Teaching multiplication facts is certainly a challenge. I surveyed more than 20,000 teachers about their biggest challenge teaching math, and the overwhelming response was that students don’t know their multiplication facts.

We’ve all experienced it. Some students easily memorize their multiplication facts, and other students memorize most of their multiplication facts but struggle with the more difficult ones. Then you have students who never manage to move beyond their 2s or 5s facts, which can create tremendous challenges.

While memorization is ideal, some students need to first become fluent before they can fully memorize their facts. Fact fluency means students can flexibly apply what they already know to efficiently solve multiplication problems. When I first started teaching, I wanted to skip straight to memorization, but I now realize that not all students can make that leap.

After many years of research and experience, I finally found a balanced system that works. Not only is it effective, it’s doesn’t create tons of extra work and stress for my students or the teacher! This system follows an intentional and systematic approach that provides consistency, engagement, and understanding.

In this post, I’ll first explain the parts of the program and then share ways you can utilize it in your classroom. I originally labeled the lessons with Monday, Tuesday, etc. because that’s the way my brain likes to organize. However, due to two years of virtual instruction, hybrid instruction, varied schedules, and so on, I’ve had to let that go. For the sake of flexibility I’ve gone with Day 1, Day 2, etc.

Teaching Multiplication Facts – Day 1-Build It

The first step for students is to SEE multiplication. On the first day of studying a set of multiplication facts, students create a grouping model for each set of facts (2×2, 2×3, 2×4, etc.). They can use any item as a manipulative. It certainly doesn’t have to be anything special.

If students complete this digitally, students drag the larger circle to indicate the number of groups. Then, students drag the squares to show the quantity in each group. It’s interesting to see who realizes they can simply add three more counters (or circles) for each fact. Students love thinking that they found a way to beat the system and don’t realize that they’ve found the relationship between repeated addition and multiplication, which is helpful for teaching multiplication facts.

Day 2-Multiple Representations

The next step teaching multiplication facts for students is to SEE the different representations of the math facts. In this activity, students sort representations of multiplication equations. There are arrays, repeated addition, and number sentences. This allows students to begin understanding what multiplication means.

In the digital version students drag a written description, array, repeated addition problem, and a multiplication equation to a single row. Each page has three rows and all the needed cards are included on that page, so students will not have to copy and paste any cards.

Teaching Multiplication Facts – Day 3-Visual Flashcards

At this point teaching multiplication facts, many students aren’t ready for traditional abstract flashcard practice. Instead, they can then use pictorial models of each multiplication problem as a flashcard. This pictorial model helps bridge concrete and abstract. I did include another set of flashcards where students can try to answer the multiplication problem using an abstract strategy.

If students are using the digital version, they create multiple representations of each set of facts. (Of course, you could have students complete both!) Students drag a square to a hundreds grid to show how to skip count. Then, they type a repeated addition representation. Students then drag blue squares to build an array of the fact. The last step is to drag an arrow to show multiplication on a number line.

Teaching Multiplication Facts Day 4-Word Problems

It’s important to teach math concepts in context. This means we shouldn’t save word problems for the end of our multiplication units. For each set of facts students should solve AND write multiplication word problems. For example, if students are working on their four facts they will solve two problems where four is a factor and write two problems where four is a factor.

The digital version looks almost identical to the paper version.

Day 5-Multiplication Strategies Booklet

When teaching multiplication facts, students should have the opportunity to WRITE about the strategies they use to solve multiplication problems. This allows students to internalize their learning. For each set of facts, students describe a strategy they could use to solve a multiplication problem. Students also represent problems using arrays, repeated addition, grouping models, and a number line.

Since the digital version Day 3 also includes quite a few of these tasks,  I included a multiplication game.  If students don’t have a partner, they can play color vs. color.

The Great Debate-Timed Test

We all want to encourage efficient strategies and automaticity, but timed tests aren’t as effective as instruction in developing these skills. Timed tests offer a good snapshot of how well students have memorized their facts, but are not a powerful teaching tool.

Rather than giving students a certain amount of problems to solve in a set amount of time, I have students time how long it took to complete all of the problems correctly. This alleviates most the stress that comes with a timed test. I also have students graph how long it took to complete the test to encourage students to look at their growth.

In the digital version, students click on a link to go to a Google Form where there is a short quiz. They must get each question correct before moving to the next question. Before beginning the quiz, students are prompted to start the included timer. Students are prompted to stop the time when they finish the quiz. Students graph how long it took them to complete the quiz.

Keeping Students Organized

I give students color coded bookmarks to use as a checklist. The bookmarks show exactly what is expected from students each week. The activities are listed in the order in which they should be completed. As students complete an activity, they check it off on the day it was completed. You can see these activities here!

You can even group the bookmarks together on a ring to keep everything in one place. There’s nothing wrong with keeping things bright and colorful when teaching multiplication facts;)

I added a few extras at the bottom of each bookmark, because these are things we do weekly, if not daily. I send home a Weekly Multiplication Game with students every Monday. These games only require dice or a set of playing cards, which I provide if students don’t have at home. This is an opportunity for fun practice and can be completely optional.

I also give students who need additional support a Multiplication Fact Booklet that will take them through each multiplication problem for that set of multiplication facts. This gives students the extra support they may need and it solidifies understanding. It’s a huge help teaching multiplication facts.

Xtra Math is another great practice tool. However, it’s important to keep in mind that it is a tool. Nothing replaces the conceptual activities where students actually work with multiplication facts.

You can read additional posts on teaching multiplication facts here and here. I’d love to hear about what you’ve tried that has been successful!

The post Teaching Multiplication Facts appeared first on Ashleigh's Education Journey.

This post is the second part of my multiplication series. It focuses on teaching multi-digit multiplication with area models, partial product, and incorporating problem solving. If you missed the first post in the series, you can find it here.

Week 4  1-Digit By 4-Digit

On the fourth week of the multiplication unit, I finally begin to focus on multi-digit multiplication. Since students should have a strong foundation, I’m able to move at a steady pace. During this first week, I keep the focus on 4-digit by 1-digit multiplication.

I try not to rush the area model lessons. I’ve found that it’s easy to introduce too much, too soon, and students end up confused. However, once students understand area models, it’s time to introduce solving multiplication with partial products.

Math Warm-Ups

This week’s Math Warm-Ups follow the same routine and format. In the SPLAT! game, there are now two blobs, which makes it interesting.

Math Check-Ups

Now that the problems are becoming more complex and more time consuming, there are fewer problems on students’ Math Check-Up. It’s important to not allow the check-ups to become too time consuming, because it’s important to avoid taking time away from your actual instruction.

Math Centers

This is students’ last week with their current set of math centers. When they finish their last activity, I have them turn in their center booklets. This is a great time to have students participate in self-reflection on their participation and effort during their centers.

Scribble Notes

As an extra review, students complete the distributive property page of their Math Scribble Notes. This guides the mini lesson for students’ lesson on Monday.

Multiplication Lessons

Once the foundations of multiplication are developed, it’s time to transition students to larger, multi-digit multiplication problems. I slowly fade away the individual squares from the arrays to allow students to build area models.

At this time, I only introduce area models and partial product. While I do teach other strategies, I’ve found that too much, toon soon can confuse students. You can read more about these multiplication lessons here.

Week 5-Area Models & Partial Product

I begin the week by reviewing 1-digit by 4-digit multiplication problems with area models and partial product. I also include word problems in our review, because it’s essential to consistently embed word problems into your instruction. I also begin to incorporate problem solving lessons into my plans, because it’s helpful for students to learn through problem solving.

Math Warm-Ups

The Math Warm-Ups follow the same format as the previous week. I love how in the count the room activity, the numbers move into the ten-thousands place, which is a great way to monitor student understanding.

Math Check-Ups

At this point in the year, rounding and subtraction are the two skills most likely to give students trouble, so I include those in the Math Check-Ups. The only other problems are one area model and one partial product multiplication problem.

Math Centers

This week I introduce brand-new centers, and these are a ton of fun! I love using a booklet for the my recording sheets and several of the center activities. This is printed front to back and folded to make a booklet. There is also a version of these centers that does not have a Halloween theme.

Candy Corn Division-This is a great way to review the concept of division. Students take a set number of candy corns and partition it equal into either equal groups or into groups with equal amounts.

Center 2-Spooky Recipes-In this activity students receive a price list of ingredients for spooky recipes.  They use the price list to answer multi-part word problems.

Center 3-Picture Puzzles-Students will think that the final pictures are hysterical. Teachers love the number sense that the activity helps develop.

Center 4-Haunted Math-This activity is similar to the game of Clue. Students use deductive reasoning and thinking about 10 and 100 times the value of numbers to determine who stole the candy from the Halloween party and where they hid it.

Center 5-Costume Shop-This is a fun problem solving activity where students determine how many of each costume was sold.

Center 6-Fall Carnival-This is another problem solving activity where students determine the cost of tickets for different fall carnival rides or activities.

Center 7-Logic Puzzles-These are a blast! If your students have never done anything like this before, you’ll definitely want to model how to solve the puzzles. Once students see what to do, they can quickly catch on.

Center 8-Geometric Haunted House-In this center students use various geometric terms to create a haunted house. I included a separate page of grid paper that students can draw their house on, as well as example images of each geometric term.

Center 9-3 Act Task-This is such an awesome and thought provoking activity. I especially love the fact that the Google Form won’t let students move forward without estimates and sharing what they know and need to know. Since it is quite challenging computation, I do let students use the calculator on their Chromebooks for the task.

Center 10-Sweet Math-I highly recommend empty bags-for the activity or use online nutrition fact sources. Students use the information to solve word problems (many multi-step) about the nutrition facts.

Multiplication Lessons

When teaching word problems, I love using a highlighting strategy that you can read more about  here. It’s also good to give students the opportunity to extend their thinking and to look at the models with a different approach.

Week 6-Area Models & Partial Product

On the sixth week of the multiplication unit, I focus on 2-digit by 2-digit multiplication. I have found that most students will breeze through this IF they have a solid conceptual understanding of multiplication.

Math Warm-Ups

A new activity is introduced to the Math Warm-Ups. Students balance equations using balance scales. There’s a lot going on in the image that activities multiplicative thinking as well as visual-spacial reasoning. I like to ask students how many dots we would need to add to a ten-frame to make the scales balanced. You can extend this by having students write equations to represent the unbalanced scales and then the balanced scales.

Math Check-Ups

You may have noticed that I skipped a Math Warm-Up. That’s simply because I’m highly intentional on making my warm-ups follow my lessons. It’s okay to skip around!

Math Centers

This is students’ second week with their Halloween themed math centers. They will continue following their booklet and complete four additional activities this week.

Scribble Notes

This week’s Scribble Notes culminate the multiplication lessons and strategies by having students share different strategies for multiplying multi-digit numbers. As always, it’s completely up to you on whether or not you teach the standard algorithm.

Multiplication Lessons

Once students are comfortable with single-digit multiplication, it’s time to move on to 2-digit by 2-digit multiplication. It’s best to follow the same process and begin with arrays and area models.

Once students are proficient with area models, they begin solving multi-digit multiplication problems with partial product. If needed, I provide students with scaffolding to help them transition to this strategy of multiplication.

Over the years, I’ve found that some students tend to strongly prefer area models and become confused with partial product. In those cases, I hold off requiring the student to use partial product. I’d rather the student be very confident and consistent with one strategy, rather than inconsistent or uncomfortable with multiple strategies. I give them time and come back to partial product later in the year.

Multiplication Assessment

Since this is a long unit, it may be helpful to break up the assessment into two separate assessments or to give part one and two on separate days.

Culminating Activity

My favorite way to culminate the unit is with a multiplication escape activity!

This is definitely a lot of moving parts, but it really does work! All the pieces of the puzzle fit together and work to compliment each other.

Let me know if you have any questions!

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These graphing worksheets and activities are a huge help in my math instruction. I kind of don’t have a unit for teaching data analysis and graphing. Technically, I do have one (here it is), but I don’t teach it lesson by lesson. Instead, I incorporate the graphing worksheets and lessons within my other math units. The reason I do this isn’t because of some profound revelation in teaching math. Instead, it’s simply because I don’t have time for an isolated math unit.

I’ve found that the best way for me to teach data analysis and graphing is to integrate it in my other math units. This often results in me adding an additional lesson to each of my math units or embedding a lesson or two into our math centers. In this post, I’ll share some of the ways I teach data analysis through graphing worksheets and activities.

Graphing Vocabulary

I do make sure students know general graphing terms. Most of these are taught in the primary grades, so that aren’t completely new to students. If you feel the need to teach more basic graphic skills as a quick review, you can do something short and easy like this Favorite Flavor of Ice Cream lesson.

Students survey classmates and teachers on their favorite flavor of ice cream. Then, students use their tally chart to create a picture graph of the top ice cream flavors.

Another introductory lesson is the My Favorite Pet survey. This is a quick and easy graphing activity to review essential vocabulary and concepts.

Once again, students interview peers and/or adults to determine their favorite pet. Then, students use that information to create a bar graph.

Line plots are most likely to be new to students, so this is one introductory lesson I don’t skip. (BTW-These names, as always, are totally made-up. While I do teach these lessons, all names and student work are made outside my actual classroom.) 

In this graphing worksheet, we list how many hours a week, books a week, or even hours a month students read. Then, students make a line plot to represent the information. It’s helpful for students to create their line plots on graph paper, because it helps them keep their Xs lined up correctly.

Graphing Worksheets – Place Value

One way to incorporate place value and teaching data analysis is through the Place Value Tower activity, which is an incredibly fun graphing worksheet.

Students build a “Place Value Tower” out of base-ten blocks. Then, they determine the total value of the base-ten blocks, as well as graph the number of one, tens, hundreds, and thousands blocks used to build the tower. To incorporate graphing, have students create a bar graph to either show the total number of ones, tens, hundreds, and thousands blocks in their Place Value Tower, or create a bar graph to show the total value of ones, tens, hundreds, and thousands in their Place Value Tower.

Graphing Worksheets – Addition and Subtraction

In this graphing worksheet, students think of as many addition and subtraction number sentences as possible that equal a given number. (If you did this later in the year, you could use multiplication.)  I like to give my students a one minute time limit for each number, and depending on the group and their need for a challenge, I may add rules such as: you can only use each digit once or you have to use 2-digit numbers.

After students have played five rounds, they complete a line plot that indicates how many number sentences they were able to generate during each round.

Math Facts

This is a great way to have students track their progress in learning multiplication or division facts. This is not a one time graphing assignment, but rather a graphing tool that students can use to monitor their own learning. I have students complete this graph independently and don’t display the graphs, so students do not compare their results with each other.

Once a week, every other week, once a month, or any set time period, give students a 100 question multiplication or division quiz. If you use Option 2, and feel like 100 questions is too much for your students, you can absolutely reduce the number of questions! It’s just important to keep the number consistent, so that graph accurately displays the data.

• Option 1-Count how many questions students answered correctly during the 5 minute test.
• Option 2-To reduce the anxiety of having a timed test, have students take as long as they need to solve the math facts. Rather than recording the number of problems correct in a minute, record the number of minutes and seconds needed to complete the math quiz.

Have students complete one section of the bar graph after each time they complete the assessment. Use the data to encourage students to focus on growth and not perfection.

Graphing Worksheets – Multiplication

This array lesson can be taught after students have experience reading and creating arrays. In this activity,  students build all of the arrays possible seven different numbers. They record their arrays on grid paper and write a multiplication equation for each array. Once students have found all of the arrays possible, they create a bar graph to show how many arrays were found for each number.

Elapsed Time

Have students complete the Where Does the Time Go survey independently. The survey asks students to estimate how many minutes they spend each day reading, watching television, playing video games, playing outside, helping around the house, and practicing an activity such as sports or a musical instrument.

Use the survey organizer to collect and display students’ data from the Student Survey. This organizer will allow students to consolidate the data and see the data as a whole. Then, have students work in groups to create a line plot that displays the data from one of the activities from the survey. Have students create their line plots on chart paper, so it can easily be viewed around the classroom.

After each group has completed their line plot, post the line plots in a central location and meet back together as a whole group to discuss the data represented in the line plots. The results are usually very surprising to students, and this is a great time to discuss goals and how it is important to use our time wisely to help us achieve our goals.

Fractions

Before beginning this lesson, students should multiple experiences using a ruler. Have students use a ruler to measure the length of ten pencils to the nearest ¼ inch and record their measurements. Then, have students create a line plot to show the length of each pencil measured.

You can complete a similar activity with paper airplanes and tie that into a STEM lesson. Give students time to create their paper airplanes and a few minutes to practice flying their airplanes. Break students into groups of four and have students take turns measuring the distance each plane flies. Each group needs masking tape to mark the starting and end point, a ruler to measure the flight, and a recording sheet. Then have students use the data in their recording sheet and the other groups’ recording sheets to create a line plot representing the data.

Graphing Worksheets – Area

Have students use cm cubes find the area of their hand. Another option for finding the area is for students to trace their hand on cm grid paper. Have students to work in groups of 4 to create a picture graph (using hands as their picture) that represents the area of each student in the group’s hand.

Groups will need to determine how many square centimeters each hand on their graph represents. Students can use the color coded hand pictures to help them create their picture graph. After students complete their graph, have them write questions that can be answered using information from their graph.

Geometry

Have students go on a geometry scavenger hunt! Students tally how many of each shapes they find and create a bar graph to show how many they found. Naturally, have students be ready to tell you what their examples are. Otherwise, the quantity of examples tends to be a bit skewed!

Each of the lessons in this post can be found here! These lessons has inspired me to think of even more ways to incorporate data analysis and graphing into our math units.

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Teaching multiplying with area models and partial product is a major component of fourth grade math. Before teaching multi-digit multiplication there are many concepts that are essential for student success. You can read more about those lessons here. A solid understanding of those concepts makes multiplying with area models and partial product much easier for students.

All of these lessons are from my 4th Grade Multiplication Unit. There are also a few free resources linked in the post!

Multiplying With Area Models

I begin by teaching students about area models. This ties in perfectly to the transitional lessons and can connect to third grade area lessons.

I start with traditional arrays and gradually fade the grids away so that students are multiplying with area models, rather than arrays.

To keep students interested, I try to be humorous about the monotony of drawing and counting each individual square. A little over-exaggeration and “accidental” mistakes counting help students see how much more efficient an area model is than an actual array.

I typically spend a couple days introducing multi-digit multiplication with arrays and area models. It’s important not to rush these initial lessons.

Partial Product

Area models tie in easily to partial product. I do like to fully teach 2, 3, and 4-digit numbers with both strategies before I introduce 2-digit by 2-digit multiplication.

To keep it non intimidating for students, I begin the partial product problems with 2-digit numbers and gradually increase to 3 and 4-digit numbers.

Students should guide how long you spend on each lesson. If you see students struggle a bit, it’s beneficial to slow down a bit. I like to utilize centers during this time by alternating instructional days. On Tuesdays and Thursdays students participate in math centers, and during that time I meet with students who need additional instruction.

Throughout the process of teaching multiplying with area models and partial product, it’s important to incorporate multiplication word problems. We continue the highlighting word problems strategy that I wrote about here.

It’s also good to give students the opportunity to extend their thinking and to look at the models with a different approach. I love using this type of activity where students determine the missing numbers.

2-Digit by 2-Digit Multiplying With Area Models

Once students are comfortable with single-digit multiplication, it’s time to move on to 2-digit by 2-digit multiplication. That’s when you typically start to see some struggle. It’s best to follow the same process and begin with multiplying with area models and arrays.

Once students are proficient with area models, they begin solving multi-digit multiplication problems with partial product. If needed, I provide students with scaffolding to help them transition to this strategy of multiplication.

Over the years, I’ve found that some students, especially students who struggle in math, tend to strongly prefer area models and become quite confused with partial product. In those cases, I hold off requiring the student to use partial product. I’d rather the student be very confident and consistent with one strategy, rather than inconsistent or uncomfortable with multiple strategies. I give them time and come back to partial product later in the year.

Algorithm

Whether or not you choose to teach students the algorithm for multi-digit multiplication is really up to you and your school or district’s exceptions. I personally focus on the algorithm on an individual basis. If/when I feel the student is 100% comfortable with both strategies, I teach that student (typically in a small group) the algorithm. If they learn the algorithm during the actual multiplication unit-great. If not, that’s fine too. I will still find time within the year to introduce it when the student is ready. I can determine this through our various spiral review.

Digital Multiplication

When I taught this unit, we were still face-to-face. However, I did make a digital version of each lessons for virtual and distance learning students. One of my favorite lessons included a variety of area model practice.

You can download this digital lesson for free here.

Multi-Digit Multiplying Area Models and Partial Product

One thing that is always a challenge for teachers, students, and parents is that we all learned multi-digit multiplication differently. As an elementary student of the 80s, I had never even heard of an area model or partial product until I needed to teach it. I can completely understand why it’s confusing for parents.

I’ve found that it can be beneficial for all when students explain the rationale for each strategy of multiplication.

Students first explain how they use arrays and area models to multiply smaller numbers. Then, they explain how the distributive property connects to those models.

Then, students explain the relationship between multiplying smaller numbers to multi-digit multiplication.

I think the most interesting part is seeing students’ view point on each model or strategy. It’s fun to see their reflection and their attitude toward each representation. That peak into students’ reasoning is so valuable.

You can download a copy of the booklet here.

Hopefully, these ideas will help you out as you teach multi-digit multiplication! All of these lessons are from my 4th Grade Multiplication Unit.

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Most curriculums and textbooks make a lot of assumptions. That’s especially true in math. For instance, in fourth grade, students jump right in to multiplying a 4-digit number by a one-digit whole number and then multiplying 2-digit numbers by 2-digit numbers. Years of experience has taught me that the majority of students are not ready to make the leap from third grade multiplication to fourth grade multi-digit multiplication. They need transitional lessons that bridge what students learned in third grade to what students will learn in fourth grade.

Students need more time and learning opportunities. Many students are not developmentally ready to fully understand concepts such as the distributive property or multiplying by multiples of ten in third grade. However, with additional instruction the following year, the missing pieces often click. These two concepts are essential for understanding multi-digit multiplication, so these transitional lessons are incredibly important. The lessons below are from my 4th Grade Multiplication Unit. If you’re looking for 3rd grade multiplication ideas, be sure to check out this blog post.

Multi-Digit Multiplication Basics: Multiples of Ten

One of the most important concepts for students is multiplying by multiples of 10, 100, and 1,000. This is extremely difficult for students who lack number sense. While it may seem obvious to us as adults, these students do not see the relationship.

Introduction

I like to introduce this concept with a digital activity, because whether I’m face-to-face or teaching virtually, manipulatives tend to get a bit out of control with this lesson.

Students first use base-ten blocks to build different multiplication situations. This allows them to see how 3 groups of 40 is the same as 3 groups of 4 tens.

Students then use place value chips (that’s what we call them) to analyze arrays. For example, there are 6 rows of 4 ten, which represents 6 x 4 tens. By doing this, students should realize that 6×4 tens equals 24 tens.

Students do the same type of task with hundreds, which helps to solidify the relationship of multiplying ones, tens, and hundreds.

Students then build arrays to represent multiplying by multiples of 10 and 100. In the digital activity, students drag the place value circles to build the array.

To give students even more multi-digit multiplication exposure, students read arrays using multiples of 10 and 100. They also build arrays be highlighting the squares within the chart.

To shift students to more abstract understanding, students drag and drop numbers to complete the multiplication equation. Can you see how helpful this will be when students get to division?

Practice

Students can also use grid paper to represent their understanding of multiplying by multiples of 10 and 100. This lesson emphasizes how the associative property allows students to multiply by multiples of 10 and 100.

You will see this concept immediately click for some students, and you’ll have others who will need additional small group instruction.

Distributive Property & Multi-Digit Multiplication

Confession: when I was teaching third grade and the distributive property was added to my standards, I had no idea what it was! Even when I first taught it, I wasn’t sure why it was important. It took a bit to see just how powerful it was.

Introduction

A great way to review the distributive property is through a matching lesson. I give students a collection of arrays, and students work in a group to match the blue arrays with two green arrays that can be combined to have the same dimensions and product. For example, students may match 8×6 with (8×3) and (8×3). This helps students see how they can break larger multi-digit multiplication problems into 2 or 3 smaller ones.

Continued Practice

In a following distributive property lesson, students complete a cut and paste activity. Assign groups different multiplication problems and have students try to make as many arrays as possible to model how to use the distributive property to solve the problem. I like to display these posters in the hallway as evidence of what we’re learning in class, as well as to allow other students to see the variety of models.

You can follow the cut and paste lesson with an activity where students break apart multiplication problems on a small section of grid paper.

Making the Transition to Multi-Digit Multiplication

Once these foundations are developed, you can begin to transition students to larger, multi-digit multiplication problems.

I often give students a large problem such as 346×7 without any guidance or instruction. I want to see what strategy they use to approach the problem. The results are often shocking, because many students instinctively go to partial product without having ever heard the term or seeing an example.

You can see in this example, how I slowly fade away the individual squares from the arrays to allow students to build area models. I like to over dramatize the annoyance of counting out all the squares and act as if using the area model will be SUCH a time saver.

As you introduce this to students, it’s important to not only use horizontal area models. Students should be flexible in their thinking and understanding and not view these models as just boxes or rectangles. They are a representation of an array. There should be meaning in the area models.

These transitional lessons make the “official” fourth grade multi-digit multiplication lessons so much easier-for you and your students!

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Teaching multiplication as a comparison is the second portion of my multiplication unit. I begin the unit by teaching factors and multiples which you can read about here. You can certainly switch the lessons around and begin with multiplication as a comparison, though.

Multiplication Situations

In order for students to understand the significance of multiplication as a comparison, they must first understand different multiplication situations. One way to do this is through an open (and then closed) sort. Give students a collection of cards with multiplication word problems and ask students to sort the cards into three groups. After students have completed their sort, ask students to share how they decided to sort their cards.

Once students share, explain that one way to sort cards is by the different types of multiplication situations. Explain each type of multiplication situation:

• equal groups
• arrays/area
• comparison

After your explanation, have students resort their cards. Allow students to share what cards they placed with each type of situation and discuss any misconceptions.

I anticipated this being an easy task for students, but I was WAY wrong! This presented a huge challenge, which let me know exactly what I needed to focus on.

You can follow that lesson by having students write word problems to represent different multiplication situations. Students can also begin exploring multiplication as a comparison by using Cuisenaire rods (physical or virtual). For example, students will determine how many times larger the orange rod is than the red rod.

Multiplication as a Comparison – Bar Model

After giving students the opportunity to explore with the hands-on lesson above, I introduce a pictorial model of multiplication as a comparison with bar models. Students can use a tape diagram to represent addition and multiplication comparison problems. When creating models, encourage students to add labels to their models to ensure understanding. Drawing the diagram without labels can confuse students and lead to inaccurate results.

A great extension to bar models is to have students write word problems that represent the diagrams. You can display a model digitally or on a sheet of chart paper.  Have students write a word problem that represents the model on a sticky note and place the sticky note on the chart. After students are finished, ask questions such as: Does this addition or multiplication?

During this lesson, I also teach students how to write a multiplication equation from a comparison statement. Students often want to switch numbers around in ways that don’t accurately represent the statement, so it’s important to focus on accurate representations.

Multiplicative as a Comparison and Additive Comparison

It’s important for students to recognize the difference between additive comparison and multiplicative as a comparison, so I often try to incorporate a problem solving lesson that allows students to apply what they’ve learned thus far. This also gives students all important practice in explaining their reasoning.

Multiplication as a Comparison – Application and Exploration

To have students put it all together, students can go on a multiplication scavenger hunt. In the scavenger hunt, students look for items that reflect given situations. For example, students may need to find an item that is two times as long as a sheet of paper. Students LOVE this lesson!

After teaching these lessons, students should continue to review all types of multiplication situations through consistent practice with word problems.

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I’m careful to make sure to have students determine the number of groups they can create, rather than me telling them how many groups they can create with a given amount. This adds the element of problem based learning to instruction.

Introducing Multiplication-Repeated Addition

After teaching grouping models, show students how to use repeated addition to solve multiplication problems. This is another important step in teaching multiplication. I try to differentiate my lessons, so that some students may be working with a total of 12, others 24, others 36, and some up to 48. I always allow students to use snap cubes during these lessons to make this a hands-on learning experience.

Before I move on to arrays, I like to spend a little time reviewing grouping models and repeated addition with a game. I’ve found that it’s easy to go too fast in these beginning lessons, and some students will need a bit more time to develop a solid understanding of the two models.

Introducing Multiplication-Arrays

Since arrays and area models are incredibly important in fourth and fifth grade, it’s incredibly important to teach for deep understanding. As I teach arrays, I emphasize the importance of EQUAL groups. Once again, I use manipulatives to give my students more hands-on learning opportunities.

The book Amanda Bean’s Amazing Dream is awesome to use for teaching multiplication as you introduce arrays. The entire book is based around the use of multiplication situations, especially arrays. I made a recording sheet to go along with the book. You could either show the book on YouTube or you could read the book and show the pictures using a document camera.

Multiplication Games

Before students move on to more challenging multiplication concepts, I like to spend a little more time reviewing through games such the one below. It’s similar to the game War, but students must read grouping models in order to determine who has the highest product.  We also play games such as the classic Multiplication Bump.

I save multiplication on a number line for last, because this model the trickiest representation for my students. For extra practice, I enjoy this Representing Multiplication activity that you can download  here. I gave every student 24 snap cubes, and they had to find ways to place the cubes into equal groups. Once they placed the cubes in equal groups, they drew a model of the cubes. Then, students wrote a repeated addition number sentence and a multiplication number sentence that represented the equal groups. This is such a great task for introducing the concept of multiplication, especially the connection between multiplication and repeated addition.

Teaching Multiplication – Problem Solving

Once I’ve taught each form of representing multiplication, we spend a bit more time working problem solving with multiplication. For instance, in this activity, students find all of the possible arrays to build the number 48. This could be differentiated by using many different numbers.

Another great way to incorporate math and literature is to read One Grain of Rice, which ties in perfectly to multiplication. After reading the book, let students try to find the total number of grains of rice that Raini gave the raja as an extension.

Another great book to tie in to multiplication is Spaghetti and Meatballs for All. In this book a family is having a get together and as more and more people arrive, they have to change their seating arrangements. In this activity, students develop a seating arrangement for 36 people, but they have to use different shaped tables. Each shape table holds a different number of people, which presents students with a bit of a challenge.

It’s also fun to have students use pattern blocks in a multiplication activity. In this lesson, students grab a handful of pattern blocks from the tub. Then students have to show how to use multiplication to find the total number of sides on all of the pattern blocks. This is fun, because everyone will have different answers. Naturally, some students will try to be cute and get a massive handful of blocks. It’s kind of funny when you see them realize how much more challenging that will make the activity.

When teaching multiplication word problems, I like to start with problems that include pictures. This allows students to visualize the problem and the meaning of each factor.

Properties of Multiplication

Once students understand the concept of multiplication, the should become familiar with the properties of multiplication, which help students learn multiplication facts, as well as multiply larger numbers. You can read about teaching the distributive property here.

It’s important to teach conceptual lessons on the properties of multiplication. It’s not super important for students to be able to name the properties. Instead, students should understand the properties.

As someone who has taught fourth grade, I cannot emphasize enough the importance of the distributive property. It is the foundation of so many fourth grade multiplication and division standards. I begin these lessons with base-ten blocks. Students use their blocks to solve problems such as 3×15. Students will build an array to solve the problem. BUT, I don’t initially tell students how to build the array. Some may only use ones and some may use ones and tens. We look at the differences and compare the two arrays. This allows students to see HOW the distributive property works. We later move on to three digit numbers.

I like to spend several days on the distributive property to make sure that students really feel comfortable with this concept.

Multiplication Facts

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There are so many great options for using digital math manipulatives. However, once school started I quickly realized that while I had access to great digital manipulatives, I wasn’t quite sure how I wanted to use them. We all know the importance of using math manipulatives in the classroom, but there is a difference (other than the obvious) in using digital manipulatives.

I recently wrote a blog post as A Guide to Math Manipulatives. It has a lot of great information and ideas so if you haven’t read it yet, be sure to check it out.

I wish I could make this an all-inclusive blog post that shares how to use every option for every standard. But, that would get overwhelming quickly. Instead, I’ve shared some of my favorites for teaching whole numbers and operations for upper elementary students. I’ll try to create a new post for digital manipulatives for teaching fractions and decimals, and operations with larger numbers a little later in the year.

You can download all of the forms you see below for FREE! Click here to grab them!

Base-Ten Blocks

One way to use base-ten blocks is to have students build numbers using different combinations of blocks. This helps students compose and decompose numbers.

When I have students complete these activity, they look at the example, then we do one together with me using the document camera, before they try this on their own. Mathematically, they don’t need the extra scaffolding, but they do need support using the manipulatives.

Another way to use base-ten blocks is to have students multiply by multiples of ten. This can also be done with place value discs, but I did learn that it was easy to overwhelm students teaching too many options and types of manipulatives too quickly. I’ve found that it’s best to slow down and introduce them over time.

Hundreds Chart

We don’t use a hundreds chart a lot in class, but I do love them for teaching multiples of numbers through 100.

I love how this feature allows students to see how multiples overlap, as well as see prime numbers.

Cuisenaire Rods

Cuisenaire Rods are by far my favorite manipulative for teaching fractions. They can also be used for teaching students to find the factors of numbers.

This is an activity I wouldn’t try with my blocks, because they don’t have any numbers written on them. I love that for fractions, but for the purpose of finding factors, the numbers are a big help.

Once again, I would show students an example and complete one together before students try this activity on their own. I wish this wasn’t the case, but there will always be technical issues: students need to zoom out, students don’t know to to drag or resize an image, etc.

Color Tiles

Color tiles are awesome for perimeter and area, but they’re also great for teaching students to build arrays. The tiles transition nicely into area models, which can be used with larger multiplication problems.

I love relating repeated addition to arrays. Anything that helps students visualize is a win! Plus, just think about the options for teaching the distributive property!

My only word of caution is to go slow. I tried to introduce all of these digital math manipulatives in one class. We did it, but like anything else they lose their novelty. Next time, I will absolutely spread out my lessons.

Snap Cubes or Linking Cubes

Linking cubes are great for teaching multiplication as a comparison. As I mentioned in my word problem blog post, students need a lot of hands-on opportunities for this tricky standard. I like giving students two numbers and have students create a model for additive comparison and multiplicative comparison.

A great extension is to give students the model and have students write a word problem to reflect the model. That significantly raises the rigor for students.

It’s exciting to think about all the options students will have once they are familiarized and comfortable with using digital manipulatives!

If you’d like a copy of the Google Slide images above, click here. I’ll send you a free copy!

I assigned the slides above in Google Classroom and the slides linked to the manipulatives. After students created the image, they copied and pasted their final product to the Google Slide. I was worried that the extra step would be complicated, but it wasn’t at all.

There are many options for using digital math manipulatives. My personal favorite is BrainingCamp, which is what I used for all of the examples. I feel like it has the most options and easy to use interface. Of course, you can use whichever platform works best for you. I have included links within the slides, but once you may your copy, you’re welcome to change them!

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It’s no secret that addition, subtraction, multiplication, and division word problems present a considerable challenge to students. Students often rush through word problems and immediately try to find an answer, rather than thinking about the meaning of the problem. Students see a word problem and want to immediately grab the numbers and do something with them. Fortunately, there are things we can do that will allow students to develop an understanding of word problems.

In this post:

• How do I teach word problems?
• Addition and Subtraction Problems
• Action and Non Action
• Types of Action Word Problems
• Types of Nonaction Word Problems
• Situational Equations
• Multiplication and Division Problems
• Equal Groups
• Multiplication Comparison
• Error Analysis

How do I teach word problems?

I’ve found that it’s best to avoid teaching key words and strategies such as CUBES. Key words are not enough to solve a problem and can often lead students to an incorrect operation or a single operation where multiple operations are needed. Other procedural strategies omit the understanding of the situation of the problem.

In order for students to fully understand word problems, they must have opportunities to solve word problems using multiple representations. This allows students to understand the actions and relationships in the problem. Options for representation include:

• Concrete-using physical representations or acting out through modeling.
• Pictorial-pictures and diagrams can illustrate and clarify the actions and relationships in ways that words, even concrete representations, can’t.
• Symbolic-symbols can be operation signs or variables.

When students begin working with larger numbers, representing situations with a physical model becomes more challenging because it requires strategic grouping of units. In these situations, students may need to provide a key or additional explanation to their representations.

As students work, it is important to note when they are not representing the actual context and the quantities but are representing only the numbers in the problem. Not matching the problem situation to a model reveals that the student has not displayed operational sense in the solution. Operation sense is understanding all of the “work” that each operation can do. This extends to the use of appropriate representations for the action or relationship in the problem so that it represents an understanding of the situation.

Addition and Subtraction Problems

I begin the year with only addition and subtraction word problems. Even though upper elementary students typically have considerable experience with addition and subtraction word problems, these lessons are incredibly important.

Action and Non Action

One of the keys to understanding word problem is to recognize action and nonaction word problems. Action does not necessarily refer to movement but that something happens to the initial quantity in the problem. This is a great time to have students take turns acting out various word problems. I have students sort word problems into action and non action word problems.

For extra practice, I have students complete the Action or Nonaction Word Problems sheet. In this practice page, students solve six addition and subtraction word problems. Then, they shade in the word problems that show action with green and the nonaction word problems with red.

This may seem like an overly simplistic lessons, but it begins to build a foundation for more specific types of word problems. It’s best to not overwhelm students with too much, too soon.

Types of Action Word Problems

The next step is to have students understand the different types of action word problems. In this lesson, students can reuse the sorting cards and sort the action cards into joining and separating problems. This is often surprisingly challenging for students.

After students sort their cards, introduce the terms result unknown, change unknown, and start unknown. Go through each of the cards to determine what type of unknown the card includes.

Place six sheets of chart paper around the classroom. Label each sheet of chart paper with: Joining Start Unknown, Joining Change Unknown, Joining Result Unknown, Removing Start Unknown, Removing Change Unknown, Removing Result Unknown.

Have students write an example of each type of word problem on a separate sticky note. After writing the word problems, students should place the sticky notes on the corresponding sheet of chart paper. Once again, I like to follow that with extra practice.

Over the years I’ve found that one of the best ways to slow students down when solving word problems is to have them color code and underline each part of the word problem. This is a game changer! I introduce it right away and try to make color coding become a habit for students. This is the expectation for every word problem they solve.

Types of Nonaction Word Problems

Students should then learn about nonaction word problems. I repeat the same process to teach the different types of nonaction problems: Comparing Difference Unknown, Comparing Greater Unknown, Comparing Fewer Unknown, Part-Part-Whole Whole Unknown, and Part-Part-Whole Part Unknown.

For extra practice, students can complete a Types of Nonaction Word Problems worksheet. These are the only word problems that I do not have students color code.

I do not make students memorize each type of problem. But, I do make sure they understand the work each number is doing within the word problem.

Situational Equations

Many times when students write equations, they write an equation that can be used to solve the problem, but it does not represent the situation of the word problem. Situation equations represent the action of the problem. Solution equations can be used to solve the problem.

Example: Some children were playing tag at recess. When another class came outside, eight more students joined them. Now there are 14 students playing tag. How many students were playing tag to begin with?

The solution equation would be 14-8=_____. However, the situation equation is ____ + 8=14.

To teach this, discuss each type of action word problem and have students notice the placement of each unknown quantity for each problem. Students should recognize that the unknown quantity naming the word problem also describes the location of the unknown in the situation equation.

Then discuss what type of solution equation would be most effective to solve the problem. Once again, follow this with extra practice.

Each of the lessons above are from my Addition and Subtraction Unit. It’s a 4th grade unit, but I did create and include a version for 3rd grade students

Multiplication and Division Problems

When I begin teaching multiplication and division word problems, I don’t stop reviewing addition and subtraction word problems. It’s important to combine all operations to make sure students are seeking meaning in every word problem they solve.

The key to understanding multiplication and division problem situations is identifying the work each factor does. Students must consider the following:

• What do the numbers represent? What do they do?
• How can I represent the quantities with manipulatives or pictures?
• What number sentence best shows what is happening in the story?

Equal Groups

To fully understand equal groups, it is essential that students see the difference between the number of groups and how many objects are in each group. The table below can be used a starting place for thinking about equal group situations. It can also be a reference for students as they write their own word problems.

While the commutative property is helpful in memorizing multiplication facts, it is not always helpful in allowing students to develop a deep understanding of operation sense and the meaning of word problems. In word problems, most situations are not commutative, and students should be able to represent the situation accurately.

Questions like the one below allow you to see how deeply students understand multiplication situations, and it does a great job of illustrating how situations are not commutative.

As we move into multiplication and division word problems, the highlighting continues. I chose different colors so students don’t confuse addition and subtraction color coding with the multiplication and division color coding. The word problems below show how this can be done digitally.

In the above problems, I have already sized the rectangles for students. However when all operations are combined, students may have to resize the highlighting rectangles.

Multiplication Comparison

Multiplicative comparison problems often pose a challenge to students, as students tend to confuse additive and multiplicative comparisons. Students need the opportunity to experience acting out and building representations for this type of problem. These comparison problems should be interwoven with equal group problems.

To give students time to reason about these word problems, they can complete sorts where they discern between additive comparison problems and multiplicative comparison problems.

Students need multiple opportunities to work with multiplication as a comparison, and I hope to have a new post ready for that next week!

During your instruction, it is incredibly beneficial to have students write their own word problems. This takes their learning to a deeper level and it gives you a greater insight into your students’ understanding.

Error Analysis

Even using these strategies, You will still see errors and confusion, especially with situational equations. This is a process and this operational sense will be developed over time with consistent practice and instruction. Once I teach students the basics of word problems, I like to have students complete error analysis problems. It’s best to only complete one problem a day to allow deep conversations. The errors found in these problems are not computational errors. Instead, they’re errors in the representation of the problem.

I included a Peardeck option for all of the error analysis activities. Of course you don’t have to use that option. It’s just a way to make the lesson even more interactive.

All of the slides above are from my Multiplication and Division Word Problem resource. You can find that resource here.

With or without the resource, I hope that this has helped provide you with ideas and strategies for teaching word problems. Let me know how it goes!

The post Teaching Word Problems appeared first on Ashleigh's Education Journey.

Strategies for multiplication are so important for students. As a math teacher, multiplication facts have always been a source of concern. We know that a deep understanding of multiplication is essential for students, but there is also need for students to have at least a moderate (ideally more) memorization of multiplication facts, as well as strategies for multiplication.

The past several years, I’ve used my own resources along with online programs such as Xtra Math to help students learn their multiplication facts. However, I developed a couple concerns with these programs.

One concern is the stress that I see it cause students. Many children, especially the ones who struggle with memorization and/or anxiety, appear to become frustrated by the timed aspect of the quizzes.

The other concern is that I don’t see a lot of math fact progress with my struggling students, even with ample practice time in class. While all students can certainly learn and improve, some students need alternative strategies for multiplication and approaches, rather than solely focusing on memorization.

This digital multiplication program provides an alternative solution. It can be used along with any of your favorite existing resources or math curriculum.

This multiplication strategies program is broken into 5 parts. Each day of the week students complete a different task based on the math facts they are currently practicing.

Monday-Build It

On Monday, students create a grouping model for each set of facts (2×2, 2×3, 2×4, etc.), and grouping models are an important strategy for multiplication. They allow students to visualize and “see” multiplication.

To create the grouping model, students drag the larger circle to indicate the number of groups. In the example, there are two groups, so the student drug two large circles to the slide. Then, students drag the squares to show the quantity in each group.

Students will repeat this process for each set of facts.

Tuesday-Matching Representations

On Tuesday, there are three matching pages for students to complete. This allows students to begin thinking about visual representations of multiplication, which is also important strategies for multiplication.

For each set of facts students drag a written description, array, repeated addition problem, and a multiplication equation to a single row. Each page has three rows and all the needed cards are included on that page, so students will not have to copy and paste any cards.

Strategies For Multiplication – Wednesday-Multiple Representations

On Wednesday, students use the slides to create multiple representations of each set of facts, rather than match representations.

Students drag a square to a hundreds grid to show how to skip count. Then, they type a repeated addition representation. Students then drag blue squares to build an array of the fact. The last step is to drag an arrow to show multiplication on a number line.

Multiplication Strategies – Thursday-Word Problems

It’s incredibly important to incorporate word problems throughout your instruction and to not wait until the end of the unit. This allows students to understand multiplication in context, which supports strategies for multiplication.

In this activity, students solve to multiplication word problems (that use that set of facts) and then they write two word problems using that set of facts.

Multiplication Strategies – Friday-Game

On Friday, students move to a more abstract multiplication with a multiplication game. The game will focus on the multiplication fact of the week. If students don’t have a partner, they can play color vs. color.

Daily Check-Up

The last item of each day with the digital multiplication program asks students complete a daily check-up. It’s a timed test with a twist.

Students click on the link to go to a Google Form where there is a short quiz. They must get each question correct before moving to the next question.

Before beginning the quiz, students are prompted to start the included timer. Students are prompted to stop the time when they finish the quiz.

Then, students slide the colored tab on the graph to indicate how long it took them to complete the quiz. Students will only take one quiz a day and by the end of the week, they can see their progress.

Implementation Ideas for the Digital Multiplication Program

If you’re teaching third grade, you may want to assign program this to all students. I would not begin this practice until after the concept of multiplication has been introduced to students. However, it’s necessary to wait until the very end of the unit to begin implementing strategies for multiplication.

If you’re teaching fourth or fifth grade, you may want to give a pretest for each set of facts. You could easily use one of the Google Forms for the pretest. Then, only assign each set of facts to the students who need the extra practice.

You could pace students with one set of facts per week and pretest as you begin each week. You could also have all students work at their own pace with some students practicing 2 facts, some 3 facts, and others 4 facts.

Whether or not you grade each page is completely up to you and your expectations. I plan to use this as an instructional tool, not an assessment.

This can be incorporated into morning work, math centers, a skills block, remediation program, or even homework. There is definitely no one right way to implement this program.

It is recommended to teach multiplication facts in the following order. However, once again, it’s entirely up to you.

• 5 facts
• 10 facts
• 2 facts
• 4 facts
• 8 facts
• 3 facts
• 6 facts
• 9 facts
• 7 facts

Rather than create a new resource, I added this digital multiplication program to the existing print Multiplication Program. That way you will have access to both. If you already own it, you can download it again (at no cost) for the update! Students will greatly benefit from these strategies for multiplication.

If you don’t already own it and would like to test it out, click the link to download the 2 facts for FREE here!

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