Chapter 10: "Developing Whole-Number Place-Value Concepts"
Starting in first-grade, Common Core State Standards (2010) expect students to develop an understanding of place-value concepts. The foundational concepts familiarize students with the base-ten system. First-graders learn that ten is really a group of ten ones, and they learn that two-digit numbers are composed of tens and ones. Until the fifth-grade, these foundational concepts are consistently built upon. Van de Walle et al. (2014) stated, “Place-value understanding requires an integration of new and sometimes difficult-to-construct concepts of grouping by tens (the base-ten concept) with procedural knowledge of how groups are recorded in our place-value scheme, how numbers are written, and how they are spoken” (p. 155). For instance, if students do not understand that two-digit numbers are composed of tens and ones, they will surely struggle with three-digit numbers. It is vital that your students truly understand these connected concepts, because they will forever impact their mathematical successes! I have compiled a handful of tips that will help you teach place-value concepts, regardless of grade-level.
- Base-Ten Language: When teaching place-value concepts, you will use specific terms to describe aspects of the base-ten system. The term grouping is popularly used to describe combining 10 ones to create one 10. It is important that students understand what the term grouping means, so they can properly apply the skill that it describes. Do not be afraid to explicitly teach the term grouping along with other terms, such as tens place, hundreds place, and thousands place. Additionally, two-digit numbers can be described in multiple, interchangeable ways. For example, you can describe 64 with variations: 6 tens and 4, 6 tens and 4 ones, or 6 tens and 4 singles. As you teach place-value concepts, it will be helpful to pick one description and stick with it. If you flip from one description to another, it may confuse your lower-level and English language learners (Van de Walle et al., 2014). Consistency reduces confusion!
- Concrete Base-Ten Models: Students typically use base-ten models to construct quantities, as they apply the place-value concepts that they have learned. Base-ten models are concrete manipulatives that help students visualize the “10 ones make one 10” relationship, the “10 tens make one 100” relationship, and so on. Van de Walle et al. (2014) stated, “Remember, though, that the models do not show the concept to the students; the students must mentally construct the ’10-makes-one relationship’ and impose it on the model” (p. 158). There are groupable models and pregrouped models. Groupable models are any single objects that can be grouped together to make tens. Plastic connecting cubes, wooden craft sticks, and plastic coffee stirrers (as pictured above) are common groupable models. They are easy for first-graders and beyond to manipulate, as well as inexpensive. However, pregrouped models are more commonly used. Pregrouped models are commercially made, and they cannot be put together or taken apart. They are purposely designed to represent ones, tens, hundreds, and thousands. The students do not group ten ones to create one ten. Instead, they trade 10 single cubes for one ten-rod. They do not group 10 ten-rods to create one hundred. Instead, they trade 10 ten-rods for a one hundred-square. According to Van de Walle et al. (2014), “Students combine multiplicative understandings (each place is 10 times the value of the place to the right) with a positional system (each place has a value)” (p. 159). So, pregrouped models are best for students who have already demonstrated an understanding of the base-ten system. I have an example of pregrouped base-ten models, if you scroll down this page a bit.
- The Hundreds Chart: The hundreds chart is popularly found in kindergarten, first-grade, and second-grade classrooms. It is used to familiarize younger students with the number sequence and patterns within the number system. Teachers beyond those grades seldom use this teaching tool, because they believe older students no longer need it for those purposes. However, research has found that the hundreds chart is beneficial for older students learning place-value concepts. Van de Walle et al. (2014) stated, “When students are exploring invented strategies for addition and subtraction, the hundreds chart can be used as a model to support students’ thinking and to support the communication of their ideas. The rows of 10 encourage students to think about using strategies based on place value and benchmark numbers- in this case, working with multiples of 10” (p. 163). So, if you teach third-grade and beyond, go buy yourself the hundreds chart! It will help your students solve problems with invented strategies that are rooted in place-value concepts.
"Virtual Place-Value"
Throughout my preservice teacher training, I have visited multiple elementary classrooms. Regardless of grade-level, I always notice iPads, laptops, SmartBoards, and other forms of technology within the classroom settings. The teachers explicitly teach their students how to operate those technological devices. Then, they personally use those devices to teach their students literacy and mathematical concepts. According to Burris (2013), “Students are no longer being asked to learn to use technology but to develop skills and learn with technology” (p. 229). Technology is not only changing the way we teach. It is changing the way our students learn. Third-grade teachers from Kennedy Elementary, located in Houston, Texas decided to research students’ educational interactions with technology. They wanted to discover how students think mathematically, while learning concepts with technological devices. They especially wanted to discover how students think mathematically, when using virtual base-ten blocks to learn place-value. The teachers compared a group of third-graders’ conceptual development with concrete base-ten blocks with a group of third-graders’ conceptual development with virtual base-ten blocks (Burris, 2013). Their results are interesting.
Just A Note: Virtual base-ten blocks look exactly like concrete base-ten blocks, except students can manipulate them with computer programs. Students can create groups of thousands, hundreds, tens, and ones with both forms. Below, on the left, I have an example of virtual base-ten blocks. If you click on the image, you will be directed to this online resource. If you teach second-grade or third-grade, you may want to try it out with your students. On the right, I have an example of concrete base-ten blocks.
- Similarities: When working with both the concrete base-ten blocks and the visual base-ten blocks, the students put together blocks from left to right. They started building with the largest digit and ended with the smallest digit. For instance, they put together two groups of hundreds and worked on from that. After building the numbers with both types of manipulatives, students correctly verbalized the numbers that corresponded with the blocks. They properly grouped the thousands, hundreds, tens, and ones, in order to generate the correct spoken number. Additionally, after using both types of manipulatives, they correctly wrote the numbers from left to right. During all of these steps, the students actively talked amongst themselves and verbalized their thought processes (Burris, 2013). All in all, both the concrete base-ten blocks and the visual base-ten blocks allowed the third-graders to successfully practice place-value concepts.
- Differences: Despite the multiple similarities between both types of manipulatives, there was one noticeable difference. Burris (2013) stated, “A distinct difference was that students using the virtual manipulatives ‘reused’ the quantity on the screen. They used the hammer and glue tool to show various representations without clearing the screen or starting over” (p. 234). However, when using the concrete base-ten blocks, the students created a quantity, disposed of those blocks, and created an entirely new quantity. They did not use the same blocks to show various representations of the same quantity. They composed multiple different quantities compared to the students using the visual base-ten blocks (Burris, 2013).
- Conclusions: According to Burris (2013), “Regarding place-value, the study suggests that students construct quantities, write numerals, and count or identify quantities similarly with concrete or virtual manipulatives” (p. 235). So, if you are trying to decide which form of base-ten blocks to use in your classroom, either one is a good choice. If you are not provided with concrete base-ten blocks and do not want to purchase them with your own money, do not be afraid to use the virtual manipulatives. Your students will love working on computers, and they will still learn place-value concepts. If you are not provided with computers, do not be afraid to use the concrete manipulatives. Your students will learn place-value concepts, as well. Whichever resource you have available will be educational!
Just A Note: Virtual base-ten blocks look exactly like concrete base-ten blocks, except students can manipulate them with computer programs. Students can create groups of thousands, hundreds, tens, and ones with both forms. Below, on the left, I have an example of virtual base-ten blocks. If you click on the image, you will be directed to this online resource. If you teach second-grade or third-grade, you may want to try it out with your students. On the right, I have an example of concrete base-ten blocks.
Classroom Activity: Place-Value Flip
When learning how to compose three-digit whole-numbers, it is important that students know the value of the hundreds place, the tens place, and the ones place. It is important that they know how to properly write the numbers from left to right, starting at the hundreds place and ending at the ones place. Yet, it is equally important that they know how to properly verbalize three-digit whole-numbers. As adults, this seems like a natural skill. But, young students often need practice with it! In the above video, I explain a simple game that will help students practice properly verbalizing three-digit whole-numbers. It can be implemented in a variety of ways inside the classroom or inside students' homes. I love how flexible it is, and I think you will love how effective it is.
Independent Practice: Place-Value Hockey
According to Common Core State Standard (2010) 3.NBT.A.2, by the end of third-grade, students should be able to fluently add and subtract within 1,000 using strategies based on place-values. In order for students to successfully accomplish these skills, they must have a deep understanding of each place-value. For students who are currently learning each place-value or struggling to retain this concept, "Place-Value Hockey" is the perfect independent practice. This game instructs students to click on the number that corresponds with the place-value (either hundreds, tens, or ones). If the students click on the correct number, they receive instant praise with entertaining sound effects! If they click on the incorrect number, they are allowed to try again. Second-grade students, who are learning the concept, and third-grade students, who need help retaining the concept, will enjoy playing this fun game! Click on "Independent Practice" to access this awesome resource.