Equation Balance

Teacher notes

A balance scale that keeps both sides honest

Equation Balance puts an expression or an equation on a two-pan scale. In Standard math, students see an expression balance its value, with every step listed in order-of-operations order (or left to right, to show why the order matters). In Solve for x, they undo an equation one inverse operation at a time, and the scale tips the moment they change only one side.

Standard math: the order of operations, live

Type the numbers on the keyboard and use the buttons for + − × ÷, parentheses and exponents. To type an exponent, press aⁿ, type the power, then press it again (or press any operation) to come back down. To change something in the middle, click there in the expression, or move the caret with the arrow keys or the ◀ ▶ buttons.

The expression sits on the left pan and its value on the right, worked out again every time the expression changes, so the scale is always level. The Steps list shows every operation in the order it was done, and the PEMDAS strip lights up the rules the expression uses.

Switch PEMDAS to Left to right to work it out with no order of operations: every + − × ÷ in reading order. When that gives a different answer, the banner shows what PEMDAS would give, which makes a good “why do we need rules?” discussion (3 + 4 × 2 is 14 left to right, but 11 with PEMDAS).

Solve for x: undo what was done to x

Choose Random for a 1-, 2- or 3-step equation. A one-step equation can use any of the four operations. Two-step equations always mix one of + and − with one of × and ÷, in either order (3x + 4 or 3(x + 4)). You can show division as a fraction or with the ÷ sign. The value of x is never shown unless you press Peek at x.

Choose Build my own to type in a value for x (it shows as dots) and build the left side with the same buttons as Standard math, plus an x button. The right side is worked out for you. Or press = and type the right side too, with x on one side or both (5x + 3 = 2x + 12). Then the value of x is optional: leave it blank and it’s worked out.

To solve, pick an operation and a number, choose Left side, Both sides or Right side, and press Submit. Turn on the x button next to the number to add or subtract an x term (like 2x) instead. The operation is written onto the pan and stays there, marked, and the Work list writes the move the way it would look on paper.

• Simplify looks through both sides, marks what can go (operations that cancel out, like + 4 − 4, or × 3 ÷ 3, and ones that change nothing, like × 1), pauses, then clears them away. It works out any arithmetic the same way, one stage at a time, until nothing is left to tidy.
• Combine like terms: press it, click the terms on the scale (each one is picked together with its + or − sign), then press it again. Numbers go with numbers and x terms with x terms, on the same side.

Extras that help the idea land

• Hint names the last operation done to x, and sets up its opposite in the controls. When x is on both sides, it sets up moving the x term off the right side.
• Take back last move and Start over make wrong turns cheap to try.
• When x is alone, a build-up and undo diagram appears under the scale: the same values, with the arrows going forward when x was built and backward when it was undone.
• Show the original with x = … puts the original equation back on the scale with each x replaced by its value in parentheses, marked with a small x tag so students can see where x was. Press it again to go back to the solved equation.
• Check the answer step by step does the same, then simplifies the left side one operation at a time, using the Standard math engine. The scale stays level the whole way.
• The Number pad at the bottom of the panel types digits on touch screens and smartboards with no keyboard.

Frequently asked questions