Minds on.

On sunny days, you may notice your shadow. Is your shadow short or is it long? How is it related to your height? Can you predict how long your shadow will be?

Shadows are created when the sun is in the sky. The length of the shadow depends on the position of the sun. Learn more about the sun’s position and the shadow length throughout the year. Watch The Apparent Path of the Sun:

 

The sun rises in the east. Your shadow will be to the:
a. East
b. North
c. South
d. West

Answer

d. West

 

When the sun is at its highest place in the sky, your shadow must be
a. behind you
b. the longest shadow of the day
c. the shortest shadow of the day
d. in front of you

Answer

c. the shortest shadow of the day

 

The shadows are the shortest on:
a. the winter solstice - first day of winter
b. the summer solstice - first day of summer
c. the spring equinox - first day of spring
d. the autumn equinox - first day of autumn

Answer

b. the summer solstice - first day of summer

 
Action.

This is the Portfolio icon. Mathematical Processes

In this activity, the Mathematical Processes Problem Solving and Communicating are the focus.

Open your document U4 Mathematical Processes and read the descriptions.

As you complete the activity, notice when you are thinking about your solution and how it is being presented.

Insert your record below the description of the process.

 

Every day, the sun rises and sets. Everyone is aware of the greater number of daylight hours during the summer season in Ontario and the fewer number of daylight hours during December. It is something that we notice.

Another aspect of light and dark that occurs on sunny days is the change in length and position of shadows. The success of various projects may be determined by the number of hours that an area is exposed to or protected from the sun’s rays. Pools, solar panels, and some landscaping items may benefit from a greater number of sunlight hours while other garden plants, beehives and living spaces may prefer the cooling effect of shade.

Providing shade in living spaces, whether they are houses, schools, park areas, or recreational facilities, offers a break from the sun and helps to reduce energy consumption for air conditioning. Planting trees is one way to provide shade.

When a tree is chosen for a particular site, e.g. a house, one needs to consider:

  • the type of tree; deciduous (one that loses its leaves in the winter) or coniferous (one that keeps its leaves/needles year long)
  • the maximum height that the tree can reach
  • the direction the house faces (e.g. north, south, east, west)
  • the location of the house (e.g. longitude and latitude)
  • size of the area that is available for planting trees

Shade On The Deck

Ms. O’Neil lives in southeastern Ontario and recently moved into a house that has a  deck in the back yard that is 12 ft. by 16 ft.

An image of the diagram of a house with the deck attached to the back of the house.

The back yard faces south and it is exposed to the sun all day. She is currently planning some landscaping for the property. She wants to include at least one tree that will provide shade on the back deck in the afternoon, generally the hottest time of the day. 

She knows that the shadow cast by objects changes in length and direction as the sun moves through the sky.  Watch the following video to see the direction of the shadow and its length change during the day.

 

Ms. O’Neil knows that she must leave a minimum of 3 feet between the edge of her property and any permanent objects, including plants, that she places on the property; also trees should be planted at least 15 feet from the house. She also wants to have space between the tree and the deck for people and equipment to pass.

Where do you think the tree should likely be planted to maximize the number of hours that the deck will get shade?
a. SE corner of deck
b. midpoint of deck
c. SW corner of deck

Answer

The SW corner of the deck is likely the best area to cast the shadow towards the deck.

 

Ms. O’Neil realizes that she needs specific information about the sun’s movement. The sun’s daily motion has been recorded for hundreds of years. The sun’s position is described using two angular values at different times of the day:

The azimuth - the position of the sun using north as 0°.

An image showing a shadow from up above with north, east, west and south indicated on the image.

The altitude - the angle of elevation.

Learn more about the azimuth and the elevation.

The azimuth and altitude are recorded according to your location.   Ms. O’Neil found a website that specifies the azimuth (position of the sun), the altitude (the angle of elevation) and the shadow length for an object height of 1 metre. When she opened the website, she realized that she needed to insert her address in the top left corner, below Computational path of the sun for.

When the screen refreshed, she recognized the street names around her house. 

She selected the date, June 21st, since the shadows at midday should be their shortest length for the year, and she printed the data for her location. She plans to use this information to look at shadow lengths but she needs to decide upon the type of tree that she will have planted.

She researched trees that grow in eastern Ontario, using a reference from the Ministry of the Environment and Climate Change, and noted that many choices will be 25 m tall when fully grown. She plans to plant a Balsam Poplar because it is a fast-growing tree. She can purchase a 12 ft. tree which she estimates will grow to 60 ft. in 10 years.

Where should Ms. O’Neil plant the tree in her yard so that she will have shade on the deck in the afternoon while still be following the requirements for her property?

This is the discussion icon. Shade on the Deck

Reread Shade on The Deck and state the problem.

  • What are you asked to determine?
  • What information have you been given?
  • What information will you need to use?
  • What values are given and what values might you need to determine?
  • What questions do you have?

Share your understanding of the problem with your classmates in a discussion. Include in your post:

  • your list of given information and what you will use
  • the information that you need to determine
  • questions that you have

Choose one classmate and read their understanding of the problem, the information, and their questions. When reading the work of other students, make comments, ask questions or answer their question.

Possible sentence starters are:

  • You wrote that … I didn’t consider that. Is that important?
  • You asked … I think we need to …
  • You added … How do you know this information?
  • I noticed … I wonder if …?

 

This is the dropbox icon. Finalize and Submit your Plan

Use the feedback as you Make a Plan to solve the problem.

Your detailed plan should include:

  • organized steps that you will follow to solve and answer the problem
  • strategies that you may use e.g. draw a diagram
  • information that you are given
  • values that you may need to determine
  • mathematics that you may use

Submit your finished plan to your teacher.

 

Ms. O’Neil’s Plan

Ms. O’Neil made a plan to follow to determine where she should plant the Balsam Poplar tree in her backyard.

Her general plan was to make a scale drawing of her property, make scale diagrams of the shadow lengths of the tree at different times of the day, and then use the scale shadow lengths against the scale drawing of the property to test out different possible locations.

Her plan:

  1. Make a scale drawing of the house and deck. Include the edge of the property.
  2. Select different times during the day to look at shadow lengths, e.g. 10 a.m., 1 p.m., 3 p.m. and 5 p.m.
  3. Use the azimuth information to show the sun and shadow positions at the selected times during the day.
  4. Use trigonometry to determine the length of the shadow at the selected times for a 12 ft. tree and a 60 ft. tree.
  5. Using the same scale as the house drawing, create the shadow models for the 12 ft. tree and the 60 ft. tree on the azimuth sketches.
  6. Place the shadow model on top of the house drawing to determine a good placement for the tree.
  7. Check that the chosen placement meets the criteria stated. Make adjustments if necessary.

This is the quiz icon. Learning Goals and Success Criteria Self-Assessment

Now that you've had a chance to make a plan and solve a problem, it's time for you to consider whether or not you are meeting the learning goals and success criteria of the acitivity.

Complete the following self-assessment.

 

Carry out the Plan

Ms. O’Neil made the scale drawing of her house and the deck, using a scale of 1 cm = 4 ft.

An image of the scale diagram of a house with the deck attached to the back of the house.

To determine the length on the scale drawing, she used the equation, y = ¼ x where x is the actual measurement in feet and y is the drawing measurement in centimetres.

She made another sketch showing the direction of a shadow at different times. Starting at north, she used a protractor to measure clockwise to the azimuth reading for 10 a.m. (104.68°), 1 p.m., which is 13:00 on a 24-hour clock (177.12°), 3 p.m. (236.45°) and 5 p.m. (265.69°). She extended the lines through the centre (the tree) since the shadow will fall along the same line.

An image of the direction of the sun’s rays and the shadows that are created if there is an object at the centre, C. The shadow lines are not scaled to the length of the shadow at that time.

She needs to do some calculations before she can add the shadow lengths at the selected times. She finds the angle of elevation (altitude) for the selected times, so she can use a trigonometric ratio to determine the length of the shadow.

At 10 a.m. the angle of elevation is 46.84°. She draws and labels a triangle, letting x represent the length of the shadow in feet. Next, she chooses the tangent ratio since she has the side opposite the indicated angle and needs to find the side adjacent to that angle.

An image showing the calculations to determine that the shadow line must be 2.8cm.
An image of what the shadow looks like from directly above. Various times are indicated on the image including 10 am, 1pm, 3 pm and 5 pm.

She adds this shadow to the azimuth line for 10 a.m.

She completes the calculation for the 1 p.m. where the angle of elevation is 68.01°. She noticed that this was the highest angle of elevation.

An image showing the calculations that determine that the shadow line is 1.2 cm.

She adds the shadow to the azimuth line for 1 p.m.

An image of the direction of the sun’s rays and the shadows that are created if there is an object at the centre, C. The shadow lines are not scaled to the length of the shadow at 1 pm.

She completes the two other calculations for the 12 ft. tree and adds shadows to her sketch.

An image of the direction of the sun’s rays and the shadows that are created if there is an object at the centre, C. The shadow lines are not scaled to the length of the shadow for 3pm and 5pm.

At 3 p.m. the shadow length is 7.6 ft. and at 5 p.m. the shadow length is 15.3 ft.

This is an image of the shadow lengths on grid paper.

The shadow lengths can also be shown on grid paper that has the same scale as the drawing of the house.

If you place the tree shadows on top of the house drawing, you can see where the shadows might be the first year, when the tree is 12 feet tall.

This drawing indicates that the placement of the tree means that shadow will reach the deck at 5:00 pm.

An image showing the shadow from up above, indicating how the shadow reaches the deck.

This drawing indicates that the placement of the tree means that shadow will not reach the deck at all.

An image showing how the shadow of the tree does not reach the deck.

Practice

Complete the calculations when the tree is 60 ft. tall. Make a shadow diagram drawn to scale, 1 cm = 4 ft., of the house and the deck. Use a ruler to measure the required lengths and a protractor to measure the required angles.

Check that your calculations and shadow drawing are correct.

Use the scale diagrams to determine a good place to plant the tree. A scale diagram of the shadow lengths for the 12 ft. tree is available.

You can move the shadows around on top of the house drawing as shown in the image. Remember the restrictions that she has for placing the tree.

This is a real-life scenario where mathematics provides you with information to make an informed decision, but it doesn’t involve a “right answer”, just correct calculations that you can use to make an informed decision.

Advise Ms. O’Neil where to plant the tree. Describe the position with respect to the SW corner of the deck, e.g. plant the tree 5 ft. west and 5 ft. south of the SW corner of the deck. Justify your recommendation.

Look Back at The Solution

Ms. O’Neil chose to plant the tree 4 feet south and 2 feet east of the SW corner of the deck. As she reviews her work, she notices that the position is reasonable. The tree is more than 15 ft. from the house and there is 4 feet between the tree and the rear property line. It is a position that will cast shadow onto the deck for most of the afternoon once the tree starts to grow. She also likes the fact that there is space between the deck and the tree so people or equipment can easily pass. See the shadow lengths and scale drawing.

This is the dropbox icon. Directions

There are two parts to this assignment:

Part 1:

Read the details about the other work that Ms. O’Neil is planning for the deck. Notice that measurements are not given for the objects at this time. They will be given in the assignment.

  • With your partner, ensure that you understand each problem and make a plan for how you will solve each one.
  • Submit a copy of your plan to your teacher.

Part 2:

Submit a report to Ms. O’Neil with the details of what she needs to do to finish the work on her deck. You can choose to do a written report, a slide presentation or a filmed presentation.

  • Open the assignment, U4A4 Assignment. Notice the values that you need to complete the calculations are included.
  • Work independently to answer the problems according to your plan.
  • Record any places where you chose to revise your plan and explain why.
  • Look back at your solution and review your methods.
  • Prepare your report that includes the calculations and answers to the problems. Include the revised plan as an appendix to your report.

Submit the report and revised plan to your teacher.

 

This is the Portfolio icon. Important Book

You have been creating The Important Book to clarify important concepts and skills in the different parts of the course.

Open your document. As you make your plan, perhaps you have something written that gives you a suggestion about where to start. Maybe there will be something that you want to add.

Keep your document open as you complete the assignment. You may want to add to or change some of the information in your book.

Save your document to your Portfolio.

 

This is the dropbox icon. More About the Deck

Ms. O’Neil is making more plans for the deck. She will be placing some potted plants outside and erecting a parabolic trellis on one side. She also needs to install a ramp for wheelchair access to the deck.

Ms. O’Neil realizes that it will take a number of years for the tree to grow and provide the desired shade on the deck. In the interim, she will move some of her larger potted plants outdoors for the summer. She purchased three new plant pots and will repot the plants before moving them outdoors.

Problem 1: Buying Potting Soil

She notices that the top half of the pot is cylindrical and the bottom half is a hemisphere.

An image of the pots to be used.

The pot will be placed on a base to keep it standing upright. She wants to determine how many bags of soil to purchase and how much the soil will cost.

Problem 2: Placing the Pots for Shade

She moved a table to the deck and wants to arrange the potted plants in a way that the table will be in shade during the hours of 3:00 p.m. to 7:00 p.m.

An image of the scale drawing of the house and deck with the table.

If she places the pots 2 ft. from the west edge of the deck, will the table be shaded? Use the azimuth and altitude for Ms. O’Neil’s location.

Problem 3: Access to the Deck

The deck is above the ground and has 2 stairs between the ground level and the deck level along the eastern edge of the deck.

A side view of stairs showing the 7 inch rise and the 11 inch run.

Ms. O’Neil wants to add a ramp to make it accessible for a friend who uses a wheelchair.

If the angle of elevation of the ramp must be no greater than 5°, how long will the ramp be? Is the yard large enough for the ramp to be built?

Problem 4: Parabolic Trellis

She received a garden trellis from a friend. She would like to place it along one edge of the deck but she doesn’t want it to stop people from walking onto the deck.

An image of a garden trellis in a parabolic shape.

If the arbor is anchored to the ground, can two people who are 6 ft. tall, walk under the arbor, side by side, at the same time? Allow 24 in. for each person’s width.

Submit your assignment with your revised plan to your teacher.

 
Consolidation

This is the Portfolio icon. Mathematical Processes

Complete your entry for the Mathematical Processes Problem Solving and Communicating in the document U4 Mathematical Processes.

What did you notice about the choices that you made as you worked with your partner, following a plan?

Save to your Portfolio.

 

This is the Portfolio icon. The Important Book

You have revisited many of the concepts that you learned in the first three units of the course.

Where have you used The Important Book as you completed the activity or the assignment? Is there anything else that you want to update on your slides?

Review the rubric and save your document to your Portfolio.

You will be submitting your final version at the end of the unit.

 

As you completed the activity and the assignment, you used many of the skills that you have studied throughout the course to solve some possible problems that you may experience in the future. The situations or dilemmas that we often experience in everyday life can be resolved using a mathematical relationship.

Understanding similar triangles and trigonometry allows for calculations to be completed in many examples where angle measurements are a piece of the information.

Items may be described in the imperial system and/or the metric system; it is beneficial to understand both systems and to convert between the systems. Measurement formulas are regularly used in our lives; they can be used to calculate different amounts. Knowing how much to purchase often means you don’t have to return to the store to buy more or you don’t have to store excess amounts of a material purchased in error.

Using relationships to help determine the placement of objects often means that an object won’t need to be positioned elsewhere. For living things, such as plants and trees, getting the right amount of sunlight and shade may be necessary for them to thrive.

This is the quiz icon. Learning Skills and Work Habits Self-Assessment

Consider the learning skills and work habits you demonstrated as you completed this activity. You are to complete this self-assessment of your learning skills in order to move on to Activity 5. Your teacher will read your self-assessments and may use them to help you set next steps in your learning. The Learning Skill you will focus on for this assessment is independent work.

 

Congratulations, you have completed Unit 4, Activity 4. You may move on to Unit 4, Activity 5.

 

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