This unit, emphasising the need for standard units of volume, explores the context of trading with people on an imaginary island called Smati, where their measures are sloshes, dribbles, and plops.
- Recognise the need for a standard unit of volume.
- Estimate and measure to the nearest litre.
- Relate the litre to familiar everyday containers.
Capacity is the amount of internal space within a container. The attribute of internal space is important to housing liquids and gases, though sometimes the volumes of luggage are given in units of capacity, e.g. 40L backpack.
The base unit of capacity is the litre. The litre is the amount of space equivalent to a cube that measures 10cm x 10cm x 10cm. One litre of water has a mass of 1 kilogram at sea level. Prefixes are used to create smaller and larger units from the base unit. The most common examples of these units are the millilitre (1/1000 of a litre), the microlitre (1/1 000 000 of a litre), the kilolitre (1000 litres or 1 cubic metre). In european countries it is common to encounter cL, centilitres (1/100 of 1 litre or 10mL).
The learning opportunities in this unit are very practical. Differentiation of tasks to meet the needs of students can occur in a variety of ways:
- providing capacity measurement tools that have easy-to-read scales, and not those with few measurements displayed
- providing containers that have manageable capacities, e.g. 500mL rather than 285mL
- physically filling containers using trusted measures, like 100mL, and asking questions aimed at anticipation of the result
- allowing the use of calculators for conversions between units of measurement, e.g. 2.5 L = 2500mL
Tasks can be varied in many ways including:
- manipulating the complexity of the comparisons required among containers
- providing supportive templates for students to record results in
- encouraging flexibility in the way students choose to solve problems, and who they work with.
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Use containers that students are familiar with from their daily lives. Give purpose to the mathematics through context. Trading with a Pacific Island nation will mean more to students if they have whānau in such places or have visited them. Look for videos of Pacific Island nations to help student to interpret the context. Also, consider when students use units of capacity, for tasks like cooking, catering at a party, providing beverages for a sports team, or watering plants.
- Plastic cups of different sizes
- Ice-cream containers (2L)
- A range of plastic soft drink bottles of different sizes (300 mL, 500 mL, 600mL, lL, 1.25L, 1.5L, 2L)
- Sticky labels
- To begin the week we will use some simple experiences that will reinforce the students’ existing knowledge of units of capacity and will provide an assessment of their initial knowledge.
- Pose the problem:
Marama is planning her birthday party.
She wants to invite eleven friends so there will be twelve people at the party.
How many bottles of soft drink will she need to buy so that each person can have one drink?
- Discuss this problem with the class. Students will have ideas like: It depends on the size of the bottle. (What sizes can you get? 600 mL, 1L, 1.25 etc.) It depends on the size of the cups/glasses.
(How much does a cup hold? 200mL, 250 mL, 300 mL, etc.)
It also depends on how much you are going to fill each cup.
- Produce a 1.5 litre bottle full of water and a 250 mL cup. Get the students to estimate how many full cups of drink they believe could be poured from the bottle. Ask individual students to give their estimates and explain their thinking. Record the estimates on a dot plot with each dot representing an individual student's estimate.
Pour one cup of water from the bottle and invite the students to change their estimates if they wish. If the dot plot has been made with sticky dots on a whiteboard they can easily be rearranged.
Confirm the actual number of cups the bottle holds by pouring until it is empty. Compare the actual result to their estimates on the dot plot and invite them to explain what they notice.
- Remind the students of Marama’s birthday problem and tell them that they are going to solve her problem in groups. Each group will be given a full bottle of water, an icecream container (for catching spillage), and a cup. They are to work out how many bottles Marama will need, by estimation first, then confirm their estimation by pouring. Note that different groups will use bottles of different sizes. Remind the students to record their results for reporting back.
- Allow the groups time to attempt Marama’s problem then bring them together for reporting back. Focus on their methods of estimation, particularly how they may have applied the results from the introduction.
- Ask if there is an easier way to find out the number of cups that can be poured from each bottle. Using comparisons from one metric unit to another may help the groups solve Marama’s problem. For example, a student may say that you can get ten 250mL cups from a 2.5 litre bottle since 2.5 litres equals 2500 millilitres.
- The groups then reform to use metric units to solve Marama’s problem with the bottle and cup from another group. For example, given a 1.5 litre bottle and a 250 mL cup they should work out that the bottle will hold six cups (6 x 250 = 1500mL) with 50 mL to spare. Marama will need two bottles in this case. The students’ answers can be checked, as before, by repeating pouring.
- Get the students to write a sentence or two in their book/journal telling you what they now know about the metric units of capacity. Their reflections can be used to gauge their initial knowledge. Invite ideas from them to make a chart about metric units. Key comparisons are 1000 millilitres (mL) = 1 litre (L), 500 mL = 0.5 litres, 250 mL = 0.25 litres, etc.
- For the next three days we put the students in situations where they need to apply their knowledge of the metric units for capacity. They will be required to make sense of an unknown system and compare it to our own.
- Tell students that there is a tiny island nation, called Smati, in the South Pacific Ocean. The people of Smati have a special elixir that helps people to remember their multiplication and division tables. (Wouldn’t you like some of that?) You are going to go on a trade mission to Smati so New Zealand can trade our milk for their elixir. Unfortunately, on Smati they do not use litres and millilitres to measure capacity. They use sloshes, dribbles, and plops. Before you can go to Smati you will need to figure out how big each unit is so that you can talk their language (of capacity, that is).
- Tell the students that you have some bottles of elixir from Smati which give some clue about how many millilitres or litres a slosh, a dribble and a plop are. You will need to have prepared a measurement jug and the following full bottles of "elixir" for each group:
600 mL labelled 12 dribbles, 1 litre bottle labelled 4 plops,
1.5 L labelled 1 slosh.
In their groups students are to find out how they can change measurements in dribbles, plops and sloshes into metric units. Remind them to record their results.
- While the students are investigating circulate around the groups to monitor their progress. Important guiding questions are:
Which Smati unit do you think is biggest? Why?
Which Smati unit do you think will be easiest to compare? Why?
How do you think dribbles, plops and sloshes are related?
How could you find out how many millilitres are the same as one plop?
- After a while bring the class together with the intention of making a conversion chart for Smati measures to metric measures. Get different groups to report back and look for common findings. These can be confirmed by checking with the bottles:
12 dribbles = 600 mL, so 1 dribble = 50 mL
4 plops = 1 litre = 1000 mL, so 1 plop = 250 mL
1 slosh = 1.5 L = 1500 mL
- Ask: How many dribbles would make a plop? (5)
How many plops make a slosh? (6)
How many dribbles make a slosh? (30)
- Give each group a 2 litre milk bottle. Tell them that it needs to be relabelled so that Smati people will know how much milk it holds. Send them away to figure out the new label. Report back on their ideas. Note that many answers are possible like, 1 slosh and 2 plops, 40 dribbles, 8 plops. Discuss which label might be best.
- Have a range of plastic bottles of many different sizes for the students to use. Provide sticky labels.You may need to discuss why bottles are certain sizes. This brings up ideas of suitability for purpose (technology curriculum) and tidiness in terms of units of capacity (348 mL is not a common bottle size!).
- Tell the students that they must come up with a new range of milk bottles for the Smati market. Each bottle must be labelled with how much milk it holds in dribbles, plops, and sloshes. Consumers in Smati like the capacities to be both functional and easy to recognise. Discuss what milk is used for.
How much milk does your household use? What sized bottles does your family buy? Why is that size good for your needs?
- Once each group has its range of milk bottles another group can adopt the role of the Smatian customs inspectors to check that each bottle is the capacity that is stated.
- Now for the trade mission to Smati! Tell the students that when you go to Smati the trade rules are half as much elixir for of the same volume of milk. So, if you give them a two plop bottle of milk, they will give you a one plop bottle of elixir. Use the milk bottles the students have created and pretend they are taking this milk to Smati. They must find out how much elixir, in millilitres and litres, they will receive back for the milk they are taking. This may be done by acting out with one group as New Zealanders and the other as Smatians!
- It is good to trade with Smati as kiwi students need the elixir desperately to help them remember multiplication and division facts. However, pricing the bottles of elixir for the supermarket might get quite complicated. Provide bottles of different shapes and capacities for students to use.
- Remember that New Zealand exchanges milk for elixir so the prices need to be comparable.
What should be true about the price of a bottle of elixir?
Students should say that elixir is twice as expensive as milk for the same volume.
What does milk cost?
- You might look online for the current price of milk. That price will vary depending on the size of the bottle. A 1L bottle might cost $2.65 while a 2L bottle might cost $4.90.
Is that the same amount for every 100mL? How could we find that cost per 100mL?
- Discuss how dividing the price by the number of mL gives the cost per millilitre. Multiplying that unit cost by 100 gives the price per 100mL. Confirm that smaller bottles usually cost more per 100mL than larger bottles.
- Pose the problem:
A Supermarket is now importing Smarti elixir. They wonder how much they should charge for a bottle.
How could they work out what to charge?
- Students might say that the price depends on the size of the bottle.
Take a specific bottle as an example (say a 3L juice bottle)
Ask: It says that this bottle holds 2 sloshes. Does that information help? What other information do we need?
- Students might suggest converting the Smati measure to millilitres. (1 slosh = 1500mL so 2 sloshes equals 3000mL)
So we should charge the cost of 3000 mL of milk? Have we forgotten something?
3000mL of elixir was traded for twice as much milk, 6000mL. How much does 6000mL of milk cost?
- Students need to convert 6000mL into 6 litres and calculate the cost of three 2L bottles.
The pricing of bottles takes two steps:
- Each group chooses a set of five bottle in various sizes and shapes. They work out the Smati measure of capacity for each bottle and label it accordingly. e.g. 4 Plops for 1L.
- Groups exchange sets of bottles and the receiving group must work out the correct price to charge for each bottle full of elixir. Someone in the group acts as the recorder so that the calculations are clear.
- After a suitable time ask the groups to offer one bottle with a displayed price to the class collection.
Do the prices look right?
- The discussion might raise issues about estimating the capacity of bottles, especially those that are irregular. Students might also question some calculations.
Which prices look correct? Which prices are possibly incorrect?
- Check on dubious prices (be the Consumer watchdog or Fair Go).
- If time permits investigate how long a bottle of elixir from Smati will last.
To improve your basic facts it is recommended that you drink 20mL of elixir per day.
How many days do these bottles of elixir last at that rate?
- Get the students to bring along the oddest shaped plastic bottles they can find. Shampoo and fruit juice bottles are good examples. With the whole class choose three or four bottles of different shapes but similar sizes. Label them A, B, C, D. Tell the students to rank them in order of capacity by sight and record their beliefs. Pour 100 mL of water from each bottle and mark the level with a felt pen (some bottles will not be very transparent).
- The students now revise their order and estimate the total capacity of each bottle. Discuss their methods of estimation, particularly their use of the 100mL as a benchmark and observations about the shape of the bottle at different points. Get a student to pour the remaining contents of each bottle into a measuring jug to check.
- Put the students into groups of four with several different odd shaped bottles. Their task is to make marks on each bottle to indicate fair shares of the bottle into quarters. They may use a capacity measure to do this if they wish. This will provide excellent assessment information about their ability to apply the metric units. For example a 600mL bottle could be marked in divisions of 150mL.