common theme throughout this course is “math on the job”. In this activity students examine how a game warden might predict an animal population by tagging animals. Students simulate tagging “animals” and then make predictions by examining the ratio of tagged animals: not tagged animals. Scatterplots are drawn as another method to make predictions.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

Planning Notes

The teacher will need to provide students with objects to simulate the fish. Some suggestions include; toothpicks, popcorn kernels, macaroni, different-flavoured fish crackers, bear crackers, beads.

The teacher will also need to provide:

· a permanent marker for each group for “tagging” toothpicks, popcorn kernels or macaroni;

· containers to hold the “fish” such as bowls, bags, baggies, and envelopes. The investigation will be more effective if the students can’t actually see into the container;

· baggies for nets if using crackers (hygiene);

· graph paper.

Students examine similar sampling techniques in both the applied and academic science programs.

Prior Knowledge Required

Students should be able to solve proportions, construct scatterplots and lines of best fit and use them to make inferences. This could be by hand or by using the graphing calculators.

Teaching/Learning Strategies

Teacher Facilitation: Students should work in pairs to simulate the tagging method used by game wardens, naturalists, and conservation officers to estimate the number of animals, birds, or fish in a certain area. The teacher will provide each pair of students with an envelope of “fish”. It would be best if each pair received a different amount, approximately 50-70 to get accurate results, and the amount should be unknown at this point.

There should be some discussion about the method of random sampling and how that is being used here by mixing up the fish each time a sample is chosen. The teacher should have the students discuss and decide how many fish should be tagged in this case to get accurate results. To avoid confusion later the number of fish the students tag should be something other than 20 (the number in each sample that will be collected).

Other examples of “math on the job” could be brought out here by mentioning the use of random sampling in quality control.

Student Activity: Go Fish!

How might you count the number of fish in a lake? Estimating is a method that can be used when it is impractical or impossible to count the actual number. In order to predict the size of a large population we sometimes examine the characteristics of a smaller sample. To estimate the number of fish in a lake, a game warden will catch and tag a number of fish and then release them back to the lake. After several days when they have had a chance to mix with the other fish in the lake, the warden will catch a sample of fish. The warden then examines the ratio of tagged fish caught: total fish caught and compares this to the number of tagged fish placed in the lake.

Simulate this activity with the “fish” in a “lake” provided by your teacher.

1. “Catch” 20 “fish” and tag them by colouring them with your marker. (If using crackers replace the crackers caught with a different coloured flavour) Replace the tagged fish back into your lake.

2. Shake up your lake to simulate the fish swimming around.

Method A

1. Catch a sample of 15 fish from the lake.

2. Use the table provided to record the number of tagged fish and the number not tagged and then return them to the lake.

3. Shake the lake and repeat a total of ten times. Record your results each time.

Trial	Fish caught	Number tagged	number not tagged
1	15
2	15

10	15

4. Calculate the average number of tagged fish and the average number not tagged.

5. Use ratios to estimate the total number of untagged fish in your lake.

Average # tagged fish in samples	=	Average # tagged fish in lake
Average # untagged fish in samples	=	Average # untagged fish in lake

6. Use the total number of untagged fish that you just found and the total number of tagged fish to calculate the total number of fish in the lake.

Teacher Facilitation: For this method students will be collecting samples of different sizes each time and then graphing their results on a scatter plot of untagged vs. tagged. Please note that setting the graph up in this way allows for the slope of the graph to be comparable to the ratios used above. The graph can be constructed with paper and pencil or with the help of the graphing calculator. Instructions for the calculator are given below.

Method B

In this method you will be collecting samples of different sizes and recording the information below.

1. Collect a sample of five fish and record the number tagged and untagged.

2. Return these fish to the lake and shake. Collect a sample of ten fish this time and record the values below.

3. Complete the rest of the chart, paying careful attention to the changing sample size.

Trial	Fish caught	Number tagged	Number not tagged
1	5
2	10
3	15
4	20
5	25

You will now use your graphing calculator to plot this data.

· Press STAT and EDIT.

· Enter the number of tagged fish in L₁ and the number not tagged in L₂.

· Press 2^ND STATPLOT and ENTER on PLOT1.

· Turn the PLOT ON, choose the first type of graph, set your x-list as L₁ and your y-list as L₂, and choose the first mark.

· Press ZOOM and select 9:ZoomStat.

Now you will draw a line of best fit for your data.

· Press STAT and cursor over to CALC

· Choose 4:LinReg

· Type L₁, L₂, Y₁ (note: to get Y₁ press VARS, cursor to Y-VARS and press ENTER twice)

· Your screen should read like the screen shot to the right.

· Press GRAPH to see your line.

Now you will use your graph and the TRACE button to estimate the total number of fish in the lake. To do this you will need to change your WINDOW setting so that it is larger.

Once you have changed your window:

1. Use the TRACE key and your line of best fit to determine how many untagged fish there are when there are 15 tagged fish.

2. Use these values to predict the total number of fish in the lake.

3. Compare your prediction using this method with your previous prediction. Account for any differences in your results.

4. Use all of the information that you have collected to make a final prediction about the total number of fish in your lake. Justify your prediction.

5. Count the actual number of fish in the lake. How does your prediction compare with the actual number. Account for any differences.

Teacher Facilitation: The teacher could have students find the slope of their line of best fit and help them make connections between this slope and the ratio of untagged: tagged calculated in Method A.

Homework: It would be appropriate to assign textbook questions that practise the component skills of this activity. A question that requires clarity of communication could be assigned as well. An example follows.

A game warden would like to determine the number of deer in a particular area. He believes there are somewhere between 300 and 450 deer. Explain how the warden could use tagging to get a more accurate estimate.

Assessment/Evaluation Techniques

No formal assessment needs to be done at this point. Students should be getting comfortable with this sampling technique for assessment at a later point. The teacher may want to use this activity as an opportunity to observe students and assess their skills in teamwork using the rubric from the Grade 9 profile. Formative feedback on the clarity of their communication would also be appropriate.

Resources

Alexander, Bob. Minds on Math: 9. Canada: Addison-Wesley Publishers Limited, 1994.
ISBN 0-201-56015-1

Activity 1.3: Aerial Photography

Time: 75 minutes

Description

Students develop a method of counting large crowds by dividing an area into smaller sections, counting the number of people in a random section, and use ratios to extend to the whole.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

Planning Notes

· A photo of a crowd will be needed. Suggested resources for this photo:

· school “rooftop” or aerial photo – Some schools have these done for the yearbook.

· photos of large crowds from books or Internet

· Each student pair will need a die and a spinner (students could make their own spinners).

· Students will examine similar sampling techniques in both the Grade 10 applied and academic science programs. The teacher may wish to co-ordinate timing or methods with the Science Department.

Prior Knowledge Required

Students should be able to solve proportions.

Teaching/Learning Strategies

Teacher Facilitation: Organize the students to work in pairs. Have students draw a grid over the picture or create a template on an overhead sheet to place over the picture. If the teacher does not provide the students with spinners have the students make them.

The spinners will be used to choose a letter from A to F and the die will be used to choose a number from 1 to 6.

If the crowd in the picture is not spread very evenly throughout, some discussion should occur about how to deal with this:

· choose representative examples from different areas of the photo;

· divide the entire photo into sections and have different students count the people in different sections to get an estimate of the total.

Student Activity: Aerial Photography

How would you determine how many people were at a Rock Concert from a photo?

1. Draw a six by six grid on your picture, or on an overhead sheet to place over your picture. Label it as shown below.

A
B
C
D
E
F
	1	2	3	4	5	6

2. Use your spinner to select a letter from A to F and your die to select a number from 1 to 6. This letter combined with the number gives you a random section of the grid to work with.

3. Count the number of people in this portion of the grid.

4. Use this value to estimate the total number of people in the picture. Do you feel this is an accurate count?

5. Use the spinner and die to choose another section of the grid. Count the number of people in this section. Use this value with your count from Question 3 to estimate the number of people in this crowd. How does this answer compare with your answer from Question 4?

Teacher Facilitation: Have students share their findings with the class. Since different sections would be counted, it would be interesting to see how close their results would be.

For the homework suggested below, samples might be selected by throwing a hoola-hoop or piece of cardboard with a cutout in the centre and counting all the weeds in the area where it lands.

Homework: In testing a new chemical for effectiveness in killing weeds, it would be necessary to find an area containing many weeds, count them, apply the chemical, and count the weeds again some time later. How might this be accomplished without counting every single weed? Try and devise a random sampling technique different from the one used in class. Depending on the time of year, students could actually investigate the above problem by answering the question “How many dandelions are there in our school yard? ” Students may wish to discuss environmental issues and the mathematics of persistence of herbicides.

Assessment/Evaluation Techniques

Photocopy a piece of bread and give students 20 minutes to determine the number of holes in a piece of bread. Try to find a piece of bread where the holes are fairly evenly distributed. Be sure to make the tools (spinner, die, and template) that they have used in the previous exercise available. This quiz provides the teacher an opportunity to assess the students’ ability to conduct an Inquiry using the techniques used in this activity. Criteria for assessment could include the steps of an inquiry, and guidance could be provided to the student as follows:

Inquiry: How many holes are there in your piece of bread?

Explore: Gather some data that will help you calculate an answer. Use a sampling method.

Hypothesize: If you had to guess right now, how many holes would seem reasonable?

Form a model: Write the ratio you could use to calculate the required number of holes.

Manipulate: Solve your proportion.

Infer/Conclude: Answer the question, explaining your reasoning. Explain why your answer does or does not agree with your hypothesis.

The written report could also be assessed for Communication using clarity of reporting and integration of narrative and mathematical forms as the criteria.

Activity 1.4: A Distorted View

Time: 150 minutes

Description

Students draw a picture and then shrink it in two ways: i) keeping the same proportions and ii) distorting the proportions. They examine the problem of changing proportions when “big screen” movies are put on a TV screen. This provides students with an example of the distortion of figures resulting from irregular scales.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Overall Expectations: PRV.01.

Specific Expectations

PR1.02 – draw and interpret scale diagrams related to applications (e.g., technical drawings);

Planning Notes

The teacher may want to provide students with examples of simple logos for students to use as motivation for their pictures. The teacher may want to talk with the art teacher as some art classes may have worked with distorting or “morphing” pictures.

Prior Knowledge Required

Demonstrate facility in operations with percent, ratio, and rate

Teaching/Learning Strategies

Teacher Facilitation: In this problem, students will be examining the difficulties involved in putting a regular size movie (ratio 16:9) on a TV screen (ratio 4:3). Students have likely seen the dark bands above and below a movie when they rent films at home or perhaps have seen a film where the picture has been distorted somewhat. A discussion about these examples would be helpful before completing this exercise. In the activity below the students will take a picture 16 by 9 and shrink it proportionately to 8 by 4.5. They will then take the same picture (16 by 9) and shrink it disproportionately to 4 by 3 (the proportions of a TV).

Student Activity: Distorting the Picture

1. Draw a rectangle with width 16 cm and height 9 cm.

2. Draw a simple picture or logo inside this rectangle.

3. Divide the rectangle into 4 equal columns and 2 equal rows (a total of 8 pieces).

4. Draw a new rectangle with width 8 cm and height 4.5 cm.

5. Divide this rectangle into 4 equal columns and 2 equal rows

6. Copy each “block” of your original diagram into the corresponding block on your new rectangle.

7. How does your new picture relate to your old picture? Is it in proportion to your original one or is it distorted? Explain.

8. Draw a new rectangle with width 4 cm and height 3 cm.

9. Divide this rectangle into 4 equal columns and 2 equal rows.

10. Copy each block of your original picture into the corresponding blocks of this rectangle.

11. How does your new picture relate to your original picture? Is it in proportion to your original one or is it distorted? Explain.

Teacher Facilitation: Students will likely need help setting up the proportions to do the calculations below. This could be done prior to the next step or the teacher could help students, as they need it.

Student Activity: A Distorted View

The 16 by 9 rectangle that you drew is in the same ratio as the screens in many movie theatres. The 4 by 3 rectangle that you drew is in the same ratio as most TV screens. Why would this create a problem?

One possible way of solving this problem is to distort the movie as you did with your picture above. Another possibility is to make the movie image have the same width as the TV screen. In this case the movie will not be tall enough to fill the screen. You may have seen this happen when you rent videos and watch them on your TV at home.

1. Consider a TV screen that is 40 cm wide by 30 cm high and a movie with a width to height ratio of 16:9. If the proportions of the movie are maintained and the movie image has the same width as the TV screen you will get dark bands at the top and bottom of the screen.

a) Use proportions to calculate how high the movie picture will be if you make the movie fit the TV screen’s width, while maintaining proportion.

b) What will be the size of the dark bands on the top and bottom of your screen?

2. Suppose the movie was reduced to fit the height of the TV screen while maintaining proportions. In this case parts of sides of the movie would be cut off.

a) Use proportions to calculate how much of the movie would be cut off if your TV screen were 40 cm by 30 cm.

b) What percentage of the total area of the movie is this? Do you think it is reasonable to cut off this much of the movie or will the viewer miss too much?

Teacher Facilitation: The teacher may wish to mention that there are TVs available now called HDTV that are in the ratio of 16:9. Discussion could develop about how a normal TV show would appear on these screens. The teacher may want to walk students through the activity below that looks at the differences in the TV and movie ratios graphically.

3. Set up a graph with width as your independent variable and length as your dependent variable. Use widths of 0 cm to 64 cm.

4. Pose the question: If your movie was 16 cm wide, how high would it be? (Use ratio 16:9.) Plot this point on your graph.

5. Repeat step 4 for movie widths of 32 cm, 48 cm, and 64 cm. Draw a line of best fit for these points.

6. On the same grid, plot the points for TV screen widths of 16, 32, 48, and 64 and their corresponding heights. (Use ratio of 4:3) Draw a line of best fit for these points.

Teacher Facilitation: Pose and discuss the following questions with students:

· What do each of these lines show?

· What does the gap between them represent?

· Find the slope of each line? What is the significance of these slopes.

Homework:

1. Optimal Viewing

Did you know that there is actually an optimal distance for you to be from your TV for ideal viewing? The ratio of the size of your TV screen to the distance you should sit from it is 1:6. Television sizes are stated as the distance diagonally across the screen.

a) Measure the size of your TV and calculate the ideal viewing distance for it.

b) Is it possible for you to sit this distance from your TV? Why or why not?

c) What size of a room would you need for a 32-inch TV?

d) If the same viewing ratio applies to movie theatres why do they put seats up close?

2. Fish Story

Your friend Jill has just returned from a weekend of fishing and has photos of her “big catch”. She tells you that she caught a lake trout 100 cm long but she had to let it go. Finding this hard to believe you ask for some proof. She passes over a picture showing her standing up, holding the fish up next to her. You study the picture carefully, ask Jill how tall she is, and make some careful measurements on the photo. You claim that Jill has just told you a big fish story. How would you convince her that you are right?

Teacher Facilitation: The above question is intended to be a simple proportion problem. It is not intended to imply that Jill is holding the fish out in front of her (closer to the camera), thus causing an illusion that the fish is larger.

Assessment/Evaluation Techniques

The Fish Story could provide an opportunity for formal assessment on Communication and Application. The application of ratio and proportion could be assessed using an objective marking scheme. Communication could be assessed using a rubric having as criteria: degree of clarity of explanation and appropriate use of mathematical vocabulary.

Accommodations

When assessing communication for ESL students, teachers should:

· allow students to work with classmates with similar backgrounds;

· clarify the phrase “fish story”.

Some students may need help with the Imperial conversion on the 32-inch screen.

Resources

Alexander, Bob. Minds on Math: 9. Canada: Addison-Wesley Publishing Limited, 1994.
ISBN 0-201-56015-1

Activity 1.5: Design Problem

Time: 75 minutes

Description

Students design and draw a scale drawing of a property development. The building will need to be constructed to meet certain by-laws and regulations.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

PR1.02 – draw and interpret scale diagrams related to applications (e.g., technical drawings).

Planning Notes

· The teacher will need to spend some time discussing how to make proper calculations to draw a scale diagram.

· Provide students with grid paper to make the drawing easier once the scale is established.

· Plan for access to a photocopier or secure a supply of NCR (no carbon required) paper.

Prior Knowledge Required

Demonstrate facility in operations with percent, ratio, and rate and solving problems involving perimeter, area.

Teaching/Learning Strategies

Teacher Facilitation: In the first part of this activity students should attempt the problem on their own, coming up with a design. Students should hand these in to be assessed. The teacher will need to photocopy the students’ plan or have them complete it on NCR paper so that they can hand in a copy and keep a copy to share with their groups.

Student Activity: A Design Problem

Your group has been hired by a property developer to draw a scale diagram of a lot design for an office building with a rectangular footprint. The developer would like his property to have ample parking but still be nicely landscaped.

· The lot size is 84 m by 90 m. Use a scale of 1: 600 to draw a scale plan of this lot.

· The city regulations for office buildings state that they cannot cover more than 40% of the total area of the lot.

· Total area of lot: _________________

· Maximum floor size of office building: ______________

· You client has decided that he wants his building to cover this maximum area. He also wants the length of the building to be 48 m. What will the width be? __________________

· Draw this building to scale on your lot. The city requires that all buildings be at least 6 m from the perimeter of the lot. Keep in mind you will need to add parking and landscaping in the steps that follow.

· The driveway into your lot needs to be 9 m wide. Indicate where the driveway will be on your lot.

· Local by-laws state that the following requirements must be met for parking:

· The size of a parking space is to be 3 m by 6 m.

· Any aisles between rows of parking and at the end of each row must be 7 m wide.

· The ratio of wheelchair spots to regular spots must be at least 1:40. Wheelchair spots are 4 m by 6 m.

· Add parking spaces to your design. You must include at least 50 spaces. Try a few different layouts until you find the one that provides the most number of spots but still leaves room for landscaping.

· Add landscaping to your plan. What is the ratio of landscaped area to total area for your design?

Complete a report for your client explaining your reasoning for the placement of the building, arrangement of parking spaces and proposed landscaping. Include any calculations necessary to show that all by-laws and city regulations are met.

Teacher Facilitation: In the second part of this activity students should share their designs in small groups and together come up with an optimal design using the best ideas from their own plans. These plans could be put on poster board and posted around the class.

The teacher could have the student write a journal entry in which they compare their original plan to the final plan that the group came up with. Have the student answer “What could you change/add to your plan that would improve it to the level of the plan that your group made”?

Assessment/Evaluation Techniques

The teacher could assess the students’ Knowledge and Application on their individual work by marking it using an objective marking scheme. Teamwork skills, and the exploration and hypothesizing steps of Inquiry could be assessed while students work in groups.

Accommodations

Provide students with 1-cm grid paper. A simpler design problem with fewer variables could be presented. Encourage tactile learners to cut out a shape for their building and move it to various locations on the diagram until they are happy with its placement.

Activity 1.6: Master Mixer

Time: 75 minutes

Description

Students solve a variety of ratio and proportion problems contained in connected contexts.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

Planning Notes

· The teacher will need to spend some time discussing how to set up “mixture” problems.

Prior Knowledge Required

Demonstrate facility in operations with percent, ratio, and rate

Teaching/Learning Strategies

Teacher Facilitation: Students should work individually on this activity. The teacher may want to show the students some techniques for working with “mixtures”. For example, setting up a table:

	Gas	Oil	Total
ratio
actual values

Student Activity: A Day in the Life of a Master Mixer

Master Mixer’s alarm clock wakes him up at 6:30 am. He jumps out of bed and looks outside – yes, today indeed will be a good day for a swim! Before heading out to work MM goes out back to check the chlorine level in his pool. Using his chlorine testing equipment he discovers his chlorine level is 1.0 ppm (parts per million) but it should be 2.5 ppm.

If his pool holds 36 000L of water:

a) Calculate how much chlorine is in the water now.

b) Calculate how much chlorine should be in the water.

c) How much chlorine will he need to add?

After adding the necessary chlorine, MM heads out the front door and notices a work crew pouring new sidewalks. He waves good morning and heads over to pose a question to the crew. MM wants to put a birdhouse on a pole in his front yard. To secure the pole he will need to pour some concrete but he needs to know how to make the proper mixture. The work crew tells him that his mixture needs to be 2 parts sand, 2 parts gravel, 1 part water, and 1 part cement.

a) MM will need to dig a somewhat cylindrical hole that is approximately 95 cm deep and is 30 cm wide to anchor the pole. Use the formula V = πr²h to calculate the volume of concrete required.

b) Use the mixture ratio provided by the work crew to calculate the volume of sand, gravel, cement and water MM will need.

MM heads off to work at a small photography shop that specializes in developing black and white photos. First job of the morning is to set up the solutions he needs for developing. He actually needs three different mixtures to complete his developing:

Developing Solution: 1 part developer + 9 parts water

Stop Bath: 1 part stopping chemicals + 10 parts water

Fixing Solution: 1 part fixer + 4 parts water

a) He first starts to mix the developer. He wants to make 1.8 L of developing solution. How much developer and water will he need?

b) MM discovers he is almost out of stopping chemicals. He has only 190 ml of chemicals left. How much water should he mix with it?

c) MM would like 3 L of fixing solution. What amount of fixer and water will he need to make this?

After a full day at work Master Mixer returns home and jumps in the pool for a relaxing swim. While floating around in his pool he notices that his grass has grown too long – time for a mow. He climbs out of the pool and heads to the garage to get the lawn mower and notices that he needs to add some gas before he gets started. MM’s lawn mower requires a gas to oil mixture in the ratio 40:1.

If MM pours 1.5 L of gas into his lawnmower tank, how much oil should he add?

After mowing the lawn Master Mixer heads inside to get a glass of refreshing ice tea. Looking in the fridge he finds none so he’ll have to make it. MM finds the ice tea mix in his cupboard and reads the label: “3 tbsp mix for 225 ml of water”. He looks at his juice container and sees that it holds 560 ml of fluid. MM is tired of figuring out all of these ratios and just throws in 12 tbsp with the 560 ml of water.

How did MM do with his guessing? Will his ice tea taste OK? Explain.

After a hard day of mixing, MM heads to bed, thanking his mathematics teacher for all of those ratio lessons!

Homework: Students could be asked to make up their own stories that include at least three examples of mixture, solution, or other ratio/proportion problems. Students should also provide solutions for their problems. Questions from the textbook can be used to reinforce these skills.

Assessment/Evaluation Techniques

Have students exchange stories with a partner. Each student should work through their partner’s story and then compare answers. This is an opportunity for both self- and peer formative assessment. Students should reflect on their ability to Communicate - read and interpret mathematics, and write clear solutions. The teacher could use this as an opportunity to assess Communication by listening to the discussions between partners.

Activity 1.7: Trip to Florida

Time: 75 minutes

Description

Students use ratios, rates, and unit conversions to calculate the cost of a trip to Florida.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

Planning Notes

· The teacher may want to give updated rates of exchange or have students use the internet or newspaper to research up-to-date rates.

· The teacher will need to provide a map with a clearly labelled scale so that students can calculate the distance from their hometown to Daytona Beach, Florida. Suggested resources for maps, hotels, and exchange rates are listed at the end of this activity.

Prior Knowledge Required

Demonstrate facility in operations with percent, ratio, and rate.

Teaching/Learning Strategies

Teacher Facilitation: Students should see several examples of currency conversions before starting this activity. Teachers may want to take advantage of the cultural make-up of their class when discussing currency by having students share types of currencies from their (or their parent’s) home country. Work with students on converting Canadian dollar values to various other currencies as well as the reverse. Students will also need to see some examples of converting units such as litres to US gallons, miles to kilometres.

Student Activity: Trip to Florida

Part 1

You work for a travel company that is organizing a trip for students to go to Florida for March Break. Your boss has given you the following information and wants you to determine how much you should charge each student for the trip.

Some conversion rates you may need:

1 US gallon = 3.784 litres

1 mile = 1.61 km

1 US$ = __________CDN$

1. The students will be taking a bus. Use a map to determine the approximate distance from here to Daytona, Florida. Show all calculations.

2. If the bus drives at an average rate of 90 km/h, how long (in hours) will the drive be?

3. The bus they are taking consumes about 35 L/100km of diesel fuel. Use this value to calculate the amount of fuel they will need for the return trip.

4. The tank holds 450L of fuel. How many times will they need to fuel up?

5. Use the following information to calculate the total cost of fuel (in Canadian $):

· diesel fuel in Ontario is selling for 52.4¢/L

· diesel fuel in the US is selling for $1.09US/gallon

6. The trip will be leaving at 4:00 on Friday afternoon and return at 11:00 am the following Saturday. The bus company will send two drivers on the bus so that they can take turns driving. This means that there will be no need to stay overnight somewhere on the way there or back. Use this information to determine how many nights they will be staying in Florida.

7. Use the Internet or look in the travel section of the newspaper to get rates for hotels. You can put 48 people on the bus plus the two bus drivers. Assume two people per room. Calculate the total cost of the hotel rooms for the trip. Since you are travelling as a group the hotel will give you a 20% discount. (Make sure you change the room rate to CDN dollars if it is given in US$)

8. Each bus driver will be paid a flat rate of $850 CDN for the trip, including all expenses.

9. Calculate the total cost of the bus and hotel. Use this value to determine how much you will charge the students for the trip. Remember that you will want to make a profit.

10. Write a report for your boss showing detailed calculations for the total cost of the trip. Include in your report the amount of profit you expect if the trip sells out.

Part 2

Upon returning from the trip one of the students has some American money left over. She decides to head to the bank to exchange it. A friend tells the student that the value of the Canadian dollar is much lower than it was last week so it’s a bad time to exchange her American dollars. The student is confused because she thought it would be a good time to do this. Should the student exchange her money now or wait for the value of the Canadian dollar to rise.

Teacher Facilitation: Part 2 is an example of an inquiry problem, in that it is open-ended as no values are given. The teacher may want to give the students an opportunity to explore the problem in small groups or lead the class in a discussion about how to investigate this problem. Once students have been assisted in exploring the problem, students should write up a solution to the problem.

Homework should include questions involving currency exchanges going in both directions and unit conversions.

Extension: If computers are accessible, the teacher may want to make use of spreadsheets or TI-83 Plus™, graphing calculations and have students set up a table showing the value of various denominations of Canadian money in other currencies. For example:

Canadian $	US $	Yen	Euro $	Mexican peso	Pound
1
5
10
15

Discuss for whom this table would be useful: travellers, tour operators.

Assessment/Evaluation Techniques

· Part 1 provides an opportunity for the teacher to assess the students’ Knowledge by checking their solutions to the conversion, rate problems. In addition, the teacher could assess the students’ ability to integrate narrative and mathematical forms of Communication in their report.

· Part 2 provides an opportunity to assess the students’ ability to apply their Knowledge. Students’ ability to identify the misleading conclusions of their friend could be assessed for Application. The steps of Inquiry could be assessed for all students if teachers did not lead the class through the exploration step. If part of the inquiry was teacher lead, then the teacher could assess the exploration and hypothesizing steps for the students who asked and answered questions..

Resources

Exchange Rates
http://www.finance.yahoo.com

Maps:

· CAA

· Atlases from geography department

· http://www.media.maps.com

· http://www.clients.mapquest.com/aol/visitors/mqtripplus (This site allows the student to enter the start location and final destination and it will output a list of driving directions as well as a total distance. If used the student will not have the opportunity to work with a scale on a map but it may save time if necessary.)

Hotels:

· CAA

· Internet – most search engines have a Travel channel that will allow you to get hotel rates.

Activity 1.8: Pet Vet

Time: 75 minutes

Description

Students examine applications of ratios, rates, and mixture problems. The context is veterinary science.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

Planning Notes

· This activity gives a very specific example of proportional reasoning on the job. Students could be asked to investigate where their parents might use ratios and rates in their jobs and bring these ideas to class to share.

Prior Knowledge Required

Facility in operations with percent, ratio, and rate.

Teaching/Learning Strategies

Teacher Facilitation: As students work, circulate around the classroom and provide hints as needed. For example, in 2. c), a hint might be:

First convert the amount of anesthetic needed from mg to grams and then use the proportion:

2 g	=	number of grams calculated in b)
100 ml		amount of solution dog needs

Student Activity: Pet Vet

Animal Health Technologists (AHTs) assist veterinarians by taking x-rays, blood work, setting up IVs, prepping animals for surgery, and inducing anesthesia. AHTs need many mathematical skills to do their jobs and must be able to do some calculations quickly.

Part A: Conversions

When an animal comes into the clinic one of the first things checked is its pulse. The AHT will take the animals pulse for 10 seconds but then needs to change this to beats/min to record on its chart.

Convert the following:

beats /10 seconds	beats/min
21 beats/10 seconds
32 beats/10 seconds
37 beats/10 seconds

Often an owner knows his pet’s weight in pounds but the vet needs it in kilograms to determine medicine dosages. Convert the following weights from pounds to kilograms.

1 pound = 0.45 kilograms

pounds	kilograms
21 pounds
85 pounds
	34 kg

Part B: Food and drugs in proportion to animal size

1. Most pet food packages include instructions about how much food to feed the animal based on its weight. Suppose a package suggests 10 grams of food for every kilogram the animal weighs. Calculate the amount of food required to feed an animal that is 39 pounds.

2. The vet has asked you to prep a dog for an operation where anesthesia is required. According to the owner, the dog weighs 87 pounds. The anesthetic your clinic uses comes in powder form. You have already mixed it to create a 2% solution. This means that your solution has 2 g of anesthetic per 100 ml of solution.

a) Convert the dog’s weight into kilograms.

b) Your clinic usually gives 12 mg of anesthetic per kilogram of animal weight. Calculate how many milligrams of anesthetic must be given to the dog.

c) You have the 2% solution already made up. How many millilitres of this solution needs to be given to the dog?

3. Explain why the veterinarian would use weight to determine the amount of anesthetic required as opposed to the length or height of an animal.

4. If the solution of anesthetic that your clinic had were a 5% solution instead of a 2% solution, how much solution would be given to the dog?

5. Create a graph that shows the amount of anesthetic solution required for dogs weighing 0 kg to 50 kg. Plot both the 2% and 5% solutions on the same axis.

6. Use the graph to calculate the amount 2% and 5% solutions that would be required for dogs with the following weights:

a) 15 kg

b) 45 pounds

c) 38 kg

7. An animal weighing 56 pounds comes into the clinic. The AHT calculates that they should give the animal 20 ml of anesthetic solution. Is the calculation correct? Check it by doing the calculation yourself and by examining the graph you constructed.

8. Why would a graph similar to the one you create in Question 5 be useful for the AHT to have in their office?

Assessment/Evaluation Techniques

This activity addresses no new expectations, but re-visits many expectations from the unit. Therefore, the questions in this activity could be used for review or for gathering assessment data, whichever is more appropriate for the class. These questions allow for assessment of Knowledge, Application, and Communication that can be scored using the same types of schemes and criteria that have been emphasized through the unit.

Accommodations

The teacher may choose to work through this set of problems with the students if structured review is needed at this time.

Activity 1.9: Catering

Time: 150 minutes

Description

This activity is a summative assessment for the first unit. Students apply their skills with ratio, rate and percent to calculate costs for a large group using data for a smaller sample.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Specific Expectations

PR1.02 – draw and interpret scale diagrams related to applications (e.g., technical drawings);

Planning Notes

· An overhead (aerial) photo of a residential neighbourhood is needed. You may be able to obtain one for your area through the municipal offices. A web site that contains a similar photo is provided under resources.

· NCR (no carbon required) paper will allow teachers to collect copies of estimates while students continue their work.

Prior Knowledge Required

The students will have completed the activities in this unit.

Teaching/Learning Strategies

Teacher Facilitation: Students may work in pairs, handing in individual submissions. The teacher should circulate and prompt students as needed, recording any help given to determine level of achievement. As students will be making cost calculations for 300, 400 and 500 people, they may want to do this on a spreadsheet.

Student Activity

Part A: The Catering Company

1. Your company has recently moved to a new area and you want to send some flyers out to each household in the surrounding neighbourhood to announce your arrival. You will need to determine approximately how many copies of your flyer to make. Use the aerial photo of the neighbourhood to approximate the number of flyers required. Explain your method.

2. Your advertising campaign works and you are flooded with phone calls. Many of the people calling want estimates for various sized crowds, and they want them fast. You have the following information on a paper by the phone.

· lasagna that serves 8 costs $12

· a tray of sandwiches that serves 15 people costs $60

· lemonade that serves 12 costs $3.50

· 7 pans of salad serves 100 people, each pan costs $7.00

· 3 lb. of coffee serves 100 people at a cost of $4.50/lb

· cupcakes $2.75/dozen

· butter tarts $ 3.60/2 dozen

Organize this information in a way that will make it easier for you to make quick estimates for any size crowd.

You will be read a series of five “phone calls” and will be required to write down an estimate in a limited amount of time. If you cannot give an estimate in the time available, you will lose that person’s business (marks). If you provide an estimate but have made an error you will have an opportunity to return that call and explain your error with the possibility of earning back the business (some marks).

Teacher Facilitation: Give the students 10-15 minutes to study the above information and make it more user friendly. Provide students with NCR paper and then read out the following “phone calls”. Provide students with just a few minutes between the phone calls to write down an estimate. Collect the estimates and mark them while students are working on the remaining portions of this activity. Return any poor estimates, with a written version of the phone call, to students to correct and hand in full solutions. If some students got all their estimates right, they could be given a more complex question to earn more business (marks).

Phone calls:

1. Hi, my name is Mrs. Ivanov. How much would it cost, approximately, to have your company supply lasagna and salad for 100 people? My number is 555-2313.

2. Good afternoon. I was wondering how much it would cost to have your company supply our celebration dinner with lasagna, coffee and salad. We are expecting about 30 people. Please call me back at 555-1212.

3. Hello, my name is George. I need some sandwiches, lemonade, and cupcakes for a lunch meeting on Friday. How much will it cost for your company to provide this for 25 people?

4. Hi! We will require food for a staff party with approximately 220 people attending. We’d like to have lasagna and salad for dinner. We’d also like cupcakes and coffee for dessert. Could you give us an estimate for this meal?

5. We are serving snacks at a meeting next week. We would like to offer lemonade, coffee, cupcakes and butter tarts. We will be expecting approximately 280 people with half drinking coffee and the other half lemonade, and half preferring cupcakes and the other half butter tarts.

Student Activity

3. Your team has been hired as the catering company for the year-end awards banquet for the local high school. The school expects approximately 400 people to attend.

a) The following graph displays dessert preferences at the last banquet you catered. Determine the number of each type of dessert you should bake for the banquet and the total dessert costs for the crowd. Cupcakes are $2.75 for one dozen, butter tarts are $3.60 for 24 and cookies are $5.40 for 48.

b) Your recipe for lasagna for eight people is :

175 g ground beef 340 ml tomatoes

780 ml spaghetti sauce 175 ml water

250g cheese

Determine the quantities of each ingredient required for 400 people.

Part B: The Banquet Planning Team

1. There are 400 people attending the awards banquet and your principal wants to know which room would be best for the event; the cafeteria or gymnasium. Determine the minimum size of the banquet room needed using the following information. Include a floor plan of the layout of the tables using the scale ratio of 1:200.

· The tables are 2 m by 5 m.

· You must provide spacing of 2 m between the tables so that people can walk between them when they go to the front to get their awards.

· Each table will seat 12 people.

Which room at your school would be best suited for the banquet?

2. As part of your decorations you would like to put 10 helium-filled balloons on each table. These balloons can be purchased in Canada for $0.75 each. Alternatively they can be shipped from the United States at $0.45 US each. Assume the currency rate of exchange to be $1US = $1.49Can. Determine where the balloons should be purchased and the savings. Include Ontario sales tax in your calculations.

Assessment/Evaluation Techniques

Students should submit individual solutions and could be assessed in the Categories of Knowledge, Communication, and Application. Most of this could be marked using an objective marking scheme. However a rubric could be used for Communication, using criteria: ability to read and interpret mathematics, ability to integrate narrative and mathematical forms of communication, and degree of clarity in explanation and justifications

Accommodations

Teachers may have to make the services of ESL support staff available for the “phone call” part of the activity.

Resources

Aerial photos
http://www.captainjack.com/dm/grimes.jpeg