These number sense activities are a blast for students! Most upper elementary teachers notice a lack of number sense in their students, and that lack of number sense makes high levels of math extremely difficult for children. We’ve all seen the blank stares when students are not able to grasp or connect to the concept we’re teaching. Maybe you’ve had students answer math questions with numbers that seemingly come out of thin air that are not even remotely reasonable. This is typically due to a lack of number sense.

Number sense isn’t something you can teach in a single math unit. Instead, students must be exposed to opportunities that allow them develop number sense. In this post I’ve shared a few of my favorite number sense activities.

I’ve always been a teacher who is laser-focused on the standards. I could easily attach every activity my students completed to a specific standard. But, I finally came to the understanding that I could not limit myself to just the written content standards. Sometimes we have to provide students with the instruction they need before students can access those standards.

Number Sense Activities – The Essentials

Below are essential components to upper elementary number sense. This isn’t an all inclusive list. This is just the areas where I’ve observed the greatest weaknesses and the areas that allow students to make the greatest gains. I’ll keep the overview extremely brief, so you can get to the fun stuff!

• Unitizing – the understanding that you can count a large group of items by counting equal groups of items from within the large group. Multiplication begins with students’ ability to unitize. Unfortunately, unitizing is area of struggle that often goes undetected, and students become stuck as additive thinkers and never progress to thinking multiplicatively.
• Spatial Reasoning – Spatial relationships give students the ability to manipulate numbers in ways they may not yet be able to do with abstract numerals.
• Benchmarks-Benchmark numbers are not limited to five and ten. In upper elementary, students begin to work with one and one-half as a benchmark number.
• One/Two More or Less (Units) – Knowing one/two more or less allows students to be flexible thinkers and aides in mental computation. It’s important to incorporate this with multiple units. For example what is 2 ten thousands less than 431,388, or what is 2 unit fractions more than 4/5?
• Parts and Wholes – Students recognize that a number can be decomposed into many different parts.

Number Sense Math Centers

These number sense activities are perfect for math centers and are not your traditional math centers. Instead of being focused on a specific standard, these centers are designed to provide opportunities for developing number sense.

I included an optional booklet which students can use to record all of their responses. I love having students keep everything in one place, and this seems to simplify recording sheets. Of course, I also included traditional recording sheets if that’s what you prefer.

You can find the 4th grade math centers here and the 3rd grade math centers here.

Hundreds Chart

In the first activity, students complete four sets of hundreds charts. In the first chart students work with numbers 400-500, and then each hundreds chart has students work with progressively larger numbers. By the last hundreds chart, students work with numbers 703,500-703,600.

Difference to 100

This number sense activity lets students develop their spatial reasoning and benchmark number sense. They take turns rolling two dice to build a number. Students shade in the total amount rolled. Then, they use the grids to determine the difference from 100. The 10 by 10 grids are outlined into four quadrants of 5 to allow for easier visualization.

Find the Number

This activity also allows students to focus on spatial reasoning. Player 1 secretly chooses a number between the two whole numbers on the number line and places a star on the number line where the number would most likely be found. Player 2 tries to guess the mystery number. After each guess, Player 1 writes the guessed number on the number line in the correct location. When the mystery number if found, students change roles.

Picture Puzzles

Picture puzzles are a great activity for developing number sense. I had always assumed they would be too easy for students, but I tried them out at home with my own kids. It was really surprising to see how challenging this was for them. I tried to use pictures that older students would find humorous, because I always want to make it fun.

In this particular puzzle, students are looking for numbers that are one ten more than the previous number.

Line ’em Up

The object of this number sense activity is for students to be the first player to play all of the cards. Players take turns laying out one card at a time to build four rows of cards in the middle of the table or floor. If Player 1 puts down 5 groups of 10, player 2 can then play the 6 groups of 10 or the 5 groups of 100, 5 groups of 1,000 or 5 groups of 10,000. When the player lays down the card, s/he must say the total “5 groups of 10 is 50”. All numbers must be placed in order in each row. If a player does not have a card that can be played, they must skip their turn.

I love to mix these number sense activities with more traditional standards-based math activities.

Number Sense Activities – Roll and Review

Calculator Cards

Number Sense Activities – Vocabulary

Math Journal Prompts & Number of the Day

Your students will love these number sense activities! You can find the 4th grade math centers here and the 3rd grade math centers here.

You can read about developing number sense through math warm-ups here.

The post Number Sense Activities appeared first on Ashleigh's Education Journey.

These number sense routines can be a game-changer for your students. Last year, I wrote a blog post solely dedicated to teaching number sense. That post shares a more general overview of what number sense is and why it’s important. This post moves directly into ways to improve students’ number sense. I’ve shared some of my favorite ways to use math warm-ups to develop and improve upper elementary students’ number sense. You can find the lessons here.

As an upper elementary teacher, it’s easy to identify students who lack number sense. Unfortunately, it feels like the quantity of students lacking number sense is increasing every year, and it makes teaching advanced math concepts incredibly difficult. Without number sense, it is difficult for students to compute and to find relationships within numbers or equations. In fact, most common error patterns point directly to a lack of number sense. This is also one of the primary reasons students struggle with problem solving, as students solve problems without finding meaning or making sense of the question.

Number Sense Routines

The good news is that none of the number sense routines below require any prep. If fact, simplicity is the name of the game. There’s no need for elaborate set-ups or fancy materials. Students gather together as a whole group (virtually or in-class) and complete one number sense activity a day.

Once students are gathered, display the problem of the day by opening the included link that takes you to Google Slides. You can also display the PowerPoint version of the file, whichever works best for you! If you use the PowerPoint version, it is important that the problem is displayed in slideshow mode, because this will allow the transitions to display appropriately. 

The power in these lessons come from the class conversations, so this is not something that can be completed independently. Students need to share and discuss their thoughts and the conversations will definitely need your guidance.

Number Sense Routines – Quick Images

Quick images are pictures of quantities that are organized to encourage subitizing and unitizing, and they are the perfect addition to your number sense routines.

Unitizing is the counting and quantity principle that refers to the understanding that you can count a large group of items by counting smaller, equal groups of items from within the large group. This is a HUGE shift in thinking for students, and is often not naturally developed. Not surprisingly, unitizing is area of struggle for students as old as high school that often goes undetected.

However, its importance cannot be overstated. This is the foundation of our entire place value system. This is also what helps transition students from additive thinking to multiplicative thinking, which is an essential transition.

When completing a Quick Image activity, show the class an image for a few seconds and have students try to determine the quantity shown. After showing the image, I have students raise their hand to share the quantity, and I record their responses on our dry erase board. Once students share their quantities, show the image again for verification and to discuss the strategies for identifying the quantity.

Ten frames are great for these activities.

You can also incorporate base-ten blocks and other materials such as dice or dominoes. It helps me to see this number sense routine in action, so I have a short video that gives you can idea of what it would look like presented to students.

Counting

Another number sense routine are counting activities, which aren’t just for young students. Counting helps students recognize the relationship among numbers and allow students to become more proficient with math concepts. Students need repeated practice with counting to become fluent with counting sequences and to develop their number sense. One of my favorite ways to teach this is through a count around the room activity. Each person says a number as students count one-by-one around the room. To teach the procedure for this activity, start with simple numbers and sequences, so you can focus on expectations and procedures. This does work best with students sitting in a circle, but as long as students know the order in which they are counting any seating arrangement will work.

After students are familiar with the routine, begin asking questions such as, “If we are counting by fives what number will Ashleigh say? How do you know?” You may want to record the sequence of numbers as students orally count around the room.

As the year progresses, students can work with larger numbers and more challenging tasks.

There are so many options for this activity, and it’s perfect for fractions. Here’s another video for a more interactive overview for this number sense routine.

Measurement Word Problems

We all know what a challenge word problems can be for students, which is why I didn’t want to exclude them from these number sense routines or warm-ups. This is definitely not the only word problem instruction or practice that my students receive. I teach through word problems as often as possible, and I explicitly teach different parts of word problems. You can read about that here.

During our warm-ups, I incorporate measurement conversion word problems. That way, if I don’t have time to squeeze in measurement conversions before testing, my students will have experience a years worth of practice already. I start the year with somewhat simple word problems and progressively increase the level of difficulty as the year goes on.

We only complete one of these word problems during our warm-up. I don’t want to rush through the explanation in order to squeeze as much as possible in. Instead, I slow down and have students focus on sharing what information they need in order to solve the problem.

If you ever feel like any of these are too difficult, you can certainly edit the numbers in the PowerPoint file.

Which One Doesn’t Belong – Number Sense Routines

Students need experiences comparing groups of items by looking at similarities and differences. Christopher Danielson wrote the book Which One Doesn’t Belong and has further developed the idea on his blog. In this number sense routine students are asked to look at four images or numbers and determine which one doesn’t belong. Of course, students must also share their reasoning. Students love this routine because there are no wrong answers, as long as the student can justify his/her reasoning. The focus isn’t on the answer, as much as on the student being able to communicate his/her justification of their selection.

This is even more fun when combined with Peardeck. This lets you (and students) see what was chosen.

This is especially great for helping students develop fraction sense.

Empty Number Line

Number lines are great for your number sense routines. An empty number line is a line that is presented without markers or numbers. It acts as a visual representation for recording and sharing students’ thinking strategies, as well as providing a stage for students to think about number. The model helps students develop relations between numbers and visualize operations.

In this activity, choose a mystery number, but do not tell students the number. Show students a number line with two numbers that are at the endpoints and place a question mark on the mystery number’s location on the number line. Then, have students try to guess the mystery number. After each guess, record the number they guessed where it lies on the number line.

As with the other activities, the progressively gets more challenging as the year progresses. Plus, you can absolutely change the numbers to make the activity work with your students.

This is also particularly useful when teaching fractions and decimals. The activity provides students with the opportunity to think about the size and relative location of numbers.

Number Sense Routines – Error Analysis

I sporadically included error analysis problems into our weekly number sense routine. In the error analysis problems, I included common errors in modeling solutions in work problems.

These are unique because the errors aren’t in the computation. The errors are in the representation of the problem. For example, in the problem below 32 divided by 8 does equal 4. However, the model does not represent the equation.

Parts and Wholes

Parts and wholes are another part of our number sense routines. This is the basis to compose and decompose numbers. One great way to teach parts and wholes is through a target number activity. The purpose of the activity is to help students think about numbers in flexible ways by having them think about how the number can be represented, composed, and decomposed. In this activity, give students a number such as 100 and have students think of all the ways they can make 100. As students develop an understanding of the routine, add constraints such as using three addends or only with multiplication and division.

As students become more proficient, they may decompose numbers into expanded notation or unit fractions. Doing these activities may allow you to see incorrect generalizations that can be addressed and retaught correctly. For example, many students try to use the commutative property of addition with subtraction. This will allow you to see what math concepts need extra time and what skills to address with students.

This is an activity where you do not need any special resources or materials. I ask the question on my whiteboard, and my students write on their personal white boards or a sheet of notebook paper. As we discuss the problem, I do have some students write their responses on the whiteboard.

I have created a YEAR’s worth of math warm-up activities to make this easier to implement. I love this because it’s all digital which saves me SO MUCH time! Of course, we all want to tailor our instruction to our students’ needs, so I’ve made the resource editable. You can click here for a FREE week!

The post Number Sense Routines appeared first on Ashleigh's Education Journey.

Many upper elementary teachers notice a severe lack of number sense in their students, and that makes high levels of math extremely difficult for students. Number sense refers to “a well organized conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms”. There are five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers, and referents for numbers and quantities. This post shares some strategies I use to develop number sense in upper elementary students. It’s certainly not too late to develop number sense with third, fourth, and fifth graders. Before we get started, I’d love to share a brief introduction to number sense video by Jo Boaler.

Hopefully, you’ll be able to use some of the ideas and strategies below to develop a number sense with your upper elementary students. More than likely, you’re already doing several of these things!

Browse the Number Sense Strategies:

1. Number Talks
2. Math Facts
3. Conceptual Instruction
4. Communication
5. Time

Number Sense Strategy 1: Number Talks

I heard about number talks several years before I ever tried implementing them. I convinced myself that they were a complete waste of my time and just another buzz word in education. However, I saw other math teachers that I greatly respected begin implementing number talks and sharing the success they were seeing. I was in fear of missing out, so I reluctantly gave number talks a try. To be honest, I did not have any positive expectations.

I am now happy to share that I was completely wrong! Number talks now have a valuable place in my classroom. My favorite number talks resource is Number Talks by Sherry Parrish. It is a long, content heavy book, but it’s not necessary to read cover to cover. I started by selecting the essential portions and teaching strategies as I learned them. One of my favorite things about number talks is that I don’t need any special resources or materials. I use the book for my lessons and my dry erase board. That’s it!

I quickly realized that number talks are an amazing routine that will help improve students’ number sense, and this is as important for upper elementary students as it is primary students. I’m sure we’ve all had students who have relied on rote algorithms to complete math problems, but these students did not understand what they are doing. We’ve all seen the same error patterns over and over again (forgetting to carry the one, subtracting from bottom to top, etc.) and these errors show how the lack of number sense impacts computation. When we develop number sense, students understand the underlying concepts of math operations.

When it’s time for our number talk, I have students sit on the carpet in front of my whiteboard. My rug isn’t quite large enough for my whole class, but we improvise the best we can. When students come to the rug, they don’t need to bring anything with them, since everything is done mentally. I use my Number Talks book to select 2-4 math problems for my students to solve for the day. I follow the order of suggested lessons in the book, but I’m sure you can skip around too! Next, I write one of the math problems on the board. I typically write the problem horizontally to discourage students from using the algorithm.

As students think about the problem, they hold their fist to their chest, and when they determine the solution, they replace their fist with a thumbs up. While students are waiting for the rest of the class to solve the problem, they should think of other strategies they could use and indicate the number of strategies they’ve though of by holding up two, three, or even four fingers to their chest. I like this procedure, because I can see who has solved the problem and who is thinking. It also prevents students from becoming intimidated by students who solve the problems quickly.

I ask students to share their answers, and I write all of the answers on the dry erase board. I then have students share how they solved the problem. As students explain their thinking, I write what they are telling me they did. After we go through the problem, students share if the strategy worked or not.

I explicitly teach the strategies from the Number Talk book before I expect students to apply that particular strategy, which is completely different from how I typically teach math. In a normal math lesson, I want students to construct their own rules and strategies, but this is a bit different. As I teach each strategy, I add them to our number talk anchor chart. The only bit of an issue that we ran into was that my students were coming up with too many different strategies. Our number talks could have gone on forever if I let them. However, I used this to guide discussions on the importance of selecting the most efficient strategy and how to determine which strategies were efficient.

To keep track of what problems I had used and who shared each day, I created forms for each strategy used for each operation. This is a great addition to a data notebook or teacher binder! You can download a free copy at the link below.

I can say with complete confidence that number talks are playing a significant role in improving my students’ number sense. They’ve also opened my eyes to why some students continue to make the same errors again and again and why some types of problems cause students so much difficulty. Even if you only have time for them two or three days a week, I strongly encourage everyone to at least give it a try!

Number Sense Strategy 2: Math Facts

As an upper elementary teacher I do focus on the memorization of math facts. I’ve found that higher levels of math are extremely difficult for students who do not know their basic math facts. However, I’m not a proponent of isolated memorization or high pressure timed tests either. Many of us have standards that include ‘fluency’ as a goal, and this fluency comes about when students develop number sense. But isolated practice and repetition will not develop number sense in our students.

Research tells us that the best mathematics classrooms are those in which students learn math facts through engaging activities that focus on mathematical understanding rather than rote memorization. When students focus on memorization, they often memorize facts without number sense. This means they are prone to making errors and are not flexible in their thinking. The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense. There are three strategies I use to help students learn their math facts and develop number sense.

One strategy is through my Multiplication Fact Booklets. As students work through their booklets, they gain a conceptual understanding of each multiplication fact. They are able to solve the fact with multiple strategies, begin to observe patterns, and develop a mathematical vocabulary.

Another strategy I enjoy is incorporating meaningful multiplication games into my instruction. Games are effective and eliminate the high stress that timed tests bring to many students. Math games can be played for homework, morning work, math centers, or early finishers. I’ve created a set of Weekly Multiplication Games, and we focus on one game a week.

Xtra Math

When I first started teaching, I had a firm rule that for me to consider a student fluent with a math fact, s/he must be able to solve the math fact in 3 seconds or less. Over the years I’ve learned that some students will be slower when memorizing a fact or recalling a fact, but they still have exceptional math potential. I’ve shifted my mindset to allow for flexibility and reasoning in students’ thinking. I use Xtra Math as a multiplication fact practice tool, and I customize the program so my students have six seconds to solve a problem, rather than the defaulted three seconds. This gives my students the opportunity to apply multiplication strategies they’ve learned and builds math confidence with my students. Of course, if a student is sailing through without effort, I can adjust the time to 3 or even 1.5 seconds.

Flash Cards

I also like to put a spin on multiplication flash cards. With this twist, students match the cards with the same numerical answer. This activity encourages an understanding of multiplication as well as rehearsal of math facts. This is a great way to practice math facts while developing number sense. You can download a free set here.

Number Sense Strategy 3: Conceptual Instruction

To help our students continue to develop number sense and make sense of math, our math instruction must be conceptual, rather than a series of procedures. By far, the book with the biggest impact on my math lessons is Teaching Student-Centered Mathematics by John Van de Walle. The first time I read it, Van de Walle stepped all over my toes. I have many notes in the margins from the first time I read the book where I questioned everything I read. I’ve since reread the book several times, and acknowledge that I had a lot to learn when I first started teaching. I still do! This book challenges me to continually look at my math lessons and to find ways to improve those lessons.

When I started teaching, I often focued on algorithms, tricks, and shortcuts without first ensuring conceptual understanding. Algorithms and shortcuts are not necessarily bad. However, the key to using them correctly, lies in the sequence of events that occur in learning math concepts. Our math instruction must first ensure that students’ conceptual understanding is deeply embedded. When students have truly mastered a concept, and they should be able to show all the detailed steps in a process, explain why those steps occur. Once students reach this point, we can expose them to more efficient ways to express or perform those same processes. When we teach algorithms and shortcuts first, they become the means to get to the answer. When we bypass conceptual understanding, it is harder for students to understand more complex topics as they advance in school.

Over the years, I’ve modified my math instruction to allow students to construct their own understanding. For example, when using maipulatives, I no longer structure my lessons so that students are directed in exactly how to use the manipulatives. When I do this, my students are blindly following directions and look like they understand. But a rote procedure with a model is still a rote procedure. When getting answers rather than solving problems becomes the focus, students will gravitate to the easiest method available to get answers.

This does not help a student’s number sense, because the student is not reasoning with numbers. We must allow math to be problematic for students. A problem is a task where students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific solution method. These problems focus students’ attention on ideas and sense making.

An example of a task that would be problematic for students is this divisibility task. In this assignment, students must add a digit to the ones place to create a number that is divisible by three, and students explain how they chose that number. Yes, I could simply have students memorize divisibility rules, but there is no application or problem in that lesson. We could make a cute craft or sing songs, but without authentically understanding the rule, students will not be able to apply what they have learned. Of course, I’m not criticizing crafts and songs! They’re a tool, but they are not enough for conceptual learning. You can download the divisibility task here.

Number Sense Strategy 4: Communication

My students ask me all the time why they have to write so much in math! Students should be able to think through what methods make sense and discuss the use of different methods. Communication of math ideas help students solidify their understanding of mathematics. Math can be thought of as a language, and communication plays an important role in making mathematics meaningful.

I occasionally have students write about math through Math Journal Prompts. I had my prompts for the year bound into books for my students, and they regularly write in their journals. The prompts are not word problems or multi part problems. They’re unique in the sense that they’re opened ended and encourage metacognition and evaluation.

I also try to regularly incorporate writing in my math tasks. I have students explain how they solved the problem using multiple representations. Getting a good response is typically like pulling teeth at the beginning of the school year. But with extensive modeling and practice, students always come around.

Number Sense Strategy 5: Time

I was hesitant to add time to this post, because it’s something we don’t have a lot of control over as teachers. As I write this, I recognize that I’m preaching to the choir. I get it. I’m teaching full time, so I’m right there with you. However, it’s hard to mention the topics above without at least mentioning time. We all know that there isn’t enough time in the day to do all that we need to do, and there isn’t enough time in the school year to effectively teach all of our standards.

One thing we have to look at is our daily schedule. I’ve heard of classrooms with math blocks of 25 and 30 minutes. With that short of time frame, there is no way to adequately teach math standards and build number sense. Many of us have no control over the time allotted for each subject, but if possible, I recommend at least an hour a day for math. With our daily time crunch, it’s important to eliminate as much down time and transition time as possible.

Every year I feel I have to move so fast through math standards that my students don’t have time to truly understand and conceptualize the standards. Conceptual instruction takes longer than procedural instruction, so I’m not able to rapidly move through the standards. This leaves me in a panic after Christmas, because I have little time left to teach so many things. I’m learning to spend the bulk of my time on my essential standards or power standards. Those are the standards that it’s important for students to master for success in math in later years. For me, those are my place value, addition, subtraction, multiplication, division, fraction, and decimal standards. I absolutely teach my other standards, but I don’t go as deep as I would like because of time constraints.

In my experience, math is a subject that is often short changed. Whether it’s professional development, early intervention programs, RTI, or after school tutoring, there always seems to be less of an emphasis on math. I teach reading too, so I’m completely on board with supporting students as much as possible with reading, but I don’t think math should be neglected either.

I’m learning, not there yet, to not waste my energy complaining or worrying about what I can’t control. I’ll continue to advocate for high quality math instruction, but if I only focus on what’s wrong, I won’t make positive gains. I tutor one afternoon a week after school, because I know it’s a need for my students. A coworker and I have set aside time for biweekly professional development, because we’re not satisfied with where we are in our instruction. We want to continually improve. These are things I can do something about, and the rest….well, maybe one day.

I’ve found that the best way to instill number sense is through my math warm-ups. This incorporates each of these strategies, using only about 10 minutes a day-at most! If you’d like to learn more about my warm-ups, be sure to check this out. I also have a post with more information on math warm-ups.

The post Developing Number Sense appeared first on Ashleigh's Education Journey.