Steep Streets

Purpose

The purpose of this unit is to engage the student in applying their knowledge and skills of measurement to investigate gradients in practical situations. 

Achievement Objectives
GM5-10: Apply trigonometric ratios and Pythagoras' theorem in two dimensions.
GM5-1: Select and use appropriate metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time, with awareness that measurements are approximate.
GM5-8: Interpret points and lines on co-ordinate planes, including scales and bearings on maps.
GM5-9: Define and use transformations and describe the invariant properties of figures and objects under these transformations.
Specific Learning Outcomes

Students develop their skills and knowledge on the mathematics learning progressions measurement sense, using maps and measuring tapes and/or supplied measurements to find, describe and use the steepness of a street. Students will be able to describe the steepness of a street in terms of a gradient or an angle of elevation. 

Description of Mathematics

Students will apply their understanding measurement and algebraic skills, investigating the gradients, distances and angles of elevation in the context of steep streets and access ways.

Activity

Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

  1. There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
  2. It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
  3. The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the reinforcing ideas or even the extending ideas activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the building ideas activity together, peeling off to complete this activity and/or to attempt the reinforcing ideas activity when they feel they have ‘got it’.
  4. It is expected that once all the students have peeled off into independent or group work of the appropriate selection of buildingreinforcing and extending activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

Introductory session

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of each student’s location on the learning progressions):

Stephanie lives on a very steep street. She says it feels steeper than 45° (from horizontal). At the steepest part there are steps in the footpath. These steps are shown in the image below. Use the tape measurements to decide whether Stephanie is correct in saying that her street has an elevation that is greater than 45°.

   
   
The street is so steep that it
needs steps in the footpath.
The height of the step is difficult to
determine measuring from the front,
so the tape measure is photographed
from the back as well.
The tape measure is laid out to find
the depth of the step.

In this activity, the teacher(s) will be able to locate their students on the measurement sense learning progression by observing their methods of problem solving to find unknown lengths and angles. Students might be more comfortable using purely practical techniques, or may use techniques based on similar triangles, or utilise their knowledge of Pythagoras and trigonometry. This activity integrates mathematical skills and knowledge with the science learning area, the physical world.

Mathematical discussion that should follow this activity could involve:

  • How accurate did you need to be to answer just this question?
  • What are some of the practical ways that you could solve this problem?
  • How might your knowledge of similar triangles be useful in solving this problem?
  • How could you calculate the angle required to solve this problem?

Session 2

Focusing on finding angles of elevation and using these to solve problems.

Activity

The steps on the footpath of a very steep street have a height of 7 cm and a depth of 51 cm. What is the angle of elevation of the footpath in that section of the street?

Discussion arising from activity:

  • What are some of the practical ways you could solve this problem?
  • How can this problem be solved purely by calculation?

Building ideas

A footpath is being designed for a 25 m section of roadside that has an elevation of 12°.

  1. Construct a scale diagram of this section of the roadside to represent this problem. The diagram should be a right angled triangle.
  2. Use your diagram and the principle of similar triangles to find the depth the steps will need to be if they each have of height 10 cm.
  3. How many steps would be needed to cover this section of the footpath?

Reinforcing ideas

  1. What depth of steps with the height 10 cm would be needed for a street with an elevation of 12°?
  2. What depth of steps with the height 15 cm would be needed for a street with an elevation of 12°?
  3. What depth of steps with the height 15 cm would be needed for a street with an elevation of 8°?
  4. What depth of steps with the height 10 cm would be needed for a street with an elevation of 15°?

Extending ideas

  1. How many steps of height 10 cm would be needed for a 25 m section of a roadside that has an elevation of 12°?
  2. How many steps of height 15 cm would be needed for a 25 m section of roadside that has an elevation of 12°?
  3. How many steps of height 15 cm would be needed for a 25 m section of roadside that has an elevation of 8°?
  4. How many steps of height 00 cm would be needed for a 25 m section of roadside that has an elevation of 15°?

Session 3

Focusing on finding gradients and using these to solve problems.

Activity

The steps on the footpath of a very steep street have a height of 12 cm and a depth of 42 cm. What is the gradient of the footpath in that section of the street?

Discussion arising from activity:

  • How can this problem be solved purely by calculation?
  • What are some of the practical ways you could solve this problem?
  • What are some of the different ways a gradient may be expressed?

Building ideas

A footpath is being designed for a 25 m section of roadside that has a slope of ‘1 in 3’.

  1. Construct a scale diagram of this section of the roadside to represent this problem. The diagram should be a right angled triangle.
  2. Use your diagram and the principle of similar triangles to find the depth the steps will need to be if they each have of height 10 cm.
  3. How many steps would be needed to cover this section of the footpath?

Reinforcing ideas

  1. What depth of steps with the height 10 cm would be needed for a street with a gradient of ‘1 in 4’?
  2. What depth of steps with the height 15 cm would be needed for a street with a gradient of 0.25?
  3. What depth of steps with the height 15 cm would be needed for a street with a gradient of 0.3?
  4. What depth of steps with the height 10 cm would be needed for a street with a gradient of 0.3?

Extending ideas

  1. How many steps of height 10 cm would be needed for a 25 m section of a roadside that has a gradient of ‘1 in 4’?
  2. How many steps of height 15 cm would be needed for a 25 m section of roadside that has a gradient of 0.25?
  3. How many steps of height 15 cm would be needed for a 25 m section of roadside that has a gradient of 0.3?
  4. How many steps of height 10 cm would be needed for a 25 m section of roadside that has a gradient of 0.3?

Session 4

Focusing on using the scale and contour lines of a topographical map to determine the slope or gradient of a street. The topographical map and satellite image are downloaded from www.topomap.co.nz .

Activity

The steepest street in the world is called Baldwin St. It is in North Dunedin. Use the street map of North Dunedin, to locate Baldwin street on the Satellite image and the topgraphical (contour) map.

    

Discussion arising from activity:

  • Which lines on the topographical map are contour lines?
  • What are contour lines used for on a map?
  • Can you tell which streets are steep from the street map?
  • Can you tell which street are steep from the satellite image?
  • The scale of the topographical map is shown in the blue grids where each ‘square’ represents 1 km2. Discuss why it might be useful to have such a scale. What other ways could scale be shown?
  • Why aren’t the blue ‘squares’ perfectly square and aligned with the horizontal and vertical edges of the map?

Building ideas

The scale of the topographical map is shown in the blue grids where each square represents 1 km2.

  1. Obtain a digital image of the topographical map, either from a scan or screenshot of this map, or by downloading the map of North Dunedin from www.topomap.co.nz .
  2. Zoom the map out (or in) until the side lengths of the blue ‘squares’ are a tidy 10 cm.
  3. Measure the length of Baldwin street.

Reinforcing ideas

The scale of the topographical map is shown in the blue grids where each square represents 1 km2.  The orange contour lines mark every 20 m change in elevation.

  1. Find the length of Baldwin St.  
  2. Count the number of contour lines (and parts thereof) Baldwin St passes through and use these to measure the height of Baldwin St (from top to bottom).
  3. Find the (average) gradient of Baldwin St.

Extending ideas

Use the 1 km2 per square scale and the contour lines on the topographical map to find the average gradient of Baldwin St.

Session 5

Focusing on finding and using the angle of elevation of a steep street.

Activity

A Baldwin St home-owner wants to build a carport with the roof parallel to the street. They know that Baldwin St rises from 30 m above sea level at the bottom of the road to 100 m at the top. They have measured the length of the street to be 375 m. At what angle to the horizontal does the carport roof need to be?

Discussion arising from activity:

  • Using the measurements for the full length of the street gives the average angle of elevation of the street. How might this be useful for the home-owner’s building project?
  • What could the home-owner do to ensure their carport roof is parallel to the part of the road it is situated on?

Building ideas

The steepest street in Wellington is Weld Street. Some of it is too steep for traffic, so just turns into steps for pedestrians. The total length of Weld St is 1.2 km with a rising from 136 m above sea level at the bottom of the street to 192 m above sea level at the top.

  1. What is the change in altitude you would experience if you walked up the full length of this street?
  2. Weld St is a relatively straight street. Find the average gradient of this street?

Reinforcing ideas

Filbert St in San Francisco has a gradient of 0.315.

  1. What is the change in altitude over a 100 m length of this street?
  2. If steps are made on the footpath of this street with a depth of 50 cm, what height do they need to be?
  3. Find the angle of elevation of Filbert St.
  4. Compare the angle of elevation of Baldwin St (19°) with that of Filbert St.

Extending ideas

A very steep street in Canada, Côte St-Ange has a 33% gradient.

  1. Express this gradient as a decimal number.
  2. What is the change in altitude over a 100 m length of this street?
  3. If steps are made on the footpath of this street with a depth of 50 cm, what height do they need to be?
  4. Find the angle of elevation of Côte St-Ange.
  5. Compare the angle of elevation of Baldwin St (19°) with that of Côte St-Ange.

Session 6

Focusing on

  • comparing and using gradients,
  • using right angle triangle problem solving techniques to find unknown lengths.

Activity

The Wellington cable car runs in a direct line from the CBD to the suburb of Kelburn where a street would be useful but the land is deemed too steep. The cable car operates on a gradient of 1 in 5.06 

Compare the gradient of the Wellington cable car with that of Baldwin St which is has a 35% gradient.

Discussion arising from activity:

  • How can these gradients be compared when they are in different formats?
  • Which format of presenting a gradientis the more useful to visualise steepness?
  • Which format of presenting a gradientis the more useful for calculation?
  • Which is steeper, the cable or the street?

Building ideas

A cable car operates on a gradient of 1 in 5 and gains 120 m in elevation.

  1. Construct a right angle triangle that represents this situation using a suitable scale. How high is your triangle?
  2. Measure the hypotenuse of the triangle you have constructed.
  3. Use the scale and length of the hypotenuse of your triangle to find the distance the cable car operates over.

Reinforcing ideas

The Wellington cable car operates on a gradient of 1 in 5.06 and gains 120 m in elevation. Find the distance travelled in a single cable car journey.

Extending ideas

A pedestrian stairway is to be built beside a cable car which operates on a gradient of 1 in 5 and gains 120 m in elevation. Each of the steps is to be 40 cm deep.

  1. How high will each step need to be?
  2. How many steps will be needed in total?
  3. What is the distance covered by the cable?