Teaching Algebraic Expressions and Equations
Teaching Algebraic Expressions and Equations
Algebra fails early for a small number of reasons, and the two biggest are inherited from arithmetic: an equals sign that means write the answer here, and a letter read as a label for an object. This page covers a grades 7 and 8 sequence that repairs both.
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The teaching problem
The equals sign students bring from arithmetic
For six years the equals sign has been an instruction. It appears at the end of a calculation and means write the answer here. An equation asks something else, whether two expressions have the same value and for which numbers, and nothing about the symbol signals that change. Which is why students write chains like 3 + 4 = 7 + 2 = 9 without discomfort. The letter causes the second failure. Told that a stands for apples, students read 3a as three apples, then combine 3a + 2b into 5ab, because five pieces of fruit is a sensible answer to the question they think they were asked. That is not carelessness. It is a consistent reading of what they were taught, and it has to be replaced rather than corrected.
A sequence that works
Five lessons from pattern to solution
Expressions come before equations, and substitution comes before simplification, so students always hold a way of testing whether what they have written is true. The solving lessons arrive last.
- Expressions from a growing patternStudents build a pattern of matchsticks or tiles, describe the tenth term in words, then in symbols, and compare two correct expressions that look nothing alike.
- Substitution before simplificationGiven values for the letters, students evaluate expressions and find that two different-looking ones agree for every value. Equivalence gets a meaning before any rule for collecting terms.
- Collecting terms that matchWhy 3x and 2x combine while 3x and 2y do not, tested by substitution each time. The apples-and-bananas reading is confronted directly rather than quietly avoided.
- Balance: two sides, one truthEquations as a question about values. Students solve by keeping both sides equal, record every move, and check the solution by substituting back into the original equation.
- Brackets, negatives and checking backExpanding with a negative outside, the classic sign errors, and equations with the variable on both sides. Every answer is verified before it counts as final.
Where it goes wrong
Sign errors and other predictable traps
Keep a list on the wall and add to it. Conjoining: 3x + 2 written as 5x, because an expression that is not a single term feels unfinished. Fruit-salad algebra: 3a + 2b as 5ab. The distributive slip, where minus one times x minus 3 comes out as negative x minus 3, and 2(x + 3) as 2x + 3. Dividing one term of a sum but not the other. Treating cancel as a license to delete matching letters wherever they appear. Almost all of these get caught by substitution, so make checking a required step rather than good practice, and mark it. Error-analysis tasks expose the reasoning better than another page of correct exercises.
What's in the download
Inside the files
Editable Word and PowerPoint plus print-ready PDFs, with answer keys throughout.
- Pattern and expression building tasks
- Substitution check sheets
- Equation practice at four levels
- Error-analysis worksheets with keys
- Editable slides for the balance lesson
- Word problems that build equations
Good to know
Frequently asked questions
What should students be able to do before this unit?
Operations with negative numbers, order of operations, and enough fluency with multiplication that arithmetic is not a distraction while something new is being learned. Fractions inside expressions come later, so a class still shaky there can start. If negatives are weak, expect the bracket lesson to take two sessions, because most sign errors there are arithmetic errors wearing algebra's clothes.
How does it line up with standards?
The content covers writing and evaluating expressions, equivalent expressions, and solving linear equations in one variable, including brackets and variables on both sides. That matches the expressions and equations strand for grades 7 and 8 under Common Core, and comparable content in UK and Australian curricula. It is written to fit alongside those rather than approved by any board, so compare the contents list with your scheme of work.
Is there support for students who fall behind quickly?
Worksheets come at four levels of difficulty, so the same lesson runs with more or less structure without splitting the class into different topics. The substitution checks help here too. A student who cannot yet solve an equation can usually verify whether a given value works, which keeps them inside the lesson and builds the idea that a solution is something you can test.
Fix the equals sign first
Students who can test an answer by substitution stop guessing. Solving methods land much faster once that habit exists.
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