The purpose of this unit is to engage students in applying their knowledge and skills of measurement to investigate gradients, angles, elevation in a range of practical situations.
This unit provides an opportunity for students to apply their skills and understanding of measurement to solve food technology problems involving the interdependence of variables when scaling the components of a recipe.
To ensure maximum engagement and participation in this unit, you should consider your students' prior knowledge in the following areas:
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, whilst encouraging differentiated, student-centred learning.
The learning opportunities in this unit can be further differentiated by providing or removing support to students, and by varying the task requirements. Ways to differentiate include:
With student interest engaged, mathematical challenges often seem more approachable than when presented in isolation. Therefore, you might find it appropriate to adapt the focused contexts presented in this unit. For example, you might investigate the elevation and steepness of streets in your local area, or in places that are of increased relevance to your students.
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.
Following the introductory session, each subsequent session in the unit is composed of four sections: Introducing Ideas, Building Ideas, Reinforcing Ideas, and Extending Ideas.
Introducing Ideas: It is recommended that you allow approximately 10 minutes for students to work on these problems, either as a whole class, in groups, pairs, or as individuals. Following this, 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.
Building Ideas, Reinforcing Ideas, and Extending Ideas: Exploration of these stages can be differentiated on the basis of individual learning needs, as demonstrated in the previous stage of each session. 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’.
It is expected that once all the students have peeled off into independent or group work of the appropriate selection of building, reinforcing and extending activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.
Importantly, students should have multiple opportunities to, throughout and at the conclusion of each session, compare, check, and discuss their ideas with peers and the teacher, and to reflect upon their ideas and developed understandings. These reflections can be demonstrated using a variety of means (e.g. written, digital note, survey, sticky notes, diagrams, marked work, videoed demonstration) and can be used to inform your planning for subsequent sessions.
The relevance of this learning can also be enhanced with the inclusion of key vocabulary from your students' home languages. For example, te reo Māori kupu such as hoahoa āwhata (scale drawing, scale diagram), putu (degree - angle and temperature), koki rewa (angle of elevation), koki (angle), rōnaki (gradient, slope), mahere (map), roa (length), ōrau (percent), tau ā-ira (decimal number), and koki hāngai (right angle) might be introduced in this unit and then used throughout other mathematical learning.
This activity is intended to motivate students towards the context and to inform teachers of students' understandings.
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. |
Observe your students' use of problem solving methods when tasked with finding unknown lengths and angles. Use these observations to locate their position on the measurement sense learning progressions. Students might use purely practical techniques, 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).
This session focuses on finding angles of elevation and using these to solve problems.
Introducing Ideas
Building Ideas
Reinforcing Ideas
Extending Ideas
This session focuses on finding gradients and using these to solve problems.
Introducing Ideas
Introduce the following problem to students: 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?
Discuss, drawing attention to the following points:
Building Ideas
Introduce the following context to students: A footpath is being designed for a 25 m section of roadside that has a slope of ‘1 in 3’.
Provide time for students to work through the following tasks:
Reinforcing Ideas
Provide time for students to work through the following tasks:
What depth of steps with the height 10 cm would be needed for a street with a gradient of ‘1 in 4’?
What depth of steps with the height 15 cm would be needed for a street with a gradient of 0.25?
What depth of steps with the height 15 cm would be needed for a street with a gradient of 0.3?
What depth of steps with the height 10 cm would be needed for a street with a gradient of 0.3?
Extending Ideas
Provide time for students to work through the following tasks:
This session focuses 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.
Introducing Ideas
Discuss, drawing attention to the following points:
Your students might benefit from, and be engaged in, exploring other topographical maps, street maps, and satellite images relevant to their local area, as an extension to this task.
Building Ideas
Introduce the following context to students: The scale of the topographical map is shown in the blue grids where each square represents 1 km2. Provide students with 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.
Have students zoom the map out (or in) until the side lengths of the blue ‘squares’ are a tidy 10 cm, and then use this visual to measure the length of Baldwin street.
Reinforcing Ideas
Extending Ideas
This session focuses on finding and using the angle of elevation of a steep street. Within this session, students are required to draw on their knowledge of metric measurements, percentages, and decimals.
Introducing Ideas
Introduce the following context to students: 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?
Discuss, drawing attention to the following points:
Building Ideas
Reinforcing Ideas
Extending Ideas
This session focuses on comparing and using gradients, and using right angle triangle problem solving techniques to find unknown lengths.
Introducing Ideas
Introduce the following context to students: 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
Discuss, drawing attention to the following points:
Building Ideas
Introduce the following context to students: A cable car operates on a gradient of 1 in 5 and gains 120 m in elevation.
Provide time for students to work through the following tasks:
Reinforcing Ideas
Introduce the following problem to students: 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
Dear parents and whānau,
Recently, we have been exploring how angles of elevation and gradients can be found, and, with the use of problem solving methods, be used to find unknown lengths, angles, and to solve practical situations.
Ask your child to share their learning with you.
Printed from https://nzmaths.co.nz/resource/steep-streets at 2:07am on the 20th May 2024