# Weighty Problems

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Purpose

This unit comprises six problems for students to apply and interpret measurement of mass. Students are also introduced to the concepts of net and gross mass.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
Specific Learning Outcomes
• Select the appropriate standard unit of measurement for a specific application.
• Measure masses with appropriate measuring devices.
• Measure net and gross mass.
Description of Mathematics

Mass is the force created by gravity acting of on an object. Mass is felt as weight, a force that pulls the object towards the centre of the Earth. Mass is measured in units based on grams, and tonnes. Larger or smaller units are created by combining or equally partitioning these units. One kilogram is a combination of 1000 grams (kilo means 1000). One milligram is 1/1000 of a gram and one microgram is 1/ 1 000 000 of a gram. The units for mass come from the mass of water. One cubic metre of water has a mass of 1 tonne, or 1000 kilograms. One millilitre of water has a mass of one gram.

Note that in the New Zealand Curriculum document, “weight” and “mass” are used interchangeably. In a science context, the definition of  “force created by gravity acting on an object” would often be equated with weight, not mass. Consider the scientific knowledge of your students (e.g. are they studying forces in science). It may be more appropriate to define mass as the amount of matter in an object (measured in kilograms) and weight using the adorementioned definition (measured in Newtons, N).

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

• letting students attempt problems using physical materials as much as possible, so they develop a ‘feel’ for the benchmark units
• directly modelling measurement with tools, like digital scales for mass
• providing opportunities for students to copy the correct use of tools
• clarifying the language of measurement units, such as “kilogram” as a mass that is made up of 1000 grams
• clarifying the meaning of symbols, e.g. 45g as 45 grams, and 45kg as 45 kilograms; 45t as 45 tonnes
• encouraging students to work collaboratively (mahi tahi) to share and justify their ideas
• easing the calculation demands by providing calculators where appropriate.

Tasks can be varied in many ways including:

• reducing the complexity of the numbers involved in the station tasks, e.g. whole number versus fraction dimensions. The choice of measurement units influences the difficulty of calculation as well as the level of precision
• allowing physical solutions with manipulatives before requiring abstract (in the head) anticipation of measures
• creating or using models of standard units, e.g. 1 litre of water for the mass of 1 kilogram
• reducing the demands for a product, e.g. less calculations and words, and more diagrams and models.

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Use the interests of your students to create contexts that will engage them. Students may be interested in the mass of rugby players. Students from large whānau, or who prepare food for large numbers of people, may relate to measuring quantities to scale up recipes. Carrying heavy objects was a major problem for pre-European Māori. How did they carry heavy loads, or move waka? Counting on Frank by Rod Clement may inspire some students to look for eccentric ways to apply measurement to their daily lives. For example, the human body is 60% water, by mass. How much water is in their body?

Te reo Māori vocabulary terms such as maihea (weight / mass), karamu (gram), manokaramu (kilogram), and tana (tonne) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials
• Station 1: Copymaster 1, \$1, \$2 and 50c coins, scales (preferably digital and able to weigh in grams), metre ruler
• Station 2: Copymaster 2, calculator
• Station 3: Copymaster 3, calculator
• Station 5: Copymaster 5, 1L measuring jug, five different sized plastic containers, scales, eyedropper
• Station 6: Copymaster 6, can of dog food, supermarket bags, calculator
Activity

The following six stations provide a range of problems for students to apply and interpret measurement of mass. Consider what would be the most effective method for introducing these to your class. You could work on them as a whole class and provide support to groups of students. Alternatively, you could use another relevant igniting activity to introduce the context for learning, before directing students to work on one or more of these stations independently, or in small groups. These stations could serve as the basis for learning in different sessions, or could be used as one session. At the conclusion of these stations, students draw on the problems presented and create their own stations to be used in other lessons.

#### Station 1: A kilo of coins

You have won a prize which can be just one of the following:

• 1 kilogram of \$1 coins
• A 1.5 metre long trail of \$2 coins (lying flat and touching)
• A 0.5 metre high stack of 50c coins

1 kg of \$1 coins (1000 ÷ 8 = 125 coins, so \$125)

1.5 metre of \$2 coins (1500 ÷ 26.5 = 57 coins, so \$114)

0.5 metre stack of 50c coins (500 ÷ 1.7 = 294 coins, so \$147)

#### Station 2: Largest Lasagne

This problem could be adapted to reflect food that is meaningful to your students (e.g. the largest tray of pani popo).

The world’s largest lasagne was made in 2012 at a restaurant in Wieliczka, Poland.
It weighed 4865 kg and measured 25 m x 2.5 m.

The ingredients were:

2500kg of pasta, 800kg of mince, 400kg of mozzarella cheese, 100kg of peas, 100kg of carrots, and equal amounts of white sauce and tomato sauce.

1. How much did the white sauce and tomato sauce weigh?
2. What would be the size of a 500g piece from the lasagne?
3. How many people could be fed with the whole lasagne?
Show how you arrived at your estimate.

1. The other ingredients total 3900kg so the sauces must weigh 4865 – 3900 = 965 kg.
500L of each sauce was used. Does that sound right?
2. A 500g piece would be about 1/10 000 of the whole lasagne. One way is to cut both the length and width into 100 parts, since 100 x 100 = 10 000. A single piece would measure 25cm x 2.5cm. That’s a bit skinny so 12.5cm x 5cm might work better.
3. The lasagne was actually cut into 10 000 pieces so that’s how many people were fed. Each piece had a mass of 0.486 kg or 486 grams. That is a good serving of lasagne.

#### Station 3: Weighing Tonnes

Konsihiki was the largest active sumo wrestler in the world with a mass of 287 kg. Now he is retired.

How many Konishikis weigh as much as 1 tonne?

Make a table of tonne weights using objects in the classroom. Remember that 1000 kg is a tonne.

 Object Mass Number in a tonne Konsihiki 287 kg School bag 5 kg 200

The number of Konshihikis in 1 tonne equals 3.48, about 3 and ½ of him.

To find how many of any object make 1 tonne, divided 1 000 by the weight of the object in kilograms. For example, if a schoolbag weighs 5kg then 1 000 ÷ 5 = 200 make 1 tonne.

#### Station 4: Jumbo facts

Find out facts about the mass of very large animals and make a report about these animals for the class. To get you started here are some facts about the Blue Whale, which can be seen in New Zealand waters.

The blue whale is the largest animal living on Earth. It can reach up to nearly 30 metres in length and weigh up to 180 tonnes (t). Their tongues alone can weigh as much as some elephants and their hearts are huge, weighing a whopping 180kg. They have the largest babies on Earth. When they are first born they can be 8 metres (m) in length and weigh 4000kg. Imagine a jet engine that registers at 140 decibels. A blue whale, when it calls, registers at 188 decibels. Compare the facts about the Blue Whale with the large African elephant

The African elephant is the biggest animal on land. Fully grown the male can be 7 metres long, 3.2 metres tall at the shoulder and have a mass of 6500kg. Its tusks can weigh as much as 100kg each. The largest pair of tusks on record are in the British Museum and weigh 133kg each.

What combination of animals could be equal to the elephant's weight?

For example, it takes 6500 ÷ 5 = 1300 big domestic cats to weigh 1 elephant or 130 big dogs.

How many rhinoceroses, lions, giraffes, or hippopotamuses weigh the same as an elephant?

#### Station 5: Mass of water

Measure out one litre (l) of water.

1. What is the mass of one litre of water?
If 1L = 1000ml, what is the mass of 1mL of water?
2. For each container, estimate the capacity of the container, measure it to check, estimate the mass of water when the container is full, and find the mass of the water using scales.

 Container Estimate capacity Measure actual capacity Estimate mass of water Measure actual mass A B C D E
3. How many drops of water are needed to fill each container?
4. What is the mass of a single drop?

1. 1 litre (l) of water has a mass of 1 kilogram (1000 grams). 1 millilitre (mL) of water has a mass of 1 gram.
2. Answers depend on the size of the containers. Here is an example:

 Container Estimate capacity Measure actual capacity Estimate mass of water Measure actual mass A 400 mL 450 mL 390 g 450g
3. About 20 drops make 1 ml of water. Find the capacity of the container in mL then multiply by 20 to get the number of drops.
4. A single drop has a mass of 1/20 of 1g, that’s 0.05g.

#### Station 6: Frank’s arms

Counting on Frank by Rod Clement (1990; Harper Collins Publishers: Sydney) has some great ideas for measurement investigations. You can view readings of the book on YouTube if you cannot source a copy of the book. One of the ideas introduced in the story is about Frank carrying a trolley load of cans to the supermarket.

1. Trolleys measure 60 litres or 80 litres.
What do those measures mean?
2. How heavy do you think Frank’s load of cans is?
3. How many cans are you able to carry in a reusable supermarket bag?

How did you work that out?

Here is another task based on Counting on Frank that you may choose to use.