Many tips and tricks that we teach our elementary students as rules of mathematics, are introduced as a way to help students recall a procedure rather than truly promote their conceptual understanding of the content. However, many of these rules learned early on don’t hold true as students start to learn more advanced content in middle and high school.
An article in Teaching Children Mathematics, 13 Rules that Expire, by Karp, Bush and Dougherty addresses some of these common misconceptions. Let us know if you see these rules that expire in your classroom, and how you address them.
The first rule we are going to talk about is, "Just add a zero!"
When you multiply 4 by 30 what strategy do you use?
Consider these possible strategies for solving this problem:
| Strategy A | Strategy B |
|---|---|
4 times 3 is 12.
Then add a zero and you get 120.
|
4 times 3 is 12.
12 times 10 is 120.
|
At first glance one may think both of these strategies are appropriate. However, use the same strategies to multiply 0.4 by 30:
| Strategy A | Strategy B |
|---|---|
0.4 times 3 is 1.2.
Then add a zero, so 1.20.
|
0.4 times 3 is 1.2.
1.2 times 10 is 12.
|
The strategy of adding a zero to the right of the number when multiplying by a multiple of 10 only applies to whole numbers, and can’t be generalized. Additionally, utilizing this trick of “adding a zero” isn’t mathematically sound, and does not support students in reasoning and justifying their answer.
Let’s take a look at the mathematics behind Strategy B for each of the above problems.
| 4×30 | 0.4×30 | |
| 4×3×10 | 0.4×3×10 | Decomposition or Partitioning into Factors |
| (4×3)×10 | (.04×3)×10 | Associative Property of Multiplication |
| 12×10=120 | 1.2×10=12 |
Elementary students can and do use the properties of operations when computing; it’s our job as teachers to help students see and understand the value of the mathematics behind each strategy.
Cluster problems are one way to support students with using facts and combinations they likely already know in order to solve more complex computations (Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M., 2016). Here’s a set of cluster problems that lead to 34 x 50. Consider how these problems are related and the rich discussion you can have with students about the properties of operations they used to get their final answer.
4×5
3×5
3×50
30×50
34×50
Karp, K.S., Bush, S.B., & Dougherty, B.J. (2014). 13 Rules that Expire. Teaching Children Mathematics, 21 (1), 18-25.
Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Boston: Pearson.
1) Number and Operations, and 2) Geometry and Measurement. According to the NRC (2009), conceptual development within number and operations should focus on students’ development of the list of counting numbers and the use of counting numbers to describe total objects in a given set. It is recommended that teachers provide students with opportunities to “subitize small collections [of objects], practice counting, compare the magnitude [size] of collections, and use numerals to quantify collections” (Frye et al., 2013). Conceptual development in geometry and measurement should support the idea that geometric shapes have different parts that can be described and include activities that model composition and decomposition of geometric shapes. 
