MathReps

Understanding Subtraction with Pictorial and Expanded Models

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Why Use Multiple Models?

By using a pictorial model alongside expanded notation, students get a visual and numeric understanding of subtraction. This helps them move beyond memorized procedures to truly grasp why and how regrouping works.

Let’s break it down step by step!

Step 1: Set Up the Problem

We’re working with 736 – 274.

  • 736 is represented using pictorial models in a place value chart:
    • 7 hundreds
    • 3 tens
    • 6 ones
  • Below, we note in the corner of each place value column what we’re subtracting: 200 + 70 + 4 = 274

Step 2: Subtract Using the Pictorial Model

Subtract the Ones (6 – 4):

  • Cross off 4 ones (I like to use x to represent 1s).
  • That leaves 2 ones.

Regroup the Tens (30 – 70):

  • Uh-oh! We don’t have enough tens to subtract 70.
  • So, we regroup 100 into 10 tens (since 100 = 10 tens).
  • Now, we have 13 tens total (13 tens = 130)
  • We subtract 70 (or 7 tens) from 13 tens, leaving 6 tens (or 60).

Subtract the Hundreds (700 – 200):

  • After regrouping, we have 600 left in the hundreds place.
  • Subtracting 200 leaves us with 400.

Final Answer: 462

Step 3: Solve Using Base 10 Expanded Notation

Now, let’s represent the numbers in expanded form:

  • 736 → 700 + 30 + 6
  • 274 → 200 + 70 + 4

Subtracting step-by-step:

  • 6 – 4 = 2
  • 30 – 70 (not possible, so we regroup from the hundreds)
    • Moving 100 over to the 10s place, making it 130 – 70 = 60
  • 600 – 200 = 400

Final Answer: 462

The Power of Dual Modeling

Using both pictorial and expanded form models side by side helps students see:
✏️ Why we “regroup” in subtraction
✏️ How place value plays a role in regrouping
✏️ That both methods lead to the same solution, reinforcing accuracy

When students see and practice both models together, they build deeper number sense and mathematical confidence!

Try This in Your Classroom!

You can grab a free MathReps template at MathReps.com and start using this strategy with your students today!

💡 Bonus Resource: Wipebook offers workbooks with four MathReps for second grade, providing structured practice all year long. Check them out!

Watch the Full Tutorial Video

Effective MathReps for Addition with Regrouping

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Today, we’re diving into second-grade math with a MathRep that focuses on addition with regrouping.

In our previous posts, we explored using MathReps for addition and subtraction without regrouping. Now, we’re taking it a step further and tackling regrouping, helping students make the jump from pictorial models to expanded notation and the base 10 system.

So, buckle in—let’s get started!

Step-by-Step Guide: Regrouping with MathReps

Set Up the Numbers

For this example, we’re adding 682 + 234, and we’ll use a pictorial model to visually represent the numbers before transitioning into the expanded form.

Solve Using a Pictorial Model

1️⃣ Start with the Ones Place:

  • 2 + 4 = 6
  • No regrouping needed here, so we write 6 in the ones place.

2️⃣ Move to the Tens Place:

  • We add 8 tens + 3 tens = 11 tens, which equals 110.
  • Since 10 tens = 100, we regroup by circling ten tens and moving them into the hundreds place.
  • This leaves 1 ten (10) in the tens place.

3️⃣ Move to the Hundreds Place:

  • We now have 6 hundreds + 2 hundreds + 1 regrouped hundred = 9 hundreds.

Bringing it all together:
916

Transition to the Base 10 Model (Expanded Notation)

Now, let’s break it down using expanded form:

  • 682 → 600 + 80 + 2
  • 234 → 200 + 30 + 4

Adding the place values:

  • Ones: 2 + 4 = 6
  • Tens: 80 + 30 = 110 → Regroup into 100 + 10
  • Hundreds: 600 + 200 + 100 = 900

Total: 916

Alternative Thinking: Flexible Number Sense

Instead of regrouping immediately, students can leave the tens as 110 and think of the sum as:

800 + 110 + 6 = 916

This approach challenges students to see numbers flexibly before applying the standard algorithm, deepening their conceptual understanding.

Why This Works

MathReps provide a structured, visual approach that helps students bridge the gap between pictorial models and formal mathematical notation. By working through regrouping in different ways, students develop number flexibility and problem-solving confidence.

Get Your Free MathReps!

Want to try this in your classroom? Download your free MathRep template at MathReps.com and explore even more math resources.

💡 Bonus Resource: Check out Wipebook for workbooks featuring MathReps—each workbook includes four MathReps to last the entire year!

Watch the Tutorial Video

Subtraction Made Simpler With MathReps

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I’m here to show you how MathReps can make subtraction practice easier and more engaging for your students. In our last post, we explored how to use a MathRep for addition, and today, we’ll focus on subtraction.

This step-by-step guide will show you how to use the MathRep template in your classroom, helping students bridge the gap between visual models and abstract concepts.

See this MathRep in action!

Set Up the Numbers

For this example, we’ll work with 658 and 234. The subtraction problem is set up in a pictorial model, where the number 658 is represented visually, and 234 is broken down into ones, tens, and hundreds to subtract from each place value.

Use a Pictorial Model to Subtract

In the pictorial model, each section represents a place value: ones, tens, and hundreds.

  • Subtract the ones:
    • Start with 8 ones and subtract 4.
    • Cross out or erase 4 ones, leaving 4 ones.
  • Subtract the tens:
    • Start with 50 tens and subtract 30 (3 groups of ten).
    • Cross out 3 groups, leaving 20 tens.
  • Subtract the hundreds:
    • Start with 600 hundreds and subtract 200 (2 groups of 100).
    • Cross out 2 groups, leaving 400 hundreds.

Combine the remaining values:
400 + 20 + 4 = 424

Transition to the Base 10 Model

Now, transfer the pictorial model into a base 10 model by writing the numbers in expanded form:

  • 658 → 600 + 50 + 8
  • 234 → 200 + 30 + 4

Follow the same subtraction process:

  • Subtract the ones: 8 – 4 = 4
  • Subtract the tens: 50 – 30 = 20
  • Subtract the hundreds: 600 – 200 = 400

Bring it all together:
400 + 20 + 4 = 424

Why This Works

This activity helps students make meaningful connections between the pictorial model and the base 10 system, reinforcing their understanding of subtraction and place value. By bridging these two models, students gain confidence in solving subtraction problems and build a deeper understanding of the math concepts.

Get Started Today!

Ready to bring this subtraction strategy to your classroom? Head over to MathReps.com to download your free MathReps template and explore more resources for effective math practice.

With MathReps, subtraction doesn’t have to be overwhelming. By providing students with a structured and visual approach, you’re setting them up for success in their math journey. Let us know how it works for your students!

Engage Students with 2nd Grade MathReps Activities

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I’m excited to share how MathReps can transform daily math practice into a fun and engaging activity for your students. Whether you’re a seasoned teacher or just starting your journey, MathReps are a fantastic tool to build fluency and confidence in math concepts.

For today’s demonstration, let’s dive into a 2nd-grade MathRep and explore how you can use it in your classroom. This template can also be found on Wipebook Workbooks and Kami!

Blank 2nd grade MathRep for addition or subtraction
Filled in example of the 2nd Grade MathReps template using addition.

Step-by-Step Guide: Using MathReps in Your Classroom

1. Set Up the Numbers

For this example, we’re working with 237 and 422. Begin by deciding whether your students will add or subtract these numbers. For this demonstration, we’ll focus on addition.

2. Use a Pictorial Model

Start by setting up the numbers in a pictorial model.

  • Creating a 100’s chart will help students represent each digit in the numbers.
    • Each x will represent the ones (I prefer x’s over dots as it can become hard to distinguish individual dots)
    • Each l will represent the tens
    • Each square will represent the hundreds
  • Count the ones, tens, and hundreds in each number:
    • Ones: 7 ones + 2 ones = 9 ones
    • Tens: 30 tens + 20 tens = 50 tens
    • Hundreds: 200 hundreds + 400 hundreds = 600 hundreds

Add these values together:
659

3. Transition to the Base 10 Model

Once students understand the pictorial model, encourage them to transfer the information to a base 10 model by breaking down the numbers into expanded form:

  • 237 → 200 + 30 + 7
  • 422 → 400 + 20 + 2

Have the students add the expanded forms:

  • Add the ones: 7 + 2 = 9
  • Add the tens: 30 + 20 = 50
  • Add the hundreds: 200 + 400 = 600

Bring it all together:
600 + 50 + 9 = 659

Why It Works

This activity bridges the gap between pictorial models and abstract math concepts. Students not only practice addition but also develop a deeper understanding of place value and number decomposition.

Get Started Today!

Ready to try this out in your classroom? Visit MathReps.com to download your free MathReps template and discover more resources for math practice.

By using MathReps, you’re helping students build foundational skills in a way that’s both structured and engaging. Give it a try, and let us know how it works for your students!

Check It Out In Action!

2 New Video Series Coming: Google & Math

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I’ve been diving into the world of videos lately, and it’s exhilarating to see the endless array of tutorials available at our fingertips. To contribute my own unique perspective, I’m launching TWO engaging series that I can’t wait to share with you. The first will unveil practical tips and tricks for using Google Tools effectively—because there’s always something new to discover! The second series will delve into math strategies tailored for both students and teachers, exploring everything MathReps has to offer. Get ready to enhance your skills and insights!

I’m not sure when these will launch, but know for certain that they will be showcased here. The first Google tool I will be looking at is Slides. I plan to look at different components and dig into them. The goal is to keep the videos quick and engaging. Personally, I hate watching longer videos for the one piece of information I am looking for. I feel that by creating several Quick Tips, viewers can learn what they need to, quickly. As you can see from the image here, I’ll be covering Word Art in one of the first episodes.

The series based on MathReps will contain information on MathReps, Math Eduprotocols, and other math strategies. Like the previously mentioned series, the videos will be bite-sized nuggets for viewers. I’m unsure when I’m starting that series, but I am just as excited to get it going. I have several thoughts: looking at MathReps and how to implement them, teacher tutorials on different math strategies we teach students and other helpful math tidbits.

  1. For Google Tools Series:
  • What’s one feature in Google Slides you’d love to master or learn a quick tip about?
  • Do you have a favorite Google Tool that you’d like to see covered in future posts or videos?
  1. For Math Strategies and MathReps:
  • Have you ever used MathReps in your teaching or learning? What’s been your experience?
  • What’s your go-to math activity that engages students most effectively?
  1. For Both Series:
  • Which series are you more excited for—the Google Tools or the MathReps and Strategies? Why
  • Is there a specific topic you’d like to see covered in either of these series? Share it here, and I’ll consider adding it!
  • Do you have any tips or shortcuts for Google Slides or math strategies that you’d like to share with others?

Engaging Kids in Math: Using Play-Doh for Multiplication

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This school year, I have had the privilege of collaborating with several 3rd-grade teachers. During one meeting, I shared an innovative idea from a 4th-grade teacher, Ben Cogswell. He introduced the use of Play-Doh to help his students understand multiplication through hands-on activities. I really like how he incorporated MathReps into this lesson. In the accompanying image, you can see students recording their findings in a MathReps, along with an instructional video they can reference for guidance.

Students using Play-Doh and MathReps to assist in their understanding of Multiplication and Division.

Fast forward a few months into 3rd-grade. While meeting with one of the groups, one teacher, Jensen Hall, was excited to share that she had purchased some Play-Doh for her students to use as they begin learning about multiplication. Then, about a week later, she sends me this message with images:

Message: I meant to send this to you earlier but I'm so excited about it I had to make sure I didn't forget again (sorry it's after hours). I introduced multiplication today using the play doh and it was a massive success! Such a great idea you shared!!
Students showing off their work.

Taking a closer look at the image, the red arrow points to where a student wrote out the problem on their desk. Using equal sets, students were able to model how to solve the problem. While the faces have been hidden, I can assure you that they were all smiling and clearly having a good time.

We all want our students to be excited and engaged with math. In the rush to fit everything into a school day—assessments, end-of-the-year testing, and much more—we often forget how to make learning fun and hands-on. Starting with concrete models is essential when teaching new skills like multiplication. Taking the time to begin with concrete models will ultimately save us time later and allow students to truly understand these foundational skills, setting them up for future success.

The story and images have been shared with permission from the teacher.

Let’s (Line) Plot Something

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If you are a 3rd-grade teacher looking to engage students with the concept of line plots, you are not alone. The task of transferring data onto a number line may seem straightforward to adults, but for young learners, it can be quite challenging. The 3.MD.B.4 standard, a supporting cluster in the Common Core Standards, plays a substantial role in reinforcing the understanding of fractions and measurement. This means that as students delve into the world of line plots, they are simultaneously immersing themselves in the intricate connections between fractions on number lines, and measurement. It’s a perfect illustration of how math is both messy and beautiful, all intertwined in a way that connects to the real world.

Putting It All Together

Here is an example of what this integration could look like. This MathRep integrates the fractions on a number line and measurement. The information is collected in the upper left quadrant. Students then fill in the information on the line plot. In the upper right quadrant, they can record their mathematical observations. Much like in the previous blog post, the observations serve as a low-floor, high-ceiling activity—one that all students can access and be successful in.

Teacher Set Up

The MathRep method grants teachers flexibility. Its primary aim is to offer consistent learning experiences for students while minimizing the teacher's preparation workload. When implementing this approach, teachers can start by providing all the necessary data in the upper left quadrant and should ensure variation in the data to allow the line plot to start at different points. For example, if a line plot ranges from 0 to 2, it's beneficial to vary it so that it may start at 5 on one occasion and 9 on another. In 3rd grade, students typically work with whole numbers, halves, and fourths. However, the provided data may not always include fourths, so students should adjust the intervals accordingly. As students become proficient in recording data on a line plot, they can then be tasked with measuring objects, recording the data, and accurately placing it on the line plot. No matter where in the process you find yourself, be sure to encourage students to record all mathematical observations in the upper right quadrant.

Outcomes

The outcomes of this MathRep further build students’ number sense, understanding of fractions, measurement, and data interpretation. It also illustrates how math concepts are interconnected and can be taught together instead of isolated. Another simple yet powerful activity. Have you tried it yet? What are your thoughts?

Measurement & Data Excite Me!

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When you think about data, the first thoughts that come to mind might be how dull or uninspiring it can be. But what if there was a way to turn that perception around and make learning about data an exciting journey for elementary school students?

Enter the newest MathRep. This template is designed to engage students while teaching bar graphs and picture graphs to young learners.

In this blog, we will delve into how this new MathReps template is exciting elementary math students. It offers educators a fresh and dynamic method to ignite enthusiasm for data interpretation among their students. Get ready to explore the possibilities and discover how this MathRep can make a real difference in the classroom!

Picture This

This is an example that aligns to 3rd-grade standards: 3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.

You’ll notice that the same information is used to complete each of the graphs. This leaves quadrant 4. It can have questions about how many more and how many less, but why not allow students to observe the data and make their own observations. Leaving it open-ended like this allows all students to be successful. In addition, you will have students making observations that go beyond how many more/less.

The Setup

When first introducing this MathRep activity, it is recommended to provide students with the information located in the center of the paper. Subsequently, they can proceed to create the graphs themselves. Once they have gained proficiency in graph creation, the teacher can fill in the graph and task the students with completing the remaining sections, including the center. For a more advanced approach, the teacher can fill in the ‘Mathematical Observations’ square (quadrant 4), leaving the rest to the students. This adaptable strategy can effectively challenge students at various proficiency levels, guiding them toward a deeper comprehension of the material.

What Will You Do?

The power of this MathRep lies in its ability to enable students to interact with data in multiple ways simultaneously. This allows students the opportunity to make connections between different representations of information, leading to a deeper understanding. Teaching skills in isolation should and has been discontinued according to the Common Core Math Clusters, as math is intertwined with all aspects of learning. This MathRep illustrates these connections, preparing students for future success.

Number Paths for Students

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Primary teachers have been embracing the concept of number paths as a valuable tool to foster fluency and number sense among young learners. A number path is a linear visual representation of numbers, which serves as a versatile aid in helping primary students comprehend and engage with numbers more effectively. As I have been collaborating with a group of primary teachers, it has become evident that they recognize the significance of number paths in facilitating a deeper understanding of numerical concepts. This has led to an increasing demand for additional number paths, reflecting the enthusiastic adoption of this resource within their educational practice.

Why?

After spending a few weeks working with small groups, the teachers quickly realized the potential of this tool for the whole class. They noticed an improvement in their students’ abilities, prompting another teacher to request number paths from 21 to 40 for practicing addition and subtraction with larger numbers. These number paths are completely customizable, so whatever your needs are, you’re covered.

Create

Initially, I created the prototype with cardstock, pipe cleaner/craft sticks, and a bead. That proved not to be the best. So, for the teachers, I laminated the cardstock, switched to a piece of yarn, and changed out the bead style. They work fantastically. I also made sure to take a piece of packing tape to the back to secure the yarn after it was knotted.

Result

It was a huge success! Students can easily manipulate the beads, allowing them to learn about numbers in a hands-on fashion. If you would like a copy of the templates, feel free to download them. They are in the Kinder and 1st-grade MathReps slide decks. I have also created some number path games that I will post about later this week.

How are you using number paths in your primary classrooms, and how are your students responding?

Download the template

MathReps Deliver Results

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For years, I’ve questioned the effectiveness of MathReps and pondered whether the skills transferred to everyday math tasks. Known for overthinking and constantly questioning things, I’ve been intrigued by recent data from teachers showcasing the positive impact of MathReps on their students.

There are numerous reasons to celebrate these results. Firstly, this is a multi-year comparison illustrating a significant improvement. The 2024 year initially showed a much lower proficiency level than the 2023 year. However, it is important to note that their overall gains were greater in 2024. Secondly, the teaching staff remained consistent over these two years; there were no departures or new additions. Remarkably, one particular class saw an astounding increase from 5% to 90% proficiency in just 4 weeks. It’s worth noting that this exceptional progress happened after the teacher conducted the final assessment earlier than planned due to external circumstances. Subsequently, the same group of students took the SBAC IAB for Fractions, which resulted in 14% below the standard and 86% at or above it. This is particularly noteworthy as the class had only been 5% proficient with fractions a few weeks earlier. While the teacher deserves full credit, she acknowledged that regular completion of MathReps played a role in this achievement. Furthermore, the IAB takes the students’ knowledge of the subject matter and requires them to use that knowledge in context. This further supports knowledge transfer is happening. This is just one example of the success that can be achieved through the use of MathReps.

Why do I bring this up? Students are struggling, and teachers are noticing. Too often students come without the basic skills necessary to succeed with the expected standards. I witnessed this many years ago (pre-pandemic) and created the first MathRep. I noticed that my 5th-graders could master the expected skills during the first trimester. Then, the skill expectations shifted to fractions in the second trimester. This is where many of my students lost the skills they acquired in the first trimester. So by the time the third trimester rolled around, all was forgotten from the beginning of the year. Needless to say, this is less than ideal, especially when state testing is right around the corner. So MathReps were born!

So, after several years and an ever-growing library of MathReps I have started hearing from countless teachers from across the county witnessing the same success I am seeing with teachers in my district. I love that they also share stories about their students and how they have gained confidence through MathReps. All this is to say, if you haven’t tried them, you should!

What are MathReps?

MathReps are carefully crafted templates that incorporate interconnecting skills/standards. MathReps allows students to make connections between skills and concepts seamlessly. They are customizable and versatile. The repetition allows students to master the skills without the cognitive load of navigating a new ‘review page’ each day. The format remains the same. The teacher changes the number each day, allowing students to work through a series of skills. The teacher provides immediate feedback by walking around the classroom or checking the work as a whole, leaving zero paperwork for the teacher to deal with at the end of the day.

MathReps explanation video

Where to Start

With this data, it’s a no-brainer. MathReps work. If you’re looking to get started, head on over to MathReps.com and find a template that’s right for you! No need to worry about the cost, it’s FREE! The templates are all teacher-generated. That’s right, made by teachers just like you for students just like yours.