Math Fluency Archives - Tanya Yero Teaching

How to Implement Dot Talks In Your Classroom

April 7, 2019

Dot talks are an off-shoot of the daily classroom number concepts called “number talks.” Dot talks can be incorporated into number talks, and can be used with students of any age. But because subitizing is such an important developmental skill, they are uniquely suited to meeting the needs of kindergarten minds.

Why We Implement Dot Talks

Subitizing is the ability to look at a number of objects and “know” how many there are. Some subitizing is perceptual, such as when we recognize this pattern of dots on a die as five.

Conceptual subitizing is recognizing smaller sets within a larger number of items and quickly adding them together.

Looking at the 6 side on a die, we see 3 and 3, which we know is 6.

What are Dot Talks?

Dot talks are short, routine class meetings that help children build number sense, articulate their thinking, and appreciate different perspectives. They should be scheduled as a regular part of your day and flexible enough to expand with the children in your class or group. When dot talks are first implemented, especially if it is early kindergarten, they will take only a few minutes. Later on, as children learn the process, build their skills, and develop confidence, you may be struggling to keep it under 20 minutes.

Beginning training for dot talks is almost painfully simple. You present a card with 1 dot on it for a moment, and ask the students, “How many dots are there?” They will see 1 dot and can prove their thinking by touching the dot and counting, “one.” You can vary this first lesson by presenting dots in different places around the page or card. This fosters the additional critical understanding that sets can change shape or orientation, but only change if something is added or taken away.

Starting with the set of 2 dots, more conversational possibilities open up. Two can be subitized (I just know it), but it can also be decomposed (I know 1 and 1 is 2). Presenting the dots in different locations around the page does not change their “two-ness” or their “1 plus 1-ness.”

What Do Dot Talks Look Like In The Classroom?

Starting this routine in the beginning of the year is ideal (but not mandatory) because the complexity of numbers and placement of dots can grow naturally with the children’s development. The early stages of dot talks address Counting and Cardinality standards 4 and 5, understanding the relationship between counting and cardinality, and counting to answer “How many?” As the number of dots you are using increases, so does the range of activities you can introduce with them. Here are our favorite nine dot-talk activities for the kindergarten classroom:

How many dots are there? Hold the dot card up briefly. Students look at the card, but don’t count the dots (of course, they will in the beginning). Ask, “How many dots are there?” Record or discuss students’ observations. Follow up with, “How do you see them?” Students will explain how they saw groups that they added up to the whole. Dot talks start out very simple! Your students are learning how to do Dot Talks and learning how to complete the question-answer process. Over time, as the numbers get more complex and the children develop their skills, a single dot talk card can take as long as 15 minutes to complete! Dot talks for 1, 2, and 3 dots can be used for training your class in the process. You will model strategies, self-talk, and reasoning aloud. Early on, each number can be presented in a variety of arrangements so children can experience a variety of strategies. You can certainly buy specially made dot-talk cards. In the beginning, it is also easy to use a marker and an index card!
Which number doesn’t belong? Starting this activity early on is great, because the visual representation is easy to differentiate in small sets. For example, the difference between 2 dots and 4 dots is clear. This activity requires a set of 4 cards. 3 of the cards have the same number of dots. 1 card is different. Children briefly look at each card, and then it is flipped. Children “see” the difference by subitizing the dots on each card. Children share their thinking and how they knew which card didn’t belong. This activity quickly builds confidence in young learners, which in turn builds excitement for dot talks, and math in general.
How did the number change? This activity is presented in sets of 2 cards each. Children look at the first card briefly. The card is flipped, and a different number of dots is displayed. Children explain how the number changed. For example, there may be a new dot on the 2^nd card. One dot may have been subtracted. Again, children rely on subitizing to come up with the answer and develop their reasoning/articulation skills through the reporting process. This activity supports Operations and Algebraic Thinking standards 1, 2, and 5.
Secret Partners addresses the Operations and Algebraic Thinking standards 1, 2, and 5, as well as standard 3 which asks children to decompose bigger numbers. Starting with 2 dots, children report the “secret partners” that make the bigger number. For 2 it is 1 and 1, but as the numbers get bigger, the possible addends increase rapidly! This activity really illustrates students using different strategies, which is another important part of dot talks.
How many more to 10? 10-frames are great because they form a number-sense foundation of 10 that is easy (over time) for students to subitize. Using dots in a 10-frame, children find the missing addend to ten, Operations and Algebraic Thinking standard 4. After plenty of practice, the sum total goes up to 20! With plenty of visual support and practice, your students will master this activity.
Strategies to count scattered dots addresses Counting & Cardinality standard 5. It also provides an opportunity for your students to utilize and then share different strategies for solving the dot card. Starting with small numbers, students look at how changing placement doesn’t change numbers. As the numbers get more complex, students use and share strategies to group scattered dots. Subitizing works best with well-organized dots. This activity provides a challenge.
How many dots are there? with 10-frames is introduced well into the year, as the numbers start to spill into the teens. These cards can be used in conjunction with subitizing cards. Students can compare and contrast subitizing larger numbers without a frame and with a frame. This activity addresses Counting & Cardinality standards 4 and 5 as well as NBT standard 1.
Dice Dots Addition cards present dots on dice. The dots are somewhat separated and address composing numbers, number sense, and addition fluency. Children continue to build a conceptual bridge between the strategies they use for the dot cards and using formal operations.
How Many Groups of 10? This activity is designed to grow with your students. Presented mid- to late in the year, students can build on their hard word developing an understanding of numbers to 100! Counting and Cardinality standard 1 is a monster, and this activity can take your students to the finish line.

Students use their subitizing skills to count full 10-frames for groups of 10. It also helps children get familiar with talking about groups of 10.

I hope these activity ideas give you a sense of how powerful dots talks and number talks can be in your classroom. As an integrated part of your math instruction, dot talks push children’s thinking, knowing, and reasoning skills. If you are looking for another math element to take your classroom math practice from great to superb, dot talks might be for you. Check out our Dot Talks resource in our TpT store for print and ready resources!

Education, Math, Math Fluency

How to Teach Decomposing Numbers Conceptually

March 27, 2019

Primary teachers feel loads of pressure to ensure that their students are able to understand and master basic mathematical concepts. Because the subject of mathematics begins with a foundation of knowledge that builds upon itself over years of schooling, students must have a firm grasp of the foundational skills of math in order to be successful in later years.

As an upper elementary teacher, I cannot tell you the number of times that I have encountered students who still do not fully understand basic mathematical concepts, such as decomposing numbers. Students who struggle to add and subtract often do not have the understanding of how single-digit numbers can be broken up into even smaller numbers. This deficit not only affects their arithmetic skills, but it also affects their ability to grasp higher-level concepts, such as decimals and fractions.

The key to teaching decomposing numbers is giving students the opportunity to play with numbers rather than simply providing a simple algorithm or forcing the ever-so-popular “skill and drill” approach to teaching.

Now, you may be hesitant like I was at first to spend valuable instructional time allowing students to play with math manipulatives, but believe me, it is absolutely essential for them to be able to have this time to explore and come to their own conclusions about numbers. When students are able to manipulate objects and express their observations, their brains make more neural pathways in relation to math, which then allows them to excel in math in the future.

Day 1

Start with a five-frame and double-sided counters. I’ve found that it is best to laminate the five-frames so they last longer and can be easily wiped down. Give each student their own frame and exactly five counters. Have them place all of their counters on the five-frame (one in each box) and tell you what they notice.

Some prompts you may want to use are:

“How many do you have?”

“How do you know there are five?”

“Are there any empty spots on your frame?”

Give the students time to ponder and discuss these questions with a partner, table group, our as a whole class.

Then, dive a little deeper. Notice if any of the students used both sides of the double-sided counters. Did any of them use some yellow and some red on their frame? Did any use all of the same color?

It’s not time yet to get into making these into addition statements, but have them state what they did out loud. Some students may already be able to verbalize that 2 yellow and 3 red counters make 5 total, but others still need to manipulate, observe, listen, and talk about this concept without thinking of equations.

Tell the students that you are going to talk more about this the next day.

Day 2

On the second day, give the students the same materials – a five-frame and five counters. Ask them to put one counter in each box, but this time, ask them to pay attention to the colors of their counters and see if they can come up with a different way to show five than they did the previous day.

You will have varied responses to this – some students may continue to put all of their counters with the same color showing, some may do an ABAB pattern, and others may lump a few yellow together and a few red together. All of these are still acceptable for their exploration phase.

Again, prompt them to discuss what they notice. Today, also prompt them to state how many counters they have. You may give them a sentence frame to help initiate discussion:

“I have ___ red and ___ yellow counters. I have 5 counters total.”

This discussion will take yesterday’s lesson to the next level as they are now differentiating between the colors of the counters and recognizing that the total is still five, regardless of what colors they used.

They should now be ready for the “Shake and Spill” game. You can use a recording sheet for this game, or you can continue with student discussions. Either way, the kiddos are going to get the hang of decomposing the number 5.

Demonstrate for the students how to play. Take all five counters in your hands and shake them around before dropping (spilling) them on the desk. You may want to give them a small Dixie cup to shake their counters in and a small box or piece of paper for them to spill in.

After shaking and spilling, they should arrange their counters on their five-frame, keeping all of the yellows together and all of the reds together. For example, if they spilled and had 2 yellow counters and 3 red counters, they would group them together on the frame rather than creating a pattern out of them.

Students should then use the same sentence frame as before:

“I have ___ red and ___ yellow counters. I have 5 counters total.”

They can repeat this activity for the remainder of their math lesson, discussing their results with a partner and/or recording their work on a simple “Shake and Spill” recording sheet. Please note that if you use a recording sheet, the students will also need a yellow and red coloring utensil to record.

I like to tie it all together at the end of the lesson by making an anchor chart with all of the combinations that they created. The students can then share all of the ways that they were able to make 5, and you can record them for later reference. It’s great to be able to put them in a logical order as well so that they can see the pattern in yet another way.

0+5=5

1+4=5

2+3=5

3+2=5

4+1=5

5+0=5

Day 3

On Day 3, you can introduce the ten-frame. Provide each student with his or her own frame. Before giving them the counters, have them tell what they notice about the ten-frame. They may state that it is bigger than the five-frame. Some students may even say that it can fit 2 five-frames. Facilitate conversation around their observations.

Then provide them with ten counters. Go through the same process with having them place the counters on their frame (one in each box) and telling you what they notice.

Their verbal responses will be more advanced now as they have been exposed to noticing the color of the counters. If you are hearing them chatter about their different colors, celebrate with them! They are having “ah-ha” moments and making connections to larger numbers!

Again, have them discuss what they notice about the frame, how many counters they used, and how they know that there are ten counters on the frame. This helps develop recall when they see ten-frames in the future.

On Day 4, provide each student with a ten-frame and counters. Have them fill out their frame with all of their counters, paying attention to the color of their counters. Prompt them to discuss what they notice about their counters, and provide them with a sentence frame if needed. At this point, they may not need one since they have been using a sentence frame for a couple of days, but I always post it just in case:

“I have ___ red and ___ yellow counters. I have 10 counters total.”

Build on this connection by playing “Shake and Spill” with 10 counters. Again, they should group the same color counters together in order to visually see an equation prior to looking at equations abstractly. You can have them discuss their findings and observations, or you can have them record their work on a recording sheet.

I prefer to transition from discussion to a recording sheet because it allows them to not only see the number relationship with manipulatives, but it gives them a chance to transfer it onto paper and then actually write out the number sentence, making even more connections for them in the process.

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End the lesson with another anchor chart for their reference. Many of them will notice that there is a pattern that can be made for making 10 that they saw with the “Making 5” anchor chart.

The concept of decomposing numbers is invaluable to your students and the rest of their mathematical careers. If they can make connections with single-digit numbers, you have set them up to be successful with larger numbers.

Education, Math, Math Fluency

How To Integrate Number Sense Into Your Math Lessons

November 11, 2018

There are a lot of components to a math lesson. You look for activities with engagement and complexity to assess understanding. Sounds simple enough but there are several layers to each of those factors making it hard to find the perfect balance. So when I work with teachers on implementing number sense during their math class they start to panic.

“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.

Math, Math Fluency

Week 1: Teaching Strategies that Focus on Number Sense

October 23, 2017

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 use the questions above to develop number sense with my students.

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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Math, Math Fluency

5 Ways to Foster Meaningful Math Fluency

October 19, 2017

I wish I had a time machine. I would go back and change some of my routines I did in my classroom. Without a doubt I would modify the way I taught and assessed math fluency. I was an advocate of timed tests and rote learning of math facts. I sat in parent conferences and defended my stance on the importance of timed fluency tests. Thru professional development and trial and error I modified my fluency curriculum. I came to the conclusion that timed tests and merely recalling math facts did not support the core foundation of my math pedagogy.

I teach math conceptually. I believe that math curriculum is comprised of 30% procedural and 70% conceptual understanding. My main goal is to teach my students to be problem solvers; to work smarter, not harder. But that philosophy wasn’t being used by my students when it came to fluency.

The importance of math fluency hit home one day when I had my students check and grade their own timed tests. I have never noticed how many of them would have to write out 100 minus however many they got wrong to calculate their score. By 4th grade my students should have been able to mentally manipulate numbers within a 100. The time I spent adding rigor to my math block was being wasted if my students couldn’t carry that deep thinking into fluency.

Current trends in education focus on growth mindset, project-based learning, and critical thinking, yet many educators limit student growth by only using timed fluency assessments.

I developed a new process for teaching math fluency. My “5 Steps to Building Meaningful Math Fluency” outlines how I define, teach, and assess math fluency. Each step provides details and examples of how you can start implementing meaningful math fluency in your classroom.

Week 1-Teaching strategies that focus on number sense (CONTAINS A FREEBIE!)