The purpose of this unit is to engage the student in applying their knowledge and skills of measurement and algebra to investigate a physical system.

Students develop their skills and knowledge on the mathematics learning progressions measurement sense and using symbols and expressions to think mathematically.

Students will apply their understanding measurement and algebraic skills, investigating the conservation of mechanical energy in the context of catapults.

### 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.

- 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.
- 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.
- 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’. - 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.

### Introductory session

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

Investigate, using a rubber band, an eraser and a protractor, the path of a projectile launched from a slingshot. To replicate a slingshot, stretch a rubber band stretched between a splayed thumb and forefinger. What is the optimum firing angle of a slingshot to achieve the:

- maximum height of the projectile?
- maximum horizontal distance of the projectile?

In this activity, the teacher(s) will be able to locate their students on the measurement sense learning progression by observing their management of the quantities under investigation. This activity integrates mathematical skills and knowledge with the science learning area, the physical world. In this unit of learning activities, the SI units of measurement have been used to 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.

Mathematical discussion that should follow this activity involve:

- Describe the shape of the path of the projectile.
- What factors other than the launch angle affected the distance the projectile would cover?

### Session One

Focusing on problem solving involving quantities that are defined by a physical property; the conservation of mechanical energy.

**Focus activity**

A projectile is drawn back in slingshot, with elastic potential energy of 40 J. List and quantify the energy transformation present at the launch of the projectile.

Discussion arising from activity:

- What are the types of energy present?
- Is mechanical energy conserved?
- Will there be further energy transformations during the flight of the projectile?

**Building ideas**

A projectile is drawn back in a slingshot. The projectile is launched horizontally.

- Sketch the shape of the path of the projectile.
- Assuming there is no friction present, describe the energy forms present when the projectile falls to the height it was launched at.

**Reinforcing ideas**

A projectile is drawn back in a slingshot. The projectile is launched at an angle of 45° to the horizontal.

- Sketch the shape of the path of the projectile.
- Assuming there is no friction present, describe the energy forms present at the top of the path.
- Assuming there is no friction present, describe the energy forms present when the projectile falls to the height it was launched at.

**Extending ideas**

A projectile is drawn back in a slingshot, with elastic potential energy of 40 J. The projectile is launched at an angle of 45° to the horizontal.

- Label the diagram below to show how the initial velocity of the projectile has a horizontal speed equal to its vertical speed.

Once it has been launched, assuming there is no friction present, the horizontal speed is constant. It is the vertical speed that changes because of the force due to gravity.

2. Use the conservation of energy to quantify the energy form(s) present at the top of the path.

3. 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.

### Session Two

**Focus activity**

To find the amount of elastic potential energy that can be stored in a stretch rubber band, measure the extension (in m) of the band when various weights (in N) are hung from the band. Record your results in a table. You should have data for at least four different weights.

- How did you know the number of N of each mass? How does this differ from mass (in g or kg)?
- How did you measure the extension of the band?
- Does this relationship appear to be linear or non-linear?
- Would the point (0, 0) lie on the line/curve of this graph?

**Building ideas**

Graph the data collected in your investigation to find the relationship between weight (N) and extension (m) of the rubber band:

- Draw a set of positive axes occupying at least half a page of quad paper.
- Choose a suitable scale for the weight (N) that will use as much of the vertical axis as possible. Label that axis and mark on the scale in N.
- Choose a suitable scale for the extension (m) that will use as much of the horizontal axis as possible. Label that axis and mark on the scale in N.
- Plot the data you collected.
- Rule the line of best fit for your data.
- Describe the relationship shown by this graph.

**Reinforcing ideas**

The area under the Force (N) vs Extension (m) graph, from 0 m to a given extension, x gives the elastic potential energy stored when the spring or elastic is stretched by x.

- Construct a graph of the data from your investigation.
- Give the relationship shown by this graph
- The unit of energy is the joule, J. What is the equivalent to J in terms of the newton, N and the metre, m.
- Find the Elastic Potential Energy, in J, stored in the rubber band at the maximum extension.

**Extending ideas**

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. Based on a graph of the results of your investigation:

- Write a general rule for the Elastic Potential Energy, Ee, stored in the rubber band in terms of the weight force suspended, F, and the amount of extension, x.
- Write a rule for the Elastic Potential Energy, Ee, stored in this rubber band in terms of only the amount of extension, x.

### Session Three

**Focus activity**

Place a small projectile in a rubber band slingshot. Extend the rubber band by 5 cm. Launch the projectile vertically and estimate the height the projectile reaches. Repeat sufficient times to find an average height.

- How did you ensure accuracy in your measurement of the extension of the rubber band?
- How did you ensure accuracy in your measurement of the height reached by the projectile?
- How might you know that you had made a sufficient number of repeats for this investigation? (hint: The differences between mode, mean and/or median could be considered.)

**Building ideas**

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}.

- Measure the mass of the small projectile that you are using in your investigation.
- Use the results of your investigation to find the gravitational potential energy gained by the projectile when the slingshot was extended by 5 cm.
- Estimate the elastic potential stored by the slingshot when it was extended by 5 cm.

**Reinforcing ideas**

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.

- Investigate to find the height the projectile reaches when the rubber band is extended by 10 cm and the slingshot is aimed vertically.
- Use the results of your investigation, and the mass of the projectile, to find the gravitational potential energy gained by the projectile when the slingshot was extended by 10 cm.
- Estimate the elastic potential stored by the slingshot when it was extended by 10 cm.

**Extending ideas**

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.

- Find the height reached by a projectile launched from a vertical slingshot, with the rubber band extended 3 cm, 6 cm and 9 cm.
- Find the gravitational potential energy gained by a projectile launched from a vertical slingshot, with the rubber band extended 9 cm.
- Find the elastic stored energy provided by a projectile launched from a vertical slingshot, with the rubber band extended 9 cm.
- Is the relationship between the elastic stored energy in the slingshot and the height reached by the projectile a linear or a non-linear relationship? Justify your answer.

### Session Four

**F**

**ocus activity**

Do these data indicate a linear or a non linear relationship between extension and maximum height reached?

- How did you organise the recording/display of your data?
- How did the way you organised your data assist your conclusion about whether or not the relationship is linear?

**Building ideas**

Graph the data collected in your investigation to find the relationship between height reached (m) and extension (m) of the rubber band:

- Draw a set of positive axes occupying at least half a page of quad paper.
- Choose a suitable scale for the height reached (m) that will use as much of the horizontal axis as possible. Label that axis and mark on the scale in decimal values of m.
- Choose a suitable scale for the extension (m) that will use as much of the horizontal axis as possible. Label that axis and mark on the scale in decimal values of m.
- Plot the data you collected.
- Is the relationship shown linear?

**Reinforcing ideas**

- Construct a graph of height reached (m) of the projectile against extension (m) of the rubber band.
- Draw in the curve of best fit for your data.
- Describe the relationship shown by this graph as linear or non-linear.

**Extending ideas**

- Graph your data to describe the relationship shown by this graph as linear or non-linear.
- Use your graph to predict the height a projectile will reach when the rubber band is extended by 8 cm.
- Test your prediction.
- Comment on the similarities/differences between your predicted and tested heights.

### Session Five

**Focus activity**

The elastic potential energy of a rubber band slingshot depends on the extension of the rubber band by the rule: E

_{e}= ½ kx

^{2}. What shape will the graph of elastic potential energy against extension form? (nb k is a constant for a given spring or rubber band)

- Could you answer this question by inspection of the rule? What clues did it give about the shape of the graph?
- Could you answer this question by testing various values of x? Would you need to assign a value for k?

**Building ideas**

The elastic potential energy stored by extending a slingshot of extension, x, is described by the equation Ee = ½ kx2.

- Find the elastic potential energy that is stored in a slingshot of k = 40 N/m extended by 0.075 m.
- Find the elastic potential energy that is stored in a slingshot of k = 40 N/m extended by 0.15 m.
- What was the effect on the amount of elastic potential energy of doubling the extension of a slingshot?

**Reinforcing ideas**

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.

- Use the conservation of energy to find an expression for h, the height the projectile will reach, in terms of k, x, m and g.
- Given that the quantities k and m are constant for this investigation and g is always a constant value, comment on the shape of the graph of h against x.
- Use your expression for h to find the height a 0.05 kg projectile will reach if launched from a slingshot of k = 30 Nm-1 that has been extended by 0.12 m. Use g = 10 ms
^{-2}.

**Extending ideas**

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.

What would the effect be on the height that a projectile reaches, if the extension of a given slingshot is doubled? Justify your answer.