Okay, your tower will not lean, but the multiple tower is a great visual for children learning to use landmark numbers to compute more efficiently. In more traditional style cases multiples and factors are taught but not in such away that allows students to make use of the concepts. In many cases, if students know that multiples of six include six, twelve, and eighteen...then they can pass the test. But what really is a multiple? What is a landmark multiple and how can multiples be used to help compute more efficiently.
More progressive style mathematics program help children construct a better conceptual understanding
of various ideas so that they can better use them in other areas of mathematics. In the case of multiples, if children know that 210 is a landmark multiple for 21 then they might be able to mentally figure 21 x 12 is only 42 more than 210 rather than have to rely on a procedural algorithm to hopefully arrive at the correct calculation. But how will they realize this unless they discover it with guidance?
In this lesson, children begin listing multiples of 21 as the teachers write the responses from the bottom of a six foot long piece of chart paper heading toward the top (building the tower). The teacher charts all multiples of 21 in a neat column until he reaches the number 210. The teacher then asks, "How many 21's are in 210?" Many of the children can see that there are ten 21's in 210 and this serves as the perfect time to illustrate the two problems on the board 21 x 10 = 210 and 210 / 10 = 21. The children see the connection between division and multiplication in this example. The children continue on by providing more multiples of 21. The teacher stops at the number 315 and asks the class how many 21's are in 315. The children will hopefully see that after 210 there are five more multiples in the tower and so now can say there are 15 21's in 315. Ask them to explain how they arrived at their answer when they give it for clarification. Next, continue building the tower. The teacher reminds students to tell him when they reach the next multiple ending with zero. Eventually as the tower is constructed up to the number 1,029 children have you circle the numbers 420, 630, and 840. Announce to the children that these are landmark multiples because they can be used to solve various multiplication problems rapidly. Each represents the final number in a group of ten multiples. Thus, 840 / 21 can easily be found to be 40. By the same token, 21 x 42 can easily be discovered to be 882.
After the large tower has been charted and multiples of ten circled on the chart, have children find the answers to various problems using the multiple tower chart. Some examples might include: 21 x 17, 882 / 21, or 273 / 21. These more
complex division and multiplication problems can be easily solved by using the tower's landmark multiples.
The goal would be to help children to construct and internalize the ideas in this lesson so that in the absence of the multiple tower they can use the tools mentally to solve more complex problems.