- Level 7
- Number and Algebra
- Patterns and algebra
- Introduction of variables using letters to represent numbers
Patterns and algebra • Level 7
Introduction of variables using letters to represent numbers
VCMNA251
Teaching Context
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Teachers have “a responsibility to ensure that students’ first experiences of using letters in algebra lay the foundation for a coherent structure of algebraic knowledge” (MacGregor & Stacey, 1997, p. 18)
At this level, it is important that students have a good introduction to algebra, stressing that the variables (letters) stand for numbers and that an equation represents balance ie the values on the left are equal to the values on the right. Variables or pronumerals are introduced to students as a way of representing numbers using letters. Students may have previously seen symbols such as squares used to represent unknown numbers but will now be introduced to common pronumerals such as x, y and t.
Students can easily confuse the meaning of letters in algebra. They may think that the letters stand for something such as a unit of measurement or represent a secret code to solve. Using symbols to introduce the concept of variables eg 3 + ? = 5 and then writing the expression as 3 + x = 5 can help students understand that the variable represents an unknown number.
Emphasising the origin of the term "pronumeral" can help to make it very clear to students that these letters stand "for a number".
pro - for
numeral - number
When asked to write an algebraic expression to represent the situation '6 doughnuts cost $12', students might write 6d = 12 and read this equation by saying '6 doughnuts cost 12 dollars', pointing to the symbols as shown below. The equation is good, but it does not mean what the student says it means.
If this is an algebraic equation, the accurate interpretation is '6 times the cost of a doughnut in dollars is equal to 12 dollars'. d stands for the number of dollars, not the doughnut, as the student may think.
Whenever students read equations using a verb other than ‘equals’ for ‘=’, question them carefully about their meanings for letters. Errors of this type are a major cause of difficulty in many later applications (eg with linear programming in senior mathematics).
Another common misconception students may have when first introduced to variables is the idea that the value found for a certain variable (eg x) remains constant for all future questions involving that variable. Ie the belief that if x = 3 for one expression then x always equals 3.
This can be clarified by using a table of values or plotting a linear relationship and showing how the variable (x) can take on an infinite number of values. E.g y = x + 1
Students may have preconceived ideas that algebra is difficult and unrelated to the mathematics they have done in the past. Discussing how algebra involves many of the skills they already use; finding patterns, using inverse operations and finding unknown values, can help to reduce any anxiety around the topic.
Victorian Curriculum
Introduce the concept of variables as a way of representing numbers using letters (VCMNA251)
VCAA Sample Program: A set of sample programs covering the Victorian Curriculum Mathematics.
VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.
Achievement standards
Students solve problems involving the order, addition and subtraction of integers. They make the connections between whole numbers and index notation and the relationship between perfect squares and square roots. They solve problems involving all four operations with fractions, decimals, percentages and their equivalences, and express fractions in their simplest form.
Students compare the cost of items to make financial decisions, with and without the use of digital technology. They make simple estimates to judge the reasonableness of results.
Students use variables to represent arbitrary numbers and connect the laws and properties of number to algebra and substitute numbers into algebraic expressions. They assign ordered pairs to given points on the Cartesian plane and interpret and analyse graphs of relations from real data.
Students develop simple linear models for situations, make predictions based on these models, solve related equations and check their solutions.
Online Resources
Teaching ideas
- Diagnostic items
- Intro to pronumerals
- Initial expressions
- What’s my own value of n?
- An infamous problem
Diagnostic items
Tab Content
Mathematical focus and overview of developmental stages
Misconceptions and common errors
References
Mathematical focus and overview of developmental stages
Misconceptions
References
Students who had not learned any algebra, and students in the first years of algebra were given these problems. A Victorian study (Stacey & MacGregor, 1997) found that the following responses were common in both of these groups, although more common amongst students who have not learned algebra. These responses reveal algebraic letters interpreted as abbreviations, as well as alphabetic codes etc. The study showed that students' interpretion of algebraic letters according to position in the alphabet is easy to fix, if attention is drawn to it.
David is 10 cm taller than Con. Con is h cm tall. What can you write for David's height?
Response | Explanation |
---|---|
10 + h, C+10, | Correct |
h10 | Add 10 onto h. |
Uh, Dh , D | Abbreviated words "Unknown height", David's height , David. |
18 or r | h is the 8th letter of alphabet, therefore 10 more is the 18th letter, r |
110 | Think of a reasonable height for Con, add 10 |
g | Choose another letter or adjacent letter for David's height. |
h = h +10, h = +10 |
Sue weighs 1 kg less than Chris. Chris weighs y kg. What can you write for Sue's weight?
Response | Explanation |
---|---|
y - 1 | Correct |
24 | y is the 25th letter of alphabet, therefore 1 less is 24 |
1y | Whereas 10h means 10 more than h to some students, 1y means 1 less than y as in Roman numerals |
Uw | Abbreviated words 'Unknown weight'. |
x | Choose another letter or letter before y for Sue's weight. |
Further to this, there are diagnostic tests available on the SMART tests website.
Values for letters
Mathematical focus and overview of developmental stages
This quiz tests students' understanding of the algebra conventions for the values that pronumerals (algebraic letters) may take. Each time a letter is used in an equation it represents the same number. If there is a second letter, it may represent either the same value as the first letter or a different value. The quiz uses only very simple algebraic expressions involving addition.
Stage 1 | These students know that letters can stand for numbers, and are able to correctly substitute into very simple algebraic expressions but they believe that the values that letters can take are in some way related to their place in the alphabet. |
Stage 2 | These students interpret an algebraic letter only as a place holder for a number in a number sentence, so they allow one letter to have several values in one expression. |
Stage 3 | These students appreciate that each time a particular letter is used in an equation it stands for the same number, BUT they over-generalise to “different letters must be different numbers”. |
Stage 4 | These students know that in one algebra question, a letter must stand for only one number and that different letters can stand for the same number. |
Misconceptions and common errors
A | Students often give a letter a value related to its place in the alphabet, such as b = 2. |
C | Students believe that the values of consecutive letters must be consecutive numbers. |
O | Students believe that if one letter is before another in the alphabet, its value must smaller. |
R | When the same letter is used more than once in an expression, these students recognise that it has the same value but state this value separately for each occurrence. |
Detailed explanations of developmental stages and common misconceptions
Stage 0 students do not correctly substitute into very simple expressions involving addition. Because of this, further diagnosis is not reliable.
Stage 1 students know that algebraic letters stand for numbers and are able to correctly substitute into very simple expressions involving addition but they may believe that the values that pronumerals can take are in some way related to their place in the alphabet. The codes A, C, O (see below) show the misconception. Some students in this code do not have an alphabetical misconception - their pattern of reponses may be due to arithmetical errors, guessing or omitting items.
Stage 2 students interpret an algebraic letter only as a place holder for a number in a number sentence, so they allow one letter to have several values in one expression. For example, in the equation x + x = 12 these students would accept as a solution x = 10 and x = 2. They are treating the pronumeral as they would have treated the boxes that they used in missing number sentences: ? + ? = 12 in primary school.
Stage 3 students know that each time a particular letter is used in an equation it stands for the same number, BUT they over-generalise to “different letters must be different numbers”. For example, they consider that x = 3, y = 3 is not a solution to x + y = 6. If algebra was a secret code, with numbers allocated 'code names', then different code names (letters) would have to have different numbers associated with them. But algebra is not like a code in this way. Some still show some evidence still of seeing variables as separate place holders for numbers. For example, these students might look at x + x + x = 15 and say that x = 5 and x = 5 and x = 5. They do not always accept the single solution that x = 5 because they see each x as a separate place holder.
Stage 4 students know that in one algebra question, a letter must stand for only one number and that different letters can stand for the same number.
Misconceptions and common errors
Code A - Students often give algebraic letters values based on position in alphabet (e.g. a = 1). For example, they could substitute a = 1 and c = 3 into the expression a + c = ? and correctly give the value 4. These students may give the same answer,4, even if told that a = 7 and c = 5 because a is the 1st letter in the alphabet and c is the 3rd. This, and the other alphabetical misconceptions, is a result of confusing algebra with a code where each letter stands for a particular number, often related to its place in the alphabet.
Code C - Students believe that if pronumerals are consecutive letters, their values must be consecutive numbers. Thus for the equation x + y = 11, x = 5 and y = 6 is the only solution. They think x = 7 and y = 4 is not a solution.
Code O - Students believe that if one pronumeral is before (or after) another in the alphabet, its value must smaller (or larger) than the value of the second pronumeral. Thus for the equation m + t = 11, they think that m = 3 and t = 8 is a solution but that m = 7 and t = 4 is not a solution.
Code R -When the same pronumeral is mentioned multiple times in an expression, these students recognise that it has the same value both times but need it to be written for each occurrence. Thus for y + y =10 they prefer to write the solution as y = 5 and y = 5 and will reject y = 5 written only once.
Teaching suggestions
For teaching ideas, see these parts of the Mathematics Developmental Continuum. The first reference gives information on missing number sentences. The second reference explains about the meaning of letters in algebra.
http://smartvic.com/teacher/mdc/structure/st325ip.html
http://smartvic.com/teacher/mdc/structure/St42504P.html
Stage 0, 1, 2 and 3 students will benefit from activities that involve finding values that could satisfy number sentences with a pronumeral repeated or number sentences including more than one pronumeral. For these students it will help to explain the rules governing the allowable values. In particular Activity 2 in the linked section of the Mathematics Developmental Continuum is most helpful.
http://smartvic.com/teacher/mdc/structure/St42504G.html.
It is important to remind students of the puzzles and codes which they may have solved in the past that rely on associating letters with numerical values that are related to alphabetical positions (or vice versa). Algebra is NOT like this. There are different rules. The alphabetical misconceptions are often easy to fix with just a little explicit attention.
Stage 4 students do not need further attention to their understanding of this facet of algebraic notation.
References
Booth, L. (1988). Children's difficulties in beginning algebra. In A. Coxford & A. Shulte (Eds.) The ideas of algebra, K-12. 1988 NCTM yearbook. Reston, VS: NCTM.
Fujii, T. (2003). Probing students' understanding of variables through cognitive conflict problems: Is the concept of a variable so difficult for students to understand? In N. Pateman, G. Dougherty, J. Zilliox (Eds.), Proceedings of the 27th annual conference of the International Group for the Psychology of Mathematics Education, (Vol.1, pp. 49-65). Hawaii: PME.
Küchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's Understanding of Mathematics :11-16 (pp. 102-119). London.: John Murray.
MacGregor, M. & Stacey, K. (1997) Students' understanding of algebraic notation: 11-16. Educational Studies in Mathematics. 33(1), 1-19.
Steinle, V., Gvozdenko, E., Price, B., Stacey, K., & Pierce, R. (2009) Investigating Students' Numerical Misconceptions in Algebra. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 2) (pp. 491 - 498). Palmerston North, NZ: MERGA. http://www.smartvic.com/smart/research/Publications/Steinle_MERGA2009proceedings.pdf
Letters for numbers or objects
Mathematical focus and overview of developmental stages
Knowledge of using pronumerals correctly involves being aware that pronumerals stand for numbers. Students hold a range of common misconceptions relating to the recognition of this fact. Some students think algebraic letters are abbreviations for words or things and that algebra is a sort of shorthand.
Stage 1 | These students generally do not interpret algebraic letters as standing for numbers. |
Stage 2 | These students sometimes interpret algebraic letters as standing for numbers. |
Stage 3 | These students consistently interpret algebraic letters as standing for numbers. |
Misconceptions
LO | These students have a tendency to interpret algebraic letters as objects (instead of numbers). |
SAC | These students also have a tendency to interpret algebraic letters as objects but also, when writing an equation, use the solution(s) as the coefficient(s) of the pronumeral(s). This is a type of interpreting algebraic letters as objects. |
Detailed explanations of developmental stages and common misconceptions
Stage 0 students are below Stage 1. They do not as yet interpret algebraic letters as standing for numbers.
Stage 1 students rarely interpret algebraic letters as standing for numbers.
Stage 2 students sometimes interpret algebraic letters as standing for numbers.
Stage 3 students consistently interpret algebraic letters as standing for numbers. They have a correct understanding of the meaning of a pronumeral.
Misconceptions
LO (letter as object) This group of students frequently uses algebraic letters to stand for objects rather than numbers. They have little understanding of the meaning of letters when used in algebra as pronumerals. They usually think that algebraic letters are abbreviations for words. For example they may think that 2c + 3d represents 2 cats and 3 dogs. They may interpret a + b = 90 as "the apples and the bananas cost 90 cents".
SAC (solution as coefficient) Like LO students, these students also interpret algebraic letters to stand for objects rather than numbers, but in a slightly more sophisticated way. These students are likely to translate the information
"I bought a apples and b bananas for 90 cents. The apples cost 15 cents each and the bananas cost 20 cents each"
into the equation 2a + 3b = 90. They intend this equation to mean the true statement "2 apples and 3 bananas cost 90 cents", without realising this is not a valid equation. They have in effect found a solution to the equation first, then used the solution in the equation. They do not understand the usefulness of equations to solve problems.
Teaching suggestions
Both LO and SAC students suffer from the very common 'letter as object' misconception. LO (letter as object). This is misconception often arises from teaching. For example many teachers teach addition of 'like terms' by givien examples like this: 2a + 3b represents 2 apples and 3 bananas and 6a + 4b represents 6 apples and 4 bananas and so (2a + 3b) + (6a + 4b) represents 8 apples and 7 bananas (i.e. 8a + 7b). While this way of thinking will initially be helpful when gathering like terms, it is incorrect and leads to long term problems. This sort of thinking and teaching is sometimes called fruit salad algebra, because fruit is so often used as the example.
Stages 0, 1 and 2 The simplest teaching suggestion is not to teach students the misconception that letters in algebra stand for objects or abbreviated words. When teaching, do not use "fruit salad algebra". Algebraic expressions are always about numbers and number relationships. In addition, students need plenty of opportunities to write equations that describe situations in the real world, so they can see how to find the equation about the numbers involved and learn to avoid the easy false alternatives.
Prefacing any formulated equation with the statement "Let n be the number of ......" is a very good habit, as is ensuring that students write this in their own algebraic work.
It may also be useful to use variables that are not the initial letter of the name of the items involved. For example, "Let w be the cost of a banana".
Time spent formulating expressions and equations before focusing on solving equations is also beneficial and will allow you to see errors as they occur.
For teaching ideas, see this part of the Mathematics Developmental Continuum, including Activity 3.
http://smartvic.com/teacher/mdc/structure/St42504P.html
Stage 3 Students who use letters to represent numbers need many opportunities to formulate algebraic expressions and equations.
References
Hart, K.M. (Ed.) Children's Understanding of Mathematics 11 - 16. John Murray, London
Chick, H. (2009). Teaching the Distributive Law: Is Fruit Salad Still on the Menu? In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia (Vol. 1). Palmerston North, NZ: MERGA. http://www.smartvic.com/smart/research/Publications/Chick_MERGA2009proceedings.pdf
MacGregor, M. (1986). A fresh look at fruit salad algebra. Australian Mathematics Teacher, 42(3), 9-11.
MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational
Studies in Mathematics, 33, 1-19.
Stacey, K. & MacGregor, M. (1999). Implications for mathematics education policy of research on algebra
learning. Australian Journal of Education, 43(1), 58-71.
Intro to pronumerals
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Early experiences with algebra often lead to students developing the misconception that letters stand for objects or abbreviations. For example, f stands for a flower and 5f stands for 5 flowers. In the misguided words of a Year 7 student: “I like algebra because it is all about flowers, animals and things”.
The examples below show teaching that contributes to this common misconception. Rather than use the pronumeral f, which encourages the letter-object/abbreviation interpretation, it would be preferable to use n . This reinforces the concept of pronumeral as number, that is, n is the number of flowers. In this case n = 5. Students should become used to using the word pronumeral instead of ‘letter’, where the word ‘pronumeral’ means ‘for a number’.
‘Fruit salad’ algebra of the type shown below has occurred frequently in textbooks, reinforcing students’ misconception that a letter stands for an object. In this example, a and b are clearly being used as abbreviations for an apple and a banana rather than a pronumeral to represent the number of apples and the number of bananas. It is this type of example that contributes to students’ misunderstanding in algebra.
If we have two bags, each containing 3 apples and four bananas, how many apples and bananas do we have? | |
To avoid promoting this misconception in students, introduce the idea of pronumeral / variables using examples that show that it is the number of something that is being considered, not a finite entity.
Introduction to pronumerals lesson
Lesson objective: to support students to understand that pronumerals are used to represent currently unknown NUMBERS or variables.
Learning objectives: Students will be able to write simple algebraic expressions and explain the situation represented by these expressions. They will use some of the basic 'grammar' of algebraic expressions (Such as 5d = d + d + d + d + d = 5 x d) and will understand how these basic 'grammar' laws are the same as the arithmetic laws that they are already familiar with (eg. in the above example, that 5 x d is the same as adding d five times).
Materials needed:
- small cups (plastic shot glasses that you can buy at supermarkets or bottle shops are perfect for this activity)
- envelopes
- small articles to put in the cups - you can use M&Ms but counters or beads would substitute well
Lesson outline:
Concrete representation
- show students a cup and then fill it with the small articles
- ask students how many articles are in the cup - hopefully they will suggest that they don't know but may try to have a guess
- reinforce that the number of articles is unknown at this point. You could have a discussion as to how you might establish how many articles are in the cup and what this would involve. Bring students around to the idea that this would / could be quite a laborious process.
- Ask students how we might represent the number of articles in the cup - don't lead them to anything in particular. Once the class has decided on an appropriate representation (be it a letter or a picture or any other symbol they have come up with) write this on the board using a clear expression such as 'x'= the number of beads in a cup
- get another cup and fill it with articles. Ask students how many in this cup. Students may identify that it is likely that there will be the same number of articles in this cup as in the first cup. Put the cups side by side and ask students how many articles there are together. Support students to identify that there are now twice the number of articles than there were in the first cup.
- Ask students how we might express the number of articles now. Support students to identify at least two ways of representing this - ie. 2 times 'x' and 2 + 'x'
- At this point, a discussion as to which is correct / more correct and why may help students to make the connection between their prior knowledge about the number system and what you are asking them to do with the unknown. You could run this as a number talks activity as well if you really wanted to bring out student reasoning and develop a strong conceptual understanding of the algebraic representations.
- At this point, it is worth playing around with different possible scenarios. For example, how might students write a representation for 4 cups full of articles and 1- additional articles? What about 3 cups full and one half of a cup? What about 3 cups take away 2 cups? For each scenario, model the situation with the concrete materials or ask students to do so themselves or in small groups / pairs and then draw their representations.
Pictorial representations
- provide students with a number of expressions using the same 'pronumeral' as outlined above. You may want to keep to fairly simple expressions or you could make them more complicated based on the group of students and how they are responding. Have students draw the scenarios to match each expression. For more advanced groups / students, it could be worth starting to play with the idea of collecting like terms.
Abstract
- In this section, it is time to introduce the formal 'rules' and language of algebra.
- Start by defining the words 'pronumeral' and 'expression'.
- Then make explicit connections between the scenarios that have been used and the arithmetic laws that relate - for example, the amount of articles in 3 cups is three 'times' the number of articles in 1 cup, hence, the expression used to show this is 3 'times' the pronumeral being used to represent the number of articles in one cup. You can also show that this is the same as if you added them together three times.
- You can also discuss the addition of the separate articles (outside of the cups) and discuss how this is expressed differently and why.
Extension
The envelopes can also be used as receptacles for the articles and the introduction of a new pronumeral to demonstrate the number of articles that could be held by an envelope. This can then lead to new expressions and expressions that contain both envelopes and cups and separate articles. you might even push students to outline a relationship between the number of articles in a cup and the number of articles in an envelope.
What to watch out for
Be mindful of your language - remember that the pronumerals do not stand for cups or envelopes but rather THE NUMBER OF ARTICLES IN a cup or an envelope. Support students to use correct language in their explanations as well - be explicit about this.
Initial expressions
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This activity supports student understanding that algebra is not a new phenomenon and they have engaged with algebraic ideas before.
Write an equation up on the board using symbols that the students can easily solve:
E.g 5 + ? = 10 or 8 - ? = 4
Discuss with the students how they worked out what numbers the symbol represented. Discuss aspects such as ‘Is the symbol always the same value?’, ‘How can we check that we have the correct answer’ and any other ideas that the student raise.
Now write up the same equations but using any variable.
E.g 5 + p = 10 or 8 – p = 4
Discuss if the answers are different now that we are using a letter to represent the unknown value? If students are having difficulty, show how the letter represents exactly the same meaning as the symbol in the first example – the unknown number. Discuss how any letter (called a variable or pronumeral) can be used to represent numbers in algebra but x and y
are most commonly used.
Give students examples to write up using variables to represent given scenarios.
E.g 5 more than x, 10 less than r, the difference between y and z.
What’s my own value of n?
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This activity can be used as an introduction to a variable.
All introductory algebra activities need to reinforce that a letter stands for a number. Sometimes it is an unknown fixed number, sometimes a variable number, but always a number.
To teach algebra, it is often useful to have each student follow the processes with their own number. For example, in a lesson about 2(n+3) = 2n + 6, it is helpful to have all students working out 2(n+3) and 2n + 6 for different numbers. In this way, students can soon see that the two expressions are equal for many different numbers.
There are many easy ways to allocate personal numbers. Using the letter n to stand for the number of letters in their first name, students write down their value of n, eg, Anna would write n = 4, Christopher would write n = 11. This can be extended to other contexts such as the number of legs of various animals, so students might choose a dog (n = 4) or an octopus (n = 8) or an ant (n = 6).
Initially there is no harm in keeping to the letter n, as it reinforces the concept of the letter being a number. However, it is also important that students see that we can use any letter if we define it as a number, eg let 'a' be the number of books in my locker.
An infamous problem
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This is one of the most famous problems in mathematics education. It is answered incorrectly by many students (at all levels, including university) if they think about letters as abbreviations or things, or as units and do not carefully think about the relationships between the numbers involved.
Display the following problem and allow students time to think about it. You may like to do this activity in pairs or groups.
Discuss the answers discovered by the class.
The very common error is to write 6s = p, instead of the correct equation 6p = s. Since S is the larger number, there is a strong mental tendency for it to be associated in the equation with the 6 times. Students explaining their wrong equation 6p = s will probably not use the word ‘equals’.
To convince students, use a numerical example. If there are 100 professors, then there will be 600 students and 6 × 100 = 600. Encourage students to check their own equations with numbers – good checking skills are an essential part of Learning in Mathematics.