Basic Operations Archives - 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.

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.

Looking for the Perfect Christmas Center Activity? Try These Christmas Math Logic Puzzles!

Shelley Gray — Mon, 28 Nov 2022 20:28:52 +0000

Logic puzzles are an amazing way to challenge your students to think about basic operations in a different way. So often we give students worksheet after worksheet where they must simply solve problems to find answers. But these Christmas-themed logic puzzles will challenge them to think in a new way. You can expect to hear amazing math chatter in your classroom as students reason, strategize, and think critically to find the value of each symbol.

How To Complete These Christmas Logic Problem Puzzles

To complete the puzzle, students will quickly catch on that they must begin with symbols that they KNOW for certain. For example, in the puzzle below, we begin with solving for the Santas, since we know that 5×5=25. Now that we know the Santa equals 5, we can solve the second row. The tree equals 15, since 15+5=20. Lastly, we can solve the bottom row. Since we know the tree equals 15, we can conclude that the two lights together must be 60, so each light bulb equals 30.

Don’t Be Fooled – These Logic Puzzles are Tricky!

When deciding which puzzles to use with your students, remember that if your students are not used to doing these types of puzzles, they will very likely find them challenging. Your grade 3-4 students might be sufficiently challenged with the numbers to 20 puzzles, even though they work with numbers in the thousands in other math activities. I recommend making several levels of puzzles available to your students and allowing them to choose the ones that challenge their brains “just enough.” I typically find that students are eager to work to their full potential and will generally choose the ones that are best suited to their abilities.

Free Christmas Logic Problem Puzzles

To get you started with these puzzles in your classroom, I’d first like to give you a free sample so that you can take them to your classroom and see how students respond before committing to purchasing an entire package or the full multi-grade bundle. In the free sample, you’ll receive one puzzle from each package that is available. Download your free sample below!

Full Packages and Bundle

Each package of Christmas Logic Problems comes with 20 puzzles. Simply laminate and cut to ensure they last for years to come. Make copies of the recording sheets and you’ve got one of your Christmas math centers completely taken care of!

The post Looking for the Perfect Christmas Center Activity? Try These Christmas Math Logic Puzzles! appeared first on Shelley Gray.

Three Easy Ways to Transform Scary Mad Minute Drills into Something Purposeful

Shelley Gray — Sat, 06 Aug 2022 15:23:01 +0000

Yes. Learners need to know their math facts. Why? As they get older and mathematics becomes more rich and complex, working memory needs to be freed up for problem-solving. Retrieval of these facts needs to be automatic. This is just like a reader recognizing sight words as a part of literacy development. Unfortunately, curricula expectations and Common Core standards all over Canada and the United States prescribe that learners be able to “recall” or “know from memory” certain math facts by certain grades and the emphasis on timing (read: speed) and pacing becomes too heavy.

Remember Mad Minutes? When I was in Grade 4 we had one minute to recall 50 multiplication facts. My teacher actually coached me on how to improve my speed by alternating the direction I answered the questions in so I wouldn’t waste time resetting my arm from right to left! While I did commit my facts to memory that year, it wasn’t until I started teaching and tutoring that I realized how problematic drills like these are and how they reinforce the narrative that fast = best instead of the importance of fact fluency.

In his 2015 TEDTalk on mastery, Sal Khan uses a great analogy to highlight the importance of self-paced mastery in learning. He describes how silly it would be to build the first and second floors of a home if the foundation is still wet and not to code and then argues that it is equally detrimental when we as educators “artificially constrain” the time learners have to master something – especially when we feel pressure to (or are mandated to) move on before they are ready.

Given the tumultuous landscape that education has been (come on, all life has been) since the COVID-19 pandemic, and with many people going on about learning loss, I wish we could let go of our grip on timing. I know, I know… easier said than done and the district/administrator/state tests are coming for you by the end of the next school year. But this brings me to my main point about drilling math facts if you are going to drill at all: timing should not be the end goal. When it comes to memorization, it’s not about when, but how (Fosnot & Dolk, 2001).

Disclaimer: you shouldn’t be thinking about drills or rote practice with math facts unless you have already been teaching for understanding.

That being said, if you are looking for some simple repetitive practice, here are three simple shifts you can make to modify those scary mad minute drills into something more purposeful:

DRILL SELECTIVELY; NOT EVERYTHING ALL AT ONCE

By grouping and sequencing instruction and practice around small clusters of related facts, rules, or properties, we create more opportunities for learners to make connections between operations and see patterns and relationships between numbers. An example of this would be a drill solely including questions related to multiples of 2, 5, and 10.

BASE ACCURACY ON THE ATTEMPTED QUESTIONS AND USE A STOPWATCH, NOT A TIMER

If the learner attempts 30 out of 50 questions, score their drill out of 30 not 50. Also, have the clock run up and not down. If they answered 17/30 correctly in 6 minutes and 28 seconds, co-create a SMART goal for the next drill that focuses on accuracy first. When they can correctly answer all of the questions, then set a goal to reduce the time. Celebrate progress, personal bests and let the learner record their score.

USE ERROR FEEDBACK TO INCREASE FLUENCY

Look at the questions that the learner answered incorrectly or didn’t attempt. Look for instructional opportunities. With the clock turned off, review appropriate strategies they could use next time and bring it back to sturdy conceptual understanding using concrete manipulatives and visualization of special relationships.

So, to drill or not to drill? How do you boost fluency and work towards automaticity in your classroom?

References

Cholmsky, P. (2011). From acquisition to automaticity: The Reflex solution for math fact mastery. Retrieved July 28, 2022 from https://www.reflexmath.com/images/Reflex_White_Paper.pdf

Fosnot, C. T. & Dolk, M. (2001). Young Mathematicians at Work: Constructing Number Sense, Addition and Subtraction. Portsmouth, NH: Heinemann.

How to Teach the Doubles Facts So Your Students Actually Understand

July 15, 2022 No Comments

What are the doubles facts in math? How do you teach them? Why are the doubles addition facts important? By the time you finish reading

Fun Multiplication Games for 3rd and 4th Grade (help them master the facts)

July 12, 2022 No Comments

How can you make learning multiplication fun while also allowing your students to master the multiplication facts? WITH GAMES! Multiplication games are an effective way

How to Teach Math Effectively Using the Concrete Representational Abstract Model

July 16, 2020 7 Comments

WHAT IS THE CONCRETE REPRESENTATIONAL ABSTRACT MODEL? The CRA Model is an instructional approach for teaching math. It consists of three phases: Concrete Representational Abstract

The post Three Easy Ways to Transform Scary Mad Minute Drills into Something Purposeful appeared first on Shelley Gray.

How to Teach the Doubles Facts So Your Students Actually Understand

Shelley Gray — Fri, 15 Jul 2022 17:23:45 +0000

What are the doubles facts in math? How do you teach them? Why are the doubles addition facts important? By the time you finish reading this article, you’ll have an excellent understanding of all things “doubles” and also leave with some free resources and fun games to get you started teaching this important math concept in your classroom!

WHAT ARE THE DOUBLES FACTS?

Doubles refers to adding the same number twice, for example 5+5. Generally when we refer to the doubles addition facts in elementary math, we are referring to facts from 1+1 to about 12+12.

The doubles are not so much a math strategy as a set of facts that students should know. When kids know the doubles, they are able to use them as a building block for other mental math strategies such as near doubles. But that’s another article.

THE DOUBLES FACTS IN REAL LIFE

Doubles are generally an easy set of facts for students to remember because they are all around us in real life. Think of the wheels on a car (2+2), the legs on a spider (4+4), fingers and toes (5+5), or the eggs in a carton (6+6).

REMEMBERING THE DOUBLES FACTS

Your students will have an easier time with more advanced mental math strategies and even with multiplication by 2 if they just know the doubles math facts. But that doesn’t mean you have to resort to memorization only.

It is important for students to work with math facts in a hands-on way as they build their understanding.

One of the best ways to work with the doubles facts is with ten frames. Have your students build the fact on the ten frames with different colors.

You can also have students build doubles using a rekenrek, or even simple colored cubes.

FREE Doubles Fact Booklets

to help your students master the doubles addition facts

Get it for free here.

LOOKING BEYOND THE DOUBLES FACTS

To understand why the doubles math facts are important, it helps to have a vision for where you are headed. Here are some other concepts that rely on the doubles.

NEAR DOUBLES

Near doubles are facts like 5+6, 10+11, or 3+4. For these facts, we can use what we know about doubles to solve a new fact. For example, for 3+4, we can think, “I know that 3+3=6, so 3+4 is just one more!”

MULTIPLICATION BY 2

In third grade when students learn to multiply, the 2 times table is generally the first set of multiplication facts they learn. We teach these facts by making a connection to the addition doubles. For example, to solve 2×3, we can think, “2 groups of 3” or 3+3.

DOUBLES FACT PRACTICE

There are lots of simple ways to practice the addition doubles, but some of my favorite involve games.

The first one is called Dice Doubles. This takes 1 minute and is something I used to do with my classes all the time to increase automaticity. My favorite part of this game is that each student is only competing against themselves, trying to increase their own score each time.

See how to play Dice Doubles HERE.

The second game is another dice game called Doubles Race to 100. It’s a fun partner game that is sure to raise the energy level in your classroom!

See how to play Doubles Race to 100 HERE.

FREE DOUBLES FACT BOOKLETS

These booklets will be fun for your students to complete. Not only do they focus on the addition doubles, but also on how we can make different connections using 1 more, 1 less, 10 more, and repeated doubling!

Get this free booklet here.

Not Sure How To Teach Rounding Numbers? Make it hands-on! (Try This!)

“Four or less, let it rest. Five or more, add one more.” Right? Wrong!! Do you find that rounding is confusing for your students? Do

Teaching the Counting On Addition Strategy? Don’t Forget to Do This.

Counting On is a beginning mental math addition strategy. It is generally taught as an introductory mental math strategy and is usually simple for students

Effective Mental Math Addition Strategies to Master the Addition Facts

So you’re ready to implement mental math addition strategies this year, but aren’t sure where to start! All the different ways to teach addition facts

The post How to Teach the Doubles Facts So Your Students Actually Understand appeared first on Shelley Gray.

The Best Way to Teach Multiplication Facts: 12 Steps to True Understanding

Shelley Gray — Tue, 12 Jul 2022 20:46:47 +0000

Teaching multiplication is no easy task! Encouraging students to memorize the facts only allows a small percentage of them to be successful. Even then, those good memorizers may not truly understand what multiplication means. Rather than teaching multiplication for memorization, we want to teach it for understanding. Once understanding is built, it’s easier to become automatic with the facts!

If you are teaching multiplication for the first time, or if you’ve taught it before but have students who don’t seem to be grasping it, here are 12 simple steps to teaching multiplication for true understanding.

STEP 1: SUBITIZE TO ENCOURAGE MULTIPLICATIVE REASONING

We can help our students develop multiplicative reasoning long before multiplication is even introduced. Subitizing with ten frames, tally marks, dot patterns, or base ten blocks can help students get a sense of what equal groups looks like. For example, the dice pattern below shows 8 equal groups of 3. We can figure out the total number of dots by adding 3+3+3+3+3+3+3+3 but this is also an opportunity to introduce language such as 8 groups of 3.

See more about subitizing for multiplication HERE or get these Multiplication Subitizing Cards HERE.

STEP 2: CONNECT MULTIPLICATION TO REPEATED ADDITION

Multiplication is a shortcut for addition, and we want our students to be able to see that connection. When we first teach multiplication, students must know that this is not a brand new foreign concept. It is simply an extension of what they already know. This will help multiplication seem less intimidating.

A problem-solving context is an effective way to help students see these connections. The problem below can be solved using addition, but we can also introduce multiplication as a potential method to solve it. Rather than introduce the times symbol immediately, we’ll start with equal groups.

Multiplication should be connected to repeated addition in a way that makes sense for students.

STEP 3: EQUAL GROUPS

Students must understand multiplication as equal groups. If your students have already built some multiplicative reasoning (See Step 1), this will be easier to understand. Provide lots of opportunity for students to work with equal groups in a problem-solving context. This will ensure that these equal groups are a more concrete concept, rather than too abstract.

FREE MULTIPLICATION ACTIVITIES

to help your students master the multiplication facts

Grab them here!

STEP 4: SKIP COUNT

Skip counting is one of the beginning strategies for multiplication. We can skip count using equal groups and then transition to skip counting on a number line.

STEP 5: USE MANIPULATIVES

This step should be used through all of your multiplication teaching, but is so often missed when teachers go directly to multiplication worksheets involving pencils and paper only. Manipulatives provide rich experiences for your students to build their understanding of multiplication, and will ultimately allow more of your students to be successful. Any object can be used for students to create equal groups, such as coins, unifix cubes, base ten blocks, or any other counters you may have in your classroom.

One of my absolute favorite ways to have students represent multiplication is using a rekenrek, which you can read more about HERE.

STEP 6: USE ARRAYS

Arrays are one of the most under-utilized tools for teaching multiplication.

We can use arrays to show students how different facts are connected. For example, the image below shows how 3×4 and 4×4 are connected. 4×4 is simply one more group of 4! Arrays can be built with physical objects (concrete) and then also drawn to represent that physical formation (representational).

STEP 7: TEACH THE COMMUTATIVE PROPERTY

The number of multiplication facts students need to learn can be overwhelming, but it helps to understand the commutative property! When students understand that the product of 3×4 is the same as 4×3, it considerably decreases the number of facts we need to remember! Arrays are one of the best ways to show students the commutative property. Here we can see that when this 3 by 4 array is rotated, it becomes a 4 by 3 array!

STEP 8: MAKE MULTIPLICATION RELEVANT

Multiplication is not one of those concepts that students are never going to use – rather, multiplication will be used all the time in real life. By incorporating real-life scenarios, we can make multiplication relevant and more interesting to students. Real-world problem solving also helps students develop a more conceptual understanding because they can visualize what the multiplication problem really means. How can we do this? We can introduce problems, rather than simple expressions. We can talk about what the problem means. We can also use project-based learning that helps students make those real-life connections.

STEP 9: HELP STUDENTS MAKE CONNECTIONS

Learning multiplication is so much easier when students realize the facts are connected and related to one another. 4×6 and 8×6 do not have to be understood as completely unrelated facts. Rather, 8×6 is simply the double of 4×6! This becomes apparent when we use pictures or visual models. Look at the arrays below. This makes it clear that if we double 4 groups of 6 (24) then we get 8 groups of 6 (48).

These connections are EMPOWERING for students, because it means that they can literally solve anything!

STEP 10: TEACH THE FACTS IN A STRATEGIC ORDER

When I first started teaching multiplication, I taught the facts in regular numerical order. Little did I know, this was completely WRONG! I was making multiplication more difficult than it had to be for my students. When you teach in a strategic order that makes practical sense, it helps students make connections between sets of facts. I recommend the following order of teaching:

x10

square facts (ex. 3×3, 4×4, 5×5, etc)

STEP 11: TALK ABOUT MULTIPLICATION!

It’s important for students to see that there are many different ways to think about and solve problems. Often math is thought of as a black or white, right or wrong subject. But it can be fascinating to see that it’s actually a flexible subject that allows us to think in different ways! One of the best ways to give your students the opportunity to practice flexibility is by having them participate in number talks or math talks.

Through listening and sharing, students are exposed to different ways of thinking, strategizing, and reasoning. This is powerful.

Number talks can feel intimidating for teachers because can feel like you are not prepared enough. But that’s the beauty of it. The role of the teacher is to listen – listen to how students think, listen to how they explain, and listen to how they strategize.

We can learn so much by simply listening.

Here’s an example of a math talk you might have with your students. Think of all the opportunities for rich discussion with this slide. We can see how we double and then double again. We can see how 10×4 is half of 20×4. We can see how 20×4 is 4 times greater than 5×4. We can see that 20×4 has the same product as 10×8.

To get started with a math talk, simply draw a multiplication picture on the board and see where the conversation takes you! Or find out more about these Math Conversations slides, ready-made for the entire year HERE.

STEP 12: REINFORCE MULTIPLICATION ALL YEAR LONG

Multiplication is not just a “unit” that you teach and then move on to other things. Multiplication is an essential operation that is important to your students’ success in future grades. It is highly relevant to their real life and will be used in any job they have. Because of its importance to your students’ future success, multiplication (and the other basic operations) must be reinforced all year long.

This can be done by having a Math Facts center during your daily station time where students work on math fact understanding and fluency.

Here are some free math activities for your math facts center to get you started.

Regular reinforcement can also be done by playing games to practice math fact recall and by spiralling the curriculum and returning to those key concepts repeatedly.

I hope this article helps to empower you to teach multiplication for true understanding, rather than for memorization only. Although it may seem complex, you make a life-changing difference for your students by making small changes and beginning to implement the suggestions above.

If you’d like a completely done-for-you system for teaching multiplication, check out The Multiplication Station, used by tens of thousands of teachers. Or grab a free sample of it here!

Fun Multiplication Games for 3rd and 4th Grade (help them master the facts)

How can you make learning multiplication fun while also allowing your students to master the multiplication facts? WITH GAMES! Multiplication games are an effective way

Here’s A Fun Multiplication Math Game Your Students Will Love

Once your students learn the multiplication facts, it’s important to continue to reinforce them. But this can be easier said than done when you have

What’s the Multiplication Area Model and How Do You Teach it? Comprehensive Guide

The area model is a strategy for multi-digit multiplication, allowing students to build conceptual understanding before transitioning to the traditional algorithm. But there is more

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Four Powerful Ways to Visualize Division (so it makes sense)

Shelley Gray — Tue, 03 May 2022 11:58:19 +0000

Understanding what division really means is essential to being able to divide accurately and efficiently. Whether you are teaching basic or multi-digit division, using visual models helps students understand it on a deeper level to build their conceptual understanding.

Here are four models that will help your students build their understanding.

Visualizing Division With Arrays

Arrays are an essential visual for both multiplication and division. When we teach multiplication, we can show students how an array represents a number of equal groups, and the product is the total number of objects in the array.

When we teach division, we can show students how an array represents a total amount of objects, and we are dividing them into equal groups, or groups of a certain number.

Relate to Multiplication

Arrays can help students see how multiplying and dividing are closely related. Now, after using the arrays to visualize multiplication and division, we can relate it to fact families. This transitions to showing students how they can use multiplication to solve or check division problems.

Find the activity sheets pictured above in The Division Station: a Self Paced Station for Division Fluency

Equal Groups Using Manipulatives or Drawings

Making equal groups works well to model how a set of objects can be divided equally. As students work with larger numbers, they can use this same strategy – making equal groups with tens or hundreds instead of only using ones.

Bar Models for Division

Bar models can be used as a transition from more concrete, hands-on experiences to a more abstract representation. Bar models will continue to be useful as students progress to dividing decimals and fractions in older grades.

Division as Repeated Subtraction on a Number Line

Repeated subtraction shows the opposite of repeated addition as multiplication. We can use number lines to make this repeated subtraction more visual.

When students learn simple repeated subtraction, strategies such as partial quotients and the standard algorithm make more sense, as we have already built this understanding of how subtracting and dividing are related.

Remember, our main goal any operation is to build understanding, not build human calculators. Although speed and accuracy is still a goal, we have to build the understanding first.

FREE MULTIPLICATION ACTIVITIES

to help your students master the multiplication facts

Get them here!

The post Four Powerful Ways to Visualize Division (so it makes sense) appeared first on Shelley Gray.

Three Ways to Use Ten Frames for Upper Elementary {Powerful!}

Shelley Gray — Wed, 06 Apr 2022 17:00:07 +0000

Ten frames are a staple in primary classrooms. These two-by-five rectangles can be used to help young students develop number sense within ten. But their uses extend far beyond basic addition and subtraction! Here are three ways that upper elementary students can benefit from using ten frames in the math classroom.

Using Ten Frames to Reinforce the Base 10 System

In our base ten number system, each place value is ten times greater than the place value to its right. Ten frames can be used to model this abstract math concept.

For this activity, you will need a ten frame and place value tokens. If you don’t have place value tokens, you can easily have students make them. For the ones in the picture below, I used a dry erase marker to write on the ten frame dots that came with this set.

Begin by modeling a digit in the ones place. For example, to model the number 3, a ones place token would be placed in three of the boxes on the ten frame.

To make a number ten times greater, exchange the ones place tokens for the tens place tokens. By making the place value ten times greater, the new number represented is 30.

Continue by exchanging the tens place tokens with hundreds place tokens (creating 300), and so on.

Using Ten Frames to Model Multiplication and Division

While ten frames are most commonly used to teach addition and subtraction, they are also incredibly useful for multiplication and division!

For this activity, you will need a ten frame. You can also use either a dry erase marker, counters, or dice. You will be modeling that multiplication is repeated addition- or adding up equal groups. Similarly, division is repeated subtraction, or splitting a number into equal groups.

For multiplication, model a problem such as 4 x 5. This can be expressed as “four groups of 5.” Have students model this expression on their ten frames. Ask students to develop strategies for finding the product. This could include skip counting as they point to each box on the frame. They may also be able to visualize other strategies, such as adding two groups of five plus two groups of five.

For division, students begin with the total number of counters. For example, when working with the expression 20 ÷ 5, students will each get 20 counters. Now, this expression can be interpreted as 20 divided evenly among 5 groups, or 20 divided into groups of 5. Students can explore both these interpretations on their tens frame to find the quotient.

Looking for ways to incorporate real-life into your math class? Real -life math projects show your students how math is used in the real world, incorporating relevant, interesting topics! Get your kids hooked on math projects with a FREE Camping Trip Math Project HERE!

Developing Computation Strategies

As students get older, they encounter longer and more complex mathematical situations. Developing computational fluency with addition and subtraction is more important than ever. When students struggle with computation errors it can be useful to get back to the basics and spend some time “seeing the math” inside a ten frame.

Designate each addend as a separate color. In this example, you see that 7 is green, while 5 is blue. Have the student model each addend in a separate ten frame. Then, use the ten frames to introduce new strategies for addition.

For example, if the student is still counting on to add, try a make ten strategy. Take three blue counters from 5 and use them to fill the frame that shows 7. Now, instead of adding 7 and 5, we are adding 10 and 2.

We can extend this same idea to an expression like 70+50 by using our place value tokens again. In the first picture, we see 70+50 modelled with each addend in a separate ten frame. In the second picture, we took 30 from the 50 to make 100. Now we are working with an easier expression – 100+20.

Another strategy that can be modeled with ten frames is creating doubles. By moving one green counter to the second ten frame, the problem changes from 70 + 50 to 60 + 60. With repeated exposure to these manipulatives, students will internalize these flexible strategies. This will lead to more successful computation within their grade level problem-solving!

It’s time to think of ten frames as more than just a tool for primary grades. They can help our intermediate students make connections with abstract math concepts and model new strategies!

The post Three Ways to Use Ten Frames for Upper Elementary {Powerful!} appeared first on Shelley Gray.

13 Simple and Quick Alternatives to Timed Math Fact Tests (Easy to Implement)

Shelley Gray — Tue, 01 Mar 2022 18:23:09 +0000

Timed tests. Chances are good that those words evoke some sort of emotion in you. For some (students and teachers alike), timed tests bring fear and anxiety. Others may recall the motivation and excitement in trying to beat the clock. While timed tests may have once been the norm for students to learn basic multiplication facts, many teachers are ready to try something new. Is your goal to teach multiplication facts in a way that will benefit all students, rather than only the ones who can be successful with timed tests?

Here are some ideas you can try in your classroom to help students develop automaticity with basic multiplication facts – without the stress of timed tests.

Timed Test Alternatives to Get Kids Moving and Grooving (Kinesthetic Ideas)

Skip Count Ball Toss

Put students in groups of 2-3 (or play as a whole class). Choose a number to skip count by and have students skip count aloud as they pass the ball from one player to the next.

Jumping Jack Facts

Choose a set of facts to recite as students do jumping jacks – “2 times 1 equals 2,” “2 times 2 equals 4,” etc.

Stand Up or Sit Down Products

Say a multiplication fact and have students stand if the product is even or sit if the product is odd. Then say the answer together.

Timed Test Alternatives to Get Kids Speaking and Listening (Auditory Ideas)

Math Talks

Engage students in conversations about how they remember multiplication facts or about different ways they can solve a multiplication problem. If you can do this consistently as a number talk routine – even better!

See more about the slides shown here.

Choral Skip Counting

Count by 2s, 3s, etc. together as a class as a math warm-up.

Roll two dice and skip count by the sum.

Products Out Game

Have the whole class stand in a circle and count from 1 to 50. Choose a number and every time a student says a product of that number, they sit down. The last person standing is the winner. For example, if the number 4 is chosen, the students who say 4, 8, 12, 16, 20, 24, etc. will sit down.

Timed Test Alternatives to Get Kids Seeing and Drawing (Visual Ideas)

Anchor Charts/Multiplication Charts

Hang posters or anchor charts showing vocabulary and multiplication strategies. It can be even more effective if you have students help you make the anchor charts – or if they can make them themselves!

Find these posters HERE.

Manipulatives

Use counters or base ten pieces to make arrays or equal groups.

Drawing Arrays Game

In pairs, have students roll two dice and draw the array on a piece of graph paper. Draw as many arrays as possible until the page is filled. The last one to make an array wins.

How Many Ways?

Have students brainstorm all the possible ways they can think flexibly about a multiplication fact and write their ideas on sticky notes. Have students reflect on the different ideas during a whole-class discussion or number talk.

Timed Test Alternatives to Get Kids Reading and Writing (Linguistic Ideas)

Word Problems and Real Life Situations

Giving students a context for why they are multiplying helps them make sense of the problem and how to solve it. For example, there is a big difference between asking a student to solve 2×3 versus asking them how many socks there are in 3 pairs. Real-Life Math Projects can help students understand where multiplication is used in real life.

Flashcards

The act of creating flashcards is often as important as using the flashcards once they’re finished. Have students create visual flashcards and draw a picture that represents each fact.

Alternatively, use a traditional set of multiplication flashcards, but have students sort them by facts they know and facts they still need to work on.

Multiplication Notebooks

Give students a place to record and reflect on the multiplication strategies they are learning. This can be done in a regular classroom notebook, or in a pre-made booklet like the one shown below.

Find this Multiplication Equation of the Day booklet HERE.

Not every idea will work for every student, but when we give kids the opportunity to learn and practice in many different ways, we allow them to learn in a way that works best for them. When kids understand that there are many different ways to learn and practice multiplication facts, it can relieve the stress and anxiety that so often comes along with timed tests, while boosting number sense and flexibility with numbers.

How do you practice math facts in your classroom? I’d love to hear in the comment section below!

The post 13 Simple and Quick Alternatives to Timed Math Fact Tests (Easy to Implement) appeared first on Shelley Gray.

What’s the Multiplication Area Model and How Do You Teach it? Comprehensive Guide

Shelley Gray — Sun, 12 Dec 2021 21:26:08 +0000

The area model is a strategy for multi-digit multiplication, allowing students to build conceptual understanding before transitioning to the traditional algorithm. But there is more to teaching the area model than you might think.

Free Area Model Activities to Help You Get Started!

Yes, please!

This is where we're headed. But it's not where we begin.

When we use the area model, we decompose the factors into tens and ones, use them to represent the side lengths of our rectangle, and then multiply the parts. The sum of the parts gives us the area, or the product of our equation.

In this example, we multiplied 10×10 to make 100, 10×3 to make 30, 2×10 to make 20, and 2×3 to make 6. Then we add up all four of these parts to find the total area of the rectangle, or the product. 100+30+20+6=156.

But this is too abstract for students who are just learning.

The example above is where we are eventually headed, but it’s too abstract for some students during the learning phase. We want our students to understand WHY we are putting the numbers where we are putting them. So we need to start in a more conceptual way.

As we know from the CRA model, ALL students will benefit from working with concepts concretely, and then connecting that understanding to more abstract representations.

So how can we make the area model more concrete?

Using Base Ten Blocks To Teach The Area Model

With base ten blocks, students are able to touch, move, and manipulate concrete models to solve an equation. This is where the understanding is built. Not to mention, it’s fun!

Step 1: Use the factors to represent the length and width of the rectangle.

For example, if we are solving 13×14, our side lengths become 13 and 14. We represent these with base ten blocks.

Step 2: Fill the rectangle.

Now it’s time to figure out the area of the rectangle, which will become our product. First, we see that 10×10 is 100, so we place a hundred block in the rectangle.

Next, we see that 10×4=40, so we add 4 tens.

Now let’s keep multiplying. 3×10=30, so we add 3 more tens.

Lastly, 3×4=12, so we add 12 ones.

Step 3: Add the parts.

Now we can see that the product of 13×14 is 182.

Thinking Deeper About the Area Model

When we work with a strategy like the area model, it’s important to work with it in different ways so the focus isn’t always on finding the product. Some ideas are:

Create different equations that have the same product.
Given the product and one side length, find the missing side length (factor).
Given the product (area), find both factors.
Give students a variety of base ten blocks. Have them create different area models using the same number of blocks. How does the product change? Why is this?

Transitioning to Representational and Abstract

Once students have had adequate opportunity to build their understanding with base ten blocks, we can transition to pictorial and abstract representations. It’s important that students see the connections between what they were doing with the base ten blocks and this new learning.

But isn't this more work than long multiplication?

In some cases – yes! But we have to remember – our goal is not to create human calculators that simply find answers to problems. We have computers and calculators for that! Our goal is to help kids think in flexible ways, reason, and strategize. We need kids to be able to truly understand and to solve problems in different ways.

What Comes Next?

The area model transitions beautifully into another strategy called partial products. This is essentially the same strategy, but we record our thinking differently. Partial products is set up like the traditional algorithm, but as you can see we are multiplying by place value, exactly as we do in the area model.

If you plan to teach the traditional algorithm, it should come after students have used the area model and partial products extensively to build their overall understanding. Remember – math is not all about the answer. It is about the ability to think in flexible ways. Strategies like the area model allow our students flexibility in their thinking, and give students the opportunity to really SEE the math.

Don't miss these free area model activites!

Help your students use base ten blocks to work with the area model, and transition to more abstract representations with the free activity sheets!

Please send them over!

The post What’s the Multiplication Area Model and How Do You Teach it? Comprehensive Guide appeared first on Shelley Gray.

Four Practical Tips to Transform the Way Your Students Understand Multiplication

Shelley Gray — Tue, 29 Jun 2021 15:16:32 +0000

I’m on a mission to help you empower your students so they truly understand multiplication at a deep level and are able to achieve true fluency.

Multiplication is so much more than simply memorizing the facts.

These videos will help you begin to think differently about the way you teach multiplication – so your students can finally build that deep understanding that leads to true fluency.

I’m happy to announce that enrollment is open for my new course – How To Teach Multiplication. See more details below the videos!

VIDEO #1: One Small Tweak For Big Results

VIDEO #2: Re-thinking the CRA Model

VIDEO #3: Are you using this tool to its full potential?

Video #4: Rethinking Multiplication Tricks

Like the ideas in these videos? The full How to Teach Multiplication course is your complete road map to teaching basic multiplication for true fluency. And of course, it comes with a ton of bonus materials so you can implement the ideas without any extra research and planning.

Wondering if this course is right for you? To help you decide, I’m hosting a free, 45-minute masterclass called, “The 5 Part Framework for Teaching Multiplication for True Fluency.”

In this free masterclass, I’ll be sharing the exact same framework that I teach in depth inside the full program. The masterclass will give you a great overview of everything that is included in the full course so you can make the decision whether it is right for you. Additionally, my goal in this masterclass is to provide you with actionable tips and strategies you can use immediately – even if you decide not to join the full program.

Register for the free multiplication webinar here.

The post Four Practical Tips to Transform the Way Your Students Understand Multiplication appeared first on Shelley Gray.