Algebra balance scale

What the balance model shows

An equation says two things are equal, like two pans on a scale holding the same weight. If you take 3 away from one pan, you must take 3 from the other or the scale tips. Solving an equation is doing fair moves to both sides, one at a time, until x is alone on one pan.

This tool has five equations to choose from: x + 5 = 9, 3x = 12, 2x + 3 = 11, 4x − 2 = 10, and 5x + 1 = 2x + 13. Each one comes with a few operation buttons, such as − 3 or ÷ 2. Pressing a button applies that step to both sides and redraws the scale with the new equation. Undo steps back, so trying a move that does not help costs nothing.

Why a picture beats a rule

Many students learn "move it to the other side and change the sign". It works until it does not, and then they have no way to tell why. The balance gives a reason for every step: you are not moving anything, you are doing the same thing to both sides so they stay equal. That reason still holds for equations with x on both sides, fractions or brackets.

The order of steps also becomes visible. With 2x + 3 = 11, subtracting 3 first leaves 2x = 8, which divides neatly. Dividing by 2 first is still allowed, but it leaves x + 1.5 = 5.5. Seeing both routes on the scale shows why undoing the addition first is usually the cleaner choice, without making it a rule to memorize.

Ideas by grade

• Grade 6: one-step equations. Common Core standard 6.EE.B.7 covers equations of the form x + p = q and px = q. Use x + 5 = 9 and 3x = 12.
• Grade 7: two-step equations. Standard 7.EE.B.4a covers equations of the form px + q = r. Use 2x + 3 = 11 and 4x − 2 = 10.
• Grade 8: x on both sides. Use 5x + 1 = 2x + 13 and talk about which x term to remove first.
• Any grade: checking. After solving, substitute the value back into the original equation to confirm both sides match.

A 10-minute equations session

• Pick x + 5 = 9. Before pressing anything, ask: "What would you do to both sides to leave x on its own?"
• Press the chosen button and read the new equation aloud together.
• Move to 2x + 3 = 11. Ask the student to plan both steps on paper first, then carry them out on the scale.
• Deliberately press a step that makes things worse, such as + 3. Ask why the equation is still true even though it looks harder, then undo.
• Finish with 5x + 1 = 2x + 13 and ask the student to explain each step in a full sentence.

The aim is talk, not speed. A student who can say "I subtracted 3 from both sides so the left pan only has the x's" understands the method far better than one who simply arrives at x = 4.

From the scale to paper

Once students can predict what the scale will show, have them write each step as a line of working. Eurekami's AI tutor checks each line of a student's working and points to the first line that breaks, and every number in its replies is checked by a math checker before the student sees it. Lessons on solving equations use the same balance picture.