Game Play, Hands-On Math, KP® Ten-Frame Tiles, Teaching and Learning
Serious Play #5
Imagine this…you walk into a noisy, bustling classroom where students are spread out in small groups at tables, on the floor, at the counter. They’re everywhere, really. They’re huddled around game boards and decks of cards, playing mathematics games that engage them in mathematical thinking, discourse, and strategy.
As you reflect on what you’ve witnessed, what did you notice? What math concepts were being reinforced? Were all the students playing the same games, or was there a variety? Did you see evidence of different levels of thinking?
After several years of using mathematics games in my own classroom, I noticed that that most of my games centered on arithmetic and math-fact practice, almost always at the abstract level (e.g., using the symbols of mathematics). It was at this time that I started looking for games that included representations other than numerals. Could I find games that incorporated concrete manipulatives? Were there games out there that incorporated pictures, diagrams, or other graphic representations? It turns out that there are plenty of options if you know what you’re looking for! These days, when selecting great mathematics games, I look for a balance in concepts, DOK (Depth of Knowledge), thinking strategies, and representations.
For today’s post, let’s look at representations that promote thinking at different levels. Below you will find three games, each addressing one of the C-P-A levels: concrete – pictorial – abstract. Notice that they all exhibit the characteristics described in last week’s blog post that make mathematics games fun.
1) Mathematics Games That Use Concrete Representations
Since students often manipulate objects when exploring a concept, games in this category typically include a game tboard or concrete materials. Race to 100 provides a concrete way for students to focus on grouping (and ungrouping) tens as they build numbers to 100.
Click here to download Race to 100.
When playing this game, students work as partners rather than as competitors. As they repeatedly group ten ones into one ten, the “concreteness” of the game promotes visualization of place value.
2) Mathematics Games That Use Pictorial Representations
Ruler Races involves the use of a 12-inch ruler marked in fractional parts of inches. Students are encouraged to think of the ruler as a number line marked in fractional parts. They physically identify different points on the fractional number line as they explore flexible ways to think of fractions and fraction addition.
Click here to read the directions for Ruler Races.
Although one may think of a ruler as a concrete object, it is not mathematically concrete. It really is a pictorial representation of the 1-inch units (or 1-cm, or whatever unit is being used) used to measure objects. Some teachers even use rulers as number lines, which is the case in this game.
3) Mathematics Games that Use Abstract Symbols
Salute has been a favorite in my classroom for years! Students practice inverse operations (addition/subtraction OR multiplication/division) during game play. Students alternate between finding sums/differences and related missing addends/factors.
Click here to read the directions for Salute.
This game uses the numbers (abstract representations) on the playing cards to generate new problems for each round, and the students use mental math strategies to determine the missing values.
So…think about the games you use in your classroom. Do they offer students the opportunity to engage in mathematics beyond math-fact practice? Do your mathematics games include various levels of representations? How might you incorporate more concrete and pictorial games into your repertoire? Please share your thoughts in the comments box, below.
Next Steps for Teachers: Set up the games in this blog post (and others) and provide them for students to play. Also, take inventory of the games you already have in your classroom – do they provide concrete and pictorial opportunities as well as abstract practice?
Next Steps for Leaders: Lead your teachers in a conversation about this post. Have them list the mathematics games available in their classrooms and categorize them by C-P-A. Then walk through the classrooms to see how students are responding to the math games that use different levels of representation.
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, Teaching and Learning
Becoming a teacher-turned-inventor never crossed my mind earlier in my career. However, since founding KP Mathematics, this is the path I’ve traveled. KP Ten-Frame Tiles, an embodiment of the base-ten number system, came to life as we searched together to find a solution to the problems presented with pre-grouped manipulatives such as base-ten blocks.
As physical and pictorial models, base-ten blocks do an adequate job of illustrating place-value-based quantities. However, everything falls apart when using base-ten blocks to demonstrate the action of grouping and ungrouping tens and magnitudes of ten.
In her article, Choosing the Right Tool, Stacy Boote explains that base-ten blocks have a static form that renders them less useful when demonstrating operations such as division due to the need to group and ungroup. In contrast, when used to demonstrate the area model of multiplication, they will suffice because the action of grouping occurs after the model has done its job.
“When using the area model, exchanges between place values do not occur in the creation of the rectangle or when finding the areas of the four inner rectangles. Renaming occurs at the end of the process, when calculating partial products. Therefore, using materials that can be easily manipulated is not an area model requirement” (Boote, p. 479).
This regrouping issue that exists for long division also appears when representing the operations of addition and subtraction. The need to manipulate the materials for exchanges between place values is an important process. And when using base-ten blocks, the trades that must take place for grouping and ungrouping (aka, carrying and borrowing) obstruct student understanding rather than support it.
Using groupable manipulatives, such as KP Ten-Frame Tiles, craft sticks, or unifix cubes, requires less cognitive load because the actions children demonstrate are relevant to the mathematics they are learning. When students are “regrouping,” they literally group and ungroup rather than “trade,” which is the only option with pre-grouped manipulatives such as base-ten blocks.
So, you may ask, how do we resolve this issue? The answer is simple…use groupable manipulatives when the mathematics being modeled requires the action of grouping and ungrouping. KP Ten-Frame Tiles, craft sticks, or unifix cubes are a few options. It’s okay to use pictures of base-ten blocks to help kids visualize quantities, especially given this is the most common model used in testing situations. But when students need to manipulate to represent the movement between places, groupable models are the way to go.
If you’d like to learn more, I recommend reading the groupable vs. pre-grouped manipulatives section in Teaching Student-Centered Mathematics, or the article linked above, Choosing the Right Tool. And take a look at Peggy’s series of journal entries, Beyond Base Ten Blocks.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
After reading this post, please join in on the conversation! What have been your experiences with pre-grouped manipulatives such as base-ten blocks? What have been your successes and challenges? We always love to hear from you!
Hands-On Math, Professional Learning, Teaching and Learning
Why Making Math Visible Matters
By Kim Rimbey
Math is the study of patterns.
Math is a way of thinking. Math helps us describe the world—quantitatively and abstractly.
Math has many expressions.
But here’s the thing: If we can see it, THEY can see it.
When it comes to teaching and learning math, getting the thinking out of our heads and into the world is essential. Visible thinking helps us communicate clearly, uncover misconceptions, and build shared understanding.
That’s exactly why we wrote Seeing the Math You Teach—to support teachers, parents, and students in making their math thinking visible. “Seeing = Understanding.” With the understanding that there are multiple solution paths to any problem, the book offers a variety of visual strategies that help learners represent and explain their ideas.
Number Lines: More Than Meets the Eye
While prepping for a recent conference, I took a deep dive into one of our most powerful tools: the number line.
We often start with number tracks—like snap cube trains—to help young learners count units. Then, we shift to number lines with tick marks. But this shift calls for a shift in focus, too. It’s the spaces between the tick marks that matter. That’s where the “jumps” happen. That’s where the math lives.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
And no matter what the intervals—1s, 10s, 25s, millions, fractions—every point on a number line represents a distance from zero.
This is true even if zero isn’t shown. Even if we’re looking at just a portion of the number line, the meaning of every number still depends on how far it is from zero. And it’s true for open number lines as well. Even though the jumps may not be proportional, they still represent the distance traveled from zero. It’s all about relational thinking.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Quick Teacher Check-In: Do your students understand that every value on a number line—visible or open—is a measure of its position relative to zero?
As students start representing their thinking using number lines, we need to help them see what they’re showing—even when zero is nowhere in sight.
Letting Students SEE and BUILD the Math
Here’s the hard truth:
We can’t just tell students what to see. They need to explore, discuss, make mistakes, and revise.
And for that to happen, we need to create learning environments that encourage deep thinking and collaborative sense-making.
Enter…Vertical Non-Permanent Surfaces (VNPSs)
VNPSs—think whiteboards, whiteboard cling film, windows, chart paper, etc.—are erasable vertical spaces where students stand, collaborate, and represent their thinking.
Peter Liljedahl’s Building Thinking Classrooms popularized this approach, and I’ve seen firsthand how transformative VNPSs can be.
For Students
- Interact and record their thinking
- Erase, rework, and revise as they learn
- Represent ideas in multiple ways
- See other groups’ strategies
- Stay on their feet and activated
For Teachers
- Easily scan every group’s work
- Step in briefly to nudge, hint, or extend
- Plant seeds and move thinking forward
- Select and sequence work to meet learning goals
- Spotlight key ideas—like number lines and distance from zero
But What If I Don’t Have Enough Whiteboards?
Good news—VNPSs aren’t about expensive equipment. Here are some ideas for where to find inexpensive VNPSs (from the Building Thinking Classrooms update posted on June 2, 2025 – used with permission – click here to subscribe):
Your Next Steps
For Teachers
Two big ideas, two next steps:
- How might you talk about number lines in a way that conveys “distance from zero” as the central idea?
- How might you incorporate VNPSs (or use them more intentionally) to support visible, collaborative math thinking?
- Which visual do you use to support your students’ thinking? For more ideas on number lines and many other visuals, check out my newest book, Seeing the Math You Teach.
For Leaders
Same two ideas, new lens:
- How might you build shared understanding during PLC/planning sessions around the idea that number lines represent distances from zero? How might you facilitate doing math together?
- What barriers can you remove to ensure teachers have access to VNPSs and the tools to use them effectively?
Final Thoughts
Let’s keep helping students see the math we teach.
Let’s help them own their thinking.
Let’s keep learning—together.
Interested in Workshops by Kim Rimbey?
Kim offers workshops on a variety of topics, including Building Thinking Classrooms, Ten-Frame Mathematics, Mastering Math Manipulatives, Small-Group Math Instruction, and See the Math You Teach: A New Vision for Math Teaching and Learning. View Kim’s PD Catalog to learn more.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, Problem Solving, Seeing The Math You Teach, Teaching and Learning
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing the Math You Teach – Part 1
By Kimberly Rimbey, Ph.D., NBCT
Have You Ever Really Seen the Math?
Did the way you learned math suit you? Were the concepts connected, or were they more about following steps and rules without really understanding why they worked?
For me, dividing fractions was always a mystery. I could recite the steps: flip the second fraction and multiply. But I had no clue why it worked. And why, when I divided two fractions, did the answer sometimes get larger? Isn’t dividing supposed to make things smaller?
If you’ve ever felt this way, you’re not alone. Many of us were taught to do math, but not necessarily to see it.
And that brings me to Ted.
Have You Met Ted?
I met Ted a couple years ago at a workshop I led in his district. Ted had been teaching fourth grade for 15 years. He was confident, experienced, and deeply committed to his students. But recently, changes to his state’s standards had shaken his confidence.
One shift in particular? Multi-digit multiplication.
Instead of simply teaching students to multiply using the standard algorithm, the new standards asked Ted to teach multiplication “using strategies based on place value, properties of operations, and the relationship among operations.”
Ted was puzzled. How do you teach strategies you’ve never seen? And even if your curriculum includes those strategies, how do you use them effectively if they weren’t part of how you learned math?
This led Ted to a moment of self-reflection: his students could follow steps, sure. But did they really understand what those steps meant?
Seeing the Math You Teach
Rather than jumping into “show and tell” mode, I invited Ted and his colleagues to engage with a two-page spread from a visual math resource I was working on with my co-authors – Seeing the Math You Teach. We explored multiple strategies for multi-digit multiplication. No algorithms. No formulas. Just visuals and reasoning.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
Take a moment to imagine what Ted saw on these two pages. Think: base-ten blocks, area models, number lines, and symbolic representations all working together to build meaning.
What do you notice? What are you curious about?
As you examine visuals like these, you begin to uncover the connections:
- Place value models that show how tens and hundreds grow
- Arrays and area diagrams that highlight the structure of multiplication
- Number bonds that reveal flexible thinking about numbers and their relationships
- Symbols that begin to make sense after the concepts take shape visually
And just like that, the math starts to come alive.
Beyond Elementary: A Universal Invitation
While the examples in this post come from elementary classrooms, the ideas are universal. Whether you’re helping first graders explore base ten or guiding high schoolers through slope-intercept form, visual models support conceptual understanding.
They help students connect the why behind the how. They turn math from mystery into meaning.
So let’s explore together.
Visual Strategies in Action
Let’s look at how some common tools and representations connect concepts across K–12.
- Number Bonds. Great for developing part-part-whole reasoning and flexible number sense. Later, they laid the groundwork for decomposing expressions and functions.
- Base-Ten Manipulatives. Perfect for understanding place value, regrouping, and decimal operations. With the right scaffolding, they even support Algebra Tiles!
- Area & Array Models. Help students visualize multiplication as repeated addition and structured grouping. Also helpful for algebraic expressions and factoring in secondary grades.
- Number Lines. Useful for addition, subtraction, fractions, decimals, and integer operations. Use Linking Cubes as physical number lines in early grades. Extend into coordinate graphing and slope visualization in later grades.
- Symbols. A powerful tool, especially when paired with visuals. They offer precision and efficiency, but only after meaning is established – whether you’re connecting quantities to numerals, building understanding of comparison symbols, or establishing ways to record exponential growth.
Whatever your grade level or content area, visual representations offer a bridge to understanding.
Seeing = Understanding
Too often, in mathematics, we are in a rush to move to the abstract. In doing so, we overlook the visual.
-Peter Liljedahl
Seeing the math you teach means making ideas visible.
It means sketching. Moving things around. Building understanding.
And when we see it, our students can too.
Next Steps: What Could You See More Clearly?
Teachers: What’s one concept you teach that students struggle to grasp? Could a visual representation help uncover the why behind the steps?
Leaders: How might you use ideas like those above to spark the conversation with your teacher teams? How might you subtly use these questions to help your teachers see the math in new ways?
Everyone: The book is available now, both from Corwin and Amazon!
In the next post, we’ll explore ways to use Seeing the Math You Teach to support both teachers and students to really see the math.
Until then, keep seeing the math—and helping your students see it too.
Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.
Hands-On Math, Problem Solving, Professional Learning, Seeing The Math You Teach, Teaching and Learning
When You See the Math, They Will See It Too
by Chryste Berda
If math has ever made you feel small, you’re not alone. Many of us have internalized the message, intentionally or not, that math is a fixed ability: you either have it or you don’t. But what if making math visible could be the very thing that transforms that feeling of smallness into strength?
In my classroom, we lived by just two rules:
- Math makes sense.
- Everyone will be good at math.
I picked up these rules at a math conference years ago, and they became my mantra from that point forward. But recently, a moment reminded me that these rules are deeply connected—and that one cannot exist without the other.
When Rule #2 Is Broken
I was screening number sense for a colleague when a 9-year-old student caught my attention. She got the problem 14-9 wrong and then sighed in frustration.
“No, that’s not right, but I don’t know why this is so hard for me! I’m just not good at math.”
In that one sentence, I heard it: Rule #2 had already been broken for her.
But here’s what I’ve learned: Everyone will be good at math only WHEN we teach it in ways that make sense. That’s Rule #1. And when we break Rule #1, we can’t keep Rule #2.
Making Math Visible Is the Key
Just the day before, I had been working with our district’s special education teaching team. The focus? Making math accessible by making it visible.
Making math visible doesn’t just mean using manipulatives or diagrams – it means illuminating the thinking, patterns, and connections behind the math. It means helping learners see what’s going on under the surface.
One teacher pulled me aside during that workshop. In a hushed voice, she said:
“I’m not a math person, but I want to be—for my students. The way you’re showing us how to make the math visible? I’m actually learning the math for myself!”
She followed up this comment by asking if we could continue to collaborate as she begins this with her students. You see, math becomes visible not just through clear explanations, but through shared exploration – with colleagues who remind us we’re not in this alone because in Rule #2, “everyone” doesn’t just mean all students, it literally means everyone – colleagues, parents, administrators, students, – everyone!
From Professional Learning to Classroom Confidence
This philosophy is at the heart of our book, Seeing the Math You Teach. We’ve seen how making math visible doesn’t just support student understanding–it builds teacher confidence, too.
At a recent 6th-grade professional learning community meeting, teachers tackled one of the most misunderstood standards: Mean Absolute Deviation (MAD). Before it appeared in standards, many educators had not even heard of it. And early resources? Practically nonexistent!
So the PLC team used page 195 from our book to understand MAD together. In those quiet moments of collaboration – around a whiteboard, over a problem, inside a question – something shifted. Confidence took root. You could feel the shift in the conversation as the process was illuminated and the math became visible—for them.
Rimbey, K., Basham, K, Berda, C. (2025). Seeing the Math You Teach. Corwin: Thousand Oaks, CA.
And when they brought that same resource into their classrooms, it became an anchor chart — a living tool for students. The color coding and step-by-step clarity in the resource helped illuminate the process behind MAD and made it visible to every student, much in the same way that it had become visible for the teachers. That’s the power of making math visible. It builds what I like to call mathematical swagger—for teachers and students.
Everyone Means Everyone
Let’s stop pretending there’s such a thing as a “math person.” Confidence in math doesn’t come from talent—it comes from clarity. From ideas being made visible. From sense-making being prioritized. And it doesn’t happen alone. It happens in community, through collaboration and exploration, with people who remind us we’re not in this alone. Because when you see the math, they will see it too.
Make Math Visible for Someone This Week
Take one small step this week to make math more visible—for a student, a colleague, or even yourself. Use a model. Ask a clarifying question. Share your own learning journey. Pull out that anchor chart one more time.
And if you’re ready to dig deeper, invite a colleague to co-plan with you or explore a page from Seeing the Math You Teach together. Learning is meant to be visible—and shared.
Math makes sense. And everyone will be good at math–because of you.
Chryste Berda - Chryste is energized by sharing her passion for learning with her colleagues as the district math coordinator and as a Regional VP for the Arizona Association of Teachers of Mathematics. She is intensely curious about students’ thinking and spends much of her time listening to students explain their ideas.
