The post Flexibility with Place Value appeared first on Math Coach's Corner.

]]>Just a quick post about place value and how important it is for students to be able to decompose numbers by place value in different ways. Here’s what I mean. Typically, we think of 34 as 3 tens and 4 ones. That’s certainly true, but it can also be decomposed as 2 tens and 14 ones. Why is that important? As you can see from the picture below, it’s the fundamental understanding behind subtraction with regrouping!

Kiddos need lots of hands-on practice decomposing numbers in different ways. And when you turn it into a game, it’s oh so much more engaging. Today I have a little game called Close to 200. Players roll two dice and create a 2-digit number. They then model that number using base-10 blocks in two different ways. After five rounds, the players add their numbers, and the player with the smallest difference from 200 wins. It’s okay to go over two hundred–202 has a difference of 2 from 200. The twist of strategy in the game makes kiddos have to think carefully when creating their 2-digit numbers. If they roll a 2 and a 5, should they use it as 25 or 52?

Remember that if you are using digital resources, you want to look for resources that provide movable pieces to simulate hands-on learning. For this little game, I’ve provided both a print and digital version. Download your freebie here. There are instructions in the PDF for downloading the Google Slides file. I hope you find these resources useful! Please share this post with others who might be interested.

See you on Twitter! @MathCoachCorner

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]]>The post Flexible Number Tasks appeared first on Math Coach's Corner.

]]>A sweet friend recently reminded me of an activity I created and shared a number of years ago, and I decided to dust it off and extend it! I love activities that are flexible and repeatable, unlike worksheets that are pretty much one-and-done. These flexible number tasks fit the bill.

The original task was for numbers within 20, and the extension I created, shown below, is for fractions. The activity involves rolling two standard dice. On the recording sheet, they are pictured as two different colors, but that’s not really necessary. The first roll indicates the fraction that students will be working with and the second roll indicates the task they should perform with that fraction. For example, if they roll a 4 and a 3, the would use the fraction 2/8 in an inequality. For example, they might write 2/8 < 1/2 or 2/8 > 1/16. You can see that it is very open-ended. They just keep rolling and performing tasks as long as you want them to.

The fraction and whole number versions of this activity, along with a blank version so you can create your own, are part of a free activity sampler with over 15 print and digital resources that you can download using this link.

I hope you find these resources useful! Please share this post with others who might be interested.

See you on Twitter! @MathCoachCorner

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]]>The post Using Manipulatives for Multi-Digit Multiplication appeared first on Math Coach's Corner.

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The standard multiplication algorithm is probably the most difficult of the four algoritms when students have not had numerous opportunities to explore their own strategies first. As with other algorithms, as much time as necessary should be devoted to the conceptual development of the multiplication algorithm using concrete- or semiconcrete models, with the recording or written part coming later.(Teaching Student-Centered Mathematics,2018. Van de Walle, Karp, Lovin, Bay-Williams)

This post contains affiliate links, which simply means that if you purchase the product I receive a small commission.

Manipulatives, thought by some to be useful only for young children, are appropriate for students of all ages and should be used whenever new concepts are introduced. The use of concrete materials, such as manipulatives, helps students visualize abstract concepts and builds understanding. Consider, for example, using base-ten blocks to build an understanding of multiplication.

While the standard algorithm is still the end goal for multi-digit multiplication, using concrete and visual models emphasizes the importance of place value when multiplying. This leads to better understanding, which leads to greater student success.

*Direct modeling* involves using manipulatives to represent a problem. Using direct modeling, the problem can actually be solved without any written record. Consider the early stages of teaching multi-digit multiplication. Base-ten blocks are used to model 4 x 23 as shown below–four groups of 23.

Even without being told to do so, students will likely group the tens together and the ones together to determine the value. They can skip-count by tens to find the value of the tens and then count on by ones to find the product, or they can count the tens, count the ones, and combine the tens and ones. Either will arrive at a product of 92. A written record could be added showing 80 + 12 = 92. Notice that no *regrouping* of the ones is needed to solve the problem in this way, although the understanding that 92 can be composed of 9 tens and 2 ones or 8 tens and 12 ones is now a part of most place value standards for good reason.

So what’s next then? Moving from equal groups to the area model creates a stronger connection to the standard algorithm and also emphasizes place value. An area model for 4 x 23 modeled with base-ten blocks is shown in the picture below. You can see that using the area model involves decomposing the 2-digit factor into place value parts. You still see 4 groups of 23, but now they are rows. As with the last model, students can determine the product by simply combining the value of the tens and ones.

A written record can be combined with the area model to introduce the partial products method.

After ample experience using the base-ten blocks to build the area model, students can transition to a more symbolic representation, as shown below.

I want to talk for just a minute about distance learning. It’s important that we don’t bypass the concrete stage due to remote learning. It’s important to use resources that incorporate movable pieces to provide that visual experience. The websites listed below also feature free virtual base-ten blocks. Each of these websites allows you to turn the pieces horizontally to more easily create an area model (some others don’t).

I hope you find these resources useful! If you have any others to share, please drop them in the comments.

See you on Twitter! @MathCoachCorner

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]]>The post Resources for Digital Learning appeared first on Math Coach's Corner.

]]>Even as we left school in May, I think most of us assumed that the 2020/2021 school year would be unlike any other. Like many of you, I spent a good part of my summer learning more about digital learning. And while many of us connect digital learning exclusively with distance learning, the truth is that many educators were using digital learning in their classrooms long before the pandemic forced us all home. Today I’m linking to a few resources that I have found very useful for overcoming the steep learning curve many are experiencing.

One of the big concerns I’ve heard voiced about distance learning is how to continue providing students with the concrete learning that is required for developing a deep conceptual understanding of math concepts. Thankfully, I came across a great resource from Julie Smith, The Techie Teacher. She created an amazing resource with links to an incredible array of virtual math manipulatives. As a K-5 Ed Tech Consultant, she’s a good one to follow.

For many educators, the idea of planning for distance learning seems daunting. How do you build a classroom community with students you haven’t met personally? How do you keep students engaged with online lessons? How do you teach all of your standards with schedules that often feature shorter class periods? I came across this new resource from Corwin Press while browsing my Twitter feed, and I immediately ordered it from Amazon*. *The Distance Learning Playbook* was literally just published in July, and I felt grateful to get my copy! It’s not a resource you need to read and absorb all at one time, rather it can serve as your guide in the field.

*This is an affiliate link, which simply means that if you purchase it, I receive a small commission.

Finally, many of us are learning new technologies. I know I am! If there’s one thing I’ve learned from my millennial son, it’s that’s there’s a lot to be learned from YouTube videos. That’s where I found Michelle from Pocketful of Primary and her amazing videos! She has a wealth of knowledge about digital learning and a delightful on-camera presence. I’ve included one of her videos here, but be sure to check out her YouTube channel.

I hope you find these resources useful! If you have any others to share, please drop them in the comments.

See you on Twitter! @MathCoachCorner

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]]>The post New Digital Math Games! appeared first on Math Coach's Corner.

]]>The other day, I posted a picture of something I was working on to Facebook and Instagram, and I was blown away by the response! Yes, I am working to digitize my resources. This is not only in response to the increased need for resources to support remote learning in the near future, but also to accommodate the use of technology in the classroom when school returns to normal. I will be putting a lot of effort into this project over the next couple of months (and beyond, I’m sure), so be sure you are on my mailing list to receive updates!

You all know that I’m a big fan of low-prep, high-yield activities. When choosing games, I like to use familiar game formats, so I don’t have to spend time going over the directions each time I introduce a new game. The game is played the same, but the skill students are practicing changes. One of my favorite games is Capture 4. The goal of the game is to capture four spaces in a row, either horizontally, vertically, or diagonally. It’s similar to a tic-tac-toe game. So, for my first foray into digital resources, I made digital versions of my Capture 4 games for Addition and Multiplication. These particular games are SO low-prep that all students will need is the Google Slides file. No dice, no playing cards, nothing but the file.

As you see in the picture below, players take turns claiming two spaces with addends that result in a sum of 10 (since this is the board for making 10). You can see that for the first round, the blue player chose 2 and 8 and the green player chose 4 and 6. On his next turn, the blue player might choose the 9 and 1 in the same row where he has already claimed the 2. The green player would then be forced to choose the 6 on that row, along with one of the 4s, to block the blue player from getting four in a row by covering the 2, 9, 6, and 1. The green and blue playing pieces are movable, so players just slide them off the pile and over to the spaces on the board. Note that both players need to be in the same file, with edit privileges, so they can see each other’s moves and slide the pieces. Students can talk on the phone while they’re playing to provide social interaction. The resources have detailed instructions for using the digital resources with Google Classroom as well as information for getting started with Google Classroom, if you’re not using it yet.

Want to check them out? I created a free sampler with one board for Addition (Make 10) and another for Multiplication (Make 24) in both print and digital version. If you like those, check out these full versions:

- Addition with game boards for making the numbers 5, 6, 7, 8, 9, 10, 12, and 20
- Print and digital version (note, if you already own this resource, download the newest version which includes the digital game boards)
- Digital version only

- Multiplication with game boards for making 12, 18, 24, 36, 48, 56, 60, and 72.
- Print and digital version (note, if you already own this resource, download the newest version which includes the digital game boards)
- Digital version only

See you on Twitter! @MathCoachCorner

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]]>The post Making Ten Back and Forth Game appeared first on Math Coach's Corner.

]]>Ten has a special place in our number system, so it is critical that children know all the combinations for making ten. A quick look at where this skill goes in first and second grade helps emphasize the importance of mastering this critical skill by the end of Kindergarten.

A powerful strategy for learning basic addition facts is the Make a Ten strategy. This strategy can be used when one of the addends is close to ten, as in the fact shown below. To apply the strategy, children think of the number that goes with 8 to make ten, which is 2. They then decompose the 5 into 2 and 3, resulting in connecting 8 + 5 to 10 + 3. With practice, children develop automaticity with 8 + 5 and will no longer need to derive the sum, but learning through a strategy-based approach rather than through rote memorization develops number sense and stresses the relationships between facts.

The Make a Ten strategy that was applied to basic facts in first grade can be extended in second grade when children are learning multi-digit addition. Rather than making a ten, they are making the next multiple of ten, realizing that if 7 and 3 make 10 then 37 and 3 make the next multiple of ten as shown in the example below. Notice that if this problem were being solved with the standard algorithm it would require regrouping. Experience with alternate strategies such as extending the Make a Ten strategy leads to better understanding once children do tackle the standard algorithm.

A visual reminder in the classroom, such as this anchor chart, is a great way for students to master all of the combinations for ten. Use a ten-frame and two-color counters to illustrate the combinations and create the T-chart shown on the left of the anchor chart. After recording the corresponding equations on the right side, allow students to notice the patterns and make connections with the turnaround facts.

Children also need practice to develop mastery with the combinations for ten, and we want that practice to be engaging! I just made a new little Back and Forth game for practicing the combinations for 10. There are two versions of the game board–one that has ten-frame images for kiddos who need pictorial support and one that has expressions with a missing addend. Back and Forth is a fun game that’s a little like tug-o-war. Bother players use the same playing piece, but move it in different directions. The game can go on for a while, so it provides lots of practice. You can download your free copy here.

Check out these other posts with games for making ten:

- Literacy Connection for Making Ten: Ten Flashing Fireflies
- Game for Making Ten Using a Deck of Cards
- Free Apps for Making Ten
- Another Card Game for Making Ten with a Magical Twist

See you on Twitter! @MathCoachCorner

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]]>The post Real-World Math: My New Picket Fence appeared first on Math Coach's Corner.

]]>As I write, the thump of sledgehammers and the whir of buzz saws can be heard outside my kitchen window. In my small community of houses, we all have white picket fences in our front yards. It was actually part of the charm that sold me on the house. The fences, however, are now 15 years old and starting to show the wear and tear of the years. The past few months have seen many of my neighbors replacing their fences. As each new fence goes in, I find it impossible not to scrutinize the workmanship. In my mind, a picket fence requires precision, and small inconsistencies like uneven pickets or cuts really stand out to me. Last week, the *perfect* picket fence went up and went up quickly in a neighbor’s yard! Each day on my twice-daily walks I looked for imperfections, but I just could not find one. I contacted my neighbor to get the contractor’s number, and today my new fence is going in.

We often think about real-world math as a way to engage students in problem-solving. But this situation got me thinking about how essential math skills are in operating a successful small business. Stop for just a minute and think about all of the math that took place between yesterday when the contractor provided me a quote through the completion of the project and how that impacts the success or failure of this businessman.

How could you turn this into a problem for your students? Let them work in groups and take on the role of the fencing company owner.

- What information would they need to generate a quote for a customer?
*(cost of materials, cost of labor, desired profit margin, length of fencing, etc.)* - What other considerations are involved in developing a quote?
*(quote must be comparable to other companies doing business in the same area, ability to schedule and carry out the work, etc.)* - What math skills are involved in building the fence?
*(very precise measurement, leveling of the fence, spacing of posts, etc.)*

Sure, we could generate a lot of word problems for students to solve related to this situation, but giving them an open-ended challenge such as the one above immerses them in the math in a way that solving word problems can’t. Will every group think of every way that math is used? No, but when the teams all present their findings, I’ll bet you’ll have most of the bases covered. And if there is something important that none of the teams think of, a few well-placed questions from you can coax out additional ideas.

I would love to read comments about how you might use, or have used, a problem like this in your classroom. Drop a comment below or post one on Twitter! Be sure to tag me! @MathCoachCorner

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]]>The post 1st Grade Essentials: Basic Fact Strategies appeared first on Math Coach's Corner.

]]>With pages of math standards at each grade level, what is really considered essential learning? In this series of posts, I attempt to highlight the key skills at every grade level, explain those skills in layman’s terms, and provide activities and tasks for helping children build those skills. Please note that as this series grows, there will be multiple posts for each grade level.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally recommend.

Fact fluency actually begins in Kindergarten, when children learn the combinations for the numbers through 10 and are expected to fluently add and subtract within 5. First grade builds on those skills. While the standard shown below is from the Common Core State Standards, other state standards are very similar.

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

There are two important ideas buried in that standard.

- First, while 1st Graders are working on the facts within 20, the expectation for fluency by the end of 1st Grade is within 10 (remember that Kinder was fluent within 5). That’s an important distinction. It basically means they should have automatic recall for facts within 10, like 4 + 5, 7 + 1, or 2 + 8, but they should have successful strategies for adding single-digit numbers up through 20, such as 6 + 7, 4 + 9, or 8 + 6.
- Second, the acquisition of fluency with basic facts is based on learning and using strategies that emphasize the relationships between numbers, not memorization. Those strategies are actually listed in the standard, and I’ll explain them in more detail in the next section.

In order to use strategies for learning the basic facts within twenty, students need to have mastered the Kindergarten skill of knowing the combinations of all the numbers through ten. See this post for more on that topic. It’s a good idea to quickly assess a child’s fluency with the combinations to ten before trying to teach strategies based on those combinations. The post I just mentioned provides information on doing that assessment. If a 1st Grader needs work on this Kindergarten skill, it’s best to provide practice with number combinations prior to trying to introduce the strategies outlined below.

In their book *Math Fact Fluency*, Jennifer Bay-Williams and Gina Kling have a great graphic that shows how all the different fact strategies are interconnected. The idea is that children need to master the facts at each level before they can tackle the strategies below. Think of it this way, I can’t use a strategy called Near Doubles if I don’t know my Doubles.

While it may seem obvious, children need to know that when we add 0 to or subtract 0 from any number, the number does not change. So 5 + 0 = 5 and 5 – 0 = 5 as well. This is also a good time to introduce the idea of the *commutative property*, which means that the order does not matter when adding. Of course, it’s not necessary for a 1st Grader to learn the mathematical term, but you want them to have the understanding. This will help them understand that 5 + 0 = 0 and 0 + 5 = 5 as well. Important to note that the commutative property works for addition but not subtraction. Concrete materials can be used to help children visualize the commutative property.

In Kindergarten, children practice counting forward and back from any number. This helps them develop an understanding of the next number and the previous number. This first set of facts uses that counting-on/counting-back strategy. Facts in this group include facts such as 6 + 1, 2 + 7, and 9 – 0. Because counting on or back becomes inefficient with larger numbers, we only really want children using it for adding or subtracting up to 2. By that I mean that it’s very inefficient to use counting on to find the sum of 7 + 8, and there are much better strategies for that fact.

Here’s a little game kiddos can play to practice the +/- 0, 1, and 2 facts. If you don’t have ten-sided dice you can use playing cards. Use only the Ace through 9 cards, and instead of rolling the dice, turn over a card. Click here to grab yours.

Children often have an easier time learning their doubles (e.g., 4 + 4, 6 + 6, 9 + 9, etc.). That said, it’s still helpful to use concrete and pictorial representations to support the development of automaticity. Using a familiar tool like a ten-frame helps children to use the benchmark of five to see that 6 + 6 = 5 + 5 + 2. Click here to download a double ten-frame mat.

Once children are fluent with the doubles facts, they can work on Near Doubles. This strategy is sometimes referred to as Doubles Plus One, Doubles Plus Two, or Using Doubles. Using this strategy emphasizes the relationship between facts. If a child knows that 6 + 6 = 12, we want them to understand that 6 + 7 is just one more, so it must be 13. Again, using concrete and pictorial representations helps build this understanding.

You will also see how that Kindergarten skill of being about to decompose numbers in different ways supports this strategy. In this little activity for Near Doubles, one addend is decomposed to make a double fact.

One important note. There’s not just one way to use doubles. For example, one child might think about one more than 6 + 6 to solve 6 + 7. But another might think of one less than 7 + 7 to solve the same problem. That’s the beauty of using a strategy-based approach–children will use them in the way that makes the most sense to them! We want to be careful about imposing structure and rules when using strategies for adding and subtracting.

In Kindergarten, children should work extensively to master all the combinations for the numbers up through 10. Ten is a special number in our place value system, so knowing that 7 + 3 = 10 helps children understand that 17 + 3 = 20. And that if I have 47, it’s 3 more to the next ten (50). And 470 + 30 = 500. You get the idea.

My friend Kris Graham made several short videos showing math games that can be played with only a deck of cards. Kids will love this Pyramid Ten game!

Ten frames are designed to emphasize benchmarks of 5 and 10, so they are an invaluable tool for practicing combinations for 10. They also help children visualize the Make a 10 strategy. Take for example the problem 8 + 3. Using a double ten frame, build one addend on the top ten frame and the other on the bottom. Using two different colors of counters helps with visualization. To add the numbers together, slide counters from the second addend to fill in the top ten frame, making a ten. The Make a 10 strategy is best when one of the addends is close to 10. Children need lots of concrete practice like this before moving to a more abstract process. Be sure that sometimes the addend close to 10 is the second addend, so children realize that 3 + 8 is the same as 8 + 3. In the case of 3 + 8, with 3 on the top ten frame, you would want to move counters down to the bottom ten frame to make a ten.

Once students understand the Make a 10 strategy using concrete materials, they are ready to decompose and work with the numbers mentally. Check out this post to grab another freebie!

Plus 10 is simply another way to say “teen numbers.” Children begin to understand how place value works when they learn that teen numbers consist of a ten (ten ones) and some leftovers. So in the number 13, for example, we can bundle ten of the ones and three will be leftover. Again, ten frames are the ideal tool for helping children visualize this concept. This understanding needs to go beyond just saying there is a 1 in the tens place. Very young children need to *see* what that means for it to make sense and have meaning.

Here’s another one of Kris’s games, this one for making teen numbers. You can download a sheet like the one shown in the video for kids to record their numbers here.

Once children know their Plus 10 facts, they can begin to use that understanding to solve nearby facts using a strategy called Pretend a 10. This strategy is very useful for the nines facts, like 9 + 5. Children simply think, *I know that 10 + 5 = 15, so 9 + 5 is just one less, or 14. *

These strategies might be very new for you and different from how you learned your math facts. Trust the process! Children who learn their facts using a strategy-based approach learn that numbers and math make sense, rather than just seeing it as a series of procedures that must be memorized.

I would love to read comments about your experiences with or thoughts about using a strategy-based approach! Drop a comment below or post one on Twitter! Be sure to tag me! @MathCoachCorner

Bay-Williams, J. M., & Kling, G. (2019). *Math fact fluency: 60 games and assessment tools to support learning and retention*. Alexandria, VA: ASCD.

OConnell, S., & SanGiovanni, J. (2015). *Mastering the basic math facts in addition and subtraction: strategies, activities & interventions to move students beyond memorization*. Portsmouth, NH: Heinemann.

Conklin, Melissa. (2010). *It Makes Sense!: Using Ten-Frames to Build Number Sense*. Math Solutions.

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]]>The post Real World Math: Cooking for One appeared first on Math Coach's Corner.

]]>Stop for a minute and reflect on your definition of real-world math. Often, we think of real-world math as word problems that use a real-life situation as the context. My only problem with that definition is that in the real world, math is often messy. Consider, for example, word problems about scaling recipes up or down. Scaling up, for example doubling a recipe, poses no real problem. That is, since you’re multiplying, the numbers will always work. Scaling down, however, involves division and typically the numbers used in these problems are carefully chosen so they can neatly be divided. But is that what actually happens in the real world? Absolutely not.

When you are cooking for one, you are scaling down a recipe that is typically written for 2, 4, or more servings. Trust me, the quantities don’t always divide easily. Often, you have to make decisions about how much of the ingredients to use to keep the recipe in balance.

Here’s an example. I found a tasty recipe for Chicken Cordon Bleu. It’s kind of a fancy, stuffed chicken dish.

Here is the ingredient list, which is for four servings:

- 4 skinless, boneless chicken breast halves
- 1/4 teaspoon salt
- 1/8 teaspoon ground black pepper
- 6 slices Swiss cheese
- 4 slices cooked ham
- 1/2 cup seasoned bread crumbs

Think about how you would scale this recipe for one serving. Give it a try!

As I start to scale it down to one serving, some of the ingredients are straightforward. I will need one chicken breast and 1 slice of cooked ham. Even the Swiss cheese is a simple division problem–6 divided by 4 = 1.5, so I need 1 1/2 slices of cheese. Let’s tackle the bread crumbs next. I can do the division. One-half divided by 4 is 1/8. The problem is that my measuring cup set doesn’t have a 1/8 cup. So now I have to truly problem solve. My smallest measuring cup is 1/4 cup. How will I use that cup to measure out 1/8 cup of bread crumbs? Finally, there is the salt and pepper. When I do that division, I get 1/16 teaspoon of salt and 1/32 teaspoon of black pepper. How will I measure that, since my measuring spoons certainly don’t come in sizes that small? Here’s something I notice. The recipe calls for twice as much salt as pepper. If I want to keep the recipe balanced, I should keep that ratio. My smallest measuring spoon is 1/4 teaspoon. For the salt, I need to fill that spoon 1/4 full. That’s not much salt. And I need even less pepper.

Wondering about the best way to practice this skill? Actually cooking, of course! Find a recipe that sounds good, scale it down, and cook it! The conversations you’ll have with your child while preparing the recipe will provide great insight into your child’s number sense and true problem-solving abilities.

If you try this at home, I hope you’ll share your experiences in the comments or on Twitter!

See you on Twitter! @MathCoachCorner

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]]>The post Real World Math: Toilet Paper appeared first on Math Coach's Corner.

]]>Look no further than your local grocery store and current events for this real-world math problem! Here’s the problem:

Lots of good math in there! First, you have to figure out how many rolls of the 6-roll package is two-thirds of the package. Drawing a picture of the package is a great strategy for solving that first part. In the diagram below, you see that two-thirds of the package is four rolls of toilet paper. So each week the family uses four rolls of toilet paper.

The next step would probably be to draw four 6-roll packages since the problem asks how long four packages would last.

Finally, I could draw and label the rolls used each week.

Keep in mind that this is just one way to solve this problem! If your child solves it a different way, I hope you’ll share it in the comments or on Twitter!

See you on Twitter! @MathCoachCorner

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