Calculation Triangle

Three fields inside, three numbers outside: each outer number is the sum of the two fields next to it. The calculation triangle (Rechendreieck) is a well-known exercise format in primary school mathematics – this app turns it into a virtual manipulative. Children place counters in the fields, and the outer numbers change with every move. The app is in German. The first version appeared in 2012. The new edition is an open environment for exploring mathematics in primary school. LESS IS MORE At the start the triangle looks like the original: counters inside, numbers outside, nothing else. Everything else is tucked away in the buttons and settings – or an investigation card brings it along. The reduced view shows only the triangle and the gear button, similar to the first version. WITH ALL FINGERS • Tap to place a counter, with several fingers at once to place several • Drag counters to another field • Drag them out to remove them, shake to clear the triangle FOUR REPRESENTATIONS Counters only, counters with numbers outside as in 2012, counters and numbers, or numbers only. All representations are linked: children move from the material to the numbers and back. COVER AND GUESS A shutter covers a field, outer numbers can be hidden. To uncover, children first enter their guess. Every triangle becomes a task of working things out. TASKS AND INVERSE TASKS The dice sets basic tasks (inside given), mixed tasks and inverse tasks (outside given), with addition or multiplication, up to 10, 20, 100, 1000 or a custom limit. In free mode, children enter all numbers themselves and check their triangle. Unsolvable inverse tasks can be switched on – then an explanation is needed. WRITE LIKE IN AN EXERCISE BOOK From number range 20, children write their guess under the shutter with a finger or pencil into boxes. Recognition runs on the device; a button switches to the keypad at any time. 33 INVESTIGATION CARDS For grades 1 to 4: find all possibilities, work out hidden counters, change tasks and observe what happens to the outer numbers, explain whether something always works. Most cards end with a question asking for an explanation, eleven show a short video. A speaker button reads the card aloud, and a word bank with key terms and sentence starters helps children explain. NO POINTS, NO STARS No time pressure and no grading: children check for themselves whether their triangle is correct. FOR SCHOOL • Introductory videos for children and for teachers • Equations for the triangle, optionally with inner and outer sum • Collection: save triangles, compare them and print them as a worksheet • Link and QR code for a prepared triangle on all devices in class • Read-aloud for children who cannot yet read confidently • Sharing, printing and the activity log are behind a multiplication task for adults PRIVACY • No ads • No tracking • No in-app purchases • No registration • No internet connection needed • All data stays on the device Developed by Christian Urff, mathematics educator and developer of virtual manipulatives for primary school mathematics.