Hands-On Math, KP® Ten-Frame Tiles, Math, Teaching and Learning
Welcome to Part 4 of our Fluency Series! In Part 3, we discussed fluency with early number concepts such as subitizing, composing numbers within ten, finding number pairs that total ten, and making the first ten. Giving students opportunities to engage in these types of activities over time helps them internalize a strong sense of number and maintain fluency. Far too often, we leave these early number experiences behind without realizing how robust they might be if we grow them up a bit for older students.
This week, we’re going to take a look at similar number concepts, only with larger and smaller numbers. By no means is this a comprehensive survey of everything middle-grade students need to know for number sense. Rather, I’ve included a few ideas to make the case for how important number sense is and how simple and fun it can be. For this post, we’ll focus on base-ten numbers and save non-decimal fractions for another time.
Subitizing with Ten Frames – Tens, Hundreds and Thousands
Subitizing is the instant recognition of quantity without counting. Quite often, students use structures or groupings to help them organize quantities into smaller chunks. For young children, we often use a ten frame, colored counters, or colorful blocks to help them easily chunk quantities into smaller amounts so they can see relationships. For older students, subitizing can be a great way to help them recognize patterns in the number system and apply their understandings of number relationships and properties.
Young children recognize quantities on a ten frame, using 5 and 10 as benchmark numbers. For example, they can see that 7 is two more than 5 or that 9 is one less than 10. Older students benefit from seeing “tens” on a large ten frame and then stating the quantity that is represented. For example, they can see that 70 is two tens more than 50 or that 90 is one ten less than 100. Next, the experience can be repeated by placing “hundreds” on a large ten frame so students can see that 700 is two hundreds more than 500 or that 900 is one hundred less than 1000. Click here to view slides that illustrate this idea.
Make a Ten:
Make a Hundred:
Make a Thousand:
Recognizing Quantities Within Complex Figures
Grace Kelemanik introduces an advanced form of subitizing in her Contemplate Then Calculate routine. You can find it on her website, Fostering Math Practices. Basically, the teacher presents students with a configuration of dots, and the student is asked to determine the quantity without counting. Students may use principles such as compensation, chunking, symmetry, and other ideas to help them determine the quantity. Students can then represent their thinking mathematically, connecting to symbolic representations. A sample of one of these configurations appears below.
One Possible Representation: (5×3) + (4×2)
Building Decimal Fractions with “Tens”
How does one build decimal-fraction sense? Primarily by connecting base-ten ideas with fraction ideas. The digits to the right of the decimal point represent a fraction of a whole. However, we often neglect to help students make the connection between fraction and decimal notations. Here are a few suggestions:
- Help students see that the ones place is the point of origin of a base-ten number, not the decimal point. There is not a “oneths” place to the right of the decimal point. The decimal point simply identifies the ones place as the unit. Tens, hundreds, thousands, etc. appear to the left of the ones place, and tenths, hundredths, thousandths, etc. appear to the right of the ones place. In other words, magnitudes of ten emerge from the ones place, going in both directions.
- Ask students, on occasion, to write decimal numbers in fraction form. For example, 2.25 can be written as 2 25/100. This reminds students that a number that has digits on both sides of the decimal point is actually a mixed number. This also helps the students remember that decimal fractions are fractions written in base-ten notation.
- Provide students with opportunities to group and ungroup tenths and hundredths in the same way they group and ungroup tens and hundreds. My favorite way to do this is with KP Ten-Frame Tiles. Students identify the blue-covered tile as 1 whole; then, they can “unpack” to see that there are 10 tenths and 100 hundredths inside the whole. Students can then visualize the concept behind the fraction language as well as see the magnitude of hundredths in comparison to tenths and wholes. Students do this with whole numbers in the early grades, so doing this with decimal fractions in the upper grades is extremely helpful in allowing students to see all base-ten numbers as part of a system.
Number Relationships
I shared this idea in Part 2 of our Fluency Series, and this is a great place to repeat it. Understanding the relationships among numbers is an important building block for fluency with number concepts. For example, a teacher might present four numbers to her class: 23, 20, 15, 25, and ask them to select the number that is not like the others. Of course, this particular exercise allows for multiple responses:
- 23 is the only prime number.
- 20 is the only even number.
- 15 is the only number not in the twenties.
- 25 is the only square number.
As you can see, this brief encounter might spark a lively conversation as students begin justifying their responses and realizing that there is more than one correct response. (Thank you to Mike Askew for this great example.)
In this post, we barely scratched the surface on ways to build fluency with number concepts in the middle grades. The point is that students need frequent and specific opportunities to engage in activities that develop number sense to the point of fluency. Automaticity and flexibility with thinking about number provides a vital foundation for success in higher levels of mathematics. Isn’t it worth taking 5 minutes per day to help develop and maintain this fluency? I think so!
Let us know what your thoughts about these and other activities for developing fluency with number concepts in the middle grades in the comments section below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, KP® Ten-Frame Tiles, Math, Problem Solving, Professional Learning, Teaching and Learning
Because we have been talking a lot about fluency in the past few blog posts, it suddenly occurred to me the other day that I can see a surprising connection between our efforts to help students develop fluency and my experiences becoming a successful high school sprinter. Let me explain.
When I was in high school, I loved running the quarter mile, and I made it to the state track meet every year. The mile relay was my absolute favorite event on the track – four young women working tirelessly to perfect our individual races in order to combine our skills to claim the final prize of the meet. We each played an important part in creating that final success:
- We worked on our form and rigorously built our individual skills (using the starting blocks, perfecting our just-right paces, engaging in distance running, sprinting shorter distances, etc.), all the while receiving immediate and continuous feedback from our coach.
- We honed and customized our individual races by putting together our own well-practiced skills.
- And then we took that last, important step — we combined our efforts into a longer, collaborative race, blending the traits of our individual races into a more complex and challenging final event.
So here’s the connection to math fluency…Just as my coaches helped me and my teammates achieve success through a prescribed sequence of skill-building activities, we, as teachers and coaches, can use the same strategies to help our students build the skills necessary to achieve fluency.
- I equate students’ development of number sense and math-fact fluency with the smaller, individual skills we sprinters developed (our sprints, form drills, and the like).
- Then, students’ fluency with number sense and math facts become the building blocks to fluency with multi-digit operations, just as we sprinters individually put together our running skills to design our full races.
- Eventually, operational fluency supports problem solving (the “more complex and challenging event” that is the goal of fluency) by enabling students to invest their thinking in the problem-solving process without getting stuck in the mire of operations.
So here we are, having developed number sense and math fact fluency with our students, laying the groundwork for them to perform more complex operations with multi-digit base-ten numbers (and fractions). Ensuring that students achieve operational fluency is critical…and how we get there can be a challenge. Below are the five steps I have found to be most important in helping students gain operational fluency.
- Continue to reinforce conceptual understanding. Using tools such as KP Ten-Frame Tiles to reinforce and describe what’s happening in the procedures will continue to support procedural understanding (Akin & Rimbey, 2017).
- Understand that speed and fluency are not synonymous. Fluency combines accuracy, efficiency, flexibility, and appropriateness. An over-emphasis on speed alone increases math anxiety (Boaler, 2018).
- Support fluency by giving relevant, immediate feedback. If you simply give students arithmetic worksheets with problems to practice over and over, they will not thrive because there is no opportunity for them to receive necessary and meaningful feedback.
- Build fluency in a fun and motivating atmosphere (Boaler, 2018). Provide opportunities for students to work on their skills through interactive games and online activities where meaningful feedback is possible. Something as simple as having students work side-by-side with models and with white boards, one “acting out” while the other records the operation, can make these practice sessions more meaningful.
- Make math fluency meaningful. All along the way, provide students with authentic opportunities to experience application of their math skills so they can see first-hand how fluency is an asset (Akin & Rimbey, 2017).
Had my mile-relay team neglected to rigorously build our individual skills, had we not had the opportunity to practice together toward a cohesive outcome, had we not focused on our common goal, we would not have been successful in seeing our vision become reality.
So it is with math fluency – students must actively build their competence with number sense and math fact fluency, put these skills together as they practice toward cohesive strategies for operations, and apply these skills as they successfully engage in deep and meaningful problem solving.
What are your thoughts about building math fluency for operations? What strategies have you found successful? Please share your thoughts in the comments box below.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
KP® Ten-Frame Tiles, Math, Teaching and Learning
I hear these questions all the time — “What math tools should be available in my classroom?” “Which manipulatives purchase would give me the best bang for my buck?” “What math tools should I buy for my teachers so their students will be successful?”
My first thoughts probably go without saying:
- When the manipulatives stay on the shelf or in the closet, they don’t get used. They are not a good investment.
- If students don’t engage with the manipulatives in meaningful ways, the manipulative will not be a successful tool.
- If the teacher is ill-equipped to help the students use the manipulatives effectively, the impact is likely to be negligible.
My “tier one list” of math tools will help teachers stock their classrooms. The list identifies what I recommend be present in every elementary classroom.
Base-Ten Manipulatives as Math Tools
The most conventional version of base-ten manipulatives are Base-Ten Blocks. These pre-grouped blocks can often hinder students’ understanding of arithmetic because they undo place value understanding (see our previous journal entries). Instead, I prefer groupable manipulatives that help students develop a concrete understanding for the “nesting” nature of our base-ten system. By “nesting” I mean where one ten is concurrently 10 ones, one hundred is concurrently 10 tens and 100 ones, etc. KP Ten-Frame Tiles facilitate developing understanding of numbers within ten and building with tens for both whole numbers and decimals. Click here to check them out!
Counters as Math Tools
My preference is either clear colored disks or two-color counters. I prefer the two-color counters because they transition so nicely from whole numbers to integer operations. I’m not a huge fan of all the “cute” counters with which we stock our primary classrooms, such as teddy bear, dinosaur, or people counters. It’s far more meaningful for students to “pretend” that the simple counters are these other objects. Such “pretending” involves substitutionary thinking that can lay the groundwork for using variables as substitutes for numbers later on.
Connecting Cubes as Math Tools
These have been around for years. They can be used for counting, simple arithmetic, and non-standard units for linear and area measurement.
Inch Tiles, Inch Cubes, and CM Cubes as Math Tools
All of these are great for counting, early arithmetic, and transitioning to standard measurement units. I also use the 1” tiles for introducing the area model for early multiplication.
Measurement Instruments
You’ll want to be sure your students have access to rulers, balance scales, and liquid volume pouring containers. Of course, using non-standard units is a must prior to introducing students to standard units and the tools that measure them.
Pattern Blocks
Pattern blocks are incredibly useful from kindergarten through middle school. They can be used meaningfully for geometry, algebraic thinking, whole number work, or fraction concepts,
Proportional Coin Cards
It’s common for us to think of plastic (or real) coins as math manipulatives. However, there is nothing conceptual about coins that helps students see the proportional relationships among their values. Therefore, KP Mathematics has created a set of proportional coin cards, based on ten frames.
These coin cards help students develop an understanding of these relationships. Download a free set from our website by clicking here.
Of course, there are many, many other things out there. My “tier two list” includes things like geometric solids, unmarked fraction bars, Cuisenaire rods, and attribute blocks. However, most of the items on my “tier two list” can easily be reproduced with paper, scissors, and tape.
Now I’m wondering what’s at the top of your manipulatives list. How do you stock your math tools in your classroom? Please let us know by responding in the comments box below.
We look forward to continuing the conversation!
KP® Ten-Frame Tiles, Professional Learning, Teaching and Learning
A while back, I was talking with a fourth-grade teacher at my school. She questioned, “What did the third-grade teachers do last year? I just started my fraction unit this morning, and none of the kids knows anything about fractions!” Rather than explaining what I already knew (since I was the one who taught the third-grade fraction unit the previous yea!), I simply asked her to bring her class to my math lab after lunch.
When they arrived, I had several manipulatives out on the tables – representations they had used in third grade. Before diving in, though, I simply asked the students to sing a song with me. I started the tune, which was about numerators and denominators, and almost everyone joined in. The teacher, now standing in the back of the classroom, threw her hands up and simply smiled.
You see, far too often, teachers begin mathematics units where they think they should start rather than tapping into where others left off. In this case, my colleague simply began the fraction unit as her textbook directed rather than talking with the previous year’s teachers to find out where they had left off. Had she simply inquired, she would have known which manipulatives, which visuals, which vocabulary, and which instructional strategies to use in order to tap into students’ prior knowledge.
Whether you’re a teacher, a coach, or a site leader, facilitating conversations among colleagues within and across grade levels is critical to connected learning. As discussed in our previous blog post, Math “Rules” That Expire, taking time for such conversations is critical. In the case of common visuals, here are three ways to ensuring students use visuals to “see” how inter-connected mathematics truly is…
- Connections within grade levels: Invite teachers within a grade level to map out the math concepts to be taught throughout the school year. Under each concept, have them list the manipulatives, diagrams, and other visuals to be used. Finally, ask them to look for opportunities to use common visuals across the year. The more overlaps that exist, the greater the opportunity for children to see mathematics as a system rather than isolated skills.
- Connections across grade levels: Once each grade level has compiled a year-long list of visuals and mapped out the connections, ask them to engage in cross-grade conversation about how visuals might be used to connect the mathematics from one year to the next. The more connections they make, the better students will be able to tap into prior knowledge and use that prior knowledge to learn new ideas.
- Connections using common visuals: The list below includes some of the common visuals that help students see mathematical connections…
- Manipulatives: pattern blocks, base-ten manipulatives such as KP Ten-Frame Tiles, unifix cubes, two-color counters, Cuisenaire rods, fraction bars
- Diagrams: number lines, bar models/tape diagrams, number bonds, ten frames, place-value charts, arrays
- Other visuals: hundred charts, multiplication charts (may be used for skip counting, multiplication facts, arrays, equivalent fractions, etc.), spreadsheets, Excel graphs
We owe it to our students to make these connections explicit. And we owe it to our teachers to ensure they have time and space to discuss and internalize these ideas.
Please share your thoughts on the visuals most likely to help students make connections. Do you have experiences with any of the listed visuals? Do you have other favorites you would like to recommend?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
One more thing: A great example of common representations is KP Ten-Frame Tiles. They are so superior to base-ten blocks! Check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks.
KP® Ten-Frame Tiles, Math, Professional Learning, Teaching and Learning
“Multiplication makes bigger and division makes smaller.” “You can’t subtract a larger number from a smaller number.” “Improper fractions should always be written as mixed numbers.” “You always divide the larger number by the smaller number.”
We’ve all heard these statements…and I’ll bet we’ve all said them at one time or another. And every time we say these, we further perpetuate misconceptions that are difficult, if not impossible, to un-teach in future years.
Far too often, teachers inadvertently teach mathematics in an overgeneralized and imprecise way. We teach tricks that promote nothing more than memorization. The result is that our students misunderstand the very ideas we are trying to illuminate. How many teachers have had to contradict their colleagues from previous grade levels because what was taught was so painfully limited in scope?
So…how can we stop this practice in its tracks and work together to teach mathematics correctly and coherently? Here are a few ideas to get you started.
- Read Up! First, I’d like to introduce you to two recently-published articles that drive this point home: 13 Rules That Expire and 12 Math Rules That Expire in the Middle Grades. These two articles simply and eloquently outline overgeneralized, commonly-accepted, inadequate strategies we need to abolish from our vernacular. Seriously – check them out – you’ll likely find one or two or more to work on.
- Talk to Your Colleagues: Professional discourse provides foundational and at-your-fingertips opportunities for growth. After perusing one or both of the articles above, talk to your colleagues. Last month, I had a group of 40 teachers and leaders read and discuss these articles, and their conversations were robust.
- Create School-Wide Agreements: Take some time during grade-level and staff-wide meetings to discuss these ideas. Then come up with a list of five to ten school-wide agreements, identifying the concepts, vocabulary, procedures, etc. that everyone, or no one, will use on campus. This will make for a lively conversation!
I’m in the process of working through these steps in my own school district right now. Won’t you join me on this journey to abolish rules that expire by engaging in professional discourse and by creating school-wide agreements?
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Preview for Next Week: When selecting visuals and manipulatives to use within and across grade levels, choose wisely. Focus on ones that allow for connections throughout the grades. A great example is KP Ten-Frame Tiles. They are so superior to base-ten blocks! More on this topic next week…for now, though, check out Peggy’s latest paper which focuses on why KP Ten-Frame Tiles are superior to base-ten blocks from the get-go.
Hands-On Math, KP® Ten-Frame Tiles, Math
Summertime… a time to relax, a time to learn, a time to catch up, a time to dream and to plan and to prepare for next year. It’s a beautiful thing that every school year is a fresh start with a brand-new group of students. And summertime gives us a chance to reflect on what went right in the past year as we anticipate the upcoming one.
I recently received a profound reflection from a teacher about her experience using manipulatives to help her students understand grouping and ungrouping.
Although I had been using Ten-Frame Tiles in my classroom, a colleague suggested I just teach my students the robotic mechanics of regrouping, teaching them to “go next door to the tens, knock on the door, and ask to ‘borrow’ a ten.” He mentioned that teaching them the concept just got too complicated and mucked up the mechanics of the automatic actions the kids were supposed to go through every time they needed to “borrow.”
This went against my better judgment, though, because in the 80’s I had previously taught my kids to regroup using flip cards, beans, straws, and bundles of tens that they would unbundle, and cards they would flip to represent each step of ungrouping and regrouping. Teaching the concepts this way had made sense to my kids then, and they had learned how to ungroup and regroup with a firm understanding. That day, after he advised me to jump in and teach the kids to “go next door and knock on the door and ‘borrow’ a ten,” I did just that, and we had a fun lesson knocking and asking and laughing about all of it.
But, the next day, I gave the kids more problems that required them to ungroup and regroup, and, sure enough, not only could they not do the problems, there was a strong air of unrest and dissatisfaction in the room. The kids acted like the world was not a friendly place for them to be in, and they were disgruntled and downright grumpy. I didn’t know this tension was coming from their place of confusion with the math. I brought out the tiles again the next day, and we spent the whole morning learning grouping and ungrouping with the tiles.
Time really did fly by. After two hours had passed, and the kids were back in a place of true understanding, one of the kids saw the clock and said, “Wow, we have been doing math for two hours.” Not one of the students had complained or asked about the time during our lesson or even noticed how much time had passed.
But after that two hours, I had happy kids again, and not only was there an air of true satisfaction in the room, but the kids were clearly happy—peaceful, content and what seemed to me with a sense of being one with their world again. They acted as if I had turned on the lights or opened a door to a better place for them.
I asked them, “Now, is this a planet you want to live on?” (I had mentioned earlier that a planet where everyone keeps losing ten dollar bills was not a place I wanted to live!) and they immediately agreed, “Yes!” It was a place they knew they wanted to be, and it was a place that finally made sense! They have been on this planet happily ever since!
While I had known that this teacher was enthusiastic about Ten-Frame Tiles, what struck me about her reflection was her understanding of the KIND of difference these manipulatives make. To her, her students’ use of the tiles contributed directly to their emotional well-being, a perspective not typically taken into consideration or perhaps even recognized.
This reflection reminds us that children want to make sense of the math they are learning. When they do, they find a world of satisfaction, and, when they don’t, the results may have implications far more significant than low test scores. And both outcomes can have life-long implications!
We would love to hear from you! As you reflect on your work last year, what one or two things went well and are worth saying out loud? Please let us celebrate with you by leaving your thoughts in the comments box below.
