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Teaching with Jennifer Findley

Upper Elementary Teaching Blog

Multiplication Strategies for 4th and 5th Grade

Do your students struggle with their multiplication facts? Despite what I thought my first years teaching, they don’t need more flashcards. They definitely don’t need timed drills or tests. Instead, they need strategies to help them use what they do know  (easier multiplication facts or addition facts) to solve any problem. Check out this post to read about the 6 multiplication strategies that I teach my 4 th and 5 th graders and grab free printable posters to help you teach the strategies to your students.

Teaching the Multiplication Strategies

I try to teach strategies that can be applied to any multiplication problem. I have never found success with giving my students specific strategies to use with specific math facts. Typically, if a student can remember to double and then double and then double for the 8s, they are able to actually memorize the math facts.

Instead, I prefer to use multiplication strategies that are conceptual in nature and work with any multiplication problem (though some are definitely more efficient with specific multiplication facts). Not only does this just make sense, but it deepens their understanding of multiplication which leads to greater success with word problems and with division.

When Do You Teach These Strategies?

Since my time is limited with 4th and 5th graders, I teach these strategies during number talks or small groups with students who need it. I try to work one day a week with my students specifically on their math fact skills. I wish I had more time, but honestly with the pacing demands and grade-level skills hanging over our heads, one day a week is sometimes a challenge.

After introducing and modeling a multiplication strategy, I like to provide my students immediate practice using this set of multiplication strategy task cards . I either use these task cards in a math center or as part of my small group intervention lesson. The goal with introducing these strategies and then having them practice (with the task cards) is to get my students familiar with using the different strategies. This will hopefully lead to them eventually choosing the ones that are most efficient for them and for the specific problem they are solving.

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Multiplication Strategies Task Cards

Now, let’s talk about those multiplication strategies., multiplication strategy #1 – repeated addition.

This is a typical strategy that most students begin with. I like to encourage my students to add quicker (by combining) and add mentally. This strategy is a foundational one that will help the students understand the other ones. This is why this is so heavily focused on in 3rd grade.

If your students are unable to do repeated addition, they may struggle with the more advanced multiplication strategies. So even though this may feel tedious and time-consuming, it is an important foundational strategy that leads to more efficient strategies (and helps with conceptual understanding).

Multiplication Strategy #2 – Arrays

This multiplication strategy is a newer one to me and I use it in a specific way. I don’t encourage my students to necessarily draw arrays. Instead, I use the visuals of different arrays to help them see the connections and known facts “inside” a more difficult fact.

As you can see from this example, 4 x 4 can be decomposed visually into 2 x 4 and 2 x 4. This helps the students solve for 4 x 4. When using this strategy, I like to invite the students to share as many decompositions as they can find and we record the multiplication problems that match.

This strategy specifically helps students “decompose” a more difficult multiplication problem into smaller problems that they do know automatically. Using the arrays is a great way to help students visualize the decompositions. This will help the more advanced multiplication strategies make sense to the students.

Multiplication Strategy #3 – Using 1s, 2s, and 5s

The decomposed array strategy leads right into the next strategy. For this strategy, the students use their known facts (usually 1s, 2s, and 5s) to solve unknown facts.

Here you can see that the 8 x 4 can be solved by decomposing the 8 into 5, 2 and 1 and solving (some students may do 5s and 3s). And the 6 x 7 can be decomposed into 5 x 7 and 1 x 7 to find the product. This is an excellent way to get your students to solve the trickier math facts and get them ready for partial products multiplication.

Click here to read a more detailed post about this multiplication strategy and grab a few more printables.

Multiplication Strategy #4 – Skip Counting with a Twist

Like repeated addition, skip counting is another foundational strategy that students learn in 3rd grade. I like to expand on this by having students use their skip counting skills to solve the unknown multiplication facts that they are unable to skip count for (4s, 6s, 8s for example).

As you can see from the example, students can use their skip counting by 5s and 2s to solve the 7s. This is not my favorite strategy and doesn’t work with every student, but some really gravitate toward it.

Multiplication Strategy #5 – Add a Group

The “Add a Group” strategy is just like the name implies. The students use the multiplication fact that is one group less (and easier or a known fact) to help them derive the unknown fact.

As you can see from the example in the image, students can use 5 x 6 to help solve 6 x 6 by adding another group of 6 to 30. Or they can solve 3 x 8 by adding another group of 8 to the solution to 2 x 8.

As I mentioned above, I prefer to use multiplication strategies that work no matter what the problem is. However, this strategy (and the next one) do have specific multiplication facts that they work best with.

The “Add a Group” strategy (in my experience) works best when solving 3s, 4s, and 6s (using 2s, 3s, and 5s respectively).

Multiplication Strategy #6 – Take Away a Group

Similar to the above strategy, this strategy has the students “taking away a group”. The “Take Away a Group” strategy (in my experience) works best when solving 4s and 9s (using 5s and 10s respectively).

Grab the Multiplication Strategies Posters Here!

Hopefully this post was informative and you can use these strategies with your students. Click here to grab the printable multiplication strategies posters shown on this post.

Tips for Success

• Introduce each strategy one at a time and allow for direct targeted practice with that strategy. Students will eventually gravitate toward the strategies that work best for them (or for the particular problems they are solving). However, we want to make sure we are providing the much-needed practice in each strategy. Click here to see the Multiplication Strategies Task Cards that I use to help provide this practice.
• Name the strategies. Naming the strategies helps the students share the strategies that they used and also helps when students are stuck and need a suggested strategy.
• Encourage students to use their strategies when they are solving multiplication problems and especially when they are solving grade-level work with embedded multiplication facts.
• Hang the posters or provide students with copies of the strategies. This goes along with the above tip. When encouraging students to use the strategies, they may need support and reminders of the strategies available for them to use.
• Be mindful of preserving conceptual understanding. I have been known to use tricks myself but I always try very hard to preserve conceptual understanding and avoid tricks until absolutely necessary (if at all). Having a conceptual understanding of multiplication is key for students when they are using these strategies (and when they are solving word problems).

Do Your Students Understand Multiplication Conceptually?

If your 4th and 5th graders still struggle understanding multiplication conceptually, they may struggle with these strategies as they build their understandings. The strategies are conceptual so using these will help. However, if your students need remediation on conceptual multiplication, check out my Multiplication Intervention Task Cards by clicking here or on the image below.

Multiplication Intervention Task Cards | Multiplication Conceptual Understanding

More free multiplication facts activities and resources.

FREE Multiplication Conceptual Understanding Pre-Assessment

Free Take Home Multiplication and Division Facts Toolkit

Free Multiplication and Division Facts Partner Games

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December 17, 2020 at 7:15 pm

Can these be virtual or shown on the screen?

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August 28, 2022 at 10:14 am

Thank you so much for you passion being shown in in all of these wonderful resources!!

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I’m Jennifer Findley: a teacher, mother, and avid reader. I believe that with the right resources, mindset, and strategies, all students can achieve at high levels and learn to love learning. My goal is to provide resources and strategies to inspire you and help make this belief a reality for your students.

Multiplication Strategies That Will Help Your Child Master the Times Tables

Multiplication strategies make it faster and easier for kids to learn the times tables. Learn tried-and-true multiplication strategies (as well as the best order for teaching the multiplication facts) that will help your kids master multiplication. 

In math, it’s usually best to move from one topic to the next sequentially.

The addition facts come before the subtraction facts.

Multiplying whole numbers comes before multiplying decimals.

Dividing by 1-digit numbers comes before 2-digit numbers.

But when it comes to memorizing the multiplication facts, your kids will find it faster and easier to memorize the times tables  out of order . In my years of helping kids learn the multiplication tables, here’s the best order that I’ve found for memorizing the times tables, along with multiplication strategies for learning each table.

(Psst…I use a visual called a multiplication array to demonstrate the strategies in this article. Click here to learn more about multiplication arrays and to download your own printable dot array. )

Strategy: Any number times 1 equals the original number.

The ×1 facts are very easy for kids to memorize, and they’re the best place to start. Since 1 group of any size equals the original group, 1 times any number equals the original number. Done!

Strategy: Double the other number.

Next, come the ×2 facts. Since 2 × 8 means “2 groups of 8,” your child can simply double 8 to find that 2 × 8 = 16.

Strategy: Use the ×2 facts as stepping stones.

The ×3 facts build on the ×2 facts. For example, your child can use 2 × 7 to help memorize 3 × 7. Since 2 × 7 is 2 groups of 7, he can add one more group of 7 to find that 3 × 7 is 21.

• Related: Learn more about using easier multiplication facts as stepping stones.

Strategy: Double the related ×2 facts.

Just like with the ×3 facts, your child can use the ×2 facts as a stepping stone to mastering the ×4 facts. In this case, she can find the answers to the ×4 facts by doubling the related ×2 facts. For example, she can use 2 × 6 to find 4 × 6: since 2 × 6 is 2 groups of 6, she can double 12 to find that 4 ×6 equals 24.

Strategy: Use place-value.

Time to skip ahead a little! Learning the ×10 facts make the ×5 facts easier, so we’ll take a quick detour. Your child can use her understanding of place-value to memorize the ×10 facts. For example, 7 × 10 means 7 groups of 10. Place-value tells us that 70 equals 7 tens, so 7 × 10 equals 70.

Strategy: Put groups of 5 together to make 10s.

Now that your child has learned the ×10 facts, she can build on that knowledge to memorize the ×5 facts. She can find the answers by putting together 2 groups of 5 to make a 10, and then use place-value knowledge to find the total. For example, to find 6 × 5, she can think, “Every 2 groups of 5 makes 10. Since I have 6 groups of 5, I can make 3 tens. 3 tens is 30, so 6 × 5 is 30.”

Strategy: Use the ×5 facts as a stepping stone to mastering the ×6 facts.

Just your child can use the ×2 facts to figure out the ×3 facts, he can also use the same idea with the ×5 and ×6 facts. For example, 5 × 7 will help him memorize 6 × 7.

6 × 7 is just one more group of 7 than 5 × 7. So, he can add one more group of 7 to 35 to find that 6 × 7 is 42.

Strategy: Use the ×10 facts as stepping stones.

It may seem like your child has 10 new facts to learn this week: 1×9 up to 10×9. But,  thanks to the commutative property, she has already learned most of the ×9 facts. (For example, she learned 3 × 9 along with 9 × 3, and she learned 5 × 9 along with 9 × 5 .) That means she only has three new ×9 facts to learn: 7 × 9, 8 × 9, and 9 × 9.

For these facts, she can use the ×10 facts as stepping stones. But, instead of adding  (as she did with the ×3 and ×6 facts), she can subtract .  For example, she can use 10 × 8 to find 9 × 8. Since 9 × 8 is just one less group of 8 than 10 × 8, she can subtract one group of 8 from 80 to find that 9 × 8 is 72.

×7 and ×8 Tables

Strategy: more stepping-stones (and lots of practice).

At this point, your child has just 3 multiplication facts left to learn:  7 × 7, 7 × 8, and 8 × 8. Many children find these 3 facts the most difficult to memorize. He can use the other ×7 and ×8 facts that he has already learned as stepping stones to these facts. (For example, he can add 7 to 6 × 7 to figure out 7 × 7.)

Even with stepping-stone facts, most children will need a lot of practice with these difficult multiplication facts to cement them in long-term memory.  Make sure to give your child lots of practice with them to make sure these last few facts really stick!

Want an easy-to-use, open-and-go, and fun resource for teaching the multiplication facts? Check out Multiplication Facts That Stick !

6 thoughts on “multiplication strategies that will help your child master the times tables”.

I am looking for the dot array and cover up I saw on your video. Would I be able to get a copy of this? Thanks so much! Very helpful!! Susan

Glad you found it helpful! All you need to do is click the green link above and you’ll be able to enter your email to receive your own copy. If that doesn’t work for some reason, feel free to email me at the contact link above and I can send it to you directly.

Happy Math! Kate

Hi Kate. I’m a teen who is about to start tutoring a third grader who doesn’t know his multiplication facts very well. He learns best through games and hands- on work, but because of coronavirus I’ll be tutoring him over Zoom. Do you have any tips for how to teach him and keep him engaged?

Tutoring online will definitely be a challenge! Take a look at this article and this article for some thoughts on how to make math fun, but you’ll need to modify some of them for the online format.

Best wishes in your tutoring, and happy math! Kate

I’ve tried to access the multiplication tips and games, but nothing is coming to my email. I’ve checked spam as well. The contact is not available, either.

Sorry that it’s not working for you, Lisa! Here’s the link to my contact page . Please shoot me a message there with your email address and the downloads you’re looking for, and I’ll get those out to you.

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October 28, 2021

Multiplication Strategies 101: Strategies That Build Fluency

I love teaching multiplication strategies! They help my students become more fluent and strengthen their knowledge of numbers. Multiplication is more than remembering which product goes with which factors. Knowing strategies improve number sense and help students retain their multiplication facts .

When students unlock multiplication strategies, it’s like handing them a mental toolkit. They not only remember their facts but understand the ‘why’ behind them. This deep understanding helps kids hold onto those facts for the long haul.

Here are some of my favorite methods for teaching multiplication facts, and if you don’t teach multiplication, keep reading so you can see what your students need to know in order to master it in the future.

If you’re looking for fact fluency for addition and subtraction, be sure to read this blog post, where I’ll show you how to break it down for your students , and also check out the Addition and Subtraction Mini Bundle from my TPT shop.

Factors of One

Let’s start with some of the easiest factors – zero and one . These factors have their properties, making them super special. The Zero Property of Multiplication means that any factor multiplied by zero will be zero. The Identity Property of Multiplication means that any factor multiplied by one will remain the same. So, 5 x 0 = 0, and 5 x 1 = 5. 

Most students grasp these two multiplication properties reasonably quick; however, you will want to look out for students who are adding instead of multiplying. Tell your students that is a zero or one factor, they’re either not grouping the other number at all (with zero) or grouping it just once (with one).

You can use this multiplication and division game to practice these factors.

Multiplication strategies for factors of two.

Learning about factor two is fun! I teach students that the product is the same as the sum of a doubles fact for this strategy. For instance, 7 x 2 is the same as 7 + 7. Both the product and the sum are 14. Another simple strategy is to use skip counting by 2’s. 

If students are having trouble with their 2’s facts, then they are likely struggling with their doubles facts in addition. I recommend going back to model two-column or two-row arrays to show how doubles facts are similar to 2’s facts. You can use an abacus or counters for this.

Make sure you read this blog post on how to introduce addition and subtraction to your students.

Factors of 3

The three’s facts are when we begin diving into the Distributive Property of Multiplication . This is basically a fancy way of saying you can break a factor into smaller bite-sized parts. In our case with threes, you split 3 into 1 and 2. You multiply those smaller parts and boom– you’ve got your answer when you add ’em up.

For example, 4 x 3 is the same as (4 x 1) + (4 x 2) = 4 + 8 = 12. Students must have a solid grasp on the ones and twos facts before mastering this concept. 

An easy way to model this in your classroom is by giving your students counters. Have them create three columns of the same amount. Then, place a craft stick between a set of columns so that you have one column on one side and two columns on the other. 

You might want to provide counters, snap cubes, or an abacus for your students while they are learning this strategy. Teach them how to use the manipulatives first, and let them explore to find all the factors. Eventually, they will begin to make a visual in their mind and complete the three’s facts without manipulatives. 

Multiplication Strategies for Factors of Four

Double – double anyone? Teaching four’s facts are always fun. Again, we use the Distributive Property of Multiplication to teach this strategy. Students should be familiar with their two’s facts. 

For this strategy, we use a similar example as we did for the three’s facts, except we use four columns. So, 4 x 4 is the same as (4 x 2) + (4 x 2) = 8 + 8 = 16. If we’re thinking about the two’s factors being a double, then this is a double-double fact. Because (4 x 2) + (4 x 2) = (4 + 4) + (4 + 4). 

You can practice four’s facts with this game.

Factors of five.

We usually teach five’s facts right after 0, 1, 2, and 10. This is such a fun strategy to teach. Students should know that all products of five end with either a 0 or a 5 in the one’s place. We skip count by five to multiply. 

This shows up in the real world when we are telling time on an analog clock. Students can see 60 represented in the face of the clock. It also shows up when counting coins. These are two simple supporting standards you can include when teaching five’s factors.

If you have students struggling to skip count by 5’s, try refreshing their skills with this math game . 

Multiplication Strategies for Factors of Six

Now that we’re getting up to the higher numbers, the strategies become even more necessary! First, any number multiplied by six will always be an even number . Next, students can use their knowledge of ones, fives, and the distributive property to solve equations. 

For example, 8 x 6 can be thought of as, (8 x 1) + (8 x 5). The products become 8 + 40, which equals 48. You can also multiply the factor by five and then add the factor one more time. 

Be aware of students who are struggling with the Distributive Property of Multiplication . Use an abacus, counters, or other math manipulatives to assist students in understanding how to use the Distributive Property of Multiplication . 

Strategies for Factors of Seven

Sevens facts are moderately tricky for most students. I like to use the Communicative Property of Multiplication to identify other factors with easier strategies if it has one, but it doesn’t always. For this strategy, I ask students what two addends of 7 would make helpful factors to use the Distributive Property of Multiplication .

They usually always say 2 and 5, so again, we model using our counters. Let’s say we have 7 x 7. We can think of it as (7 x 2) + (7 x 5) = 14 + 35 = 49, so 7 x 7 = 49. 

This factor takes a lot of practice, so provide lots of opportunities and support with manipulatives and extra practice like these math games !

Factors of Eight

Eights facts are probably the most difficult for students to learn, but there is an excellent strategy for this factor! I call it the “Turkey Factor” because it sounds like I’m gobbling when I tell the students. I ask them to double, double, double! 

What does this mean?

Well, let’s say we’re trying to solve 8 x 8. Let’s double the factor eight to get 8 + 8, and our sum is 16. Now, let’s double 16 to get 16 + 16. Our sum is now 32. Let’s double that sum one more time to make 32 + 32. The total is 64. So, 8 x 8 = 64.

This strategy works any time you multiply a number by 8. Here’s another example: 3 x 8. Double three to get 3 + 3 = 6. Next, double the sum to get 6 + 6 = 12. Finally, double the sum once more to get 12 + 12 = 24. So, 3 x 8 = 24.

Model this strategy and allow students time to practice.

Multiplication Strategies for Factors of Nine

Nifty nines are my favorite factors to teach because they are magical!

First, I model on my hands with my palms facing up. Let’s try 3 x 9. Beginning at my left thumb, I count 1 – 2 – 3. My middle finger is down. I notice my thumb and index finger are still up to the left of my middle finger. This represents two tens. Then, I noticed seven fingers to the right of my middle finger were up. This represents seven ones. So, 3 x 9 = 27. 

Now, if I were to take the digits of the product and use them as addends, my sum would be nine. For example, 2 + 7 = 9.

Not only that, but two is one less than three. I can find 7 x 9 just by thinking about one less than 7 to get 6. Now, if I put that into an addition equation to equal nine, I’d have 6 + ? = 9. I know that 3 makes the equation true, so the answer is 63. 

Try not to teach all these strategies at once to prevent overwhelm. Instead, teach one skill one at a time and give your students time to practice and explore. They may have trouble using their fingers, so use a printable of hands to give them practice. 

Factors of Ten

I typically teach the tens facts along with my ones and zeros. They are super easy to teach students. Any number that is multiplied by ten will be in the tens place, and a zero will remain in the one’s place. 

For example, for the equation 5 x 10 = 50, the product has a 5 digit in the tens place and a 0 digit in the one’s place. Students pick up on this quickly! 

So there you have some of my favorite strategies for teaching students about multiplication. Please know that we practice these strategies a lot. We use these multiplication and division math games to apply these strategies. They are fun for math centers and homework.

I’d love to know which of these strategies you teach your students.

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A Complete Multiplication Guide: Basic Strategies for Every Learner

This article is designed to provide learners of all ages and skill levels with the basic tools and techniques needed to master multiplication. Whether you’re just starting with multiplication, looking to refresh your skills, or helping a young learner, this guide will offer valuable insights. It aims to make multiplication less intimidating and more approachable by breaking it down into its simplest elements. Continue reading the article to embark on this exciting numerical journey.

Importance of Multiplication Skills

Multiplication is a fundamental arithmetic operation that is pivotal in many aspects of mathematics and daily living. It forms the basis for more complex mathematical concepts like fractions, algebra, and geometry. Beyond academics, multiplication skills are invaluable in everyday scenarios, including budgeting, cooking, and time management. Developing strong multiplication skills at an early age equips learners with a sturdy mathematical foundation, fostering logical thinking and problem-solving abilities.

Traditional Multiplication Method: Step-by-Step Guide

The traditional multiplication method, long multiplication, is a fundamental process often taught in school to handle multi-digit numbers. Here’s a step-by-step guide to help you understand and employ this method:

• Write the Numbers: Write down the two numbers you want to multiply, one above the other, aligning them on the right side. Draw a horizontal line under the two numbers.
• Multiply by Units: Start by multiplying the digit in the unit’s place (rightmost digit) of the lower number with every digit of the upper number, from right to left. Write the result below the horizontal line.
• Carry Forward: If the product is a two-digit number, write down the unit digit and carry forward the tens digit to the next multiplication.
• Next Row: Move to the next digit in the lower number (tens place). Repeat the multiplication and carry forward process, but this time, write the result in the next row, starting one place to the left.
• Continue Process: Repeat this process for every digit in the lower number.
• Add Results: After completing all the rows, add them up. The total is the result of the multiplication.

Remember, practice is key to mastering the traditional multiplication method. Start with simple numbers and gradually progress to larger ones. With regular practice, this strategy will soon become second nature.

What is Skip Counting, and How Can it Boost Your Multiplication Skills?

Skip counting is among the most effective multiplication strategies to enhance your multiplication abilities significantly. Fundamentally, skip counting involves counting forward by numbers other than one, for instance, counting by twos (2, 4, 6, 8, and so forth) or by fives (5, 10, 15, 20, and so on).

Skip counting lays the groundwork for understanding multiplication as it helps learners visualize and understand the concept of grouping, a cornerstone of multiplication. When children skip count, they add the same number repeatedly, paving the way toward understanding that multiplication is a more efficient way of adding the same number.

For example, five groups of three can be calculated by adding three, five times (3+3+3+3+3). Alternatively, it can also be calculated by skip counting by threes five times (3, 6, 9, 12, 15). Eventually, learners realize that this process can be further simplified by multiplying 5 x 3 = 15.

Skip counting is a powerful tool in the multiplication arsenal. By incorporating this strategy, learners can develop a more intuitive understanding of multiplication, making the transition to formal multiplication methods smoother and less daunting.

Can Visual Representations Make Multiplication Fun and Easy?

Visual representations can indeed make multiplication engaging and approachable, particularly for visual learners who absorb information best when presented in a graphic or pictorial format. Arrays, number lines, and area models are a few visual strategies that can be employed to illustrate multiplication concepts.

Arrays are essentially grids with rows and columns representing multiplication equations. For instance, a 3×4 array represents the multiplication equation 3×4=12, with three rows of four squares each. This visual representation helps learners understand the concept of repeated addition and grouping inherent in multiplication.

On the other hand, number lines are fantastic tools for demonstrating skip counting or repeated addition. Multiplying 3×4, for example, can be shown as four jumps of three on a number line, reinforcing the connection between addition and multiplication.

Area models divide a rectangle into smaller squares or rectangles to represent a multiplication problem, where the length and width of the rectangle represent the two numbers being multiplied. This approach is useful for grasping more complex multiplication problems involving larger numbers.

By making multiplication visually tangible, these methods make the concept less abstract, aiding learners in grasping the essence of multiplication. Visual representations can be a lighthearted and engaging way of mastering multiplication, transforming learning into a fun-filled adventure.

How to Make Multiplication Easier: Factoring and Distributive Property

The power of multiplication lies in its simplicity and efficiency, and understanding the principles of factoring and the distributive property can make it even easier. Factoring involves breaking down a number into its multiplication factors. For example, the number 12 can be factored into 3 and 4, as 3×4 equals 12. This becomes useful when dealing with larger numbers or when simplifying fractions.

The distributive property, on the other hand, is a multiplication strategy that allows you to break down complex multiplication problems into smaller, more manageable parts. Also known as the multiplication break-apart method, it is expressed as a(b + c) = ab + ac. This means multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

Take the example of 6 x 14. This might seem like a tricky multiplication problem initially, but using the distributive property, you can break it down: 6 x (10 + 4) = (6 x 10) + (6 x 4) = 60 + 24 = 84. The distributive property allows splitting more challenging multiplication problems into ones that are easier to solve, making multiplication less intimidating, especially for young learners.

Understanding and utilizing factoring and the distributive property make multiplication easier and pave the way for understanding more complex mathematical operations. Armed with these strategies, you have additional tools to tackle multiplication more easily and efficiently.

Struggling with Big Numbers? Grasping Multiplication for Two and Three-Digit Numbers

Multiplying two or three-digit numbers might seem daunting at first, but with the right approach, it can be simplified. Learners can confidently multiply larger numbers by building on the basic multiplication strategies and understanding place value. The process involves breaking down the problem into manageable parts and using basic multiplication and addition to find the solution.

Take, for example, the two-digit multiplication of 24 x 13. You can break down 13 into 10 and 3. Now, multiply 24 by 10 and 24 by 3 separately. The multiplication of any number by 10 is straightforward; simply add a zero to the end of the number. So, 24 x 10 = 240. For 24 x 3, use basic multiplication skills to get 72. Now add the two results together: 240 + 72 = 312. Thus, 24 x 13 = 312.

The same strategy can be applied to three-digit numbers. For instance, multiplying 123 x 24 breaks down 24 into 20 and 4. Multiply 123 by 20 and by 4 separately, then add the results.

This strategy, which builds on the distributive property, can make multiplying bigger numbers more manageable. With practice, this method can significantly enhance accuracy and speed in solving larger multiplication problems and lessen the fear of multiplying big numbers. Remember, mastery comes with practice, so keep practicing these multiplication strategies until they become second nature.

Can You Multiply with Just Your Mind?

Yes, you can perform multiplication operations swiftly and accurately in your head by employing mental math techniques. Mental multiplication strategies primarily involve understanding number properties, leveraging place value, and utilizing basic multiplication facts.

For instance, when multiplying a number by 10, simply add a zero to the end of the number; for multiplying by 5, halve the number and multiply by 10. When multiplying by 9, multiply by 10 and subtract the original number.

To multiply two two-digit numbers mentally, break down the multiplication into smaller parts you can easily compute. For example, to multiply 21 by 13, you can calculate (20 x 10) + (20 x 3) + (1 x 10) + (1 x 3) = 200 + 60 + 10 + 3 = 273.

Mental multiplication might seem challenging at first, but you’ll become more adept with practice. It’s a powerful skill that can be extremely helpful in daily life, enabling quick estimations, improving number sense, and boosting confidence in mathematical abilities. Remember, the key to successful mental multiplication is practicing and understanding the underlying principles of the multiplication strategies outlined in this guide.

Tips for Faster Mental Multiplication

Mastering the art of mental multiplication can significantly speed up your math proficiency . Here are some valuable tips to help you multiply numbers faster mentally:

• Understanding the Properties of Numbers: Familiarize yourself with the properties of numbers. For instance, any number multiplied by zero equals zero, and any number multiplied by one retains its value. These basic properties can save you time and effort when multiplying mentally.
• Leveraging Simplification Techniques: Use simplification techniques to make multiplication more manageable. For example, if you’re multiplying a large number by a number ending in zero, simply perform the multiplication with the remaining digits and append the corresponding number of zeros at the end.
• Rounding and Adjusting: Round complex numbers up or down to make the multiplication easier, then adjust the answer accordingly. For instance, if you want to multiply 99 by a number, you might find it easier to multiply 100 by the number and subtract the original number.
• Breaking Down Larger Numbers: Break down larger numbers into smaller, more manageable parts. For example, for the multiplication 24 x 15, think of it as (20 x 15) + (4 x 15). This simplifies the problem into more manageable multiplications.
• Practicing Regularly: Regular practice is key to improving mental multiplication speed. Start with simpler numbers and gradually move on to larger, more complex ones.
• Using Skip Counting for Smaller Numbers: For smaller numbers skip counting can prove to be a faster technique. For instance, multiplying by 4 can be achieved by doubling the number twice.
• Understanding the Distributive Property: The distributive property can be a valuable tool in mental multiplication. It allows you to break down complex problems into simpler ones, making mental calculations more manageable.

Remember, the goal of mental multiplication is not just speed but also accuracy. Therefore, take the time to ensure your calculations are correct even as you aim to improve your speed. With these tips and regular practice, your mental multiplication skills improve significantly.

Strategies for Memorizing Multiplication Tables

One of the fundamental steps in mastering multiplication is familiarizing yourself with multiplication tables, also known as times tables. These tables provide a quick reference for multiplying small numbers and are vital for enhancing calculation speed and accuracy. Here are some strategies to make memorizing multiplication tables less daunting and more enjoyable:

• Start Small: Begin with the easier times tables such as 1s, 2s, 5s, and 10s. These tables follow simple patterns that are easy to remember. For instance, any number multiplied by 1 remains the same, and for the 10s, you simply add a zero to the end of the number.
• Use Visual Aids: Create a multiplication chart to refer to as you learn. Visual aids are highly effective when memorizing multiplication tables as they appeal to visual learners and offer a convenient reference point.
• Leverage Patterns: Look for patterns in the tables. For example, in the 9s table, the tens digit increases by 1, and the ones digit decreases by 1 as you go down the table (9, 18, 27, 36, 45…).
• Practice Regularly: Constant practice is key to memorizing multiplication tables. Make it a habit to review the tables you have learned daily, and gradually introduce new ones as you become more confident.
• Use Flashcards: Flashcards with multiplication problems on one side and answers on the other provide an interactive and effective way to learn. They’re also portable, allowing you to practice anytime, anywhere.
• Sing or Chant the Tables: Turn the tables into a song or a chant. The rhythm and melody can make the tables more memorable and fun to learn.
• Use Educational Apps: Numerous educational applications are designed to make learning multiplication tables engaging through games and quizzes. These can be especially appealing to tech-savvy learners.
• Apply Multiplication in Real-Life Situations: Apply the multiplication tables to real-life situations such as shopping or baking. This provides practice and demonstrates the relevance and utility of multiplication in everyday life.

Remember, mastering multiplication tables requires time and patience. It’s not a race, so learn at your own pace and move on to more complex tables only when you’re comfortable. You’ll have the multiplication tables memorized quickly with consistent practice and the right strategies.

Here are some frequently asked questions about different multiplication strategies to improve mathematical skills.

What are the strategies to teach multiplication?

Teaching multiplication can be made effective and enjoyable by employing various strategies. Here are a few approaches teachers can use:

• Concrete Learning: Start with concrete learning using physical objects like counters, blocks, or beads to illustrate the concept of multiplication as repeated addition. This helps students visualize and better understand the process.
• Skip Counting: Once students are comfortable with counting, introduce skip counting. It’s a precursor to multiplication, teaching children to count by twos, threes, or any number other than one.
• Arrays: Use an array model for multiplication to represent problems visually. This helps students connect the numbers in a multiplication equation and the size of the array.
• Number Line: Use a number line to illustrate multiplication. For example, for 3×4, make four jumps of three on a number line.
• Multiplication Tables: Encourage students to learn multiplication tables. Make it interactive and fun by using songs, games, or flashcards.
• Group Activities: Organize group activities that involve multiplication to promote collaborative learning.
• Real-Life Examples: Incorporate real-life examples to demonstrate the practical application of multiplication. This helps students understand its relevance and importance.
• Word Problems: Use word problems to practice multiplication. This enhances students’ problem-solving skills and allows them to apply their multiplication knowledge in various contexts.
• Incorporate Technology: Use technology like educational apps or online games that can make learning multiplication fun and interactive.
• Consistent Practice: Encourage regular and consistent practice. Like any other skill, multiplication becomes easier and more automatic with practice.

These multiplication strategies can contribute to a comprehensive and engaging approach to teaching multiplication. Remember, different students learn differently, so it’s beneficial to use a mix of these strategies to cater to diverse learning styles.

What is the standard multiplication strategy?

The standard multiplication strategy, often called the traditional method or long multiplication, involves breaking down the multiplication problem into simpler steps to obtain the final result.

Why teach multiplication strategies?

Teaching multiplication strategies is fundamental in developing a strong foundation in arithmetic, which is crucial in navigating other complex mathematical concepts. It enhances learners’ number sense, enabling them to understand and connect with numbers on a deeper level. These strategies also aid in improving computational skills, critical thinking, and problem-solving capacities.

What is the purpose of multiplication?

The purpose of multiplication is to simplify the process of repeated addition. Multiplication is a more efficient method when dealing with larger numbers or quantities. For example, instead of repetitively adding 5 ten times, it is quicker and easier to multiply 5 x 10.

How do students learn multiplication?

Students learn multiplication through a variety of methods, each serving to develop their understanding and fluency in the concept. Here are some key steps in the learning process:

• Understanding the Concept: The first step is understanding multiplication as repeated addition. This foundational concept is often taught using concrete objects such as counters or blocks, which allow students to see and physically manipulate groups of items visually.
• Learning Multiplication Facts: As students grasp the concept of multiplication, they begin to learn multiplication facts, often starting with the times tables of smaller numbers. This involves memorization and frequent practice.
• Practicing with Worksheets and Problems: Worksheets with multiplication problems provide students with ample practice opportunities. They are also introduced to word problems that require multiplication, promoting the application of their skills in different contexts.
• Utilizing Strategies: Students learn different strategies for multiplication, such as distributive, associative, and commutative properties of multiplication. These strategies and mental math techniques can help them solve more complex multiplication problems efficiently.
• Applying in Real Life: Teachers often incorporate real-life situations that require multiplication in lessons. This helps students understand the practicality and relevance of multiplication in solving everyday problems.
• Leveraging Technology: Educational apps and online games provide interactive and engaging ways for students to learn and practice multiplication. Resources like these often offer immediate feedback, allowing students to instantly correct mistakes and learn from them.

Learning multiplication is a step-by-step process that builds upon each previous step. It requires time, practice, and patience, but with the right approaches and resources, students can successfully master this important mathematical operation.

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Jennifer Scott

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Are you looking for strategies to help students who have trouble solving math problems with multiplication? If so, keep reading.

1. Provide the learner with alternative math facts . As the learner shows success, slowly introduce more components of the regular tasks.

2. Select a peer to model how to solve multiplication problems for the learner.

3. Ask the learner a question when they are most likely to be able to respond successfully.

4. Provide information to the learner on a one-to-one basis or use a peer tutor .

5. Create a math facts reference sheet for multiplication for the learner to use at their desk when solving math problems .

6. Talk about and give the learner a list of words and phrases that usually indicate a multiplication operation (e.g., area, each, times, product, double, triple, twice , etc.).

7. Assess the appropriateness of the task to determine (a) if the task is too easy, (b) if the task is too complicated, and (c) if the duration of time scheduled for the task is sufficient.

8. Get the learner to use a calculator for drill of basic multiplication facts .

9. Get the learner to check all math work . Praise the learner for each error they correct.

10. Get the learner to practice the multiplication tables each day with a peer using flash cards.

11. Get the learner to be a tutor and teach a concept they have learned to another learner.

12. Get the learner to count by equal distances on a number line. Show that the equal distances represent skip counting , which is the concept of multiplication.

13. Get the learner to independently solve half their multiplication problems each day. Let them use a calculator to address the other half of the task as reinforcement.

14. Get the learner to perform timed drills with multiplication facts as reinforcement. The learner “competes” against their own best time.

15. Get the learner to solve multiplication problems by manipulating objects and stating the process(es) involved.

16. Get the learner to talk through math problems as they solve them to find errors they are making.

17. Get the learner to use a calculator to reinforce the process of multiplication. Get the learner to solve several multiplication problems each day using a calculator.

18. Find specific multiplication problems the learner fails to solve correctly. Target those problems for additional instruction, tutoring, and drill learning activities .

19. Provide all instructions, questions, and explanations in a clear, concise manner and at an appropriate rate for the learner.

20. Make sure the learner has mastery of math ideas at each level before introducing a new skill level.

21. Consider using Alexa for the Math Classroom .

22. Try gamifying your math lessons .

23. Consider using one of the apps and tools from our many math app lists:

9 Must-Have Apps and Tools for Kids Who Hate Math

10 Apps to Teach Preschoolers Math

20 Math Apps for Students of All Ages

Advanced Math Apps, Tools, and Resources for That We Love

Elementary School Math Apps, Tools, and Resources That We Love

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High School Math Apps, Tools, and Resources That We Love

Interactive Math for the Google Classroom

Math Apps, Tools, and Resources That I Would Use If I Were Still in the Classroom

Math Manipulatives Apps, Tools, and Resources That We Love

Middle School Math Apps, Tools, and Resources That We Love

Multiplication Apps, Tools, and Resources That We Love

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Multiplicative Thinking

What strategies can I use to explore the concept of multiplication with students? Although the following strategies are not yet fully ready, you may find them helpful.

2 Digit by 1 Digit Multiplication

You will need access to one of the following:

• Physical Base Ten Blocks (tens and ones) plus a piece of paper or a whiteboard
• Printable Base Ten blocks (print in black and white OR color) plus a piece of paper
• Virtual Base Ten Blocks ( CPM Tiles were used in the videos)

Watch 2 digit x 1 digit Multiplication – Concrete – on YouTube

Watch 2 digit x 1 digit Multiplication – Mixed Concrete Abstract – on YouTube

Watch 2 digit x 1 digit Multiplication – Abstract – on YouTube

Watch 2 digit x 2 digit Multiplication – Concrete – on YouTube

Watch 2 digit x 2 digit Multiplication – Mixed Concrete Abstract – on YouTube

Watch 2 digit x 2 digit Multiplication – Abstract – on YouTube

Relational Thinking Strategies: Multiplication

When we know the strategies we want students to uncover, we become more strategic with the problems we pose, the numbers we choose, and the way we facilitate student discussions.

Students (who are allowed to solve in any way that make sense to them) often implicitly use the properties of operations and equality to simplify the problem. When we look and listen closely to student thinking, we can facilitate class conversations that make these relationships explicit.  

Keep in mind:

• If you show students how a strategy works you’re doing the thinking for them. You can use problem types, number choices, and discussion facilitation to put students into a position to collaboratively construct these relationships.
• It takes an incredibly long time to construct these strategies (and more importantly, the ideas behind them). Be patient with your students and yourself.
• Arrays are abstract for students. I’m using them here for the purposes of our discussion, but I do not recommend showing these animations to students. For each strategy you’ll see one way a student might model and use relationships. 

Let’s take a look at the strategies: Partial Products, Over and Subtract, 5 is Half of 10, and Factoring. 

Partial Products

The distributive property of multiplication over addition allows us to break a multiplication problem into chunks. Often that will be by place value, but don’t get too hung up on that (for example, I can multiply by 27s by breaking them up into 25s and 2s, since 25s are usually convenient to multiply). 

Many different number choices will help surface Partial Products. What really matters (in my opinion) is keeping it rooted in the context. Notice how easy it is to understand the distributive property when you talk about boxes and chocolates:

Breaking up a box of chocolates into 10 and 8 chocolates makes sense. Counting 10 chocolates from each box, then 8 chocolates from each box, then adding them together makes sense.  Even though it might not be initially apparent to us, directly modeling the problem (by 1s, shown below) can give students access to the properties of operations.

We won’t know how the student solved this problem until we ask. Maybe they counted by 2s (which connects to the associative property of multiplication) or considered the 10 and 8 in each group (which connects to the distributive property of multiplication).

A student who has modeled and counted by 1s has grounded themselves in the context and the magnitude of the quantities in a way that will allow them to make sense of peers’ ideas. They are primed for a “light bulb moment,” whether it is the chocolates modeled with the numeral “18” or the idea of breaking up 18 into 10s and 8s.

Over and Subtract

Like seeing negative space in art, you can shift your perspective to consider what is missing in a multiplication problem and compensate. You might be inclined to call it the  distributive property of multiplication over subtraction , but if you consider subtraction to be “the addition of the opposite,” we can see it as the  distributive property of multiplication over addition. 

To nudge this conversation, I like to pose a multiplication problem with two number sets and ask students to solve both. Or, I might do the problems back to back within the week. 

Sometimes it is hard to remember what to subtract after you’ve adjusted. When thinking about 20×5 to help you with 18×5, are we subtracting 5, 20, 18, or something else? Here again, the context makes it easy to see that we’re subtracting 2 boxes of 5 chocolates. 

5 is Half of 10

Why does 5 get its own category? Although the example here is 5×18, this idea shows up in problems like 16×23 (more on that below).

Depending on how a student describes their thought process this strategy could utilize the distributive property of multiplication over addition  (solving 10×18, then imagining splitting it into two chunks) or the  associative property of multiplication (thinking about 10×18 as 2x5x18).

To surface this discussion, you might offer contexts where the amount in each group stays the same and students consider the relationship between 10 groups and 5 groups. 

Half the boxes means half the chocolate, meanwhile the number in each box didn’t change. Context!

When you create groups of groups or break something into equal sized chunks, you’re likely using the  associative property of multiplication .  Halving and doubling (or tripling and thirding, quadrupling and fourthing, etc) relies on this property.

Students often start combining equal groups in ways that make sense to them. When we see this, we can bring it to the whole class and have them discuss the ways it transforms the problem. 

I can imagine counting 2 boxes at a time as 10 chocolates rather than 1 box as 5 chocolates. The more comfortable students are explaining the relationship between the two units (boxes and chocolates, for example), the easier ratios and proportions will be!

Mixed Strategies

We’ve looked at 4 categories of strategies, now let’s see how they might combine to help you solve a problem like 16×23. Keep in mind there are way more than 2 ways to solve this problem.

And one more:

Pam Harris says the goal is “fewer, bigger chunks.” If students consider the numbers before solving, they’re less likely to automatically break up problems into four chunks by place value (which is only a good strategy if you have other options to choose from).

The better we know the strategies (and the underlying relationships and ideas), the easier it will be to put students into a position to construct those strategies with their peers using meaningful contexts, carefully sequenced problems, and discussion.

After you’ve engaged in problem solving with context, you might enjoy the multiplication Math Flips decks, which get students talking about the important relationships in multiplication. 

Questions to consider:

• Do you (as a mathematician yourself) gravitate toward one of these strategies more often than the others? Is there one that you want to work on?
• What is the role of context in connecting student’s informal and intuitive understandings with more formal ideas (like properties of operations)?
• How can a series of related problems and number sets help students generate relationships and uncover strategies for themselves (rather than being shown how to solve)? Check out this post where Liz Romero facilitates a discussion that nudges her students to discuss the distributive property of multiplication. 
• What does this mean for your practice? Who might you talk with about this?

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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Course: 5th grade   >   Unit 8

Developing strategies for multiplying decimals.

• Multiply decimals tenths
• Developing strategies for multiplying 2-digit decimals
• Multiply decimals (1&2-digit factors)
• Multiply decimals (up to 4-digit factors)
• Multiplying decimals (no standard algorithm)
• Multiply decimals: FAQ

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Video transcript