I have written several blog posts on number talks (CLICK HERE and HERE to read more) and how to implement this practice in the classroom. This has sparked conversation amongst teachers and myself. Teachers are excited to start building a number sense program in their classroom, but one question keeps making its way back to me…. “BUT HOW DO I TEACH MY STUDENTS TO MENTALLY MANIPULATIVE NUMBERS?!?!?” 

Number sense and the confidence to mentally solve equations doesn’t come overnight for students. It takes time, repetition, and lots of exposure. For students that lack math confidence, it’s going to be even harder. More often than not students with strong number sense feel good about math and their skills. They are the sit in the front with their hands raised for every question kind of students, but no one is born with number sense. It is something that is gained – a notion that was supported early on by teachers and/or parents and picked up by the child. But every student is capable of learning number sense and no matter what grade you teach you can implement purposeful practices to build number sense within your students. 

At the start of every school year, I implement my game plan for daily number sense practice. I start with teaching my students the number sense strategies.

Here is a list of strategies I teach my students each year. Grab this FREE list below!

Consider the following tips when introducing number sense strategies to your students.

Assume Your Students Lack Exposure

I used to administer timed fluency assessments in my classroom. When my students were done we would collectively grade their tests so they could graph their growth in their math notebook. It would baffle me how many students had to tangibly write out 100 subtracted by however many questions they missed to calculate their scores. They were 4th graders! How could they not mentally calculate an addition/subtraction question within 100?! For some I knew that their comfort level would always lie in writing out the standard algorithm and solving it. But I knew others were fully capable of using mental math. I wasn’t doing my part as their math teacher. My students deserved the opportunity to be taught how to mentally manipulate numbers. 

Before I implement number talks in my classroom we spend time looking at each number sense strategy and what it looks like in specific scenarios. They need this exposure to apply this knowledge on their own later on. 

I spend a month introducing and reviewing various strategies for addition, subtraction multiplication, and division. Number sense strategies also lend themselves beautifully to place value. In time you’ll start to see your students apply the strategies on their own during your number sense routine and during the introduction of new math skills.

Take a look at the samples below from two students when asked to solve 24 x 9 using an alternative method to the standard algorithm.

Teach Your Students That Number Sense Is Fluid

Mental math is based on a few simple principles: 1.) how the child perceives mental math 2.) their comfort within specific strategies and the actual numbers within the question. It’s subjective. How I see an equation and my preference to manipulate the numbers for easy computation will differ from the next person. Inform your students of this. Number sense strategies can be intertwined together or used individually. There is no WRONG way to mentally manipulate numbers. 

PRACTICE, PRACTICE, and PRACTICE 

Number sense doesn’t develop overnight. Learning how to mentally compute numbers and how to apply this way of thinking to various math questions is challenging. Combat this difficulty with a set routine of daily practice. I start my math block with number sense practice and after about a month we graduate to number talks. My students expect and enjoy this 5-10 minute conversation that benefits all. 

Model Your Thinking And Work 

Remember that story about my students calculating their timed test scores by writing the equation out? Every week when we finish grading our tests, I model a strategy for my students. 

“If I got 19 questions wrong, that’s only 1 away from 20 so I immediately know I got an 81.” 

“I missed 27 questions. 80+20 equals 100 and 80-7 equals 73. I got a 73 on my test.” 

I want them to consider using friendly numbers as benchmarks for mental calculations and understand that addition and subtraction are interchangeable. Eventually my students catch on to mentally calculating their scores from my modeling and start doing the same.

Create A Daily Routine

After you have properly introduced mental math to your students, how will you keep up with number sense in your classroom? I use the Number Talks resources I created to launch my math block every day. My students know to expect this routine and it’s something they enjoy doing. There is a set available for grades K-6 and include 250 ready to go prompts for you both in print and digital form.

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Sign up below to receive your FREE number sense building activities. Everything you need to get started in your classroom is included.

The post How To Teach Students To Use Number Sense Strategies appeared first on Tanya Yero Teaching.

Decimals are a huge part of your math curriculum. They are connected to so many standards in various domains; place value, fractions, and measurement. Teaching this topic conceptually will give your students the exposure and confidence they need to see beyond standard algorithms and truly understand the reasoning behind their work. It’s easy to teach the algorithms for adding, subtracting, multiplying, and dividing decimals and breeze through this content, but you’re missing out on an opportunity to bring concrete understanding to these four operations. 

Here is a breakdown of how I teach the four operations in connected to decimals in my classroom.

Addition and Subtraction of Decimals:

Adding decimals – the easiest of the four, addition of decimals seems to be a nice starting point for students. Before I start on addition I really want my students to have a strong conceptual understanding of decimal place values, how they relate to one another (the hundredths place is one-tenth the value of the tenths place, but 10 times greater than the thousandths place), and be able to visually represent decimals using models. 

We use place value disks and a place value chart to represent the decimals we are adding. Students will regroup disks when needed and see how that act changes the value of their numbers. 

We also practice representing addition and subtraction of decimals with hundredth grids. Students will use different colored highlighters to represent the values being combined and cross out values when subtracting. 

Multiplication of Decimals:

I think multiplication and division of decimals are far more “fun” than addition and subtraction. There are various ways to model and represent decimals using pattern blocks, fraction tiles, hundredth grids, and base ten blocks. 

Here are some examples of the types of questions we review in class.

We work on these visual representations long before I introduce any algorithms. With a concrete understanding of multiplication with decimals your students will be able to understand the work behind the formula. 

Division of Decimals:

You want to continue modeling and requiring students to conceptually understand this operation, just like the other three. Again, we use models to represent dividing a decimal by a decimal, decimal by a whole number, and a whole number by a decimal. Your students should be able to represent each of these scenarios using a hundredth grid. I integrate word problems anywhere I can so students have practice identifying which operation to use. 

If you set your students up with conceptual understanding of decimals, this will help with fraction and measurement standards later on in the year. Next year’s teacher will also appreciate your efforts and you’re building number sense too!

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The post How To Teach Decimals Conceptually appeared first on Tanya Yero Teaching.

Then my first year turned into my second year and by the time the third year rolled around I realized I was creating robotic math students that were simply trained to take a test. They weren’t problem solvers. They lacked number sense and basic strategies that I knew they would need in the real world. They understood procedural learning more than conceptual understanding. 

What REALLY is number sense?

Number sense is a child’s set of skills when working with numbers. More importantly, number sense is one’s ability to see the flexibility in numbers. Children with number sense understand quantities, grasps basic concepts, and understand key operations and symbols. But this all leads to the bigger picture of what number sense really is. A child with a strong foundation in number sense can identify multiple ways to solve problems. Number sense supports the beauty and creativity of each child’s mind and how they interpret a question. 

The Questions You Ask (And Don’t Ask) Will Affect Your Students’ Number Sense

It’s not enough to teach your students basic algorithms and train them to memorize the steps. When I teach a new math concept the algorithm is the LAST thing I introduce. Why? I want to strengthen my students’ number sense. Math is designed to be fun and exploratory and more often than not we take the drill and kill approach. This approach leaves out the opportunity for good quality questioning. 

At the beginning of the year I introduce strategies for manipulating numbers and equations. Expect your students to come to you lacking number sense tools. You have to teach them this style of math and how it works. The posters below stay up in my classroom all year.

I spend time modeling my own thinking, providing opportunities for peer discussion, and integrating manipulatives whenever I can. With time, students will organically begin to manipulative numbers on their own.

The following pictures are samples that I have pulled from student responses when asked to solve 24 x 9. Students that have the ability to see various ways to solve one problem have strong number sense.

Below is the complete list of strategies we came up with together as a class:

24 x 9 is close to 24 x 10 which I know is 240, so 24 x 9 must be one 24 less than 240. Answer: 216

25 x 10 is 225 so 24 x 9 must be 216.

12 x 9 is 108, so 24 x 9 will be double or 219.

20 x 9 is 180. 4 x 9 is 36. 180 + 36 = 216. 

24 x 4 equals 96. 96 + 96 = 192 plus one more set of 24 equals 216.

Area model of 24 x 9

Asking questions such as “Does this mathematically make sense?” And “What does this look like in visual form?” will force your students to justify their answers and use multiple strategies. Never accept only one way of solving a problem. Encourage students to share their methods and celebrate the diversity of strategies. Push student thinking and ask meaningful questions. I used the following questions to steer my students towards the equations above.

“How can we use doubles to solve this problem?”

“How can an area model help us solve a multiplication equation?”

“How can we use multiples of 10 to help us solve this problem?”

What Does Number Sense Practice Look Like In The Classroom?

A class that is heavy in procedural questioning lacks the practice of number sense. Students are relying on the memorization of algorithms to solve problems. I provide questions, especially word problems that are rigorous and deep in conceptual understanding. These sorts of questions lend themselves beautifully to discussions that build number sense. 

I also integrate Number Talks into my math block for extra practice that is specifically designed for number sense practice. Number Talks is a quick 5-7 minute discussion that builds mental math and number flexibility. CLICK HERE to read more.

The post The Power Of Number Sense In The Classroom appeared first on Tanya Yero Teaching.

“What materials do I use? Will I have to create everything I need?”

“Do I have the time in my daily schedule to designate for this new component?”

“What does this new routine look like in my classroom with my style of teaching and management?”

I had all these questions and more when I decided to  launch Number Talks in my classroom. It took weeks of research and planning to wrap my head around the idea and then bring it to fruition as part of my math routine. Today, Number Talks flows into my math block with ease and comfort. I am well versed on the topic and so are my students. Here are some suggestions and information for launching Number Talks in your classroom. 

The Purpose of Number Talks

The purpose of Number Talks is to build number sense and mental math in your classroom. We want students to see that there is often more than one way to solve a problem. We want students to have the confidence to mentally manipulate numbers. Finally, we want students to be fluent in math facts.

Number Talks should last 5-7 minutes in your classroom and include discussion and collaboration. With repetition and practice, students will begin to see how manipulation of numbers is a strategy that can be used daily both inside and outside of the classroom. 

Set Your Students Up With The Skills They Need

First, I start with the introduction of mental math strategies. You have to provide the proper TOOLS if you expect students to view numbers differently. I spend time on how to mentally execute  these strategies and how to apply them to specific numeric scenarios. The posters below stay up in my classroom all year as a reference for students. With time, you’ll see students refer to these posters less and less. 

The only way something becomes a habit is if you practice it enough with daily repetition. Number Talks should become part of your math schedule with opportunities for peer collaboration. Let your students hear from fellow classmates about how they solved a problem. This will foster the idea that learning evolves from CONNECTING and building on the thinking of others. 

Build a Number Sense Rich Environment 

Typically, students are not accustomed to mental math and using number sense as part of their problem solving. My absolute favorite question to ask students when they present an answer to me is “Does that make mathematical sense?” Usually I’m searching for clarity from students when I ask this question or I need  to redirect their thinking. Expect your students to feel odd or even uncomfortable as you launch Number Talks in your classroom. They need time and practice to gain the confidence needed to verbally share their mental thinking and number sense skills. There is a level of PRODUCTIVE STRUGGLE that is required for Number Talks and this can be overwhelming for students. 

MODEL your own thinking and reasoning with your students. If a word problem we’re solving in class discusses 24 jars of jelly beans containing 19 candies each, I may say to my students “Well, I know that 24 times 2 is 48, so 24 times 20 is 480. 24 times 19 must be 19 less than 480.” This is all done mentally and using friendly numbers that most students feel comfortable with. Soon students will start to share their own strategies for manipulating numbers. ENCOURAGE conceptual explanations rather than procedural computation. CELEBRATE the variety of ways to solve a problem and VALUE all student contributions. 

Use MISTAKES to drive instruction. Mistakes are organically created opportunities for the exploration of other strategies. Review properties of operations, and problem solving techniques. 

What does Number Talks look like in my classroom?

We all have busy classroom schedules and there is never enough time to get everything done. Number Talks isn’t designed to add another lengthy routine to your schedule. Number Talks will be a QUICK 5-7 minute discussion that packs a strong academic punch. In the past I have used Number Talks as my morning work component or to launch my math block. I project the prompt of the day on my active board, give 1-2 minutes of think time, and then let the collaboration begin. Sometimes I allow the students to share their strategy in groups, but typically I schedule the sharing as a whole group discussion so I can reinforce key concepts.

I have designed a Number Talks resource for every grade from K-6. It has everything I wished I had when I first started Number Talks in my classroom and everything you’ll need for the entire year. There are 250 ready to go prompts for you, both in digital and print formats. CLICK HERE to read more about this resource.

The post How To Launch Number Talks In Your Classroom appeared first on Tanya Yero Teaching.

“How can I add another thing to my math block?”

“I don’t have enough time!”

“I don’t know what type of questions to ask!”

I get it. You’re holding all these reminders in your mental filing cabinet when you’re teaching a lesson. You want to target specific standards and explain key concepts. You want to ask higher order thinking questions. You want your students to really grapple with concepts rather than memorize procedural steps.

But guess what?

Building number sense will ensure you target all those wants. Number sense is not another component to your math block. Instead it is part of everything you say and do in the classroom. Simply put…it’s good teaching.

Questioning

A good math teacher is always looking for ways to sprinkle number sense into lessons, but it can be hard to find and/or write number sense connected questions.

Look for questions that are open ended, require critical thinking, and reinforce the concept that numbers can be manipulated. Be intentional in the types of questions you ask.

Example: When rounded to the nearest hundred, I become 400. What numbers could I be?

This question takes a standard (rounding) that is often taught procedurally in classrooms and requires students to conceptualize the concept of rounding based on the value of specific digits. Students must work backwards starting with a number that has been rounded to the nearest hundred. Students will first share the most obvious answers such as 399. It takes some time for the class to conclude that any number from 351 to 449 rounds to 400 if asked to round to the nearest hundred. This question is far more complex than, “Round 377 to the nearest hundred,” thus reinforcing number sense.

Strategies

Some children have been given opportunities to build number sense at a very young age. Those students will appear to naturally understand how to manipulative numbers and place value for easier computation, while others will struggle. Educate your students on the different number sense strategies. Model these strategies time and time again. Constant repetition will help your struggling students develop number sense. Provide your class with daily prompts to drive the concept home. We learn better when we are asked to apply that knowledge regularly.

Math Talk

Number sense related questions lead to great math discussions. The more open ended you can make a question the better. Open ended tasks increase discussion and collaboration amongst your students. Encourage your students to work in groups or pairs so they communicate their mathematical reasoning.

I use my Grab and Go Number Talks resource to drive home these three components of number sense. It has everything I need for the year.

The post How To Integrate Number Sense Into Your Math Lessons appeared first on Tanya Yero Teaching.

I love to talk. Do all teachers love the sound of their own voice like me?

It took me years to learn to stand back and let my students do most of the work. Thru trial and error I learned that the most authentic learning comes when I pose a question and let my students devise and execute a plan to solve it with minimal help from me.

I have my students sit in groups so collaboration is a quick transition. I use daily fluency chats so my students have practice with math talk. I want to hear rich vocabulary and concise explanation of different number sense strategies being used from my students. Not only is their understanding strengthening, but I am able to formally assess my students on a daily basis. Like writing, verbal explanation takes time for students to master. The more practice students have the more accountable talk you will hear.

For the first few weeks model your own thinking. Encourage students to find alternative ways of solving one problem. Teaching math fluency thru number sense is a great topic for accountable talk because of its diversity. There are countless ways to solve problems which naturally promotes discussion.

Below is a prompt my students completed and their responses.

Student A: I subtracted 33 from each number to get an equation of 300 – 134. Then I was able to mentally subtract because I made a multiple of 100 (300). I got the answer of 166.
Student B: I added 67 to each number, to make 400 – 234. My answer is 166.
Student C: I added 67 to just 333 to make 400. Then I subtracted 167 from 400 to get 233. Then I subtracted 67 from it to get 166.
Student D: I changed 330 to 300 and 167 to 170. 300 – 170 = 130. I then added back the adjusted amounts of 33 and 3 to equal 166.

This discussion is far more meaningful than any “drill and kill” worksheet containing math facts. Students are mastering facts and building problem solving skills.

My Math Fluency Alternative Curriculum contains resources for implementing accountable talk in your math block. Click here to view this resource.

The post Week 3 – Promoting Accountable Talk Centered on Fluency and Number Sense appeared first on Tanya Yero Teaching.

In order to effectively teach math fluency, we must define fluency. In years past math fluency has simply been defined as the speedy recall of math facts. Thus, the birth of timed assessments. We’re looking for how quickly a student can see a problem and regurgitate the answer. But does this approach support number sense and problem solving?

I built my math block on conceptual understanding, yet my approach to fluency did not support that depth of understanding. My quest for a better way to teach and assess math fluency began. First I evaluated what fluency meant to me now that I was striving for deeper thinking within this subject. I decided I wanted to teach and assess math fluency based on efficiency, accuracy, and flexibility. Efficiency and accuracy is nothing new to fluency. Timed assessments evaluate how fast a child can recall their facts and accuracy is simply if their answers are correct or incorrect. I revamped how I viewed efficiency. I merged efficiency and flexibility together. I stopped using timed assessments and became more flexible with how I assessed efficiency of facts.

When I taught new math standards I fostered the idea of approaching problems in different ways. We celebrated the flexibility of math and how different pathways could be used to solve one problem. Why not teach fluency in the same way? I started to integrate number sense practice so my students could see the flexibility of fluency.
In short number sense is the ability to manipulative numbers. For some students this comes naturally, while others struggle. Number sense starts in kindergarten. From the 10 frame to base ten blocks, number sense strategies are taught early on. Unfortunately these strategies start to take a back seat as students get older. Intermediate teachers need to implement opportunities for number sense development in their math blocks. Number sense needs to be cultivated daily. Here is a list of the number sense strategies I teach my students at the beginning of the year.

Addition:
Making tens – Manipulating numbers to make multiples of 10 or 100 to make addition easier
Friendly Numbers – Manipulating addends to make numbers that are easy to add.
Place Value – Separating addends by their place values (expanded form).
Compensation – Removing one quantity from one addend and adding that amount to the other addend.

Subtraction:
Removal – Decomposing numbers to make subtracting easier.
Place Value – Separating numbers by their place values (expanded form).
Adjustment – Adding or subtracting to make numbers that are easier to compute.
Keeping the same difference – Adding a certain amount to each number to make subtraction easier.

Multiplication:
Friendly numbers – Making a number that is easy to multiply. Next, subtract the amount you added on.
Partial products – Multiplying each digit of one factor by each of the digits in the other factors
Part-part whole – Decomposing or separating into friendlier numbers to make multiplying easier

Division:
Repeated subtraction – Subtracting equal amounts of the divisor from the dividend
Partial Quotients – Using easy multiples of the divisor and subtracting the amounts from the dividend
Part-part whole – Decomposing or separating into friendlier numbers to make dividing easier

I have taught 3rd and 4th grade, so I know the struggle is real when it comes to students memorizing x6, x7, and x8 facts. Instead of memorizing these facts, what if we taught them to manipulative an equation to suit their ability? Let’s take 48 ÷ 8. A student with strong number sense could change this fact to 40 ÷ 8 = 5 and 8 ÷ 8 = 1, thus arriving at the quotient of 6. Another could see 24 as half of 48 and divide 24 by 8 twice to arrive at the correct answer. It’s this manipulation of numbers that proves both children have number sense and mastery of certain facts. This is what we should focus on in terms of fluency; strong number sense.

It takes time. A lot of time. There will be moments when you think your students will never get on board. Stick with it. Through modeling and repetition they will catch on.

My Math Fluency Alternative Curriculum has various resources that provide ample practice of the strategies mentioned above.

Want a quick and easy resource to teach your students the different number sense strategies? Check out this FREE flipbook in my TpT store.

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The post Week 1: Teaching Strategies that Focus on Number Sense appeared first on Tanya Yero Teaching.