Before Viewing

Revise the system of place value for decimal fractions and relate this to numbers expressed as fractions and percentages. The programme focuses on tenths and hundredths.

Discuss the equivalence and meaning of numbers like 0.3, 0.30, 0.300.

Compare the sizes of numbers such as 0.3, 0.28 and 0.287.

Ask questions like: What is 0.35 plus ¹/₁₀? 0.35 plus ¹/₁₀₀? 0.35 plus ¹/₁₀₀₀? What number is ²/₁₀ more than 5, 5.4, 5.01, 4.9, 5.99? Can you find a number between 7 and 8? between 7.4 and 7.5? and so on.

During or After Viewing

Examine Lisa’s method for finding the right combination to open the lock. A further example worked through as a class would be useful for developing students’ understanding and showing the number scale used in more detail (0.00 to 9.99: 1000 possible combinations).

Discuss why it is sensible to make a guess near the middle of the remaining combinations.

After one guess (assuming it is in the middle of the range of combinations), half the combinations are left; after two guesses, a quarter; after three, an eighth; and so on. Can the students continue this sequence?

Lisa chose 5.00 (too small), 7.50 (too large) and then 6.25 (too small). What would students suggest as a possible next choice?

What if the lock had had a combination to three decimal places? Could Lisa have cracked it?

For what situations are percentages most useful? What about decimals and fractions?

What fractions seem to be the most useful? Why?

The Encyclopaedia briefly mentioned the length of the year. Every fourth year is a leap year, except that every hundredth year isn’t but every four-hundredth year is. Is the year 2000 a leap year or not?

Comments

Each guess halves the number of remaining combinations that could be correct; so with 10 guesses, 2¹⁰ = 1024 combinations can be distinguished. So Lisa should always be able to guess the combination.

For three decimal places there would be 10,000 possible combinations, requiring at most 14 guesses, since 2¹⁴ = 16,384.

Students could consider other situations where digits are combined. For example:

• Bicycle locks often have 4 rings with 6 digits each. How many combinations are possible?
• Telephone codes have changed several times in recent years to create more possible numbers. How is this achieved?
• PINs are generally 4 digits. Is this sufficient to prevent fraud?

The year 2000 is a leap year. Students could explore the history of our calendar, or calendars from other cultures, and their influence on the development of mathematical ideas. The Julian and Gregorian Calendars can be explored at: http://www.magnet.ch/serendipity/hermetic/cal_stud/cal_art.htm.

The Worksheets

Worksheet 1: Zooming In

(1) The aim of this activity is to help students develop a systematic approach to a simple problem requiring trial and improvement, which can be applied to more complex tasks. It also aims to develop their understanding of decimal fractions and degrees of accuracy. Students may find graph paper useful for constructing successive number lines that allow them to ‘zoom in’ to the value they are seeking. The construction of lines for numbers containing three decimal places is more challenging. The activity could be extended to seeking numbers between 0.000 and 9.999.

(2) This activity is aimed at helping students to develop their skills in mental calculation and confidence in moving between fraction, decimal and percentage forms. The values found in the table could be used as a basis for students to construct a set of questions and answers for a quiz, a homework, a cross-number puzzle or a game of lotto, with a focus on mental calculation using equivalent values.

(3) Conversion of these numbers from fraction to decimal forms allows examination of the relationship between fractions and decimals, methods of conversion, relative size, ordering of decimals and recurrence.

(4) This activity develops the strategies of trial and improvement that students enciountered in part 1. The problem can be solved to different degrees of accuracy; for example, to 6 decimal places, 12.623475 and 2.376525.

Worksheet 2: On and On...

This activity can be done in small groups.

Initially the limitation on the number of digits displayed by a calculator may be an obstacle to recognising recurring decimals. Students will find it useful to perform the division of sevenths manually, in order to establish that there really is a pattern and to convince themselves that it will continue indefinitely.

For any division by a number N, the recurring cycle will be of length N—1 or less. Why?

Once the idea of repeating cycles is accepted, students can explore larger denominators that give longer cycles. The limitation of the calculator display now provides an opportunity for students to make predictions and recognise when they have sufficient information to describe a cycle. ¹/₁₇ gives a cycle of sixteen digits, and ¹/₁₉ gives a cycle of 18 digits. ¹/₁₁ and ¹/₁₃ are also interesting.

Worksheet 3: Pair Them Up

The cards can be used in any combination of two sets (fractions and decimals, decimals and percentages, or fractions and percentages).

The game works best if the groups are small, since it relies on memory.

Other games could be based on:

• pairing (versions of snap)
• matching or collecting three cards (using all three sets)
• relative sizes of the values on the cards
• collecting and ordering sets

Students may wish to add to the sets of cards, or use different numbers.

Similar matching games can be played at: http://www.quia.com/math.html.

Further Ideas

Number Lines

Number lines can be used to compare fractions, percentages and decimals.

Division Grid

A division grid can be used to explore fractions and decimals. A grid labelled with both numerators and denominators from 1 to 12 is manageable. Twelves are particularly useful in developing understanding of equivalence. It is helpful if the grid can be sufficiently large to show both the fraction and the decimal in each cell. The class could collaborate to produce a wall chart. Patterns can be explored leading to appreciation of equivalence, vulgar fractions, terminating and recurring decimals, and so on.

Techniques of trial and improvement can be introduced by looking for approximations to square and cube roots.