Writing in Words and Symbols

Prior knowledge.

• Add, subtract, multiply and divide whole numbers, and record word problems with symbols
• Explain that a letter can be used to talk about ‘some number’ that we do not necessarily know

The exercises in this activity vary in difficulty from EA to AP.  The teacher will need to select those most suitable for their students.

Background

These activities are about developing an understanding of the language of mathematics. They involve developing an understanding of the conventions we use to record phrases in that language, and interpreting what these phrases mean.

Comments on the Exercises

Exercise 1
Asks students to write the phrases using symbols instead of words. For this exercise, it is not envisaged that students have been introduced to all of the algebraic conventions.

This exercise can actually be done twice. Initially it is useful to develop the idea that the letter n (note the italics) is used to mean ‘some number’ (see the attached teaching activity ‘alphabet soup’). Students can then work their way through the exercise simply using this understanding. A useful follow-up is to ask the question “is there any reason why we had to use the letter n? As long as we understand we are talking about ‘some number’, could we use another letter (that we felt like using?”) The discussion could then lead to the idea that in some situations we use n (for example with patterns), sometimes x (with graphs and functions), sometimes a etc. The exercise can then be reset, with students using different letters in different problems to mean ‘some number’.

Note that the teaching activity ‘alphabet soup’ introduces the symbol notation students need to know to record symbol phrases for number problems expressed in words (for example, recording ‘one half of six’ as ‘1/2 x 6’). Appropriate recording of such phrases, and equivalence between differently recorded phrases needs to be addressed in a number context before using this activity.

Exercise 2
Asks students to write the phrases using symbols.  For this exercise, students need to show two numbers using symbols. This can be a lot more challenging than working with one symbol on its own, so before setting the exercise, some discussion needs to be held about how to do this. Say to students “you have learned to use a letter to stand for some number that you want to work with. How do you think you write down that you want to add two different numbers?” This should prompt the idea that two different letters need to be used in the same sentence. (These could be a and b or n and r…). Students should then practice with some simple problems orally expressed in words, which they record with symbols. Some students may initially need to use the same letters for all of their questions, while others will be comfortable changing the letters they use for each problem. When marking these oral questions, discussing which letters people used can be a valuable way of bringing to the fore again that it does not matter which letters are chosen for a question, as long as they were different.

The idea that even when we use two different letters, we have a special case (convention) that the two numbers could actually be the same should be introduced at this stage. A useful way to do this is to ask what this sentence means:
x + y = 12
and then ‘what numbers can we use to make this true?’

For this exercise, the use of brackets to group and separate numbers arises. Students may or may not be ready for this additional convention, hence more is done on this in exercise 3.

Exercise 3
Asks students to look at the use of brackets in number problems. It is worthwhile setting the exercise without prior teaching as there is a note about the conventions for using brackets in mathematics above the exercise. (Discussing what a convention is may be worthwhile if this term has not been met before.)

The issue of not changing the order of the way things are written is explored in question 9. The question is structured in such a way that when completing the questions students may realise that they have not all written the same thing (some may even realise that there is a negative answer if the order is changed for 9a). This question has been provided to allow discussion about the conventions mathematicians use in this area as students are likely to bring to this work a number of important misconceptions that need to be addressed if learning is to take place.

After this discussion it may be appropriate for students to revisit exercises 2 and 3, from the context of their new learning.

Exercise 4
Asks students to look at explaining what the symbol phrases say to do to the numbers. Some questions in this exercise (like n/2 in question 8) may be a little unfamiliar to students though students should try them all without scaffolding. When setting the exercise, encourage students to write what they think the phrase means, even if they are not sure. (Putting a star next the answers they are not sure about is a good practice, and will allow students to identify where they need to take particular attention during the discussions during marking).

It is important to mark this exercise collectively, as different word phrases written by the students are likely to mean the same thing. (This can give a revision of the language used to signal the use of the different operations.) Notation like that found in questions 6, 7, 8, 9, 12, 20, 22, and 23 can then be discussed and the standard interpretations introduced.

A useful follow-up to this exercise is a teaching session where a series of algebra statements are written on the board by the teacher, and students discuss in pairs what they think they mean before feeding back what they think to the group.

Pair challenge
This activity is designed as a thinking activity and builds on the work done when marking exercise 3. Students may well find this hard, but should be allowed to work on the challenge without scaffolding or intervention. They need to think about how to record these problems before discussing ‘answers’ with the teacher. The purpose of the activity is to get students to realise that the way we record such problems is a matter of convention. With luck, some groups may ‘crack’ the challenge, and come up with similar ways of recording the problems, while others come up with a different approach. This will allow discussions around the need to have a consistent approach, so everyone has a similar understanding of what is meant. In this process students should are also developing the idea that ‘letters’, being numbers (albeit unknown ones) follow the basic rules that numbers do, an essential idea for later exploration.

A good follow-up to the pair challenge is to get students to write a worksheet for other students. On this they should explain how we have conventions around recording multiplications and divisions when we use letters, and how these work. They should then create several exercises for students to practice on – with answers of course. The students should then swap work with someone else to check that it makes sense, and that the answers are correct.

Exercise 5
Asks students touse conventions for recording multiplication and division when using symbols instead of words in phrases. This exercise builds on the challenge, and introduces using the commutative law to make sure that multiplications involving letters can be recorded so they look the same. It also looks at the use of fraction notation to record divisions. Some questions also require students to use brackets to make sure the operations are completed in the correct order.

Students can be challenged to write an exercise of their own, with answers. ‘Best questions can then be swapped and a new worksheet of these created for use next year.

Exercise 6
Asks students to write out in words what the symbol phrases ask you to do.
This exercise builds further understanding of the symbol language, and is quite advanced. Here brackets (hence order of operations) are used, and understanding of what the job of the brackets in an expression is expected. In some questions, situations where the other symbols mean that brackets are not needed are also introduced. This exercise may need to be carefully discussed once it has been completed.

Question 19 again gets students to write and try out some of their own problems. This is a good activity to share in a teaching session, so students can put up an expression, and others can write down what they think it says to do to the numbers. Here any remaining misconceptions can be addressed.

Exercise 7
Asks students to interpret phrases involving, n, for example, 2n and show on a number line.  This exercise uses a measurement context to explore the relative size of numbers. Students need to realise that in this model, length (and distance from zero) conveys meaning. For example, 2n should be twice the distance from zero as n, and when placing n + 1, it goes to the right of n, and placing this has defined the length of a unit, (so once placed n – 1 needs to be placed an equal distance from n but on its left). Doing this can actually allow you to work out what the number n is on the number line…

Later problems place n so it is a negative number, then as a decimal (or fraction), which has implications for where various other numbers lie. This exercise is well worth discussing during a teaching session, though again allow students to attempt it before this discussion is held.