The purpose of this unit of five lessons is to develop the algebraic understanding that the equals symbol, = , indicates a relationship of equivalence between two amounts.
This sequence of lessons lays a fundamental and important foundation for students to be able to read, write and understand an equation. The most common understanding of students as they come to work with equations, is that the equals sign indicates that the answer is what follows. Students develop early the perception that this symbol, =, is a signal to do something.
The essence of an equation is that it is a statement of a relationship between two amounts. This relationship is a significant one of equivalence. That amounts on either side of the equals sign are equal in value, is an essential understanding that must be in place if students are to experience success (and enjoyment) in algebra in particular, and in mathematics in general, into the future.
Many students in senior primary school and early secondary classrooms, struggle to solve even simple algebra problems because they have a process view of an equation as a procedure to follow to get an answer, rather than a structural or relational view of equivalence. The more established this understanding is, the more difficult it becomes to change the student’s process view, to that of seeing an equation as an object that can be ‘acted upon’.
Students should be immersed in a range of experiences that support them to explore the concept of equivalence and balance. During these experiences, the teacher must carefully choose the language they use and model. As equations are introduced, recorded, read and interpreted, words and phrases such as ‘has the same value as’, ‘is the same as’, ‘is equal to’ and ‘ is equivalent to’, rather than ‘makes’, or ‘gives an answer of ’, become very important. It is interesting to note that the word ‘equals’, on its own, has subtly become more synonymous with ‘makes’ or ‘gives an answer’, rather than giving the message of equivalence that it should. Note that the words ‘addend’ and ‘sum’ are introduced in Number units of work, rather than in this series of lessons.
When posing problems that position the unknown amount at the beginning or in the middle of an equation, we are challenging the student to explore the relationship statement and the operations from a different perspective. This also occurs when students are asked to find ‘different names’ for the same amount. Further, to require students to discern true statements from false, and to justify their choice, pushes them to think more deeply about the nature of the relationship expressed in an equation.
Students should have opportunities to both read and respond to equations, and to record them, having interpreted a number problem expressed in words. In developing the ‘balance’ view of an equation, students will understand the equality relationship expressed in an equation such as 6 = 6, rather than being perplexed by the fact that there is no number problem to ‘answer’. Students will also readily understand relationships expressed in equations such as 4 + 2 = 1 + 5, rather than developing an expectation that a single ‘answer’ will follow the = symbol. Instead of expressing solutions in the arithmetic ‘voice’ of ‘problem, calculation and answer’, it is important in early algebra work, for students to explain their solutions in words that make the equivalence relationship explicit.
The activities suggested in this series of lessons can form the basis of independent practice tasks.
Links to the Number Framework
Counting all (Stages 2 – 3)
Counting on (Stage 4)
These learning experiences use numbers in the range from 1 to 20, however the numbers in the problems and the learning experiences should be adapted, as appropriate, for the students.
Conclude the session by reviewing =, equals and its meaning and the meaning of the word equation. Have students explain these, and record what they say.
Students play Same Name snap, using cards from Attachment 3.
(Purpose: To recognise when amounts are equivalent (or not equivalent) and to give the ‘number name’ for the ‘same name’ expressions.)
How to play:
Student pairs shuffle the cards and deal all cards so each student has an equal number of cards. These are placed in a pile, face down in front of each student. Student One turns over the top card and places it, face up, between both students. Student Two does the same, placing their card on top of their partner’s card. If the two expressions have equal value, either student calls Same Name, states the number that the expression represents, and the correct equation using either ‘is equal to’ or ‘is the same as’. For example:
2 + 3 is placed on top of 4 + 1.
“Same name! Five! Two plus three is equal to four plus one.” or
“Two plus three is the same as four plus one.”
The caller collects the card pile, records the equation, 5 = 2 + 3 = 4 + 1 on their scoring paper, and the game begins again, with the winner of this round placing the first card.
The student who does not call, can challenge the caller if they believe the “name” is not true for either or both expressions. If they are correct, they collect the pile and record the correct equation. The original caller must erase the incorrect equation.
The game finishes when one student has all the cards, or when one student has recorded ten ‘same name’ equations.
Conclude this session by discussing learning from the games, and reviewing ideas recorded on the class chart over five sessions.