Shelley Gray

Constant Difference: A Key Mental Math Subtraction Strategy For Quick Calculations

Shelley Gray — Mon, 04 May 2026 16:18:19 +0000

Mental Math is important. The ability to calculate in your head is what allows you to know when you get the wrong change back at the coffee shop, estimate the number of nails you need for a building project, or know which grocery item is the better deal. There are many wonderful mental math strategies that are useful in real life and help build our intuition with number, but constant difference has to be one of my favorites.

What is Constant Difference?

When we use constant difference, we add or subtract the same amount to/from both numbers in a subtraction expression. When we do this, the difference remains constant. For example, 9-5=4. But if we add one to both the 9 and 5, we now have 10-6. The difference is still 4.

Or we could add 10 to the minuend and subtrahend, to make 19-15, The difference is still 4.

Why Is Constant Difference Useful?

Imagine being able to solve an expression like 172-98 mentally in just a few seconds! We can easily do this using constant difference. For this example, let’s add 2 to both. Now we have 174-100. We can solve this simply in our heads!

How would you solve this one using constant difference: 386-194?

I solved it by adding 6 to each, to make 392-200.

Building Conceptual Understanding of Constant Difference: How To Teach This Strategy

Conceptual understanding of this strategy is essential. We want our students to be able to connect the conceptual understanding to their procedural understanding to ensure that this strategy is not simply a memorized series of steps, but rather an understanding. Here are some of my favorite ways to model this strategy.

Number lines are an excellent way to show this. In this example, the orange number rod represents the difference, which is 10. That 10 (the difference) is going to stay constant, no matter where we place it on the number line. The picture below shows how we can slide that difference up and down the number line. So 22-12 has the same difference as 23-13, or 24-14, or even 31-21. We can connect this concrete idea to equations, which are more abstract, to help our students develop the understanding behind this strategy.

Another way to think of this is with ages. If we have two people, ages 35 and 12, what is the difference in their ages? It’s 23 years. Next year, on this exact day, they will be 36 and 13. What is the difference in their ages? Still 23 years! There will always be a 23 year difference between them on this day each year.

Another way to model this is through the heights of real people. Try standing two students beside one another and discussing what the difference in their heights is. If they both grow 2 cm, does the difference in their heights change or stay the same? It stays the same! What if only one person grows? The difference in their heights changes. So as long as they both grow the exact same amount the difference in their heights will stay constant.

One last way to model this is with a concrete manipulative like unifix cubes. Make two towers and ask students what the difference in heights is. Add one block to each tower. Now what’s the difference? Take two blocks away from each tower – does the difference stay the same? What if we add a block to only one tower? How does that affect the difference?

Considering Different Options

As with any math strategy, flexibility is key. We don’t ever want our students to think there is only one way to think about this strategy. Let’s consider some different ways to use constant difference for this problem: 162-85. I’ve showed several different options. Which one is most effective for you?

Subtracting a Friendly Number vs Subtracting FROM a Friendly Number

When working with this strategy flexibly, it’s important to consider whether it’s more effective to subtract a friendly number or to subtract FROM a friendly number. Which one gives you the least amount of chance for error? Personally, I prefer to make the subtrahend into the friendly number. To me, it’s easier to subtract a friendly number rather than to subtract from a friendly number. Many students will make the minuend friendly instinctively, but it’s important to have the conversation to bring students’ attention to the fact that there are other ways to think about it that may be more effective.

Using Constant Difference to Reduce Chances of Errors

We can also use constant difference to reduce the chances of errors when regrouping across zeros (if your curriculum requires your students to learn the standard algorithm). In this example, instead of doing the calculations for 1000-346, we can subtract 1 from each to make 999-345. Now no regrouping is required and the chance of making an error is reduced.

This is a great strategy to incorporate into your number talks routine. Don’t have a number talks routine yet? Here’s how you can get started.

See this strategy on Instagram below:

View this post on Instagram

A post shared by Shelley Gray (@shelleygrayteaching)

The post Constant Difference: A Key Mental Math Subtraction Strategy For Quick Calculations appeared first on Shelley Gray.

How the Friendly Number Mental Math Strategy Progresses: Elementary to Middle Years

Shelley Gray — Fri, 18 Apr 2025 14:57:55 +0000

As teachers, we know that strong mental math skills are foundational for developing number sense and math confidence. One strategy that can make a big difference for students is making a friendly number. This simple yet powerful strategy helps students approach calculations with flexibility by adjusting numbers to ones that are easier to work with. The best part is that this strategy can be used from very early years to middle years and beyond. In this post we will look at how this strategy might progress througout grade levels.

First things first, there is an important understanding that students must develop – when we add two (or more) parts, we can move objects from one part to the other without affecting the sum. In young students, this understanding is best developed with manipulatives that that they can physically touch and move around.

Now let’s take a look at this strategy in action, beginning in early elementary and progressing through to middle years math.

Using the Friendly Number Strategy With Ten Frames and Base Ten Blocks

When students are first introduced to this strategy, it’s best to use concrete manipulatives that they can physically move from one part to the other. Even though it may be easy for us to understand moving from one part to another, it can still be too abstract for kids. When we use manipulatives and students can move the actual pieces, it promotes a deeper understanding.

The ten frames below show 7+5. In the next picture, we move 3 from the 5 over to the 7 to make a friendly number – 10. Now we have 10+2.

We can also represent this pictorially on a part-part-whole model or a number line, or in an abstract form with digits and symbols.

We can also use base ten blocks to make this thinking visual. In the example below, we are using this same strategy to solve 14+38. We can physically take 2 ones from the 14 and give them to the 38. Now we have 12+40. Is this easier to compute mentally?

Friendly Number Strategy with Multi-Digit Addition

Let’s take a look at how this might look with a 3-digit plus 3-digit addition problem. Here we have 127+285. There are several different ways that we could approach this:

Take 3 from the 285 and give it to the 127 to make 130+282.
Take 5 from the 127 and give it to the 285 to make 122+290.
Take 15 from the 127 and give it to the 285 to make 112+300

and of course there are many more than this.

Which feels easiest to you? For me, the third option feels the best. Having conversations like this with your class about different ways to use the same strategy is a key to adding flexibility to your number sense routine. Notice that when we are working with numbers in this way, we have a real sense of the magnitude of the numbers – we are not simply working with single digits as we do when working with the traditional algorithm. This understanding is what will lead to stronger number sense.

Friendly Number Strategy with Fractions and Decimals

The part I love most about math strategies is that they are easily transferrable to other areas. This exact same understanding that first graders might use when adding 7+5, can be used by 7th graders when adding 3.2+4.95. How would you approach this problem? Personally, I would take eight tenths from the 4.95 and bring it over to the 3.2 to make a nice friendly number, 4. Then my expression becomes 4+4.15, which is simple to do mentally.

We can use the same approach with fractions with the same denominator.

Helping students make sense of numbers—rather than just memorize procedures—is one of the most meaningful parts of teaching math. Making a friendly number equips students with a flexible, intuitive approach that supports accuracy, efficiency, flexibility, and confidence. By introducing this strategy early and revisiting it in increasingly sophisticated ways across the grade levels, we give our students tools they can carry with them well beyond the classroom. Whether they’re adding within 20 or solving multi-digit equations, friendly numbers can be a familiar and empowering part of their mathematical thinking.

See a video about this strategy on Instagram HERE.

The post How the Friendly Number Mental Math Strategy Progresses: Elementary to Middle Years appeared first on Shelley Gray.

Three Simple Shifts to Increase Fact Fact Fluency in Your Classroom

Shelley Gray — Sun, 13 Apr 2025 17:08:12 +0000

What do you think of when you think about fact fluency? Do you think about your students who can get answers quickly? Do you think about the ones who are able to answer all the questions correctly on a test? Fact fluency is about much more than speed and accuracy. Fluency also includes flexibility and appropriate strategy use (nctm.org).

Flexibility tends to be the most difficult part of fact fluency to teach, and often feels very unnatural. Teaching for flexible thinking means that teachers must make a mindset shift from teaching for answers to teaching for thinking. However, adding the opportunity for students to think flexibly and critically will forever transform the way your students learn and understand math.

If you are ready to focus on flexible thinking in your classroom this year, here are some simple shifts you can make to get started.

Make Connections Between Math Facts

Facts are connected, but too often they are taught as isolated pieces of information. We can always use what we know in order to solve facts that we don’t know. This is the idea behind near doubles facts, for example. To solve 6+7, we could think, “6+6=12 and 1 more makes 13.”

The fact webs below show examples of how we can encourage students to look for connections when multiplying or adding. This is a great thinking task for small groups! It’s empowering for students to know they are capable of solving anything. See this post on Instagram here.

Another way to reinforce the connectedness of math facts is through number strings. A number string is a string of related math problems. There are various ways to use number strings, but here is one of my favorite ways, where the number string is used to reinforce a strategy or way of thinking.

To create a number string, it’s essential to first consider the overall goal. Let’s suppose I’m teaching a strategy to encourage students to make friendly numbers then add the rest. I want students to recognize that they can solve 67+15 by making the 67 into a 70, then adding 12 more, so I might use a number string that looks something like the one below. I will begin with simple problems that encourage students to make a friendly number (like 10 or 20) and add the rest. Students will be slowly scaffolded until I get to the final problem, 67+15.

9+6 (goal is for students to realize they can make a 10 and add 5 more)

19+6 (goal is for students to realize they can make a 20 and add 5 more)

18+5 (goal is for students to realize they can make a 20 and add 3 more)

38+14 (goal is for students to realize they can make a 40 and add 12 more)

67+15 (final goal is for students to realize they can make a 70 and add 12 more)

Modify the Way You Ask Questions

Often in math, we communicate the message to our students that math is all about getting answers, even if this is not what we are trying to convey.

Consider these two scenarios.

Scenario 1: I am teaching a lesson on multiplication by 6. I pose a question to students, wait for them to think about it, then ask for answers. Students begin by solving 6×8. Then they solve 6×5. Each time they answer a question, I move on to the next one.

In this scenario, I reinforce the idea that the goal is to get answers, since I stop the conversation after a correct answer has been given.

Scenario 2: I am teaching a lesson on multiplication by 6. I pose the problem 6×8 to students and give everyone thinking time. I ask for an answer, to which a student replies, “48.” I ask him how he thought about it and he says, “I just knew it.”

I ask for other student volunteers to explain how they thought about it. One student says, “I knew that 5×8 is 40, so I added another group of 8 to make 48.”

Another student chimes in, “I know that 3×8 is 24, so I doubled it and got 48.”

As I listen to all of these ways of thinking, I record them on the whiteboard to inspire other students.

This simple act of asking for ways of thinking now communicates to students that our answers are not as important as the way we think.

This is the entire premise behind number talks. If the idea of implementing number talks feels intimidating, just remember it does not have to be a huge overhaul of how you do things in your classroom. It can be as simple as taking five minutes to lead a discussion like the one above, and is transformative for fact fluency.

Make Computation Concrete and Visual

In my opinion, there is nothing more important in math than making it visual for students. When we teach only abstractly (digits and symbols with no concrete manipulatives or visual models), we leave gaps in understanding for many of our students. Additionally, visuals are essential even for our students who do catch on quickly to abstract explanations because they force them to see things differently.

Let’s go back to the example above: 67+15. If we want students to use the strategy of getting to a friendly number then adding the rest, we can model this with manipulatives. The pictures below show what this might look like. First, we make 67 and 15 with base 10 blocks. Then we show how we can move 3 from the 15 over to the 67. Now we have 70 and 12 – typically an easier problem to solve mentally than 67+15.

As seen in the fourth picture, we can make connections between this concrete representation and other pictorial and abstract representations, showing students how they all connect.

This is a powerful way to help students build their understanding, and it helps ensure that students don’t develop gaps in their understanding that will make math more difficult for them in the future.

Conclusion

Sometimes we get stuck, thinking that shifting the way we do things takes too much time, requires a complete overhaul of the way we teach, or is too difficult. But by making a few simple tweaks to our teaching, we can change the way our students see math and approach problems. Try these ideas this week and see how it goes!

The post Three Simple Shifts to Increase Fact Fact Fluency in Your Classroom appeared first on Shelley Gray.

Using Number Tiles to Build Number Sense, Understanding, and Metacognition

Shelley Gray — Sun, 29 Sep 2024 22:59:08 +0000

We know that flexibility is an essential part of fluency, but it can be hard to wrap our heads around how to teach it. Number tiles can help students actually SEE the computation. Students can manipulate the numbers in different ways to see relationships between operations and think flexibly. Math confidence increases because students see that they can break problems down into manageable parts, making it possible to solve any problem.

Using Number Tiles for Basic Multiplication

We know that we want students to build an understanding of multiplication and division rather than rely on memorization. But it can be difficult to know how to do this.

For the x3 facts, we can use the “double plus one more group” strategy. For example, for 3×8, we can think of the double of 8 (16) plus one more group of 8 to make 24. Some kids can understand this when we explain it verbally, but we can reach more kids, and allow them to build deeper understanding when we provide concrete practice with the concept. Here’s an example of what this might look like with number tiles.

Building these equations conceptually gives students a deeper understanding of what they are actually doing when they are computing using this strategy.

GET YOUR OWN SET OF NUMBER TILES HERE.

The x9 facts are another set that we can easily relate to something kids already know – in this case, the x10 facts. This can be conceptualized easily with number tiles. We can start with a ten, and then just cover up a group to build the understanding of the relationship between the x10 and x9 facts. This understanding easily transfers to muliplying bigger numbers like 19.

Once students work concretely with this concept, they’ll build the understanding that to multiply by 8, they could start with x10 and then subtract TWO groups. Or when multiplying x18, they can start with x20 and subtract two groups.

We can also help kids build flexibility by seeing that they can break basic facts up into smaller parts. Of course, automaticity is the goal for basic facts, but understanding is the way we get there. If a fact is forgotten there is always a way to figure it out. Here’s an example of how we could use 5 groups of 9 and 2 groups of 9 to figure out 7 groups of 9.

Using Number Tiles for Multi-Digit Multiplication and Division

We can also use number tiles to show students that they are capable of solving big problems in their heads. On the left, we see how we could solve 28×4 by breaking it up into 10 fours, another 10 fours, and then 8 more fours. On the right, we could think about 34×4 as 30×4 and 4×4. There are many ways to think about problems. This is a key piece to building number flexibilty.

These manipulatives also work beautifully for division. Not only do they make division visual, but the relationship between multiplication and division becomes clear.

What other concepts could you use number tiles for? I’d love to hear in the comments below!

GET YOUR OWN NUMBER TILES HERE

to help you teach for deep understanding and flexibility!

Click Here

The post Using Number Tiles to Build Number Sense, Understanding, and Metacognition appeared first on Shelley Gray.

Five Simple Activities to Help Your Students Build Fraction Sense

Shelley Gray — Wed, 28 Aug 2024 16:58:28 +0000

Do your students possess fraction sense? Fraction sense involves a deep, conceptual understanding of fractions as numbers. This goes far beyond simply shading parts of a whole or identifying numerators and denominators. It involves true, flexible understanding.

Errors in fraction calculations can often be traced back to poor fraction sense. For example, think of a student who might add three-fourths and three-fourths and get six-eighths.

This student is likely getting mixed up with a rule that he learned for adding the numerators only when adding two fractions with the same denominator.

But we don’t want our students to rely on a meaningless rule or procedure. We want them to actually understand fractions.

So how do we ensure that our students develop this fraction sense rather than having to rely on meaningless rules?

Here are five simple activities you can incorporate into your classroom to help your students build their fraction sense.

Use Manipulatives and Visual Models

Fractions are an abstract concept for many kids, especially since they may not have much real-life experience with fractions yet. But concrete manipulatives can be a game-changer, making fractions easy to visualize and understand.

I love the circular foam fraction pieces for students to explore concepts like fraction addition/subtraction and equivalent fractions. They also make improper and mixed fractions simple to visualize.

Another amazing manipulative for fractions is Cuisinaire rods. Look at the examples below for just a couple ways these can be used.

Another visual that I love for fractions is bar models. Bar models can be used to make simple calculations more conceptual, or as an effective problem-solving tool. See this post on Instagram for more information on how to use them.

Paper Folding

Simple paper-folding activities can take fraction concepts from abstract to concrete for your students. There is some prep involved to get these paper pieces ready for students, but the actual act of your students creating the fraction pieces will be well worth it.

To prepare for this activity each student will need several different square pieces of paper. I like to have different colors. In this picture, I used black as one whole. Then students can fold the blue piece in half. Each of these pieces represents one-half. The pink piece was folded in half, and then in half again. These represent one-fourth.

The act of folding and cutting is powerful, because it helps reinforce that one-fourth is half of one-half, one-eighth is half of one-fourth, etc.

BUT! Here’s the important part. You may be tempted to have students label the pieces as 1, 1/2, 1/4, etc. However, using them unlabelled will allow for far more deep thinking. For example, consider how these tasks might look:

If the black piece represents 4, what does the blue piece represent? How about the pink piece?
Show one-half in three different ways.
How does the orange piece relate to the black piece?

As you can see, using these pieces unlabelled will allow for some tremendous opportunities to rationalize and think deeply about fractions.

Conduct Whole Class or Small Group Fraction Talks

Simply talking about fractions can have a profound impact on students’ understanding. This can be done in a whole class or small group setting and requires no prep or special materials.

For example, you could simply draw an image on the board, like the one here, and ask students what they notice about the drawing.

If students are stuck for ideas, prompts like these could help get the conversation started:

What equations could we write to represent this drawing?
How are these circles the same and different?

If you own my Math Conversations number talk resources, you’ll also find some great fraction talk slides in there.

Think of all the possiblities that students might come up with for this question. Maybe it represents 8 groups of one-fourth.

Maybe it represents two wholes and 8 groups of three-fourths.

When we start looking at how students might group the circles to add them together, there are loads of different options.

Count By Fractions

Just like we can count by 1s, 2s, or 5s, we can count by fractions! This helps students see fractions as numbers.

I’ve made a few videos that you can use in your classroom to help your students make connections between the fractions and the visual representations. See all available videos here.

Make Fractions Meaningful and Relevant to Real Life

There is no better way to encourage true understanding than to relate fractions to how they are actually used in real life. Just think of all the amazing fraction skills we learn by baking! Here are just a few other examples of real-life experiences we can use to help our students understand fractions:

Sports statistics
Discounts at a store or restaurant
Time (half of an hour, etc.)

If you’d like a ready-made math project to help your students understand how fractions are used in real life, here are a couple of my favorites.

Or see all real-life math projects here.

The big idea here is giving our students the opportunity to actually SEE the fractions. Visuals and manipulatives make fractions so much easier to understand, and if we are teaching without them, we are likely creating gaps in understanding.

The post Five Simple Activities to Help Your Students Build Fraction Sense appeared first on Shelley Gray.

Using Bar Models to Effectively Solve Math Word Problems

Shelley Gray — Mon, 04 Mar 2024 14:08:11 +0000

Have you tried Model Drawing in your classroom? I recently read the book 8 Step Model Drawing by Bob Hogan and Char Forsten. I have always been a fan of bar models, but this book was an eye opener as to just how effective they are as a problem solving tool. The authors state that about 80% of word problems that students encounter can be effectively solved using models like these.

My mind has been spinning with all of the other potential uses for bar models beyond basic problem-solving.

As I continued to read more about model drawing, it became clear to me that they can be connected beautifully to so many different math manipulatives. I think every teacher should know about model drawing, so I decided to put together a 10-part video series with ideas you can take directly back to your classroom. These videos will incorporate both math manipulatives and model drawing for different problems including fractions, multiplication, division, and more! As the videos are posted on Instagram, TikTok, Facebook, and YouTube, I’ll also add links here so you can find them easily.

Or sign up here to have the link delivered to your inbox.

I hope you enjoy this video series! If you’d like to know even more about model drawing, be sure to check out Step-by-Step Model Drawing as well as 8-Step Model Drawing.

Day 1: Using Models and Manipulatives to Solve Fraction Problems

Bar models are naturally an incredible visual for fraction problems. Fractions can often be such an abstract concept for our students, and models can transform them to easy-to-understand.

View this video below or on these social media outlets:

Instagram

Facebook

Day 2: Using Models and Manipulatives to Solve (More) Fraction Problems

I decided to focus on fractions for 2 out of the 10 videos in this series, because models are so effective for visualizing fraction problems. The problem below is one of my favorites because it makes a 2-part, more complex problem so simple to visualize.

View this video below or on these social media outlets:

Instagram

Facebook

Day 3: Using Models and Manipulatives to Make Addition and Subtraction Word Problems Visual

Models are very well-suited to addition and subtraction problems. They not only make the problem easier to understand, but reinforce the inverse relationships between addition and subtraction. This video shows two different problems and how bar models can be used.

**Note** I neglected to make the parts of the bar different sizes in the second problem. This is my mistake, and I will be re-creating this video soon. In the meantime, please note that the pieces should be more to scale to show the differences in the magnitude of the numbers.**

View this video below or on these social media outlets:

Instagram

Day 4: Using Models and Manipulatives to Build Multiplicative Reasoning

Models are a must-use tool to develop multiplicative reasoning. They beautifully illustrate the relationship between multiplication and repeated addition, and can take complex multiplication problems from difficult to easy-to-imagine.

View this video on these social media outlets:

Instagram

Facebook

Day 5: Using Models to Make Division Problems Visual

Bar models for division are pretty straightforward, and we might neglect to use them because they don’t feel necessary. However, remember that modeling simple problems is important so that students feel comfortable with models as a tool for more complex problems. Additionally, I love how division models effectively connect multiplication and division, as well as build a foundation for fraction concepts.