Virtual integer chips

What integer chips show

Integer chips are two-color counters. On this page a yellow counter is +1 and a red counter is −1. One yellow and one red together are worth 0, which is called a zero pair. Adding or removing zero pairs never changes the value, and that single idea explains most of the rules for adding and subtracting negative numbers.

Use the buttons to add or take away yellow and red counters, or to add a zero pair. Yellow counters sit on the top row and red ones underneath, so pairs line up in columns with a dashed outline. Cancel zero pairs removes every pair at once, or you can tap a yellow and then a red counter to cancel one pair yourself. The readout gives the value, the matching number sentence and how many zero pairs there are. Challenge asks for a value made with an exact number of counters.

Why counters help

Rules like "subtracting a negative is the same as adding a positive" are hard to remember and easy to mix up. Counters let students see why. To work out 2 − (−3), show 2, notice there are no red counters to take away, add three zero pairs so there are, then take away three red. Five yellow are left, so 2 − (−3) = 5.

Common mix-ups

• Thinking −3 can only be shown with three red counters. Four red and one yellow is also −3, and so is five red and two yellow.
• Believing you can't take away what isn't there. Adding zero pairs gives you more to take without changing the value.
• Answering 5 + (−8) as 13 or −13. Pair up the counters: five pairs cancel and three red are left, so the answer is −3.
• Treating the minus sign for subtraction and the negative sign as the same thing. Counters separate "take away" from "red."

A 10-minute zero pair routine

• Put out 3 red counters and read the value. Add a zero pair and ask if the value changed.
• Model 6 + (−4): add 6 yellow and 4 red, predict the value, then cancel the zero pairs.
• Model −2 − (−5): show −2, add zero pairs until there are 5 red, take away 5 red and read the value.
• Press Challenge twice and let the student build each target alone.
• Ask the student to write a rule they noticed and test it with one more example on the mat.

Where it fits by grade

• Grade 6: use positive and negative numbers for real quantities and understand opposites (6.NS.C.5, 6.NS.C.6). In England, Years 4 to 6 count through zero and use negative numbers in context.
• Grade 7: add and subtract integers, including describing situations where opposite quantities make 0 (7.NS.A.1). In England, calculating with negative numbers is part of Key Stage 3, from Year 7.

Learning more

Negative numbers are a skill on Eurekami's skill map, and Daily Practice brings them back when they start to fade. When a student is stuck on a sign rule, the AI tutor asks guiding questions so the student does the reasoning.