Solving with Multiple Methods

Solving with Multiple Methods

If you are familiar with the Common Core Progressions documents, you may recognize the name Bill McCallum. He is a Common Core Standards author as well as an author of the Progressions. He frequently posts to the site Common Core Tools, and hosts a question and answer forum which can be very valuable for teachers looking for clarification of a standard.

He recently posted about students using multiple methods to solve (computation) problems. Much like a related post we published about turning strategies into “algorithms,” his post talks about the do’s and don’ts of having students solve with multiple methods.

 

 

 

 

 

 

 

 

His transportation analogy really helps to clear this topic up. For example, we should have multiple methods of solving strategies just as we have multiple methods of transportation, i.e. a bike, a car, a bus, and walking. We should know which is best to use and when, based on the situation. (I may walk or bike to school, but I’m not going to bike to Disney World!) And students shouldn’t be asked to solve the same problem in multiple ways if they have already determined a method that works for them. (Unless they may be comparing strategies to look for efficiency)

Check out his post, and visit the Common Core Tools site.

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Dividing Fractions for Understanding

Dividing Fractions for Understanding

Student understanding of dividing with fractions is usually regulated to something to the effect of, “You need to flip the fraction and you need to multiply.” As a teacher, I am usually excited if students can even recall that information. So why is this concept so difficult for students to understand and retain? Is it because they have no understanding of why they are flipping a fraction numerically?

Teaching division of fractions can be more meaningful to students when models are used before using an algorithm. Students need to be taught this concept using their previous understanding of multiplication and division. What does division mean? It tells you how many groups of a quantity there are in a whole amount. So phrasing these problems into those terms will help students conceptualize what they are doing with the fractions.

The following link provides a great description of how to teach division of fractions by having students draw and analyze a model of the problem.
http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-4_6-6ab_1.pdf

Fraction strips can also be used to help students conceptualize what is happening numerically when you are dividing fractions. Check out this video from PBS: https://florida.pbslearningmedia.org/resource/mgbh.math.nf.divofrac/division-of-fractions-using-fraction-strips/#.WXpNVojyuM8

References:

Virginia Department of Education(2011). Modeling Division of Fractions. Retrieved from Virginia DOE website http://www.doe.virginia.gov/testing/solsearch/sol/math/6/mess_6-4_6-6ab_1.pdf

PBS Learning Media(2017). Division of Fractions: Using Fraction Strips. Retrieved from PBS & WGBH Educational Foundation websitehttps://florida.pbslearningmedia.org/resource/mgbh.math.nf.divofrac/division-of-fractions-using-fraction-strips/#.WXpPLYjyuM9

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Are your students “Number Shoppers” instead of “Problem Solvers”? Try numberless word problems!

Are your students “Number Shoppers” instead of “Problem Solvers”?  Try numberless word problems!

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“Number Shoppers”….we’ve all encountered them: students who just plow through a story problem, pulling numbers out as if they were just grabbing their favorite snacks off the shelf and tossing them into a shopping cart; no regard for the value of the numbers and what the represent, or the action taking place in the problem. Sometimes, it seems like there isn’t even any thinking going on!

On his blog “Teaching to the Beat of Different Drummer“, Brian Bushart suggests numberless word problems as an instructional strategy to help remedy this common challenge that plagues math classrooms. He states, “In essence, numberless word problems are designed to provide scaffolding that allows students the opportunity to develop a better understanding of the underlying structure of word problems.” Numberless word problems, combined with teacher facilitated questioning, are word problems that get kids thinking before they ever have numbers or a question to act on.

Inspiration for the problem can be drawn from an already existing problem in a textbook or other resource. Then, the numbers are removed and the problem is stripped down; the problem is scaffolded by sharing additional information with students, piece by piece. The power of a numberless word problem lies in the conversation students have as each new piece of information is shared. That conversation is driven by the planned questions asked by the teacher as more and more information is revealed.

Here is one example that Bushart shares:
Inspiration:
original problem
Scaffolded Problems:
scaffolded problem
Planned Questions:
questions

Read more at Bushart’s blog page “numberless word problems“, which includes posts on how to get started on using the problems in your classroom, tips for writing numberless problems, and a bank of numberless word problems ready to be tried out with your kiddos!

You can follow Bushart on his blog and @bstockus on Twitter. He also co-moderates the Twitter chat #ElemMathChat on Thursdays at 8pm CST.

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Connecting Academics and Parents- 4th Grade Relating Fractions to Decimals

Connecting Academics and Parents- 4th Grade Relating Fractions to Decimals

Digging deeper into decimal fractions?
…there is a CAP for that!
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View the “Train the Trainer” video on “Relating Fractions to Decimals” below to ensure that your instruction has truly met the depth of the 4th grade MAFS standards. Yes, these trainings were written for the benefit of parents, but the anticipated misconceptions and instructional strategies will be useful when planning to review critical areas at the end of 4th grade to prepare students for 5th.

Follow the steps below to access the 4th grade “Relating Fractions to Decimals CAP (Connecting Academics & Parents)” training materials, including the powerpoint and ancillary resources.

• Step 1: Log into your ideas account.
• Step 2: Go to the Elementary Mathematics Icon.
• Step 3: Click on the “CAP Connecting Academics and Parents” icon.

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• Step 4: Click on “Grade 4 CAP Math Parent Workshops”.

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• Step 5: Click on 4th CAP “Relating Fractions and Decimals”.

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• Step 6: From there, explore the folder to access all the resources you would need to implement the training, including the powerpoint.
cap decimals

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Inside Mathematics Illuminates the Standards for Mathematical Practice

Inside Mathematics Illuminates the Standards for Mathematical Practice

Do you ever find it difficult to to wrap your head around what each of the 8 Standards for Mathematical Practice might look like in your specific grade level? Or do you struggle with how to incorporate the SMPs into your daily lesson plans in a way that purposefully engages students with the Mathematics Florida Standards?
smps
At http://www.insidemathematics.org/common-core-resources/mathematical-practice-standards , Inside Mathematics sheds some light on the SMPs with video clips of individual examples of each of the practice standards in classroom lessons across a variety of grade levels.

Since it is also important to keep in mind that the practices can, and should, be evident together in a lesson, there are also video clips of “Mentors of Mathematical Practice” at http://www.insidemathematics.org/common-core-resources/mentors-of-mathematical-practice that offer a view of teachers who commonly engage their students in multiple practices simultaneously.

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Reboot Your Regrouping Reteach!

Reboot Your Regrouping Reteach!

What is the one math concept that you still feel students have not quiet mastered in second grade? “Regrouping,” you say. Well before you pull another small group to line up place values and regroup to the tens, consider rebooting the regrouping instruction completely. Many 2nd graders are not ready to make sense of the standard algorithm for addition and subtraction and the standards do not require it. If students have not mastered the algorithm at this point in the year, chances are likely that the student’s number sense needs to be developed further. Check out this one minute Origo video, to see some strategies for developing a students’ understanding of addition with regrouping.

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Math Journaling in Kindergarten

Math Journaling in Kindergarten

Check out this video on using Math Journal in Kindergarten as a tool for informal formative assessment!

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Connecting Academics and Parents- 4th grade Equivalent Fractions

Connecting Academics and Parents- 4th grade Equivalent Fractions

Students struggling with using fraction models to explain WHY two fractions are equivalent?
…there is a CAP for that!
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View the “Train the Trainer” video on “Equivalent Fractions” below to ensure that your instruction has truly met the depth of the 4th grade MAFS standards. Yes, these trainings were written for the benefit of parents, but many of the suggestions will be useful when planning to review critical areas at the end of 4th grade to prepare students for 5th.

Follow the steps below to access the 4th grade “Equivalent Fractions CAP (Connecting Academics & Parents)” training materials, including the powerpoint and ancillary resources.

• Step 1: Log into your ideas account.
• Step 2: Go to the Elementary Mathematics Icon.
• Step 3: Click on the “CAP Connecting Academics and Parents” icon.

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• Step 4: Click on “Grade 4 CAP Math Parent Workshops”.

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• Step 5: Click on 4th CAP “Equivalent Fractions”.

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• Step 6: From there, explore the folder to access all the resources you would need to implement the training, including the powerpoint.
4th equiv fract

fract online resources

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3 Strategies to Keep Math Convo Moving

3 Strategies to Keep Math Convo Moving

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Most of us have some “go-to” questions to initiate math conversations…”How did you get that?” “Did anyone solve it another way?” “Does anyone disagree and why?”…
But what do we do when that conversation dies out before our students discuss and uncover the desired math ideas?

The short article “Three Follow-up Strategies to Keep Math Discussions Moving Forward” discusses the follow-up strategies of probing, scaffolding and positioning to address the most common challenges that arise during classroom discussions.

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Connecting Academics and Parents- 4th Grade Division

Connecting Academics and Parents- 4th Grade Division

Looking to build your content knowledge of 4th grade division strategies?
YES…there is a CAP for that!

View the “Train the Trainer” video on “Division Strategies” below to ensure that your instruction has truly met the depth of the 4th grade MAFS standards. Yes, these trainings were written for the benefit of parents, but many of the suggestions will be beneficial when planning to review critical areas at the end of 4th grade to prepare students for 5th.

Follow the steps below to access the 4th grade “Division Strategies CAP (Connecting Academics & Parents)” training materials, including the powerpoint and ancillary resources.

• Step 1: Log into your ideas account.
• Step 2: Go to the Elementary Mathematics Icon.
• Step 3: Click on the “CAP Connecting Academics and Parents” icon.

cap-fractions-1-2

• Step 4: Click on “Grade 4 CAP Math Parent Workshops”.

4th cap ideas 1

• Step 5: Click on 4th CAP “Division Strategies”.

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• Step 6: From there explore the folder to access all the resources you would need to implement the training, including the powerpoint.
div cap
cap div 3

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Nervous about Number Talks? Now is the time to give them a try!

Nervous about Number Talks?  Now is the time to give them a try!

number talks bubble
You most likely have heard the buzz about number talks, but the pressure of state testing may have had you feeling like you didn’t have the time to implement them in your classroom…well no more excuses!

The Mathematics Florida Standards, both the content standards and the Standards for Mathematical Practice, require that we as teachers shift our instructional practices to focus on building deep conceptual understanding and sense making in mathematics. In the article “Number Talks Build Numerical Reasoning” from the the October 2011 issue of NCTM’s Teaching Children Mathematics, Sherry Parrish says, “Classroom number talks, five- to fifteen-minute conversations around purposefully crafted computation problems, are a productive tool that can be incorporated into classroom instruction…During number talks, students are asked to communicate their thinking when presenting and justifying solutions to problems they solve mentally. These exchanges lead to the development of more accurate, efficient and flexible strategies.”

number

I think we can all agree that this sounds great, right?! But, how do we get number talks started, and what do they look like in action?

Parrish goes on to describe 5 key components of a classroom number talk:
1) Classroom environment and community. The classroom must be established as a safe, risk-free environment in which all ideas and answers are accepted and valued
2) Classroom discussions. Successful number talks are rooted in communication. When students share and discuss computation strategies, they have the opportunity to clarify their thinking, investigate mathematical relationships, build a toolbox of strategies, evaluate the efficiency of strategies, and consider and test the logic of different strategies.
3) The teacher’s role. The teacher must shift from the “holder of information” to taking on the interconnected roles of facilitator, questioner, listener and learner.
4) Role of mental math. Mental math is a critical component of number talks, because it encourages students to build on number relationships to solve problems, rather than relying on rote procedures. In addition, mental math strengthens students’ understanding of place value.
5) Purposeful computation problems. As part of the teacher’s role as the facilitator of number talks, purposeful problems that will guide students to focus on the desired math relationships must be selected. The goal and purpose of the number talk should determine the numbers and operations of the problem.

Click on the links from insidemathematics.org for an example of a number talk for 2-digit multiplication by 1-digit multiplication:

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Here are additional inside mathematics video clips from a 4th grade number talk, “Can this be true?”, including a pre-and post-talk:

pre-talk

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post-talk

For more information on number talks, take our Number Talks training offered by the Hillsborough County Elementary Mathematics Department, read Sherry Parrish’s full article, “Number Talks Build Numerical Reasoning”, or checkout Sherry Parrish’s book Number Talks: Helping Children Build Mental Math and Computation Strategies
Number-Talks

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Connecting Academics and Parents- 4th Grade Multi-Digit Multiplication

Connecting Academics and Parents- 4th Grade Multi-Digit Multiplication

Have you checked out the CAP Parent Trainings on the Elementary Mathematics Icon?
Don’t worry….it’s not too late!

While it is true that these trainings were designed with parents in mind, each training also has an accompanying “Train the Trainer” video, that you may find has some valuable instructional tips for you as the teacher. View the video for “Multi-digit Multiplication” below to ensure that your instruction has truly met the depth of the 4th grade MAFS standards. Many of the suggestions may be beneficial when planning for the reviewing of critical areas at the end of 4th grade to prepare students for 5th.

Follow the steps below to access the 4th grade “Strategies for Multi-digit Multiplication CAP (Connecting Academics & Parents)” training materials, including the powerpoint and ancillary resources.

• Step 1: Log into your ideas account.
• Step 2: Go to the Elementary Mathematics Icon.
• Step 3: Click on the “CAP Connecting Academics and Parents” icon.

cap-fractions-1-2

• Step 4: Click on “Grade 4 CAP Math Parent Workshops”.

4th cap ideas 1

• Step 5: Click on 4th CAP “Strategies for Multi-digit Multiplication”.

4th cap ideas 2

• Step 6: From there explore the folder to access all the resources you would need to implement the training, including the powerpoint.
cap 4th mult
cap mul 3

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Building Math Talk

Building Math Talk

As part of the Standards for Mathematical Practice, students should be able to construct viable arguments and critique the reasoning of others.

Check out the video below to think about how the Same or Different task can be used in your classroom to promote accountable conversation!

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Assessing Mathematical Fluency

Assessing Mathematical Fluency

It is the end of the school year. You’ve spent time and energy teaching various math concepts so that your students have a conceptual understanding of content. Students in your class have explored foundations of operations using concrete models and pictures. They’ve learned and applied strategies for solving various problems involving basic facts. Now it is time to assess their understanding. Have they really mastered their basic facts? What basic facts are they even required to master?

Basic Facts are an integral part of the elementary math standards. According to the Mathematics Florida Standards, standards for Basic Fact Fluency can be found in various grade levels.

Kindergarten students have to fluently add and subtract within 5.
First grade students have to fluently add and subtract within 10.
Second grade students are required to be fluent in addition and subtraction within 20 and know from memory all the sums of two one digit numbers.
Third grade students must fluently multiply and divide within 100 and know from memory all products of two one-digit numbers.

Traditionally, timed tests were used to assess basic fact fluency. However, many wonder if that is the best way to truly assess student understanding. Do timed tests offer limitations in truly capturing students understanding of the fluency standards?

According to the article Assessing Basic Fact Fluency, by Kling and Bay-Williams (2014), basic fact fluency can be defined as, “the efficient, appropriate, and flexible application of single-digit calculation skills”

If we simply use timed tests, can we see the flexible application of these skills? Are students missing out on opportunities to decompose numbers, or apply strategies such as make a ten, doubles, doubles plus one, or the distributive property? Without flexible understanding and efficient strategies, many students miss out on applying their math skills to greater numbers which may hinder their mental math capabilities.

Alternatives to Timed Tests suggested in the article include:
1. Interviews
2. Observations
3. Journaling
4. Quizzes

Just like when we teach reading, we learn so much more about the student as they read to us. They same should be true in math!

For more information on Assessing Basic Fact Fluency check out the link below!
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http://www.uen.org/utahstandardsacademy/math/downloads/level-2/3-4-4-assessingbasicfactfluency.pdf

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Stop Perpetuating Math Anxiety

“Do not worry about your difficulties in mathematics; I assure you that mine are greater.” – Albert Einstein

Are teachers increasing students’ anxiety levels in mathematics, especially as state testing approaches? Teachers may very well be swelling the negative feelings students harbor about math by teaching procedurally and coddling students when they struggle, thereby giving them a sign that this is too hard for them. When state testing approaches, teachers often get into panic mode and rush through reviewing concepts and skills randomly. When a sense of urgency should be high, we need to keep in mind that the brain can only hold so much information at once and it’s extremely important to keep that information organized so that students can connect to prior learning and benefit from this reinforcement. If we start to see students with a zombie look, it might be that the way teachers are reviewing is so disconnected that they are building up anxiety. How much will this affect students’ ability to perform on the math test?

“Unfortunately math continues to be taught in ways that are far removed from the research evidence on ways to teach well, and many ineffective classroom practices – timed tests, speed pressure, procedural teaching – are the reasons for the vast numbers of children and adults with math anxiety.” Jo Boaler

Read about five problems Stanford expert Jo Boaler says we can solve to end math madness.It’s Time to Stop the Clock on Math Anxiety

Sometimes the math anxiety that parents and teachers have transfers over to students. Learn how you can overcome math anxiety.
How to Overcome Math Anxiety

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Subitizing and Shapes

Subitizing and Shapes

Check out this video by Graham Fletcher on connecting subtizing and geometry! https://gfletchy.com/fresh-ideas/

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Using 5 frames to build fluency

Using 5 frames to build fluency

By the end of Kindergarten, students are required to fluently add and subtract within 5. Throughout the year, students have built conceptual understanding of addition and subtraction through joining and separating situations. They’ve also have ample experience working with Five and Ten Frames. These skills can be used to practice fluency with addition and subtraction within 5.

A Five frame is an excellent tool to use to build fluency with students on addition and subtraction within 5.
Five Frame

A quick game to play with students is “Five Frame Flash”
To play, simply flash a five frame to students for 3-5 seconds and have them come up with the number sentence of what they see. There may be more than one correct response.
For example, if you flash the image below, students might say, “Five” or, they might say, “Five and 0 is 5 or 0+5=5”
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The colored dots on the cards can be changed to encourage identifying different addends. In this example, students might say, “Four and 1 is five, or 1+4=5”
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Changing the addends helps students build fluency and understanding of the different combinations that can be made from the five frame. For example, students might see, “5-3=2 or 2+3=5”
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Dots can also removed from the five frame so that students are working on number combinations that are less than 5.
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Challenge your students when you have some free time in the classroom.
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Or when you are waiting in the lunch line.
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Or at dismissal!!!
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There are plenty of opportunities to sneak in some extra time to build fluency with your students! Find a time that works for you and play Five Frame Flash to ensure your students leave kindergarten being able to fluently add and subtract within 5!!!

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Multiplying Decimals 3 Act Task

Multiplying Decimals 3 Act Task

Earlier this year, students built conceptual understanding of multiplication of whole numbers and decimals and multiplication of decimals by decimals. They used concrete models and grid paper to model the multiplication to find the product. The standard, MAFS.5.NBT.2.7, requires students to, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.”

Many times, students can regurgitate the steps in “solving” a math problem yet, many students lack the skills to apply the learning to real life situations. Connecting Math to the real world is an integral part of learning and encourages retention of the skill. There are many opportunities for students to engage in real life math that encourages them to practice the skill of multiplying decimals.

Check out these videos from https://gfletchy.com/gassed/ where students can apply their understanding of multiplication of decimals to a real life situation.

How much will it cost to fill up the gas tank?

Show students this quick video and have them make predictions about the cost of filling the gas tank. Estimation is an underutilized skill that is beneficial to helping students decide on the reasonableness of their answer.

Show the tank capacity and price per gallon to the students and watch them solve! Note the strategies they use when solving for the cost to fill up the car.
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Once students arrive at their answer, show them the following video to discuss if their answer was reasonable.

For more 3 Act Tasks ideas, check out https://gfletchy.com/3-act-lessons/

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FREE Common Core Aligned Online Interactive Math Practice

FREE Common Core Aligned Online Interactive Math Practice

kid at computer
Checkout these two FREE interactive websites for additional math practice: www.tenmarks.com and www.frontrowed.com

Both websites allow you to set up a free teacher account. You can then enter in your student roster, assign specific standards-based practice to each student, and track their progress. Students are given a log-in and password, so they can access the program from home, too.

FrontRowEd includes a diagnostic with adaptive practice to provide students practice at their level in addition to teachers’ ability to assign targeted practice. Students can click to view a youtube video for more information on a concept, and there is an option for fact practice.

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TenMarks provides students with real-time feedback, and students can access “hints” and videos for immediate scaffolding. Students can work on building fluency in “Jam Sessions”. It also has teacher resources, including lessons with anticipated misconceptions. This program

Sample Student View:
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Sample Class Assignment Summary Report:
tenmarksreport
Sample Individual Student Assignment Report:
studentreport
Either program could be beneficial for differentiating instruction and providing students with practice responding to technology enhanced math items. Keep in mind, there may be slight variations between the Common Core Math Standards and our Mathematics Florida Standards, so you will want to make sure that all assignments align with MAFS.

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Keeping Math Fun During and After Testing

Keeping Math Fun During and After Testing

Franklin Boys
Checkout this quick and easy read “Tips to Keep Math Fun During Testing” from the tenmarks.com blog.

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