Expressions & Equations

Middle School (6-8) · Mathematics

Expressions and equations is where arithmetic turns into algebra. Across grades 6-8 students move from writing a phrase like "5 less than twice a number" as 2n - 5, to solving equations where the variable appears on both sides, to deciding whether an equation has one solution, no solution, or infinitely many. This topic focuses on the manipulation itself: what you are allowed to do to an expression without changing its value, and what you must do to both sides of an equation to keep it balanced. Sessions use spoken back-and-forth, so a learner explains each step out loud rather than silently copying a procedure.

Start a session on Expressions & Equations

What this covers

  • Translating word phrases and situations into expressions and equations, and reading an expression back in words (recognising that 3(x + 4) and 3x + 4 describe different situations)
  • Simplifying with the distributive property and combining like terms, including expressions with negative coefficients such as 7 - 2(3x - 5)
  • Solving one-step, two-step and multi-step equations, including variables on both sides and equations with fractional or decimal coefficients
  • Classifying linear equations as having one solution, no solution, or infinitely many, and explaining what 6 = 6 or 3 = 8 means at the end of a solve
  • Solving and graphing one-variable inequalities on a number line, including the reversal rule when multiplying or dividing by a negative
  • Applying integer exponent rules and scientific notation to expressions (8th grade), such as simplifying (2x^3)(4x^5) or comparing 3 x 10^8 to 6 x 10^5

Where learners get stuck

Treating the equals sign as "here comes the answer" instead of a statement of balance
Six years of arithmetic trains students that = means "compute now." So when they see 4x + 3 = 19 they want to write 4x + 3 = 19 = 16 = 4, stringing operations onto one line. Until the equals sign is understood as a claim that two quantities are the same, doing the same thing to both sides feels arbitrary.
Losing the negative when distributing, e.g. writing 8 - 3(x - 2) as 8 - 3x - 6
Students distribute the 3 but not the minus sign, because they read "- 3" as a subtraction operation attached to the 8 rather than as a coefficient of -3 multiplying the whole bracket. Saying the step aloud as "negative three times negative two" catches it far more reliably than rechecking written work.
Combining terms that are not alike, such as turning 5x + 3 into 8x, or 2x + 2x^2 into 4x^3
Like terms look combinable because addition normally produces a single number. Without a concrete meaning for x, there is nothing stopping the merge. Anchoring x to a quantity (5 boxes plus 3 loose items) restores the constraint quickly.

What a session looks like

A typical session runs 20-30 minutes by voice. The tutor starts with a short diagnostic problem to see where the current gap sits, then works through problems one line at a time, asking the learner to say what operation they want to apply and why before hearing whether it works. Wrong steps are followed through rather than blocked, so the learner sees where an error leads. Sessions usually end with the learner solving one problem start to finish narrating each move, plus one problem stated in words for them to translate into an equation.

Helpful to know first

  • Fluent operations with negative integers, particularly subtracting a negative and multiplying two negatives
  • Adding, subtracting, multiplying and dividing fractions, since coefficients are often fractional
  • Order of operations, including how brackets and exponents interact
  • Knowing the inverse of each operation (that division undoes multiplication) and being comfortable with the idea that a letter can stand for an unknown number

Questions

What is the difference between an expression and an equation?
An expression such as 4x + 7 is a quantity you can simplify or evaluate for a given x, but it has no solution because nothing is being claimed. An equation such as 4x + 7 = 23 states that two quantities are equal and asks which x makes that true. Mixing the two up is common: students often try to "solve" 4x + 7 by setting it equal to zero out of habit.
My child can solve two-step equations but freezes when the variable is on both sides. What is going on?
Two-step equations can be done by undoing operations in reverse order, which is a fixed recipe. Variables on both sides need a decision first: which side to collect the variable on, and whether to subtract 3x or 5x. There is no single recipe, so the learner has to reason about the structure. Sessions target that decision point directly rather than practising more two-step problems.
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. Adding or subtracting a negative does not flip it. The rule exists because multiplying by a negative reverses order on the number line: 2 < 5, but -2 > -5. We check this against the number line rather than memorising it, because memorised versions get applied to addition too.
Does this cover exponent rules and scientific notation?
Yes, for grade 8. Integer exponent properties, simplifying expressions with powers, and writing and comparing numbers in scientific notation are all included, since the standards group them with expressions and equations. Graphing lines and slope are handled in the coordinate plane topic.

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