Level 1 Mathematics & Statistics Internal Assessment Resource

Internal assessment resource Mathematics and Statistics 1.7A for Achievement Standard 91032

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Internal Assessment Resource

Mathematics and Statistics Level 1

This resource supports assessment against:
Achievement Standard 91032
Apply right-angled triangles in solving measurement problems
Resource title: The Rugby Posts
3 credits
This resource:
· Clarifies the requirements of the standard
· Supports good assessment practice
· Should be subjected to the school’s usual assessment quality assurance process
· Should be modified to make the context relevant to students in their school environment and ensure that submitted evidence is authentic
Date version published by Ministry of Education / December 2010
To support internal assessment from 2011
Authenticity of evidence / Teachers must manage authenticity for any assessment from a public source, because students may have access to the assessment schedule or student exemplar material.
Using this assessment resource without modification may mean that students work is not authentic. The teacher may need to change figures, measurements or data sources or set a different context or topic to be investigated or a different text to read or perform.

Internal Assessment Resource

Achievement Standard Mathematics and Statistics 91032: Apply right-angled triangles in solving measurement problems

Resource reference: Mathematics and Statistics 1.7A

Resource title: The Rugby Posts

Credits: 3

Teacher guidelines

The following guidelines are supplied to enable teachers to carry out valid and consistent assessment using this internal assessment resource.

Teachers need to be very familiar with the outcome being assessed by Achievement Standard Mathematics and Statistics 91032. The achievement criteria and the explanatory notes contain information, definitions, and requirements that are crucial when interpreting the standard and assessing students against it.

Context/setting

This activity requires students to take measurements and use to find lengths and angles in right angled triangles. The context is the height of the rugby posts on the school field, which need to be taken down and transported away for storage.

The measurements of the rugby goal as specified by the RFU are:

The cross bar should be 5.6 m long at a height of 3 m above the ground.

The upright posts should be at least 0.4 m above the crossbar. This height will vary at schools.

Possible alterations to the task

The dimensions of the trailer in the task have been set so that the upright posts will not fit in flat along the diagonal for posts that are of height 8 m above the ground. If the school’s posts are lower than this the dimensions of the trailer will need to be altered so the school posts do not fit in it.

Conditions

Students will need to work in pairs during one session to take measurements and record them. A sunny day is required for the shadows.

The students will work individually during a second session to complete the calculations.

Students may use any appropriate technology.

Resource requirements

Students will need to be provided with a tape measure of length at least 30 m, and a clinometer to take the measurements. They will also need a pole of at least 1 m length for measuring the shadow.

Additional information

The standard identifies measuring at a level of precision appropriate to the task as a ‘method’. For Achieved students must both select and use a range of methods. If there is assessor confidence that the student has contributed to the selection and making of the measurements and that the measurements are accurate to a suitable level of precision then measuring would be acceptable as one of the three methods.

Internal assessment resource Mathematics and Statistics 1.7A for Achievement Standard 91032

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Internal Assessment Resource

Achievement Standard Mathematics and Statistics 91032: Apply right-angled triangles in solving measurement problems

Resource reference: Mathematics and Statistics 1.7A

Resource title: The Rugby Posts

Credits: 3

Achievement / Achievement with Merit / Achievement with Excellence
Apply right-angled triangles in solving measurement problems. / Apply right-angled triangles, using relational thinking, in solving measurement problems / Apply right-angled triangles, using extended abstract thinking, in solving measurement problems.

Student instructions

Introduction

A set of rugby goalposts has to come down and be transported away for storage. The groundsman needs to know if he can transport the posts safely using his tractor and trailer.

The posts are vertical. They are placed in the ground with 0.5 m below the ground (the dashed part in the diagram.) The cross bar is 5.6 m long.

The groundsman intends to shift the posts using his trailer. On this sunny day use the shadows, and at least one other method, to find the above-ground height of the posts.

The trailer has a rectangular base 6 m long, 2.5 m wide and 0.5 m high. The groundsman thinks that the posts cannot be placed in the trailer along the diagonal of the trailer without sticking out as shown in the diagram.

He has asked you to confirm this, using the height calculated from one of your measurements.

The groundsman wants to place the posts in the trailer, as shown in the diagram. For safety reasons, the horizontal distance that the posts stick out beyond the trailer must be no more than 1 m.

The groundsman can increase the height of the sides of the trailer. He has asked you to investigate what height the sides of the trailer will need to be if he is to transport the posts without exceeding this restriction.

The Task

Session 1

Working with a partner use the equipment provided to make whatever measurements you require to enable you to calculate the height of the rugby posts on your school field using the methods you have selected.

Session 2

Use your measurements to:

· calculate the height of the rugby posts on your school field using the two different methods you have selected.

· use the height from one of these calculations to determine if the posts can be carried in the trailer

· investigate what height the sides of the trailer needs to be to allow the posts to be carried safely.

The quality of your reasoning, using a range of methods, and how well you link the context to your solutions will determine your overall grade.

Round your calculations sensibly, and show units.

Clearly communicate your method using appropriate mathematical statements, so that the groundsman can verify your calculations.

Internal assessment resource Mathematics and Statistics 1.7A for Achievement Standard 91032

PAGE FOR TEACHER USE

Assessment schedule: Mathematics and Statistics 91032 The Rugby Posts

The schedule assumes the following measurements were taken.

Method 1:

Distance from base of post: 30 m My height: 1.73 m Angle of inclination of top of the pole: 12º

Method 2:

Height of pole: 1 m Length of shadow of pole: 2 m Length of shadow of the posts: 16 m

Evidence/Judgements for Achievement / Evidence/Judgements for Achievement with Merit / Evidence/Judgements for Achievement with Excellence
To achieve students will take measurements and apply right angled triangles in solving these problems.
This will involve selecting and using a range of methods in solving measurement problems, demonstrating knowledge of measurement and geometric concepts and terms, and communicating solutions which would usually require only one or two steps.
Students will use their measurements to make calculations using at least three different methods.
For example
Height of posts above clinometer = 30 x tan 12 = 6.4 m
(method, side from trig)
Height of post = 16*1/2 = 8 m
(method, side from similar triangles)
Diagonal of the base of the trailer = Ö(62 + 2.52) = 6.5 m
Or diagonal of the trailer = Ö(62 + 2.52 + 0.52) = 6.52 m
(Method, side from pythagoras)
If there is assessor confidence that the student has contributed to the selection and making of the measurements and that the measurements are accurate to a suitable level of precision then measuring would be acceptable as one of the three methods.
Minor omissions and variations in rounding can be accepted. / To achieve with Merit students apply right angled triangles using relational thinking in solving these problems.
Relational thinking will involve selecting and carrying out a logical sequence of steps, connecting different concepts and also relating findings to the context or communicating thinking using appropriate mathematical statements.
Students will solve at least one aspect of the problem correctly and communicate their solutions.
E.g., solve showing length of diagonal of trailer is less than length of post or find one solution showing dimensions of trailer that meet the safety requirements and relate the findings to the context.
For example
Method 1
Height of posts above clinometer = 30 x tan 12 = 6.4 m
Height of posts = 6.4 + 1.7 m = 8.1m
Method 2
Height of post = 16*1/2 = 8 m
I will take the height of the posts as 8m
The length of the posts to be transported = 8+0.5 = 8.5 m
Diagonal of the base of the trailer = Ö(62 + 2.52) = 6.5 m
Diagonal of the trailer = Ö(6.52 + 0.52) = 6.52 m
The posts are too long to be transported without sticking out of the trailer by more than 1 m.
Alternative methods for solving these problems are possible and allowed.
Students should not be penalised for minor errors (e.g., omitting the 0.5m below ground), running arithmetic or failing to communicate clearly.
Accept answers (and follow through) to 3 s.f.
Answers must include relevant units. Answers must be related to the context of the problem.
Minor omissions can be accepted / To achieve with Excellence, students apply right angled triangles using extended abstract thinking in solving these problems.
Extended abstract thinking will involve devising a strategy to investigate or solve a problem and use correct mathematical statements.
Students will solve both aspects of the problem correctly and communicate their thinking using correct mathematical statements.
E.g., solution showing length of diagonal of trailer is less than length of post AND one solution showing dimensions of trailer that meet safety requirements explained with correct mathematical statements.
For example
Method 1
Height of posts above clinometer = 30 x tan 12 = 6.4 m
Height of posts = 6.4 + 1.7 m = 8.1 m
Method 2
Height of post = 16*1/2 = 8 m
I will take the height of the posts as 8 m
The length of the posts to be transported = 8+0.5 = 8.5 m
Diagonal of the base of the trailer = Ö(62 + 2.52) = 6.5 m
Diagonal of the trailer = Ö(6.52 + 0.52) = 6.52 m
The posts are too long to be transported without sticking out of the trailer.

This maximum horizontal length outside the trailer will be directly below the post.
This diagram shows the position of the post when the horizontal distance is 1m. I have to find h.
cos a = 7.5/8.5
a = 280
h = 6.5 tana = 3.46 m
The height of the trailer must be more than 3.46 m
Alternative methods for solving these problems are possible and should be allowed.
Students should not be penalised for minor errors (e.g., omitting the 0.5 m below ground.)
Accept answers (and follow through) to 3 s.f.
Answers must include relevant units. Answers must be related to the context of the problem.
Students should not be penalised for using an incorrect answer to one calculation in subsequent calculations, provided their calculations follow through correctly.

Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the Achievement Standard.