Simulic
English

Number line – operations with integers

MathematicsNumbers & ArithmeticAges 11–12

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Three screens explore adding and subtracting integers through money: Chips shows each +1 saving or −1 debt as a chip and can cancel opposite pairs automatically; Net worth adds up several assets and debts into a net worth on the number line; Generic lets you choose the number line range, a starting point and up to two addition or subtraction steps in a row, drawn as a chain of jumps. A fourth screen, Multiply, builds a × b as a groups of chips added to or removed from the board (using zero pairs) and as repeated jumps on the number line, so the sign rules emerge; it shows division as the inverse of multiplication, with a patterns table and a sign table. Adapted from the "Number Line: Operations" simulation by PhET Interactive Simulations, University of Colorado Boulder (CC BY 4.0); rewritten.

Lesson: Adding and subtracting integers; multiplying and dividing integers; sign rules

What it shows

Integers are modelled in two linked ways: chips worth +1 or −1, and jumps on a number line. Adding and subtracting appear as savings and debts, where opposite chips cancel as zero pairs. The Multiply screen reads a × b as a groups of b: a positive a adds groups, while a negative a removes groups from a board that first gets zero pairs, so (−3) × (−2) leaves +6. Division is the inverse: count the jumps of b from 0 to the dividend, which also works for decimals. Money amounts are made-up examples.

How to use

Choose a screen with the tabs. On Chips, use +1 Saving and −1 Debt; on Net worth, tick assets and debts; on Generic, set a starting point and up to two operations. On Multiply, set the number of groups a and the size of each group b, then use Step ▶ or Play to build the product one group at a time. Set Operation to Divide (÷) to choose a dividend and a divisor.

Parameters you can change

  • Screen Chips, Net worth, Generic, Multiply
  • Chips: cancel opposite pairs automatically
  • Net worth: turn on all items at start
  • Generic: number line range −10 to 10, −20 to 20, −30 to 30
  • Generic: starting point -30–30
  • Generic: operation 1 sign Add (+), Subtract (−)
  • Generic: operation 1 value 0–30
  • Generic: use a second operation
  • Generic: operation 2 sign Add (+), Subtract (−)
  • Generic: operation 2 value 0–30
  • Multiply: operation Multiply (×), Divide (÷)
  • Multiply: number of groups a -6–6
  • Multiply: size of each group b (divisor) -6–6
  • Multiply: dividend (Divide) -36–36

Questions to explore

  1. Why does 3 × (−2) give −6 while (−3) × (−2) gives +6 on the Multiply screen?
  2. In the patterns table, how does the product change each time a goes down by 1?
  3. Why does adding one +1 chip and one −1 chip leave the balance unchanged?

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