Why track student data in Google Sheets?

Tracking student data in Google Sheets has many practical benefits!

1. 1. Accessibility- Since it’s cloud based, you can access it anywhere! I hate lugging my school computer home, so I rarely do. But, if I’m grading at home, I can still access my student data spreadsheets from home!
   2. Collaboration- Since more than one person can access Google Sheets at once, it’s the perfect program for managing student data for more than one classroom. Tracking RtI or intervention data or grade level performance on a school or district assessment are both done easily within Google Sheets. Or, it just makes it possible for you to share your data with your administration team, principal, or instructional coach.
   3. Autocalculations- My favorite reason for using Google Sheets or Excel for my data is the built in formulas! I use it to figure out student percentages from raw scores, I use it to generate a list of students based on proficiency, and I use it to color code based on student proficiency. I always build in averages into the spreadsheet so that I can monitor my overall student performance.
   4. Visualization- If you are a visual learner, it’s super fast and quick to add graphs and charts to reflect your data. While that’s not frequently how I like to view it, it’s super simple to add in.
   5. Versatile- From just checklists for attendance, weekly assessments, formative assessments, and more, you can use Google Sheets (or Excel!) to manage virtually anything you want to track in the classroom!

I used to use Excel to track my data, back in the olden days. Over the years, I’ve shifted to Sheets. I don’t think there’s a single feature I miss in my switch, especially since I don’t utilize the full functions of Sheets or spreadsheets in general.

How do I create student data trackers in Google Sheets?

The first iterations of my data tracking were literally just places to store scores. Over the years, as I’ve learned more about options available in Excel and now Google Sheets, I’ve been better able to customize them to meet my needs. Whether you’re looking to create one spreadsheet you can use to track data in a subject area, or a powerhouse of sheets to hold everything you need, you can build the spreadsheet that meets your needs.

I could list a series of steps to follow to build your student data tracker. But, chances are, if seeing the steps was enough for you, you would have already made your own! So, I think videos are the most helpful. Many people have anxiety around spreadsheet software but I love using them! I’ve spent significant amounts of time learning new things when I want the spreadsheet to do something I don’t yet know how to do! I’m not a pro by any means but I’ve learned a few tricks or two along the years.

This video walks through setting up a pretty basic, academic, standards-based student data tracker. I teach you how to build in formulas to copy information from other cells (so you don’t need to duplicate names), auto-calculating percentages, how to set up conditional formatting to color code based on student mastery, identifying the number of proficient students, and how to build in averages.

Free Student Data Tracker Template

In the video above I walk through how to set up this student data tracker and give suggestions for ways you can modify it to better meet your needs. If you’d rather not build it from scratch, I have a free standards tracking spreadsheet template you can download. The template is exactly what’s shown in the video, and it’s ready for you to continue to customize to meet your needs!

More In-Depth Student Data Tracking

In this Part 2 video, I take that same student data tracker and build individual student reporting pages. These allow you to see how each individual student is performing, standard-by-standard or skill by skill. It’s all set up with formulas so once it’s set up for one student, it’s pretty quick and simple to create it for the rest of the students in your class!  In this video, I talk through formulas for copying cells over to be displayed, adding graps, and conditional formatting.

Student Math Data Tracking Sheets

I think it’s very important to track student data standard by standard in math. It allows you to see who is in need of reteaching or extra support. So many skills build off of a previously learned skill in math. It makes it easy to spiral standards, but also means it’s critical students master them so they’re ready for the next layer. I have standards based math data tracking sheets prepped and ready for you. They’re currently available for grades 1-4. They are prebuilt with the Common Core Math Standards, but are editable so you can change them to match the standards for your state. Or, add in additional skills you’d like to track.

My favorite part of these math data trackers are the student information sheets. These student data pages are auto-generated! They’re literally no extra work for you! They prepopulate based on the assessment scores you enter into the data spreadsheet. They display the student scores by standard, organized by domain. This makes them perfect reporting pages for parents, or to take with you to a data meeting like RtI, MTSS, or IEP case conferences.

I have math data trackers available for 1st grade, 2nd grade, 3rd grade, 4th grade, and now 5th grades. They are ready to be used, and are customizable to match your state standards, or district’s assessments. Each is tailored to your specific grade level and includes a video tutorial that walks you through how to use them. You can get yours sent to your inbox (for free) at Math Data Tracking Spreadsheets.

Phonics Tracking Sheets

Another pre-made student data tracker template I have available is my phonics tracking spreadsheets. This spreadsheet is built using the Quick Phonics Screener but is entirely editable to match whatever assessment you use. Tracking phonics data skill by skill is a great way to ensure students are making progress with their foundational reading skills. I’ve included the video tutorial below, but head to my Free Phonics Data Tracker Spreadsheet page to grab the tracker and read more about it.

Looking for a phonics assessment to use alongside one of my trackers? My Diagnostic Phonics Assessment & Custom Data Tracker allows you to assess students’ phonics skills in order to target your systematic and explicit instruction. The custom data tracker is built to match the assessment and includes both a phonics only and a phonics and fluency option.

Heggerty Phonemic Awareness Assessments Data Tracking

Do you give the Heggerty Phonemic Awareness Assessments? If so, this data tracker is for you! Designed for the Kindergarten and Primary Phonemic Awareness Assessments, this data tracker also creates individual student reporting pages. Designed similar to my Phonics Data Tracker, just enter in names and scores and the individual pages are created for you! Just head to my Heggerty Phonemic Awareness Assessments page to grab it for yourself!

You can use Google Sheets to create even more in-depth student data trackers! Do you use Google Sheets in other ways? I’d love to hear it in the comments below.

The post Student Data Trackers in Google Sheets appeared first on Tales from Outside the Classroom.

Before I continue, I should explain that I do not use the term “improper fractions” frequently in my classroom. I read the post “Why You Shouldn’t Say Improper Fractions” several years back from The Recovering Traditionalist and it made so much sense to me! There is nothing “wrong” or improper when the fraction is greater than one. However, I know teachers in future years will likely use it, and our current curriculum uses it. So, I do use it from time to time alongside “fractions greater than one”. It’s important for students to have familiarity with the term improper fractions since I know it will come up for them in the future, but I don’t use it to describe fractions greater than 1 in most of my lessons.

Fractions Greater than 1 Models

The most effective way to build student understanding of fractions greater than one is by using hands-on models. I always prefer to use student created, or atypical, math manipulatives rather than commercial options. For fractions larger than one, or even traditional fractions, I LOVE to use cookies! Soft sugar cookies are easy for students to cut apart with a plastic knife. The kids are highly engaged because they’re working with cookies- and if you’re anything like me, you’ll let students eat the cookies after they’re done. I walk through my step by step lesson in the video below. Obviously, it takes a bit longer when doing it in the classroom as I give students time to make the cuts, and build in additional talking time with a partner to process their thinking. In all, the lesson takes us about 45 minutes.

Through this interactive, hands-on lesson students should understand that when the numerator and denominator are the same, the fraction is equivalent to one whole. They also should understand that when the numerator is larger than the denominator, the fractions is representing more than 1. This lesson reinforces both the concepts of the numerator and the denominator and you also can use the same lesson to talk about equivalent fractions and comparisons, depending on how much time you want to spend, and student readiness for it.

This same lesson can also be done with Play-Doh. I’ve even done the lesson a second time, with students in need of additional reinforcement, using Play-Doh and scissors. Because students can easily form and cut circles, it makes a great tool. I prefer to use cookies, however, since they are all the same size and are more perfectly rounded than student created Play-Doh circles.

Fractions Greater than 1 on a Number Line

After working with fractions of a shape, our work moves on to fractions on a number line. For my students, this is the most challenging part of our 3rd grade fractions unit. Many of my students struggle with number sense and using number lines in general, so applying fractions with number lines is a challenge. Students should already have experience navigating fractions on a number line and partitioning lines to represent proper fractions before introducing fractions greater than one.

To begin with number lines, I connect our work with shapes. I do this to try and help my students that struggle to truly understand whole numbers and fraction relationships. I find if we begin with connecting the wholes on to the number line, and then partitioning both, it helps students see what the number line represents. I draw circles above the number line to represent that many number of shapes to match the whole numbers already displayed. I love to use the Number Line tool from Math Learning Center to help with this. Then, I partition the shapes and we talk about the fractions. Finally, I do the same with the number lines. Connecting to what students already know and understand with fractions and number lines is so important!

Frequently, my students struggle with denominators with fractions greater than one on a number line (often shapes as well). They really struggle to internalize that the denominator represents the number of fractions in each whole. I intentionally incorporate a TON of practice with number lines because I know my students will need it.

Interactive Sort

After my initial lessons with fractions greater than one, I like to continue to review the concepts we’ve learned. I use these task cards as a whole class sorting activity to review whole numbers as fractions and fractions greater than 1. Here’s how I do it:

• I place the signs for “Greater than 1”, “Equal to 1”, and “Less than 1” across the room.
• I hang up the visual model signs around the room.
• I give each student a fraction task card.
• Students begin by going to the area of the room that describes their fraction: less than 1, equal to 1, and greater than 1. This is the interactive sort part of the lesson.

After students are in place, I guide students to the correct area if needed and we discuss how we know based on a fraction’s numerator and denominator. This helps reinforce the generalizations we’ve been talking about. Then, students find their visual model match around the room. After again checking their work and guiding students that need support, we are ready to do another fraction.

I like to do a “snowball fight” with this. To do it, students gently crumble their piece of paper. When I say “go”, students start finding and tossing the “snowballs” until my chime goes off. Then, they immediately pick one up and go to the area of the room that describes their fraction.

Other Task Card Ideas

• Mix the fractions and visual models so that students have to think through both components of fractions. You would have to be intentional as you choose which cards to display you pass out the matches to what you have displayed.
• Give all of the cards to students and have them find their partners. This could be done before or after they identify their fraction as greater than, less than, or equal to 1.
• Do the same activities but instead of matching with the fraction, the match is the number line model.

This MASSIVE freebie has 104 pages! With 392 task cards, you can use these throughout your fractions unit! Task cards are included for fractions less than 1, equal to 1, greater than 1, and whole numbers as fractions. You can use them to sort and match visual shape models or number line models with fractions. You can use them to help students practice equivalent fractions or comparing fractions. Because they’re task cards, there are so many engaging ways you can use them in the classroom! In addition to the “snowball fight” activity above, you can use these to do “Solve the Room” or “Musical Math” activities.

Sign up with the form below and this will be sent to your straight away- along with ideas for how to use them within your fractions unit. I compiled these task cards with 3rd grade students in mind, but they certainly could be used with other grade levels strategically.

The post How to Teach Fractions Greater than 1: Hands-on Lesson and Engaging Sorts appeared first on Tales from Outside the Classroom.

Digital Multiplication Bump

Bump is a well-known partner game that’s typically played on a printed gameboard. You can find many versions out there- both with seasonal themes and general ones. When we first went online in 2020, I was looking for ways to continue to work on my students’ fact fluency. Since we typically work on fact fluency through classroom game play, I brainstormed how we could continue those games online. Digital Multiplication Bump came out of that need. I detailed more about it in my Free Digital Multiplication Bump post.

Since then, I’ve continued to use the digital versions in my classroom. Students can play side by side on their device. Or, they can play together by sharing one computer. Part of what I really like about it, is that it gives students a game they can play at home with their families. Since students take home their devices each night, they always have it available to them.

I have 3 different versions available: Multiplication Bump Factors to 6 Freebie, Multiplication Bump Factors to 10, and Multiplication Bump Factors to 12. The free version is great to try out and see how the game works. By focusing on the numbers on a traditional die, students get a lot of exposure with multiplication with factors from 1-6. The paid versions, however, have intentional practice with the range of factors. This is perfect for working on fact fluency with specific factors throughout the year. A great way to use this version is to play one version per week- giving kids repeated practice with a specific factor through game play. This helps to add that fact to students’ long-term memory for easy retrieval later.

5 in a Row

5 in a Row is my favorite thing I’ve ever created- and I think my students think so too. What started out as a fact fluency game, has turned into much more than that. I’m always working to add versions practicing additional math standards. My students love playing and will literally BEG for us to play when we have a few minutes to fill. My 3rd Grade Math Games Bundle has 9 different sets included with 33 total game versions! Fluency is a component of the game, so the skills included are ones where quick retrieval is beneficial. You can change the speed of the game, however, so it can be played at the speed most beneficial for your students. These are some of my 3rd graders’ favorites games!

5 in a Row is easy to play! You print the gameboards and cut twice to make them into strips. Then, launch the game on Powerpoint. That’s it! It’s so easy to prep! I keep a stack of gameboards printed in my classroom, so we’re ready to play at any time. I always have a variety of skills prepped so we can work on different skills each time we play. This makes it easy to play as a fill-in during down time, but I try to play several times a week by carving out time in my math block. Depending on the game version, you can play one game in less than 5 minutes. That includes time for board distribution and checking answers.

The best part of 5 in a Row? The super awesome randomize feature that makes every round you play different! This keeps things exciting for students, and ensures they’re working on a range of content with every game play. You can find out more about 5 in a Row and how it’s played in my blog post. But let’s take a closer look at my 3rd Grade Math Games bundle.

Multiplication & Division Fact Fluency

I have 3 different game versions that work on building multiplication & division fact fluency: Multiplication Facts 0-12, Division Facts 0-12, and mixed Multiplication and Division. Each version has four different games included so that factors are broken up into manageable chunks. They’re broken down into factors 0-5, factors 6-10, factors 0-10, and factors 0-12. I always begin playing with Multiplication 0-5 and build up students’ proficiency and confidence. I then often switch the division because making that transfer can be so difficult for many students. As students become more proficient with their facts, I switch the 6-10 version. As a third grade teacher, I don’t play the 0-10 version until the very end of the year.

I have the game versions bundled and discounted. My Multiplication & Division 5 in a Row Game Bundle is available in my TpT shop.

Place Value & Number Sense Games

I also have several versions that work on building place value and number sense skills. While we wouldn’t often think of them as fluency activities, we also want students to become fluent with them. The speed component sometimes helps with overthinking. It helps students go with their first thought. At the end of the day, though, it’s another way to give students practice in an engaging setting. The speed can be increased several seconds if students need additional thinking time so that speed isn’t really a component of the game. This helps ensure students don’t get stressed by the time component, but it keeps the game as autoplay without you having to click through.

Place Value to 1,000 5 in a Row practices standard, expanded, word form, and base ten representations with 3-digit numbers. 6 different game versions are included with 2 versions of each type: base ten blocks, expanded form, and word form. Play more than one version in a day to practice several skills.

Mental Math to 100 5 in a Row is one of my favorite number sense building games. I’ve used it in 1st through 3rd grades and have found it tough for each of the grades. Students build number sense skills as they skip count the numbers shown to identify the number. This helps build their fluency switching between numeric places, building an understanding between ones, tens, and hundreds. One of the versions includes counting by 25 to help build students’ skills with counting coin values.

Fractions Games

Fractions are a foundational 3rd grade skill. I’m always looking for additional ways to work on fractions in engaging ways because my students need lots and lots of practice. I currently have 2 different fractions versions in the 3rd Grade Bundle: Fractions of a Shape and Fractions on a Number Line.

Fractions of a Shape 5 in a Row has students identify the fraction shown by the partitioned and shaded shape. Then, they match it with the fraction on their gameboard. This free version is a great way to try out 5 in a Row in your classroom! It gives you a chance to try it out with a typically somewhat easy skill. Because it’s free, it’s a great game to use to be sure it works well for your classroom systems. You can download Fractions of a Shape 5 in a Row to try it out today.

Fractions & Mixed Numbers on a Number Line  includes 4 different fractions games. One version includes practice with fractions less than one on a number line. Another includes mixed numbers with fractions larger than 1. Each game has included fractions represented on the number line. It also includes versions with only one fraction represented, upping the complexity of the task.

Multiplication Roll & Cover

Roll & Cover is another well known game that’s pretty similar to Bump. If you aren’t familiar with it, it’s pretty simple to prep and play. Students roll 2 dice, or roll one twice, and multiply the numbers together. Then, they cover the product on their gameboard. The first player to cover their board is the winner. Students can play on duplicate versions of the same gameboard, or each student can play with a different set of numbers. The only materials needed are dice and counters. I love easy prep games that build students’ fact fluency with continued play and Multiplication Roll & Cover is great. You can download 10 gameboards for free by filling out the form below. They’ll be sent straight to your inbox!

Roll & Round

Roll & Round is a fun partner game that practices rounding to the nearest 100 or rounding to the nearest 1,000. Each partner has a different objective- rounding to 0, or to 100/1000. After rolling the dice, players try to create a number that rounds to his/her specific objective. The version that rounds to the nearest 100 only works with two digit numbers, whereas the version that rounds to 1,000 works with three digit numbers.

Roll & Round helps build students’ number sense as students have to reason through which number they should create out of their possibilities. It also helps work on rounding. A number line is included with the middle placement, to help support students that need it.

Roll & Round is ready to play upon printing. Just add a dice, or several, and go! This is great to play on laminated pages, in a sheet protector, or a dry-erase sleeve. Download Roll & Round to try it out today!

Left Right Learn

Left Right Learn is a simple, easy prep game that you’ll want to add to your 3rd grade math games collection. All you need is a game piece for each player- I love to use mini erasers! Students draw a card, solve the fact, and move given the directions on the stated card. It’s a fun way to give students ongoing practice with multiplication facts. I have two different versions. The free one practices facts 0-5. I love to use this early in the year to give students repetitive practice with these facts. I find that multiplication feels less overwhelming when they feel confident with a bank of facts. We spend a good amount of timing working with facts 0-5 before transitioning to working with others.

Multiplication Facts 0-5 Left Right Learn is available for free from my TpT store.

I also have another version that works with Multiplication Facts from 0-10. 5 different games are included. Because each card practices a range of facts, students get new practice with every play. My favorite part of Left Right Learn is how simple it is to prep. Just print, and cut out the gamecards. Of course, you can also laminate the boards and/or cards. If I print on colored paper, or cardstock, I usually don’t worry about laminating. I like to put the gamecards in a cup or bag and we’re ready to play. You can take a closer look at Multiplication Facts 0-10 Left Right Learn in my TpT store.

I hope you’ve found some new 3rd grade math games to incorporate into your independent practice. Games are a great way to build student skills in fun and engaging ways.

The post Easy Prep 3rd Grade Math Games that Make Learning Fun appeared first on Tales from Outside the Classroom.

“How do mathematics educators effectively and meaningfully incorporate algebraic ideas into K–5 curriculum? When elementary teachers are unfamiliar with early algebra, lessons designed and labeled as algebraic may become arithmetic exercises; the algebra then remains hidden from both the teacher and students in the implementation. The result is that the algebra standard is only superficially addressed.” -Earnest & Balti

Math Discourse

Building strong algebraic thinking comes from continued exploration of math concepts in different formats. Asking for students to explain their strategies when solving, and asking others to share if they did it in a different way, not only builds students’ strategy use, but also their algebraic thinking. By thinking critically about another students’ steps, students evaluate mathematics. Discuss the steps and why they work (or don’t work!). Ask students if another student’s work is correct. Asking students to share their strategies also helps other see that it’s okay, even great, they got to the same answer in a different way, or that they thought of the problem differently. Research shows that students are stronger in mathematics if they can represent a problem in more than one way

I incorporate math discourse throughout our math block, but am very intentional about our discussion during our Word Problem of the Day. It enables us to focus on one problem for 10-15 minutes, so we can go deep into the problem discussing the word problem’s context, representations, and student strategy use. Math discourse is powerful and builds deep thinking around concepts. In each of the topics I discuss below, discussion is a critical part!

Interested in these free math talk posters? Click the picture to head to my Word Problem of the Day post to grab them!

Building an Understanding of Equality and Equations

Understanding equality is a pivotal part of algebra, but truly all of our mathematics instruction. It is important that students see that the equal sign represents a relationship– that the sides have the same value. Often, students see the equal sign as an indication of something to compute or solve. There are several ways we can present our computational problems that help students build an understanding of equalities.

Have you ever presented your students a seemingly simple equation thinking they’d easily solve it correctly, but they don’t? It can be eye-opening.

9 + 6 = ___ + 5

If you gave your students this problem, how many of them would answer 15 instead of 10? My experience says, a large number – especially if there hasn’t been an investment in building algebraic thinking, or specific work with the equal sign. Every single one of my 3rd graders in one class said 15!

This type of representation can be used with any grade level of students. Very small numbers within 5 can be used with kindergarteners. Equations such as this one can start with first graders. Using manipulatives and models will help students understand how these equations work. And it can be applied with larger numbers, and other operations as well.

As time goes on, we want students to be able to build relational thinking. We want them to start working with larger numbers, and use their number sense and general math understanding to solve without computing in situations where that’s possible. For example, we want student to be able to solve the following equation.

259 + 63 = 260 + ____

This equation shows the algebraic thinking we want students to be able to perform that’s discussed in the quote. Moving beyond just arithmetic and computation and truly thinking about the numbers, equal sign, and what the equation represents. We want students to be able to quickly and easily identify that 62 is the missing number. The discourse around this representation can help students solidify and expand their algebraic thinking.

Unknowns in Any Position

One of the easiest ways we can incorporate intentional algebra practice with young students is to include unknowns in any location in an equation and moving the location of the equal sign. By working with various formats of equations, students build an understanding of what each of the numbers represents and the purpose of the equal sign. For example:

_ + 2 = 5

5 = _ + 2

Both equations show an unknown addend. Many students will solve the second equation differently than the first. With a poor understanding of the purpose of the equal sign, students struggle with solving the second example. Even students much older than 1st grade! With continued exposure and dialogue with the various formats of equations and the meaning of the operations, students develop proficiency with both the operations and equality.

Balancing Equations

Balances are a powerful tool for building students’ algebraic thinking and mathematical understanding. Balances are often first introduced in first grade when trying to build the concepts of inverse operations with addition and subtraction and an understanding of the equal sign. Students often work to identify the unknown addend that balances the equation. Often, however, the balance doesn’t continue in our math classrooms as students get older. This is an easy way presentation that helps students look at and think about things a bit differently than we typically teach. This is especially true when working with all four operations. By presenting the equation as objects in a balance, students can see how the numbers relate to each other and what the operations signify.

My Two-Step Equations Balancing Equations shown here are a great segue between solving one-step equations and two-step word problems. We often jump straight into the word problems for this 3rd grade standard, without enough practice with the computation. These balances show an equal group representation with an unknown factor. Students work to identify the unknown. These are also a great way to intentionally practice writing equations with variables for the unknown. All of the numbers are under 100 because the purpose of these is to build the algebraic thinking behind them- not the computation.

True or False – Working with Inequalities

Another similar concept is true or false equations with expressions on both sides. The key here is to try and move beyond rote computation. Can students make connections between the numbers to help them answer? Can they use their number sense to identify if something is an equality? For example:

101 + 22 = 100 + 23

We want students to be able to immediately identify that the equation above is true without having to complete the computation, just like with the previous examples above. But, by giving two expressions that need to be computed, we help students see that the equal sign doesn’t mean to compute, but shows the relationship between both expressions. There isn’t a missing number box, or a variable for an unknown. Instead, they’re applying their learning by answering a true/false question.

Be sure to give students examples with inequalities as well! A great next step is identifying what would be needed to make the equation true. I have seen kindergarten students engage in this discussion! Students likely have different responses to that question since there usually several different possibilities of changes. But that’s some of the power of working with inequalities- the discussion that’s drawn out from the responses.

My True or False Equivalent Expressions Task Cards work with smaller numbers so they can be easily used with young students but would be good support for older students that struggle with algebraic thinking. With these cards, students must add both sides of the expression to determine if they’re equivalent. They are modeled on a balance to give students exposure with balance scales and to demonstrate that both sides should be equal. As with everything else, the value comes from writing the equations and having the classroom discourse around the models.

Want to grab these? Fill out the form below and they’ll be sent to you straight away!

Writing Expressions and Equations

Another easy way to bring algebra into elementary classrooms intentionally is by having students write expressions and equations for models and word problems. Often, students explain the steps they did when solving and demonstrate their understanding of the context. But a great way to provide support for others that aren’t as successful, and to connect their existing understanding to algebraic thinking is by writing it as an equation. By doing this, students learn that variables represent unknown quantities. They also demonstrate that they understand the context of the word problem and the mathematical concepts behind them. We practice writing equations with each of our daily word problems, and again to match models that are often represented in our textbooks. To read more about the steps in my problem solving routine, I detail it in my 3 Reads Strategy for Problem Solving post.

Students need practice writing equations rather than just solving them to truly build an understanding of what’s being represented and what the equation identifies. For students needing support with writing two-step equations, or understanding the differences between addition and multiplication, my Writing Two Step Equations task cards may be helpful. Students are able to see the difference between equal groups and additional items added.

For older students, my Multi-Step Balancing Equations Task Cards are great for building an understanding of the difference between expressions and equations, working with variables, and talking about the operations. With each task card students have to complete computation on each side of the balance before finding the value of the unknown. With each of them in this set, students must identify the missing factor.

Applying Algebra – Generalizations, Patterns, & Functions

Functions include building, representing, and reasoning with relationships between numbers. With functions, students generalize relationships between numbers, represent those relationships, or functions, in multiple ways including expressions, equations, tables and graphs; and reason with those generalizations. Functions are what we think of most as algebra- applying a formula to solve.

Patterns

Work with patterns begins in pre-k and kindergarten. Their work often begins with repeating patterns in shapes. Over time, students should begin to notice patterns with numbers. Beginning in first grade, and continuing through, students should be exploring patterns they notice, such as those on the addition table or multiplication table. Have you ever asked students if the sum of two odd numbers is even or odd? That’s a pattern. It’s also a generalization. Making generalizations helps students understand “rules” of mathematics. Why things work. If we look at 3, 6, 9, 12, one may say that the rule is add 3. That tells us the pattern that we follow to find the next number. But how do I find another number in the pattern without continuing to add 3 to the one before it? I generalize.

Generalizing- To continue the pattern, how can we skip ahead without solving?

Generalizations are a fundamental component of algebra. To generalize, one must first understand the relationship between and among quantities. Reasoning with those relationships is making generalizations. One of the first generalizations we often explore with students is the commutative property of addition. Through exploration, students are typically able to understand it on their own.

Let’s go back to the example above. If I want to find the 50th number in the pattern, I can generalize that by multiplying the number in the pattern by 3 will allow me to find the 50th number. The generalization is that if n stands for the numbered place in the pattern, and I multiply it by 3, or n x 3, I will find the answer.

Algebraic Functions- Input/output tables, Rules, Function Machines

We call these so many different names. I’ve seen them as input/output tables. I’ve shared Function Machines printables. But at the end of the day, they all have the same focus: helping students understand that a function is when an action is applied to a number resulting in a new one. Functions are patterns- linear patterns. When solving, students likely are identifying a pattern to help them solve.

“If a horse has 4 legs, how many legs do 8 horses have” would be an example of a function. These Input/Output Tables Task Cards give students the rule, or the equation, they have to follow to complete the rest of the table. This can easily just become a computational activity. Connecting them to the horse example given here helps students see the real-world relevance of the table and how it can be a helpful tool for them.

With input/output tables, it’s important that the input figures aren’t sequential. Students will often quickly identify the pattern and continue it going down the output side, continuing the pattern without generalizing. It takes away the algebraic component. While it’s natural that this is where many elementary students will begin, it’s important that we get them to identify the recursive pattern- how we move from input to output- the generalization. By changing the numbers by skipping ahead, or working with larger numbers, students have to identify the pattern in order to identify the output. This is much harder work than just continuing the pattern and students will need a lot of support with it.

These Two-Step Input/Output Task Cards have students identify the rule that’s presented. These are SO TOUGH! As students practice with them and build their number sense and algebraic thinking, they become easier. I use these with my 3rd graders and it’s a great challenge for them. As older students work with linear equations, these task cards are great practice for them as they work to not only understand but also write them.

Each of the task cards shown here as examples, are just that- examples. Each of these concepts can be easily incorporated into the work you are already doing! My Accessing Algebra line is designed for the middle and upper elementary grades. These print and go task cards help you integrate algebraic concepts into your every day instruction with very little prep-work. You can see my entire Accessing Algebra line in my shop.

Finally, math should be taught with a focus on understanding, not as a set of procedures. Our presentation and ongoing instruction throughout the year helps to build algebraic thinking. Over time, students become used to looking for patterns, generalizations, and structures. Ask questions like: “Is this always true? How do you know? How can you represent that? Did someone solve that differently? Can it be explained in a different way?” Questions like this help students move away from specific arithmetic examples and into generalizations. We can connect algebra to our arithmetic to help develop more proficient math students that are ready for algebra success.

For more on integrating algebra with elementary students, Project LEAP and the book Connecting Arithmetic to Algebra (affiliate link). You may also enjoy the quick read Instructional Strategies for Teaching Algebra in Elementary School: Findings from a Research-Practice Collaboration by Earnest & Balti

The post How to Integrate Algebra with Elementary Students: Equality, Equations, and Discourse appeared first on Tales from Outside the Classroom.

This post takes an in-depth look at each of the multiplication & division word problem types. I also give some helpful links at the bottom that discuss Cognitively Guided Instruction, or CGI problem types. They use slightly different vocabulary than the CCSS but are the same set of skills. An important note- when I talk about teaching the multiplication & division word problem types explicitly, I don’t mean that they’re only practiced during a specific unit, or that we teach specific keywords for each problem type. Students need to read the problem and understand the context. Recognizing the problem type can support that work, but it’s not focused on keywords. I also do a daily word problem outside of our focused unit work. I firmly believe in spiraling the standards in word problems so students have to focus on context to solve. For more support with problem solving, check out my Why Your Students Struggle with Word Problems, and What You Can Do About It post.

An important note before we begin-throughout our exploration of both operations, we often return back to the total. We identify in our first lesson that multiplication allows us to find the total number of objects arranged in equal groups. When we begin exploring division, we know the total number of objects we have and have to find the unknown in connection with groups. Throughout our work, with arrays, area, and equal groups, we continue to come back to the concept of total to help students identify the operation needed. It is through identifying the total as known or unknown that students will truly understand the multiplication and division operations. As you will see, that’s a focus with each of the multiplication and division word problem types.

 Equal Groups as Multiplication & Division

In my Introducing Multiplication and an Working with Division posts I walk through my hands-on, engaging lessons I use to introduce each operation. When introducing both operations, we focus on equal group arrangements. It’s crucial for students to understand that multiplication and division works with equal groups so it makes sense to begin here. When we first start working with multiplication and division word problems, I ask students to identify the items that are arranged in groups, and identify what that group looks like. For example, the group might be a box or a bag. Those objects could also just be objects in other scenarios. So, identifying that an object is grouped within something helps students to understand the word problem and that an equal group scenario is present.

MULTIPLICATION – Total Unknown

These are the word problems we typically think of with multiplication. It’s an arrangement of items into equal groups to find the total number of items. Identifying that the number of groups and the number of items into each group are known, while the total is unknown, helps students solve these word problems.

QUOTATIVE DIVISION – # of Groups Unknown

When the number of groups are unknown, it is a quotative division problem. Let me clear, I don’t teach my students the term quotative. Rather, I teach them that there are two types of division. When students are able to identify that the total number of objects is known, but the number of groups is unknown, they then know that they are dividing. It’s important that students have lots of practice with both types of division.

PARTITIVE DIVISION – # in each Group Unknown

When a word problem identifies the total number of items and how the items are arranged, but doesn’t give the number in each group, these are partitive division. Think of it as partitioning. The number of groups is known so the partitions are known. Again, I don’t teach this term with my students, but it’s important that students know that division is presented in two ways.

Once both operations have been introduced and practiced separately, I like to use my Multiplication & Division Scenario Tables Task Cards to practice writing equations and understanding group arrangements in context. I find that removing all of the components of the word problem, and working through just the total and group information, helps students truly understand the operations in connection with equal groups. This helps them then apply their learning within word problems. Students get practice with different ways items can be grouped, which helps them look for and identify grouping relationships in word problems later.

Arrays as Multiplication & Division

In conjunction with teaching equal group arrangements described above, arrays should also be included. Arrays are how students are first introduced to multiplication in 2nd grade. Students should have background knowledge with rows and columns vocabulary. Arrays are how I connect what we’re learning now, with what they learned before. Therefore, I teach arrays at the same time that I teach equal group arrangements. I correlate rows and the number of groups. I always emphasize that the columns are the same as the number in each row (or the group).

It’s important to note that the CCSS lay out two rigors of array word problems: ones where the items in rows are presented, and those that give the word problem in rows and columns. Through continued exposure and practice, students understand that these are interchangeable. Array word problems are not very common, but are definitely a problem type that students should have exposure and practice with. They also lay the foundation for students to work with our next word problem type: area

Area as Multiplication and Division

Area is a critical component of our multiplication and division work. It’s a major part of 3rd grade math instruction and it’s important that students work with many word problems where they have to recognize area in context and solve. This is especially important to learn how to differentiate between area and perimeter by making connections between arrays and what is known about multiplication and division. Area should be introduced with arrays and connected to an area model. This concrete, visual example helps students connect what they already know (arrays) with their new learning (area). Unit squares are written in the standards to be this bridge. Area models continue on as you work with more the more complex rectilinear figures, and to help students understand the distributive property of multiplication. The CCSS don’t list area problems as their own line, rather, they’re included with arrays. While this certainly makes sense since arrays can be created inside the rectangle, and that’s how area should be introduced, most area word problems don’t use rows or columns when providing side lengths. I think it’s an important distinction to look at area word problems separate from arrays.

MULTIPLICATION

Solving area problems again connects back to total and known/unknown values. When students are trying to identify the total area of the shape, and are given both side lengths, students need to multiply to find the area.

DIVISION

When students are given the total area of the shape, but need to find a missing side length, they must divide.

Ultimately, if students have a solid understanding of multiplication and division word problems and their connection to total, they will likely be successful with area word problems.

Comparison Word Problems

The most complex multiplication and division word problem types are comparison word problems, in my opinion. Just as they are in addition & subtraction, the way these problems are structured, cause them to be more complex than the others. For these, it’s important to look at each of the different problem types and digest what’s being given. I use a 3 Reads Routine when working through word problems, and these problems always bring the greatest discussion as students grapple with the information that’s presented.

The multiplication and partitive division (constant unknown) examples have very similar wording since they both have the comparison statement: “6 times as much”. This is a description of the relationship between two groups-the total, and the comparison group.  The quotative (comparison) example is worded differently since the unknown is the comparison relationship: “How many times as much does”

The key to understanding multiplicative comparison problems is that the comparison is the repeating groups, whereas the constant is the group size.

In a multiplication word problem, you are told the amount in the comparison group, 6, and the constant, 4. Each bag of apples is $4, so if it’s 6 times as much, it’s the same as having 6 of the bags. Thus, 4 x 6.

In a (partitive) division word problem, you are given the total number (the cost of the box of apples). Once you partition, or divide, that cost into 6 equal groups, you are able to identify the cost of one of those groups, which is the same as the cost of the small bag of apples, and that is the same as the number of ducks.

In a (quotative) division word problem, you are given the constant, the cost of the small bag, along with the cost of the box of apples. The unknown is the relationship between them. With these word problems, the wording of the question can often throw students off.

Once students are able to decompose the word problem to identify the total, and the comparison, they’re able to connect that to what they already know about multiplication and division to help them when solving. Teaching students to closely read the word problem is key. To read more about my 3 Reads Routine you can click the image below.

Looking for additional resources? You can download my Multiplication & Division Word Problem Types reference sheet and use it to help identify and write the different word problem types for classroom practice.

Want additional multiplication & division word problem practice? I have a FREE Multiplication & Division Word Problem Task Cards set. Each of the multiplication & division word problem types discussed here are practiced during the 24 task card set. This is great practice for later in the year to ensure students have ample practice with each type. It would even be a great tool to help students discuss the different problem types and how they’re presented. To download the Multiplication & Division Word Problem Type Task Cards, just fill out the form below and they’ll be sent straight to your email.

You might also find the following links helpful.

Partitive & Quotative Division from SFUSD

CGI Resources from LAUSD

Multiplication & Division Word Problems from ND Counts

Two Ways of Thinking of Division from Langford Math

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