The purpose of this unit is to engage the student in applying their knowledge and skills of measurement to investigate gradients in practical situations.
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.
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.
This crosscurricular, 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 studentcentred 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 studentcentred differentiated learning.
(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:
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:
Building ideas
A footpath is being designed for a 25 m section of roadside that has an elevation of 12°.
Reinforcing ideas
Extending ideas
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:
Building ideas
A footpath is being designed for a 25 m section of roadside that has a slope of ‘1 in 3’.
Reinforcing ideas
Extending ideas
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:
Building ideas
The scale of the topographical map is shown in the blue grids where each square represents 1 km^{2}.
Reinforcing ideas
The scale of the topographical map is shown in the blue grids where each square represents 1 km^{2}. The orange contour lines mark every 20 m change in elevation.
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.
Focusing on finding and using the angle of elevation of a steep street.
Activity
A Baldwin St homeowner 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:
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.
Reinforcing ideas
Filbert St in San Francisco has a gradient of 0.315.
Extending ideas
A very steep street in Canada, Côte StAnge has a 33% gradient.
Focusing on
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:
Building ideas
A cable car operates on a gradient of 1 in 5 and gains 120 m in elevation.
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.