Scale Matters

Working with the learning object with students

Access the learning object "Scale Matters: Tenths". Ask the students how units of one tenth are created (By dividing one units into ten equal parts).
Show how the two variations of the game work by solving three problems from each. Choose the "Name the Number" option first then "Select the Spot" second. Illustrate how the "Choose a scale" can be used to support students, if need be, as a guide to getting better estimates.
Using specific problems focus the students on imaging the scale, firstly between the given numbers, and outside of those points if appropriate. Consider this example:

In the absence of the scale students need to work out how many tenths lie between 45.9 and 46.5. This is best done using an "up through one" strategy, i.e. 45.9 + 0.1 = 46 and 46 + 0.5 = 46.5, 0.1 + 0.5 = 0.6 so there are six tenths between 45.9 and 46.5 on the scale. The students then need to visualise the space between the points subdivided into six equal lengths. In this case it is easier to halve the space then third each part. The missing number can be found either by going up four tenths from 45.9 or down two tenths from 46.5.
Similar thinking is required for "Name the Spot" problems like that given below.

The student must find how many tenths lie between 36.7 and 37.4 and image the subdivision of the space between the points into the seven equal parts (tenths) required. They must then realise that three tenths more than 36.7 is 37.0, the required spot.
Students can then select the level of difficulty they wish. Three other "Scale Matters" learning objects are available at level 3. These involve :

• tens of thousands
• hundredths
• range of numbers (a smorgasbord)

Recognise that some students may need support with interpreting the scales in hundredths. They will need to understand that hundredths are created when tenths are not accurate enough to measure a quantity. Tenths are divided into ten equal units, and these are named hundredths because 10 x 10 (one hundred) of the units fit in one. Decimal notation can also present problems to students especially where they apply their whole number thinking to decimals, e.g. 2.43 is greater than 2.6 since 43 is greater than 6. Consider this "Name the Number" problem from the hundredths learning object.

The difference between 9.04 and 9.08 is 4 hundredths. This can be confirmed on a calculator by keying in 9.04 + 0.01 = = =… This will tell the calculator to count in units of one hundredth each time the = button is pressed. Four presses of = will provide an answer of 9.08. Learning the conventions of linking counts with numbers is as important for students with decimals as it was they first learned about whole numbers. Having realised that 4 hundredths lie between 9.04 and 9.08 student need to estimate how many tenths lie between 9.08 and the location of the missing number. Since that length is about three quarters of the distance 9.04 – 9.08, it should be three-quarters of four which is 3 hundredths. Adding three hundredths to 9.08 involves the canon law of base ten place value; Ten of one unit make one unit of the next highest place value, e.g. ten tens are one hundred, ten hundredths are one tenth etc.
So 9.08 + 0.03 gives 9 ones and 11 hundredths, 11 hundredths equal 1 tenth and 1 hundredth, so 9.08 + 0.03 = 9.11, the mystery number.
The equipment animation Pipe Decimals shows how this equipment can be used to develop a linear model of decimals. Notes on how to use Pipe Decimals are found on pages 22-24 in Book 7 of the Numeracy Project Series, "Teaching Fractions, Decimals and Percentages."

Students working independently with the learning object

Allow students to choose from the full suite of Scale Matters learning objects. This includes the Level Four objects based on integers and a range of numbers. The ultimate objective of this choice is for students to integrate these various sets of numbers, whole numbers, decimals and integers together in a coherent framework. Be aware that it is easy for students to develop misconceptions such as decimals are less than zero, e.g. 0.143 lives to the left of zero on the number line.
Once students have experience with several of the learning objects challenge them to solve Name the Number problems simultaneously (e.g. 3 different computers) and show all of the numbers on the same number line. For example, this might involve locating 8, 4.7, and -5 on the same number line. Good combinations of Scale Matters learning objects are:
Ones (Level 2), Tenths (Level 3), and Integers (Level 4)
Ones (Level 2), Tenths, and Hundredths (Level 3)

Students working independently without the learning object

It is important that students appreciate the significance of reading scales to a range of measurement tasks. While some scales are linear, e.g. thermometers, others are circular, e.g. analogue clocks, weight scales, and protractors. Some are based conveniently on ten, e.g. metric measures for weight or capacity, while others on bases, e.g. 12 and 60 for time and 360 for turns in degrees. Some involve negative measures, e.g. temperature and angles.
Provide students with a range of measuring tasks which focus them on connecting the reading of linear scales in the learning objects with the scales on the measurement devices.
Examples might be:

1. The height of five players in the Silver Ferns netball team is given in order as 1.8 m, 1.84m, 1.89m, 1.9m, and 1.93m. Use a metre ruler to mark these heights on a wall or door frame. If a player’s height was given as 1.786m, how tall would that be? Why aren’t players’ heights given as numbers with 3 decimal places?
2. On a measuring cylinder the scale marks are evenly spaced. Why is that? If you made a scale to show units of 100mL on the side of a 2.25 L plastic softdrink bottle would the marks be evenly spaced? Predict the spacing based on pouring in two lots of 100 mL. Mark the scale by pouring amounts of 100 mL into the bottle. Does the scale match your predictions? How would you measure these amounts with your plastic bottle device; 350mL, 975mL, 3L, 1.125L
   Anticipate what the scale would look like for other non-regular containers, e.g. Shampoo or Fruitjuice bottles, blocked off funnels (e.g. top end of a softdrink bottle), a sphere (puncture ball).
3. What do the marks on a protractor measure? Why do they go around the outside of a circle (or half circle). What is the largest number of the scale (360)? What parts is the scale divided into (quarters of 90°, 10°, 1°)? Why do you think these divisions were chosen? Use your protractor to draw angles that measure 45°, 110°, 350°, 400°, -30°. Why are these angles drawn the same, 60°, 420°, 780°? If 22½° is 22 degrees and 30 seconds (22°30’), how many seconds are in a degree? How would you write 60¼° and 90.6° using degrees and seconds?
4. On a metric set of weighing scales what are the units? How are kilograms related to grams? Which unit, grams or kilograms, are most useful to measure the weight of; a bucket of water? a teaspoon of water?, a pot of margarine?, you?, a litre of water?, a 2.25L bottle full of softdrink? If 1000 grams is 1 kilogram, how many grams would these amounts be? 3 kg, 5.5kg, 0.5 kg, 0.25 kg, 0.9 kg.
5. On a set of house plans the size of the lounge is 5.6 metres by 4.3 metres. Use a metre ruler to mark out a rectangle that is this size. How many square metres is that? Check out the size of rooms on house plans. Work out the areas of the rooms, in square metres, from the side lengths.