The purpose of this unit is to engage students in applying their understanding of measurement and algebra in investigating a physical system.
Students will apply their understanding of measurement and algebraic skills through investigating the conservation of mechanical energy in the context of catapults.
The SI units of measurement have been used throughout this unit ensure validity of the physical relationships used in calculations. Students may be more comfortable measuring with derived units such as mm or cm, but should be encouraged to convert these measurements into SI units, such as the m, to ensure clear and accurate mathematical communication.
This cross-curricular, context-based unit aims to deliver mathematics learning, 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 contexts presented in this unit.
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 koki (angle), teitei (altitude, height, high, tall), rahi (quantity), āhua (shape), tere whaiahu (velocity), kauwhata (graph), raraunga (data), papatipu (mass), and pānga rārangi (linear relationship) could be introduced in this unit and used throughout other mathematical learning.
The aim of this activity, which presents an opportunity to practise mathematical skills and knowledge in a science context, is to motivate students towards the context and to inform teachers of students' understandings.
As students work, observe their management of quantities. Use these observations to gauge your students' positions on the measurement sense learning progression.
This session focuses on problem solving involving quantities that are defined by a physical property; the conservation of mechanical energy.
Introducing Ideas
Discuss, drawing attention to the following points:
Building Ideas
Introduce the following context to students: A projectile is drawn back in a slingshot. The projectile is launched horizontally.
Provide time for students to do the following:
Reinforcing Ideas
Provide time for students to do the following:
Extending Ideas
Provide time for students to do the following:
Use the conservation of energy to quantify the energy form(s) present at the top of the path.
Use the conservation of energy to quantify the energy form(s) present when the falling projectile is at the height at which it was launched.
Introducing Ideas
Discuss, drawing attention to the following points:
Building Ideas
Provide time for students to work through the following tasks:
Reinforcing Ideas
Provide time for students to work through the following tasks:
Extending Ideas
Introduce the following context to students: The area under the Weight Force (N) vs Extension (m) graph, from 0 m to a given extension, x gives the elastic stored when the spring or elastic is stretched by x m.
Provide time for students to work through the following tasks, based on a graph of the results of their investigation:
Discuss, drawing attention to the following points:
Building Ideas
Introduce the following context to students: Gravitational potential energy, Ep in J, is calculated from the rule, Ep = mgh, where m is mass (in kg), h is height (in m) and g = 10 ms-2.
Provide time for students to work through the following tasks:
Reinforcing Ideas
Introduce the following context to students: Gravitational potential energy, Ep in J, is calculated from the rule, Ep = mgh, where m is mass (in kg), h is height (in m) and g = 10 ms-2.
Provide time for students to work through the following tasks:
Extending Ideas
Introduce the following context to students: Gravitational potential energy, Ep in J, is calculated from the rule, Ep = mgh, where m is mass (in kg), h is height (in m) and g = 10 ms-2. The mass of the projectile can be measured with a mass balance.
Provide time for students to work through the following tasks:
Discuss, drawing attention to the following points:
Building Ideas
Provide time for students to graph the data collected in their investigations, following the process described below, and to find the relationship between height reached (m) and extension (m) of the rubber band.
Reinforcing Ideas
Provide time for students to work through the following tasks:
Extending Ideas
Provide time for students to graph their data and describe the relationship shown by their graph (i.e. as linear or non-linear).
Provide time for students to work through the following tasks:
Introducing Ideas
Introduce the following context to students: The elastic potential energy of a rubber band slingshot depends on the extension of the rubber band by the rule: Ee = ½ kx2.
Discuss, drawing attention to the following points:
Building Ideas
Introduce the following context to students: The elastic potential energy stored by extending a slingshot of extension, x, is described by the equation Ee = ½ kx2.
Provide time for students to work through the following tasks:
Reinforcing Ideas
Introduce the following contexts to students:
The elastic potential energy stored by extending a slingshot of extension, x, is described by the equation Ee = ½ kx2.
The gravitational potential energy gained by the projectile at the top of its path is described by the equation Ep = mgh.
Provide time for students to work through the following tasks:
Extending Ideas
Introduce the following contexts to students:
The elastic potential energy stored by extending a slingshot of extension, x, is described by the equation Ee = ½ kx2.
The gravitational potential energy gained by the projectile at the top of its path is described by the equation Ep = mgh.
Provide time for students to find what the effect would be on the height that a projectile reaches, if the extension of a given slingshot is doubled? Support them to justify their answers.
Dear parents and whānau,
We have recently applied our understanding of measurement and algebra in investigating a physical system. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/slingshots at 8:54pm on the 26th February 2024