Te Kete Ipurangi
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# Measurement sense

## Solve problems involving rates.

Achievement Objectives
Achievement Objective: NA5-4: Use rates and ratios.
Teaching resources for this Achievement Objective:

## Comparing by Finding Rates

• Numeracy activities
• Number and Algebra
• Level Five

Solve problems involving rates.

## More Thinking

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Paddling down the Waikato

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Feel the Beat

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Dreaming of Millions

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number link activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Animal Antics

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Lunchtime Mardi Gras

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.

## Fair Exchanges

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Energy Levels

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## In Proportion?

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## The Caves of Anawheku

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Running Hot and Cold

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Pay Rates

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## The Right Gear

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Sailing with Maths

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number, measurement, and geometry activity from the Figure It Out theme series.

## Day Trippers

• Figure It Out activities
• Geometry and Measurement
• Level Four

This is a level 4 measurement, geometry, statistics and level 5 number activity from the Figure It Out theme series.

## The Right Money

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out theme series.

## Rates of Change

• Numeracy activities
• Number and Algebra
• Level Five

Solve problems involving rates.

## Painting by Numbers

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## The Great Race

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Tiring Teamwork

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Deb the Driver

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out series. It relates to Stage 8 of the Number Framework.
A PDF of the student activity is included.

## Coast to Coast

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number activity from the Figure It Out theme series.

## GM6-4: Deduce and apply the angle properties related to circles.

Teaching resources for this Achievement Objective:

## Hinea's other watch

• Problem solving activities
• Geometry and Measurement
• Level Six
Solve problems using the relationships between angles in a circle.
Devise and use problem solving strategies to explore situations mathematically (be systematic, guess and check, think, use algebra).

## Hinea's watch's hands

• Problem solving activities
• Geometry and Measurement
• Level Six
find an angle greater than 360 degrees that relates to a given position
solve a linear equation
devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, guess and check, use algebra)

## GM5-6: Create accurate nets for simple polyhedra and connect three-dimensional solids with different two-dimensional representations.

Elaboration on this Achievement Objective:

This means that students will use appropriate equipment to create accurate nets for simple polyhedra. Polyhedra are 3 dimensional shapes bound by polygons, for example a cube is made from 6 squares, a tetrahedron is made from 4 equilateral triangles, an octahedron is made from 8 equilateral triangles.  The range of polyhedra should include the platonic solids, cuboids, right-angled prisms and pyramids. For the more difficult polyhedra students should learn to focus on the polygons around each vertex (corner) and symmetry of the polyhedron, for example the net for an icosahedron or a dodecahedron can be formed from two halves.

Students will form three dimensional shapes drawn as plan views, isometric projections or nets. To do so they will need access to material such as lego or construction cubes. Students will also be able to represent models of three dimensional shapes using plan views isometric projections or nets where appropriate.

Teaching resources for this Achievement Objective:

## Winning Ways

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 geometry strand activity from the Figure It Out series.

## Missing Anything!

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 geometry strand activity from the Figure It Out series.

## Fair and Square

• Geometry and Measurement
• Level Five

This is a level 5 geometry strand activity from the Figure It Out series.

## Tricky Truncations

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 geometry strand activity from the Figure It Out series.

## Perfect Packing

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 geometry strand activity from the Figure It Out series.

## Ruler and Compass Constructions

• Units of Work
• Geometry and Measurement
• Level Five

In this unit we use rulers and compasses to construct perpendicular lines and to bisect angles. These constructions are used to make a variety of polygons, nets and to illustrate loci.

## Packing chocolates

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multiple-level activity is to engage students in applying their knowledge of three dimensional solids to solve a package and packing problem.

## GM5-5: Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties.

Elaboration on this Achievement Objective:

This means that students will know and apply angle properties relating to intersecting and parallel lines.

The angle properties of lines are:

1. Vertically opposite angles are equal, for example a = d, b = c
2. Adjacent angles add to 180o, for example a + b = 180o, a + c = 180o
3. Corresponding angles are equal, for example a = e, b = f, c = g, d= h
4. Interior angles add to 180o, for example c + e = 180o, d + f = 180o
5. Alternate angles are equal, c = f, d = e

Students should apply the properties of intersecting and parallel lines to problems like constructing rectangular and skewed picture frames.

Students should find and apply the angle properties of polygons. They may find the formula for the internal angles of regular polygons (equal angles) by measuring and creating a table of results or by reasoning through cutting the polygons into triangles. These angles properties are:

1. Interior angles of a polygon with n sides are ((n-2) x 180o)/n.
2. Exterior angles of any polygon add to 360o.

Teaching resources for this Achievement Objective:

## How High? and Other Problems

• Units of Work
• Geometry and Measurement
• Level Five

In this collection of activities we use ruler and compass constructions to draw a variety of shapes and to construct angles such as 90°, 60°, 45° and 15°.

## Ruler and Compass Constructions

• Units of Work
• Geometry and Measurement
• Level Five

In this unit we use rulers and compasses to construct perpendicular lines and to bisect angles. These constructions are used to make a variety of polygons, nets and to illustrate loci.

## Angles, Parallel Lines, and Polygons

• Units of Work
• Geometry and Measurement
• Level Five

In this unit, students will learn to investigate polygons using an instrument of their own construction. They should be able to find the general formula for the number of degrees in any polygon (including a triangle). Finally they will investigate an interesting locus problem related to polygons.

## Flip's flag

• Problem solving activities
• Geometry and Measurement
• Level Five
apply the angle properties of squares, rectangles and triangles
use fractions to express the relationships between the areas of rectangles, squares and triangles
devise and use problem solving strategies to explore situations mathematically (be systematic).

## Vince's problem

• Problem solving activities
• Geometry and Measurement
• Level Five

Find the centre of a circle using common material or mathematical construction tools.

## Sol's serviettes

• Problem solving activities
• Geometry and Measurement
• Level Five
Use the areas of triangles and squares to find unknown lengths
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).

## Ash in the Air

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 4 and 5 measurement activity from the Figure It Out theme series.

## Pierced polygons

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multi-level task is to engage students in an investigation applying geometric properties of polygons.

## Inside Irregular Polygons

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multi-level task is to engage students in using a given rule, to deduce another rule. This is an example of the deductive reasoning required for forming successful geometric proofs.

## Synchronised Swim Shapes

• Units of Work
• Geometry and Measurement
• Level Five

The purpose of this unit is to engage the student in applying their knowledge and skills of geometric thinking to design and describe formations mathematically and the translations and/or rotations needed to create those formations.

## GM5-4: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders.

Elaboration on this Achievement Objective:

This means that students should find the ratio of the diameter of a circle to its perimeter (π) by measuring several circles and looking for the relationship.

They should understand that the formula for the circumference of a circle can be expressed in two ways:
C = 2πr (2 times pi times the radius), or
C = πD (pi times the diameter)

They should apply the formula to find the circumference of any circle, for example the circumference of a bicycle tyre. Students should find the formula for area of a circle by cutting several circles into increasingly small sectors and reassembling the sectors to approximate rectangles, A = πr2 (pi times the radius squared). They should apply this formula to finding the area of any circle.

Students will find the perimeters of composite shapes by adding the lengths of the sides together. At Level Five composite shapes are made up of common polygons (for example rectangles, triangles, semicircles). Students will find the area of composite shapes by calculating the areas of the parts and adding them together. Students will find the volumes of prisms by multiplying the area of their cross section by their length, for example for a cylinder multiply the area of the circle by the length of the cylinder.

Teaching resources for this Achievement Objective:

## Scale Factors for Areas and Volumes

• Units of Work
• Geometry and Measurement
• Level Five

This unit of work provides students with the opportunity to discover the effect of a change of length by a given factor on area and volume.

## Round the Track

• Units of Work
• Geometry and Measurement
• Level Five

This unit is about making calculations in the real life situations of athletics.The main task is to construct a scale plan of an athletics track.

## Fences and Posts

• Units of Work
• Geometry and Measurement
• Level Five

This measurement unit concerns Pick’s Rule which applies to finding areas of polygons where all the vertices are lattice points, (in this application, the nails on geoboards). The unit centres on finding linear patterns.

## A4 containers

• Problem solving activities
• Geometry and Measurement
• Level Five
Describe the relationship between the surface area and volume of a container
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a model)

• Problem solving activities
• Geometry and Measurement
• Level Five
Construct two intersecting circles where the radius of the second circle is on the circumference of the first.
Use Pythagoras’ theorem to find the area of a rhombus
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a model)

## Penny's box

• Problem solving activities
• Geometry and Measurement
• Level Five
Find the volume and surface area of rectangular prisms
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).

## Puck's girdle

• Problem solving activities
• Geometry and Measurement
• Level Five
Find the circumference of a circle.
Understand the relationship between changes in the circumference of a circle and changes in the radius.
Devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).

## Tennis ball tubes

• Problem solving activities
• Geometry and Measurement
• Level Five
Use the formula for the circumference and diameter of a sphere to solve a problem
Devise and use problem solving strategies to explore situations mathematically (be systematic, make a model)

## Playdough balls

• Problem solving activities
• Geometry and Measurement
• Level Five
compare the volume of a sphere and a cylinder by either measuring or applying a formula.
devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).

## Karen's tiles

• Problem solving activities
• Geometry and Measurement
• Level Five
investigate how perimeter changes as area changes
see the relation between area and perimeter for similar shapes
pose questions for mathematical exploration
prove or refute mathematical conjectures

## Peter's second string

• Problem solving activities
• Geometry and Measurement
• Level Five
determine the maximum area of a rectangle with a given perimeter
determine the range of areas of a rectangle with a given perimeter
pose questions for mathematical exploration
prove or refute mathematical conjectures

• Problem solving activities
• Geometry and Measurement
• Level Five

Use area formulas of circles and squares to solve problems

## Around We Go

• Units of Work
• Geometry and Measurement
• Level Five

The focus for this unit is the measurement of circles and the relationships to be found within them. Students investigate a range of circles, each with a different radius, diameter and circumference and assemble this information in their search for relationships.

• Figure It Out activities
• Geometry and Measurement
• Level Four

This is a level 4 and 5 measurement strand activity from the Figure It Out series.

## Colossal Kiwifruit

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series.

## Round the Bend

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series.

## On the Right Track

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series.

## Gumboot Games

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series.

## Area of a circle

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multi-level task is to engage students in an investigation of the area of circles.

## Marketing Tricks

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multi-level task is to encourage students to use algebraic techniques when solving a measurement problem.

## GM5-3: Deduce and use formulae to find the perimeters and areas of polygons and the volumes of prisms.

Elaboration on this Achievement Objective:

Students should create formulae for perimeters and areas of polygons by attending to features of those polygons. For perimeters they should recognise when side lengths are equal and use multiplicative strategies rather than additive ones. For example to calculate the perimeter of a parallelogram the student should double the length of two adjacent sides, for a regular pentagon they should multiply the length of one side by five.

Students should connect the formulae for finding areas of parallelograms, triangles, and trapeziums to the formula for the area of a rectangle using partitioning and reassembling the shapes. Students should be able to find perimeters and areas of shapes where the side lengths are simple decimals in order to understand that area, in particular, is not restricted to discrete situations where the units of measure can be counted individually.

Teaching resources for this Achievement Objective:

## How High? and Other Problems

• Units of Work
• Geometry and Measurement
• Level Five

In this collection of activities we use ruler and compass constructions to draw a variety of shapes and to construct angles such as 90°, 60°, 45° and 15°.

## Square pegs in round holes

• Problem solving activities
• Geometry and Measurement
• Level Five
Draw a diagram to represent a physical situation for a general case;
Apply Pythagoras’ theorem to a general situation
Devise and use problem solving strategies to explore situations mathematically

## Penny's box

• Problem solving activities
• Geometry and Measurement
• Level Five
Find the volume and surface area of rectangular prisms
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).

## Playdough balls

• Problem solving activities
• Geometry and Measurement
• Level Five
compare the volume of a sphere and a cylinder by either measuring or applying a formula.
devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).

## Area of a regular pentagon

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multi-level task is to engage students in a task that requires them to apply geometric properties of polygons and right angles triangle techniques in solving for the area of a polygon.

## Packing chocolates

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this multiple-level activity is to engage students in applying their knowledge of three dimensional solids to solve a package and packing problem.

## GM5-2: Convert between metric units, using decimals.

Elaboration on this Achievement Objective:

This means that students will apply their knowledge of decimal place value to convert between units for the same attribute, for example between units for weight. They should know the meaning of prefixes used in the metric system that act as “scalars” on base units, for example “kilo” means one thousand, “centi” means one hundredth.

Students are able to convert between units for the same attribute (for example volume, mass) with more than 1 decimal place, for example 0.125 kg = 125g, or 17.5 cm = 0.175.

Teaching resources for this Achievement Objective:

## Ratios

• Units of Work
• Geometry and Measurement
• Level Five

In this unit students convert between the metric units for capacity, within the context of drinks. Students engage in proportional thinking as they use their knowledge of fraction, decimal and percentage conversions, to explore, compare and combine ratios, and to calculate rates.

## Body Measurements

• Secondary activities
• Number and Algebra
• Level Five

These exercises and activities are for students to use independently of the teacher to practice number and measurement skills.

## Plentiful Plankton

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series. It also has an achievement objective from the Level 5 Number strategies and knowledge substrand.

## Fat in Foods

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series. It also has an achievement objective from the Level 5 Number strategies and knowledge substrand.

## Television Views

• Figure It Out activities
• Geometry and Measurement
• Level Five

This is a level 5 measurement strand activity from the Figure It Out series. It also has an achievement objective from the Level 5 Number strategies and knowledge substrand.

## Sailing with Maths

• Figure It Out activities
• Number and Algebra
• Level Five

This is a level 5 number, measurement, and geometry activity from the Figure It Out theme series.

## The ski jump

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this task is to engage students in converting metric units to solve a problem.

## Old bones

• Rich learning activities
• Geometry and Measurement
• Level Five

The purpose of this activity is to engage students in solving a measurement problem.

## Slingshots

• Number and Algebra
• Level Five

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.