Multiply and Group: One Story Builds, One Story Breaks Up

Three rows of five woodpeckers, drawn in three outlined boxes, and then twelve koalas in no boxes at all. Those two pictures carry the two stories on this 3rd grade word-problems page. Mia sees 3 rows with 5 woodpeckers in each and the question is how many animals there are altogether; Leo has 12 koalas and makes groups of 2, and the question is how many groups. The first card hands the child the groups and asks for the total, the second hands over the total and asks for the groups. Both give a dot grid, a dashed box and a line for the answer sentence.

This pairing is the cleanest picture of inverse operations a third grader can be given. One story starts from equal groups and ends at a total; the other starts from a total and ends at equal groups, and the pictures show it, boxes already drawn on the first card and a bare row on the second. A child who has just multiplied three fives to reach fifteen has a way of thinking about twelve broken into twos that is not counting, and the moment they use their multiplication to predict the number of groups is the moment the two operations join up.

Print the PDF free and ask for both answers to be written as multiplication and division sentences on the dot grid, one under the other, before the boxes are filled. Two cards make this a short lesson rather than a worksheet session. The multiplication word problems page has two building stories and no breaking; the two ways page has two breaking stories and no building. The mixed problems page carries three stories and asks the child to sort out which is which without the layout hinting at it.