MathBait™ Mastering Multiplication Part 7: Primes and Factoring

We've had an outstanding time designing and developing MathBait™ Multiplication to help your students build a lasting fluency and solid foundation! While this is the final unit in our series, don't forget, MathBait™ offers hundreds of games, missions, lessons and activities through our digital program, The Kryptografima.

The journey thus far has taken us from skip counting to dual processing to increasing fluency. Transitioning to multi-digit multiplication, we explored the many methods for introducing and helping students understand how to find larger products - including our own MathBait™ way! Parts 5 and 6 of our program focused on Napier's Bones, an outstanding tool for helping students build a strong conceptual understanding of multiplication and place value. This totaled 65 games and activities to help your students soar! Our final unit offers 16 activities and 8 new and exciting digital games to explore.

In Part 7, we shift our focus to help students better understand what we call "number traits". In other words, how can we build a number? What is any given number made of and how can this help us with multiplication? In fact, the activities in this unit will do far more than continue to build fluency in multiplication - they are promoting a familiarity and flexible view of numbers which will benefit students in division, working with fractions, finding square roots, as well as construct a foundation for algebra. Without further ado, let's jump into Unit 7!

Prerequisites

We recommend students have completed MathBait™ Multiplication parts 1-3 prior to beginning Unit 7. Students should feel comfortable with most products on the multiplication table (up to 10×10). Students need no prior understanding of division and division will not be introduced as a method of factoring.

Goal

In MathBait™ Multiplication Part 7, students will explore how to "build" a Natural number, how to break down a number, and how to use this understanding to improve multiplication fluency, as well as to find tricky products. You can think of Unit 7 like looking under the hood of a car. We might understand how to drive the car, but do we really know all the ins and outs of how a vehicle functions? Understanding the building blocks of numbers will not only skyrocket student multiplication fluency, but also provide them with strong tools vital for algebra.

It is highly recommended that students play the games provided here multiple times over a few weeks. Spiraling back to earlier games will help continue to grow fluency and to provide students with multiple methods of accessing their multiplication facts through retrieval.

Select a lesson to view details. Click on the arrow to expand. We recommend students play in the order provided for graduated levels of fluency. Bookmark this page to easily return for spiral review or a fun change of pace.

Why is factoring important?

Before we dive into our lessons, activities, and games, let's take a moment to discuss why factoring is a key skill for multiplication as well as the pedagogy behind MathBait™ Multiplication.

Recently, we came across an article with concerning claims about mathematics education. The article specifically stated skills such as factoring do not support fluency. Specifically, it mentioned for a student to use problem solving skills to find a product such as 7×8 by decomposing it into 7×4×2 and then doubling 28 to arrive at 56 would take too long. The article (which was not based on any research, did not include citations, and was mainly a puff-piece for a specific program) very strongly insisted memorization was the only path to fluency.

They are right. When students hit higher levels of math, the time it takes to recall basic multiplication facts matters. A good prealgebra question contains 5-8 steps and getting bogged down on even just 1 or 2 steps will significantly impact student confidence and performance.

While MathBait™ Multiplication introduces students to various strategies, as they practice, play the games, and gain more and more exposure, they are building fluency. Fluency is the ability to quickly recall information with what feels like little or no effort. Our strategies are constructing a strong foundation for algebraic thinking and number sense. Students will use this understanding far beyond multiplication. That being said, students will greatly benefit from quick recall. To achieve this, we recommend playing the games in MathBait™ Multiplication often. Our activities are helping students build multiple methods for recall; as they practice, they are turning one-way dirt roads into formidable multi-lane highways. So while the article is correct in that students shouldn't need to work through a multi-step strategy to recall basic products in the long-term, learning the strategy and practicing it will not only build algebraic thinking and number sense, it will also build a lasting fluency and bolster retrieval.

They are also wrong. Fluency is vital but memorization is not the best way to become fluent, in fact, it isn't even a good way to develop these skills.

In an earlier unit, we explained why memorization is not a good path to fluency. But why is the MathBait™ method better? Why is factoring the capstone unit of our program?

MathBait™ Multiplication is not simply focused on multiplication. It has been designed by mathematicians, who understand not only what students need to know in third or fourth grade, but the key foundational skills they require to be successful in mathematics, in college, and in the workplace.

The skills we are building are strengthening the whole brain. MathBait™ Multiplication provides students with many strategies and activities to look at multiplication from different perspectives. In the short-term, students are receiving constant exposure and practice to multiplication through novel activities. This builds fluency in line with how the human brain works. It also is much more fun and productive! "Drill and kill" can extinguish student interest and curiosity in mathematics, create anxiety, and inhibit creative thinking. Rather than only paving one way for a student to retrieve this knowledge, we are constructing an interchange with many lanes and directions, allowing for flexible thinking and stronger retrieval. In the long-term, students are developing a "feel" for numbers. They are strengthening their number sense and problem solving skills, and gaining a better understanding of the patterns and relationships that exist in the world and in mathematics. They are gaining algebraic knowledge which will better prepare them for when they need to use their multiplication skills in the future.

To obtain the most success with MathBait™, we recommend playing over and over again. This is how long-term fluency is achieved. As students return to earlier games, they will find them easier to win, as their skills and understanding have increased. The more they play, the more paved each road becomes. Not only is MathBait™ fun, it is providing students with the structure needed for long-term retention and success.

Quite often, factoring is not introduced to students until they learn division. Many may be wondering why (and even how) it is included in MathBait™ Multiplication as we assume no previous division knowledge.

Here are a few reasons why we include factoring in our multiplication curriculum.

• Learning to factor is an excellent strategy for building number sense and multiplication fluency.

• Division is tricky. Mathematically, it doesn't fit the rules of a binary operation as it is not associative; (48÷4)÷2 ≠ 48÷(4÷2). Thus, it is important for students to build a strong understanding of numbers before introducing division.

• Understanding factoring will greatly help students to more easily comprehend fractions and division (for example, if we wish to know 72÷4, we could take out a pen and paper and go through the tedious steps of long division, or we can rely on our multiplication fluency to see 72=9×8=9×2×4 and thus 72÷4 is simply 9×2=18.)

• Knowing how to build a number, or what a number is made of, will drastically improve student performance and understanding in subsequent topics.

We can think of factoring as a way to rewrite a tricky multiplication problem into an easier one. Throughout this unit, we will build student number sense and flexibility when working with numbers.

In conclusion, the strategies presented in MathBait™ Multiplication are doing much more than teaching your student to multiply. They are building strong foundational understanding of numbers and relationships that will take them very far and prepare them for what is coming next. Factoring is one of these skills. Understanding how a number is built will help students to remember trickier products and use problem solving skills for products outside the multiplication table. Fluency is the result of repeated exposure. It is what happens when you get so good at something you don't need to think about it too hard to know what to do or how to do it. Memorization does not build long-term fluency. Memorized facts are only stored in short-term memory and as soon as we stop constantly practicing (or retrieving) this information, it will be one of the first things our brain kicks out when in need of more space. The long-term fluency MathBait™ Multiplication builds means that when a student forgets a product, they have gathered so many tools and developed such a deep conceptual understanding and strong underlying number sense that they have many roads to "find their way".

Let's dive into our fun and engaging methods to teach students how to factor without division!

Block Target

In this lesson, we explore how we might build a number using blocks. Students use manipulatives to help organize values to determine the factors.

Warm Up: Playing with Blocks

The goal of this activity is to encourage students to play and explore in order to find and identify patterns.

Provide students with 24 blocks. We recommend stacking blocks such as Math Link Cubes. These cubes are great for building number sense and have many applications. Alternatively, small squares may be cut from paper. If using paper squares we highly recommend using card stock/thick paper to help students more easily manipulate the pieces. We encourage the use of an item students can physically manipulate but if this is not available, students can use graph paper to simply draw their rectangles.

Begin by asking students to organize their 24 blocks/squares into rows. Explain the only rule is each row must contain the same number of blocks. After giving students a few minutes, invite them to share the rectangle they made. We picked 24 as it is a number rich in factors. Once students have determined one configuration, encourage them to find another!

Display all the possible combinations for students to see.

While students have previously been exposed to the commutative property of multiplication, the goal here is to build fluency with factors and thus we encourage students to find all possible rectangles, even if they are a rotation of an existing rectangle.

Highlight how our rectangles look like a multiplication table. In fact, we could find many of these rectangles on the table. For instance, if we drew a rectangle from the top-left corner measuring 3 squares wide and 8 squares high, we'd have the same 8×3 rectangle we just made! If possible, demonstrate for students how the number 24 would end appear in the bottom-right corner when constructing this rectangle on a multiplication table. The image below is from a previous activity in MathBait™ Multiplication and demonstrates how a 3 by 4 rectangle would align with the product, 3×4, in the bottom-right corner.

Ask students which of the rectangle they created with 24 blocks would not appear on our multiplication table? (Those with side lengths greater than 10 are not on our table as it only reaches 10).

Remind students we call the solution to a multiplication problem a product. For each of these rectangles, the product, or total squares, is 24. We call the numbers we are multiplying factors. Over the next few lessons students will learn how to find all the factors of a number.

Allow students to practice identifying the factors of 24 shown by each rectangle. For instance, the factors of the 6×4 rectangle are 6 and 4. Ask students if they can name all the factors of 24 (1, 2, 3, 4, 6, 8, 12, and 24).

Conclude by explaining how knowing the factors of a number will help us to very quickly find the product, or solve tricky multiplication problems.

Activity 1: Block Target

Once the concept has been introduced in the warm up, allow students to practice using our digital game Block Target.

In this game, students are given a target number of blocks. They use sliders to create rectangles to reach the target number. The goal is to find all possible ways to hit the target value.

Different modes make this a great game to come back to as students increase their fluency. We recommend students begin with Hints on and Count Reflections off. As they become more comfortable and confident, they may choose to try without hints as well as automatically include reflections. In addition, players may select to increase the challenge with a timer.

Play

©MathBait created with GeoGebra

Benefit

In this lesson we introduce factors to students. Through exploration, they begin to build an understanding of the many different ways we can "build" a number.

When completing these activities, it is likely students will have difficulty identifying all the factors of a given value. For instance, while a student may know 9×10=90, they may struggle when trying to determine what times 5 will also equal 90. In the next lessons, students will develop key strategies to help them find these missing numbers with ease! For now, encourage students to simply find as many factors as they can. Using Hints turned on will help students to identify the tricky factors.

Factor It!

In the previous lesson, students began to explore factors. Now, we work to more formally develop strategies for finding all the factors of a target number.

Warm Up: How Do You Know?

Replicate the warm up from Block Target using small composite numbers. Place students in small groups and provide each group with a value (6, 8, 10, and 12 are good ones). Ask students to create every rectangle they can using their target number of blocks.

Once students have time to agree and make sure they have found all the factors, ask groups to share. Allow students to explain how they know they found every factor. Announce in this lesson, we will develop a method to ensure we find all the factors and don't miss any.

Remind students the factors of a number are all the block sizes we can use to build that number. For example, the factors of 6 are 1, 2, 3, and 6 as we could build a castle that is 6 units tall using 1-blocks, 2-blocks, 3-blocks, or one big 6-block. Remember (as much as possible) to redefine words in terms students can understand rather than leaning only on the mathematical definition. If homeschooling, you can target your child directly by using analogies which specifically speak to them and their interests. If your students don't have a lot of experience with blocks and building (such as LEGO or Minecraft) consider what will speak to them. If you have musically inclined students, you could use beats reminding them if we want to clap on the 6th beat we could clap every beat, every 2-beats, every 3-beats, or every 6-beats. Make sure they can convince themselves these are the only options (counting on 5th beats for instance would skip over 6). If your students are athletically inclined, present factors as a race. Mark off 6 lines on the ground and determine what sized steps could we take to land on the 6th line. More activities like this can be found in earlier units of MathBait™ Multiplication.

Activity 1: Count This!

The goal of this activity is to help students see how a systematic method is helpful for organizing information.

Provide students with a collection of small items such as a handful of M&Ms, number cubes, or paperclips. We recommend providing somewhere between 25 and 50 items. Lay them out on a table and ask students to count the items. Watch for strategies such as clever grouping. (For more on this, check out Marco the Great and the History of Numberville).

Some great strategies are making groups of 5's or 10's and skip counting. Allow students to share their different strategies.

Next, provide students with a new group of objects (it can be the same items, but should be a different quantity). Make sure to scatter them a bit on the desk/table in front of students and announce they will not be able to touch the items this time. Ask students to determine how many objects are in front of them.

Students will find the task more difficult without the ability to organize the items. Allow students to share their strategies for counting the total without the ability to sort or group.

Conclude the activity by asking students the following questions:

• How did you know you counted all the items each time?

• What strategies made it easier to count the items?

Both parts of this activity parallel the struggle in finding factors. In the first part, when students could touch the items, organizing into smaller groups was helpful as it is easier to make sure each group has 5 or 10 and then skip count to find the total. This strategy will be important in factoring: we want to use what we know to help us find all the factors.

In the second part, we couldn't touch the items. The lesson to learn here is the importance of a systematic approach. Counting one on the top, then one on the right, then one over here, then one over there will make the task impossible! Most students will likely approach by moving in one direction as they count. Highlight the strength of this method. When we factor, it will be helpful to start with the smallest building blocks and move up systematically to ensure we counted every option.

Conclude the activity by reminding students that staying organized will help us to make sure we don't miss anything.

Activity 2: Factor It!

In this activity, students will build their first "factor trees". They will practice organizing information, identifying the commutative property, and discovering strategies to help ensure they have identified all the possible factors.

Begin by modeling how to factor systematically. We recommend starting with 24 as students have seen this value previously and are familiar with it. Some numbers, like 36, will present challenges that will be covered in a later lesson. If picking an alternative value, ensure all factors (except the number itself) are no larger than 10.

Show students the following image and explain we call this a factor tree.

Our goal is to find all the ways to build 24 with different block sizes without missing any possibilities. Allow students to guide the completion of the tree.

Begin by transcribing a 1 on the left. Ask students how many 1-blocks are needed to build a 24? Once students have determined 24 blocks are needed, transcribe the 24 in the right column in the same row as the 1.

Explain we will count by ones to ensure we do not miss any possibilities. Under the 1 in the left column, write down 2 and ask students how many 2 blocks we need. Remind students they can count by twos up to 24 to determine how many. If needed, students may use their fingers, or write tallies, to keep track as 12 is outside our multiplication table. Students should determine we will need 12 two-blocks to make 24.

Continue in this fashion with 3 and 4. Students can skip count as needed to complete the grid.

The next value is 5. Allow students to decide if they can create 24 using 5-blocks. Ask students to explain their reasoning. Some ideas are: we cannot reach 24 by skip counting by fives as we would have 20 and then 25 and 24 is skipped, or, all multiples of 5 have a 5 or a 0 in the one's place, as adding five makes another ten for every other sum. Write 5 on the left and mark an X on the right. When students develop this skill, they will no longer need to write every value, but for now, encourage students to write each number to ensure they do not miss anything.

Proceed in filling out the tree up to 24. This is important to allow students to discover a better way, rather than giving them a better way. We broke our tree into two parts to conserve space.

If possible, allow students a few minutes in small groups to discuss what they notice and what they wonder. Allow students to share and make a list of their ideas. Here are a few good ones to highlight:

• Twelve is splitting the 24 into two groups. No number larger than 12 can make 24 except for 24. Will this always happen? (Yes! For an even number, any value larger than half won't work as half is made up of two-blocks and the only thing smaller than 2 is 1. For an odd number, we can jump to a close even number - for instance, with 21 we could jump to 20 - and find the same result).

• If we think about our previous activity, Block Target, many of the rectangles made were rectangles we already had but tipped over. This is the commutative property. We see 2 and 12 as well as 12 and 2 on the table.

• After 4 on the left, all the numbers on the right are numbers we already have on the left.

Before solidifying a factoring strategy, allow students to play independently to pick up on patterns. Launch Block Factor. Play with the following settings:

• Timer: off

• Hints: off

• Count Reflections: off

If your student is in need of additional support you can allow Hints to be on.

Encourage students to work on a systematic approach to finding all the factors. As they play, have students write down their factor trees. If your students need more direction, advise them to begin by placing the vertical slider on the left at 1, create the target rectangle, and write down the pair before moving the vertical slider to 2, 3, 4, and so on, increasing the slider by one each time.

Conclude the activity with a discussion. Ask students to share their strategies and their findings. Note, it is likely that students came across difficult values. For instance, a student may recognize that 36 is even and thus can be reached by counting by twos, but is not sure how to find how many twos easily. For now, encourage skip counting. Soon we will find an amazing and magical way to discover these factors as well!

Activity 3: Remember this?

The purpose of this activity is to recall the commutative property helping students to realize they can cut their work, and their factor trees, in half with this property.

Display the image of the 24 blocks for students.

Announce that each rectangle has a twin and ask students to pair the twins. When complete, allow students to explain their choices.

Remind students of the commutative property (discussed earlier in MathBait™ Multiplication). In our rectangles, we can think of the commutative property as counting the total number of squares horizontally, or vertically. Demonstrate with the 4 by 6 rectangle.

We can find the total number of squares by using the columns. Since each column has 4, and there are 6 total, we can count by fours, 6 times, to find 24.

Or, we can use the rows. Since each row has 6 squares and there are a total of 4 rows, we can count by sixes, 4 times, to find 24.

After highlighting 4×6=6×4, ask students how this might lower the amount of work needed to build a factor tree. Work together to rebuild the factor tree for 24, allow students to determine when we have collected all the factors.

Students should recognize, the tree above is sufficient as it contains all the factors (or blocks we can use to build 24) of our target number. Ask students how we can determine when to stop for any value.

Solidify the idea that once we reach a value which is already on the right, we know we have found all the factors! If we continued after 6, we would find no pairs for 7, then a pair for 8 which is already on our tree, then again no pairs until 12 and 24. If needed, display the previous factor tree in which we attempted all values. This is great news! We can do much less work to find all the factors.

Allow students a few minutes to build their smaller factor trees independently. We recommend assigning values such as 6, 8, 9, 12, 14, 16, and 18.

Benefit

In Factor It, students learned how to find the factors of a number using multiples (rather than division). They discovered why a systematic approach and organization is helpful for ensuring everything has been counted.

At first, students began by making large factor trees before re-exploring the commutative property to realize they can find all the factors without constructing a large list.

At this point, students will still likely have difficulties fully factoring larger numbers. We recommend students continue to play Block Target to build their skills. In a later lesson in Unit 7 we will build strategies to help students easily identify all the factors of a value.

The Taxman

In this lesson, students will continue to practice factoring and building their skills and fluency.

Warm Up: Red Light, Green Light

In the previous lesson, students found they could identify all the factors of a number without writing a large, cumbersome list. In this warm up, students play red light, green light to decide when they have identified all the factors.

Place students in pairs. Students will take turns factoring with their partner. Student A will be assigned a number to factor, Student B will act as the traffic monitor. As Student A begins factoring, Student B will pay close attention. After writing each factor pair, Student B will announce "red light" (meaning to stop) or "green light" (meaning to continue).

If Student A misses a factor (runs a light), Student B will give them a ticket. If Student B announces the wrong light color, Student A will give them a complaint. When one number is factored, students will swap places. The student with the fewest slips (both tickets and complaints) will win.

This warm up allows students to continue to practice factoring while keeping in mind the markers to look for to know when factoring is complete.

Activity 1: Factor Climb

Let's build fluency! In Factor Climb, students help a mountain climber by determining how many factors the given number has; too many and our climber will overshoot the mountain.

Once students are satisfied with the number of factors, they build a ladder to help the mountain climber descend.

Students can be successful without identifying all the factors. The goal of this game is to simply practice and gain exposure. A full list of factors is given at the end of each round, allowing students to view any factors they missed. As students gain more exposure to factoring, they can replay Factor Climb earning a higher score each time!

Play

©MathBait created with GeoGebra

Activity 2: The Taxman

MathBait™ Multiplication Unit 7 premiered on April 15, 2024 - Tax Day in the US. We thought it was fitting to include this fun factor game which is a blast to play, difficult to win, and keeps students thinking about factors and growing their number sense.

We have provided a digital game, however this game may also be played with two players offline. Consider using a deck of cards and assigning the values 11, 12, and 13 to J, Q, and K respectively.

How to Play

Players begin with a set of consecutive numbers starting at 1. On each turn, a player can select any number from the list - but be careful! You must pay taxes on the number you receive. Taxes paid are all the numbers on the board which are factors of the value selected.

For instance, selecting 15 as the first number means the Taxman will collect all the factors of 15 from the board (1, 3, and 5). This doesn't seem too bad as the Taxman has only collected a total of 1+3+5=9, less than your 15, but his amount can add up quickly!

On the next turn, players can only select from the remaining values on the board and the Taxman can only collect from the remaining values. Selecting 14 would give the Taxman 2 and 7 (as 1 was removed in round 1).

Players may only select values with factors remaining on the board. That is, they cannot select an amount in which no taxes will be paid. When there are no more values with factors, the Taxman will collect everything leftover! Can you outsmart this clever bean counter?

Play

©MathBait created with GeoGebra

Benefits

In this lesson, students continued to build their familiarity with finding the factors of small numbers (under 100). Through fun activities and games, students are gaining exposure and practice. The Taxman also forces students to consider strategies to win, developing a deeper understanding of factoring. We recommend students play regularly, spiraling back to Block Target to build fluency.

Centipede

It's time for a blast from the past! The Centipede lesson was included in MathBait™ Multiplication Part 3. We've included it again here as we want to remind students of their previous understanding to make a bridge between factors and eventually factorization.

In this lesson, students will focus on a strategy of using smaller values and facts they are likely more comfortable with to tackle larger products.

Warm Up: Another Way

The goal of this warm up is to allow students to explore different ways to find familiar products.

Provide students with a multiplication fact such as 6×7. Explain that our goal is to make an expression (a number statement) as long as possible. To do this, we need to think about what two numbers we can multiply to make 6 or 7.

Many students will rush to 1×6 and 1×7. This is okay. In fact, if students do this, encourage them to continue. Write each statement below the previous:

6×7

1×6×1×7

Allow students to continue to append the product with ones (until you run out of space or are over it).

6×7

1×6×1×7

1×1×6×1×1×7...

Explain that 1 is a special number. It acts as a mirror in multiplication. Normally, when a number is multiplied it acts as a growth laser, enlarging the value; but 1 has the special ability to only reflect, like a mirror. This begins to build a strong foundation for fractions. MathBait™ uses fun-house mirrors as an approach to fractions which is not only very successful and impactful, but also enjoyable to students. Setting up the idea of 1 as a reflector (i.e. multiplicative identity element) will help students with many mathematical concepts to come.

Now, challenge students to do the same, but without our magical mirror 1. With a 1, we can extend our expression indefinitely as multiplying by 1 simply reflects what is already there, but can we extend our expression without this?

Guide students as needed to notice 6=2×3, allowing us to write this expression as 2×3×7. Note that the only way to make a 7 is 1×7, so 7 cannot be extended. If a student mentions "prime", this is a great chance to acknowledge 7 is prime, but it is not necessary to bring up prime numbers if students do not mention it.

Present a challenge. Provide students with a multiplication table to allow them to see all the products they have worked with. Ask students to first pick any number on the table and expand it as much as they can without using 1. Students can swap with a partner to check their work when they are finished. The image below shows the standard multiplication chart along with a chart listing the number of prime factors for each value. Students may benefit from displaying the image without context and asking them to share what information they think the image is trying to convey.

Discuss some of the longest and shortest expansions students found. Prime numbers will be the shortest. There are 24 values with an expansion of length 2 (such as 4 with 2×2 or 10 with 5×2), 34 values with an expansion of length 3 (such as 8 with 2×2×2 or 20 with 2×5×2), another 24 with an expansion of length 4 (such as 24 with 2×2×2×3 or 36 with 2×3×2×3), only 8 with an expansion of length 5 (such as 80 with 2×2×2×2×5), and a single value with an expansion of length 6.

Have students work together in small groups to try to identify the only value with an expansion of 6. Invite students to share their strategies. Will they simply divvy up the table and find it through force? Will they use problem-solving strategies to try to determine how to build such a number?

Play around with this idea as time permits. You can provide students with a challenge to find an expansion of length 4. Once they have found one, write it on the board forcing the remaining students to find a different number. Encourage students to develop ways to quickly build such a number (for example, starting with 2's and multiplying is a great method because 2's are small and we can multiply a good deal of 2's and stay under 100).

Activity 1: Bid A Number

The goal of this activity is to provide students with more practice finding products using smaller values.

To set up the game, place about a dozen numbers in a jar or other container. We recommend using values with 4 or more prime factors (you can use the table above to identify them). Group students into two teams. In each round, one student from each team will go head-to-head in a bidding product battle.

Call up a student from each team and select a number from the jar. Alternate which team member bids first (in round 1, the player from team 1 bids first while in round 2 the player from team 2 bids first, etc.). Each player makes a declaration on the length of the multiplicative statement they can write. The bidding continues back and forth, each player bidding more than the previous declaration, until they cannot bid higher.

When a player cannot bid any higher they announce for their competitor to "make that product". If the opposing student can successfully make the product in the stated number of factors, their team earns a point. An incorrect answer gives the opposing team the chance to steal. If the opposing team can make the product in the number of factors bid, they earn the point. If neither team can make the product, the round ends.

Play continues by selecting a new member of each team to go head-to-head. Play as time permits or set a number of rounds at the start of play. For larger classes, students can be broken into groups to play simultaneously as teachers rotate around the room.

Example Game

Josie is selected from team 1 and Armand is selected from team 2. The number selected from the jar is 16.

Josie: I can write this as a product of 2 numbers

Armand: Well, I can write this as a product of 4 numbers

Josie: Okay Armand, make that product

Armand: 16=8×2=4×2×2=2×2×2×2

Since Armand successfully created an expression, team 2 earns a point. Note students need not increase their bid by only 1 each time. In addition, students do not need to achieve the maximum number of factors. If Armand instead bid 3 and Josie could not think of a way to write 16 with more than 3 factors, Armand's team could still win the point with 4×2×2.

Conclude with a whole group discussion. Ask students why it can be helpful to think of the multiplication facts as the product of more than 2 numbers. Demonstrate to students how this can help us with some of the trickier facts, and with facts larger than 100 as well.

For instance, if you are not sure what 7×8 is, you can break 8 into 4×2. This changes the question to 7×4×2. You may be able to more easily determine 7×4=28 to know 7×8 is the same as 28×2. By doubling 28 (28+28) you can arrive at the correct product of 56. Allow students to give additional examples. By sharing the values they have trouble with, not only are we creating new strategies, but also finding connections and realizing that other students share in our struggles. This models perseverance and problem-solving which can help with motivation and confidence.

Activity 2: Centipede

Our digital game allows students to play with the idea of writing products with more than two factors.

In this game, a centipede is making its way down the screen. Players can pop the first segment by directing lasers to create an expression that equals the value shown. On the right of the screen, students will see the maximum multiplier as well as their current expression. The more values students shoot, the higher their score. For instance, if the current segment is 16, students can earn a multiplier of 2 with 4×4, a multiplier of 3 with 2×2×4, or a maximum multiplier of 4 with 2×2×2×2. Note, the order of factors does not matter.

Play

©MathBait created with GeoGebra

Benefit

This lesson reminds students of their previous skill to break down a number. This is factoring! Often a confidence booster, connecting a previous activity to a new skill or new terminology helps students to feel like they aren't learning anything new, but rather exploring something they are already familiar with.

Centipede helps students to break down a number and in fact, will allow students to earn the most points by finding the complete prime factorization. We recommend coming back to this game after The Sieve and asking students what they notice. They should find the buttons in Centipede are color-coded to highlight the primes. To gain the maximum score, students should use only the red/pink colored buttons.

You Can Do That?!

In this lesson we pull it all together. Students have practiced finding factors, developed a method for systematically identifying factors, and practiced breaking down multiplication facts into smaller chunks. Now it's time to try to factor big values the easy way!

Warm Up: Substitution

In Marco the Great and the History of Numberville, we present substitution as one of the most basic and powerful skills of a mathematician - because it is! Substitution is an amazing tool that has the ability to turn something complex into something much simpler. In this warm up, students will play with substitution.

Explain to students that at a very famous sandwich shop they offer a Classic Sub which includes:

• White bread

• Cheddar Cheese

• Lettuce

• Tomato

• Onion

They charge $3.00 for this special which is less expensive than the Build Your Own with the same ingredients. May wants almost everything on the classic sub. She doesn't like onions. Instead, May would like either olives or avocado. Do you think they will allow May to swap out the onions for olives? What about swapping the onions for avocado?

Present the menu board to students and allow them about 5 minutes to determine their solution.

Allow students to share their thoughts. Explain as needed that the shop will not allow May to switch out the onions for olives as onions are $0.25 and olives are $0.50. They certainly will not allow to switch onions for avocado. The shop will allow substitutions. A substitution is when you swap out an item for an item of equal value.

Ask students if there is anything on the menu she can substitute for the onions. Give them time and encourage creative solutions. Remind students May doesn't want onions.

Have students share their findings. Here are a few options:

• If May is okay with no cheese, she could order the Classic, substituting the cheese and onions for avocado, as $0.75+$0.25=$1.00.

• May could swap two items such as lettuce and onion or tomato and onion for olives.

• May could order no onions and extra tomato (or lettuce).

• May could swap lettuce, tomato, and onion ($0.75) for olives and update her cheese to Swiss.

If time permits, allow students to play more with this idea, creating their own menus and acting out orders and substitutions.

Conclude the activity by reinforcing the term substitution. Explain we can substitute anytime items have equal value. Tie back to the previous lesson, Centipede. It can be helpful to allow students to play a quick round of Centipede before moving forward. With substitution fresh on their mind, what do they notice?

In Centipede, we can eliminate segments by stating a multiplication fact. For instance, 56 can be expressed as 8×7. However, we can also use substitution. Since 8=2×4, we could eliminate the same segment using 2×4×7 as we replaced the value of 8 with an equivalent value of 2×4. In today's lesson, we will see how substitution can turn tricky problems into easier problems.

Activity 1: Ninety

Announce to students that in this activity they will try to find all the factors of 90. Place students in pairs or small groups and allow them to work on the problem. This is a hard value! It is likely students will get stuck. It is okay to leave values blank in the first round. For instance, a group may know that 90 is a multiple of 5, but are not able to easily determine what times 5 will produce this value. MathBait™ Multiplication assumes no prior knowledge of division, and relies on students' ability to skip count. Zealous students may try skip counting by fives up to 90, however, finding other values such as what times 6 equals 90 will become tedious! Allow students to focus on determining what values they believe 90 is a multiple of, and add this to the left side - leaving the corresponding left value blank as needed.

When time has expired, ask students to share one factor pair of 90 they feel confident about. Many students will recognize 90=9×10. Announce we will use this fact to help us determine the other pairs.

Focus attention on the 10. Ask students to find a factor pair of 10. When they recognize 2×5, add this under the original expression, replacing the 10, or alternatively, display the image below to students.

Ask students if they agree with the equation; is 90 equal to 9×2×5? Circle back to the warm up if needed. Remind students we can substitute equal values. As 10=2×5 we can replace the 10 with an equal value of 2×5.

Allow students to hypothesize how this might help us to factor a number. Show them the next image and ask if they agree it is true.

Again allow students to hypothesize how this may be helpful in factoring. Explain when trying to factor 90, we might know that 90 is a multiple of 5, it is a number we say when skip counting by fives as we say every number with no ones or exactly five ones. However, knowing how many fives is challenging. Eighteen is outside of our skip counting table and while we can check by counting by fives (most students will benefit from counting by fives to convince themselves of the validity), that is a lot of fives to keep track of! We know, as long as we are substituting equal values, we haven't changed the total.

Have students work together to try to fill in the remainder of their 90 factor tree using this idea. Most students will be able to initially construct the following table. If any of these initial values are missing, provide students with the given explanation to help build problem solving and reasoning skills.

Help students as needed to find the factors listed with a ? above. These are outside our multiplication table and thus more difficult to find. However, using substitution and the commutative property we can more easily identify each.

• Factor Pair for 2: As 9×10=90 we have 9×5×2=90. We know 9×5=45, thus 45×2=90.

• Factor Pair for 3: As 9×10=90 we have 3×3×10=90. We know 3×10=30, thus 3×30=90.

• Factor Pair for 5: As 9×10=90 we have 9×2×5=90. We know 9×2=18, thus 18×5=90.

• Factor Pair for 6: As 9×10=90 we have 3×3×2×5=90. Using the commutative property we see 3×2×3×5=90 and as 3×2=6 and 3×5=15, we know 6×15=90.

Conclude by explaining we can now use one fact we know to help us find all the factors of a number! The key here is to use substitution and the commutative property. For instance, if we are unsure what times 6 makes 90, we can factor the smaller numbers in 9×10 to try to make a 6; the product of the remaining factors will produce the factor pair we are looking for!

If time permits, encourage students to find the factors of 36, using the technique above for identifying the pairs for 2 and 3, which are outside our multiplication table.

Activity 2: Fives

In this activity, we will reinforce the ideas developed in Ninety to help students easily determine the factor pair with values outside the multiplication table.

Begin by asking students the value of 8×5. If students have completed the previous units in MathBait™ Multiplication, they will be able to identify the product of 40. Explain this product is within our multiplication table so we are familiar with it, when values are outside the table they can be more tricky. We are going to learn how to make the tricky problems easier by forcing them back into our table.

Tell students to suppose they did not know what 8×5 was, but they are really good at counting by tens and so they know 8×10=80. Ask how they might use the fact 8×10=80 to find 8×5. Allow students to share their thoughts and ideas. Highlight the previous activity, as 8×10=8×2×5. How might we use this fact to help us find 8×5?

There are in fact multiple ways to go about this!

Method 1: Halving

Students are not yet familiar with division, so this is a fun way to get us thinking about halves. Since 8×2×5=80 we know 16×5=80 as well. But our goal here was to find 8×5. Looking closely, we can see 8×5 is hidden in our expression 8×2×5=80. This tells us 8×10 is double (or two times) 8×5. Ask students how we could split 80 into two equal groups. In other words, what number could we add to itself to find 8? Students should be able to determine 4+4=8 and thus 4 tens + 4 tens = 8 tens or alternatively 40+40=80. If students are not flexible enough with addition, this might be challenging.

Method 2: Building 40

If your students are not yet ready to consider halving, stick to the methods shown in the activity Ninety. An alternative method to determine 8×5 is to break up the 8. There are many ways to achieve this. Encourage students to find and share different methods. One way is to write 8 as 2×4, which makes 8×5=2×4×5. For here, students can notice this expression is equivalent to 10×4 or to 2×20, both easier facts to find. This is a good opportunity to get them thinking about halving by noticing 4 is half of 8, or two fours make 8. This view is further helping students in factoring as they are now thinking about how to break a number into 2 times something.

For the remainder of the activity, have students work independently or in pairs on additional problems.

• 12×5

• 13×5

• 14×5

• 15×5

• 16×5

• 17×5

Every problem except 13 and 17 can be broken down into a known fact. Here are some possible solutions:

• 12×5=2×6×5=2×30=60

• 12×5=3×4×5=3×20=60

• 14×5=7×2×5=7×10=70

• 15×5=3×5×5=3×25=75 (this is particularly good for students with a solid understanding of money)

• 16×5=2×8×5=2×40=80

• 16×5=8×2×5=8×10=80

Students will likely have a hard time with 13 and 17. In the next activities we will re-introduce prime numbers so this is a great time to discuss why these values are so hard to find! We introduced prime numbers in MathBait™ Multiplication Part 2. Remind students as needed that a prime number is a tower height that can only be built with blocks which are 1-unit tall or, if we have a large block, can be built with one large block equal to the number. The values 13 and 17 are prime. We can only write them as 1×13 or 1×17, making our strategy not very effective.

Ask students if we can think of another way to determine 13×5 or 17×5, or any tricky value when dealing with a prime number. One way which we have explored previously in MathBait™ Multiplication is using the distributive property. We can find,

13×5 = (10+3)×5 = (10×5)+(3×5) = 50+15=65.

Another method, not previous discussed, is to use the common theme of manipulation. We previously found 12×5=60. We have learned this simply means we counted by 5's twelve times. Thus, thirteen times is simply one more 5, and so 13×5=60+5=65. Encourage students to "try on" both methods of working with tricky primes to find 17×5 and discuss what they liked and disliked about the process.

Benefit

This lesson focused on students using their existing knowledge to find tricky products. Students learned the power of substitution and reinforced their earlier work in Factor It and Centipede to use factoring to identify new products and to determine missing factors.

Additionally, the final exercise allowed students to recall prime numbers, preparing them for the final steps in MathBait™ Multiplication: prime factorization.

The Sieve

Before beginning this lesson, we recommend giving students 2-3 days (as needed) to simply play the games found earlier in Unit 7, as well as games from previous units. Going back to earlier games can be a great confidence booster as students will find them easier and easier to win (yet still a ton of fun!). Playing also increases fluency and familiarity. In the next two lessons, we are introducing a new concept by building on existing knowledge. Moving too fast can be counter-productive. Give students a bit of time to let factoring sink in and feel confident about factoring before beginning.

The goal of this lesson is to help students to identify prime numbers. If following our progression, students were first introduced to prime numbers in MathBait™ Multiplication Part 2. Our game Rainbow Multiplies allowed them to get a feel for prime numbers without a formal exploration on these special values. This is a great time to replay Rainbow Multiples to activate prior knowledge.

Now, we will formally define prime numbers once again and see how prime numbers are a powerful tool for factoring. Understanding prime factorization will greatly help your students in division, working with square roots and radicals, as well as many algebraic topics to come. We highly recommend taking time to build a strong understanding in prime numbers and prime factorization.

Warm Up: Does it Matter?

Working with prime numbers is so important, the topic we will introduce in this warm up is called The Fundamental Theorem of Arithmetic, which basically says there is exactly one way to build a number.

Circling back to a previous activity, present students with the number 90 and ask them to write 90 as a product of 3 or 4 factors. There are multiple options, here are a few:

• 90=3×3×10

• 90=3×6×5

• 90=15×3×2

• 90=3×3×2×5

Identify students with different factors and present them to students. Explain each of these looks different, but are they different? Allow students to share their thoughts.

Since every statement is equal to 90, each representation is equivalent - they are all 90. If multiple students figured out four factors, display the different orders (and if possible different methods). Make sure students agree these are all the same. Demonstrate how we can start with 90=9×10, find 9=3×3 and substitute to have 3×3×10 before identifying 10=2×5 and substituting to end up with 3×3×2×5. If we started with a different fact, such as 90=15×6, using substitution we could end up with 3×5×3×2. We know because of the commutative property these are the same, or we can reorder the numbers to make both expressions look the same.

Announce the most factors we can use to write 90 is 4 without using ones. (Circle back to Centipede if needed to remind students this would be our max multiplier). Ask students to justify why this claim is true. Students may or may not recall prime numbers, but should be able to conclude that each of the values 2, 3, and 5 don't have any other factors except 1.

Wrap up by telling students in today's lesson, our goal is to find the max multipliers of numbers. That is, to break a number down into the longest possible multiplication statement without using ones. Just as students found different ways to write 90, the order in which we break it down will not matter because we are using substitution and only swapping out equal values. This provides a lot of flexibility! But, in order to know we have reached the max multiplier, we have to identify when a factor cannot be broken down any further. Prompt students to recall what we call this type of number.

Based on student understanding in the warm up, decide if revisiting Build It from MathBait™ Multiplication Part 2 would be beneficial as a review. We have included it here if needed.

Optional Activity: Build It

This activity is from MathBait™ Multiplication Part 2. If needed, students may revisit this activity to recall prime numbers.

The purpose of this activity is to introduce students to the word "prime". We have multiple activities to build a deep understanding of prime numbers in both the MathBait™ Multiplication series and within The Kryptografima. For now, we are only exposing students to the idea and not worried about mastery of the concept.

Provide students with 5-6 sheets of graph paper and colored pencils. Explain their goal will be to build towers of a given height. Since any height can be made with one-blocks, we won't have one-blocks. Our first block size will be a two-block. Have students label the top of one sheet of their graph paper "My Blocks". On the page, select a color for their two-blocks, draw it out, and label it.

On another piece of graph paper have students draw out towers using their two blocks. Note, the width doesn't matter. A common LEGO is 2×2 but making all blocks have length 1 will save space. Ask students to label their towers by their height; what do they notice? Highlight that with blocks that are 2 units tall, we can make towers of size 2, 4, 6, 8, 10, ..., which are our multiples of 2 or the numbers we say when skip counting by 2's.

Ask students for tower heights they were not able to build with 2 blocks (3, 5, 7, 9, 11, 12, ...). We should notice many sizes are multiples of 3, so let's make a 3 block. Allow students to select a design for their 3 block and add it to their My Blocks page. Then, draw out the first few tower sizes they can make using only 3 blocks.

Emphasize to students that in building our towers, we cannot combine block sizes. We are looking to build only using one size of block.

Ask students what they notice. Some key ideas are the towers we can build with 3 blocks are the multiples of 3, and some blocks can be built with 2 blocks or 3 blocks, such as 6 and 12. Connect back to previous activities when we learned that we can count by 6's by skip-skip counting by 3's or skip-skip-skip counting by 2's.

Now ask students what is the next size block we need. Students may suggest 4 (as logically we have 2, 3, 4). Explain we don't need a 4 block because we can make it using 2's! We already have a tower of size 4. Students should settle on 5 and design their 5 block on their My Blocks sheet and draw a few towers of size 5.

If time permits, continue up to 10. Students will find they only need a 2, 3, 5, and 7 block as the other sizes have already been completed.

Explain that we call numbers like 2, 3, 5, and 7 prime because they are needed to build towers, or other numbers. These numbers are super powerful and we will continue to learn more about them later. For now, we just want to be introduced to what prime means.

Conclude by asking students to each give a number larger than 10 that they think is prime. Have a short discussion about their choices and how we might be able to test if a number is prime. Here are some strategies:

• If we know a number is a multiple, or a number we say when skip counting, it can't be prime because it can be built with smaller blocks.

• If a number is on our multiplication table (other than the first row and column, as everything can be built with ones), it can't be prime.

Activity 1: The Story of Eratosthenes

This is a visual activity that introduces students to the Sieve of Eratosthenes. A sieve is a systematic way of eliminating numbers. In our case, we want to eliminate any number that cannot be prime. Begin by asking students for ideas on how we could quickly and easily identify all the prime numbers up to 100.

You might be surprised when a few good ideas pop up! If students need a boost, remind them a prime number is not a multiple of any other number except itself and 1. For instance, 11 is prime because we cannot get to 11 by skip counting by anything but 1's or if we are skip counting by 11's. Hopefully students notice this means any number we say when skip counting cannot be prime. For instance, if skip counting by twos, all the numbers said after 2 can be built from twos, and thus are not prime. If students do not immediately recognize this, it is okay.

Explain to students that a mathematician named Eratosthenes created a way to find all the prime numbers using skip counting! First, he got out his multiplication chart and started with 2's. He knew 2 was a prime number. Ask students how we know if 2 is prime. (Eratosthenes knew because he could only organize 2 M&M's into a single pile with both candies or two piles each with one chocolate meaning 2 could only be written as 2×1.)

Any number that was bigger than 2 and a multiple of 2 couldn't be prime because it could be built with 2's. So Eratosthenes started skip counting, crossing off all the multiples of 2, but leaving 2 untouched. Next, he moved on to threes. Again, three is prime so he left it alone, but he crossed off every multiple of 3 on his board. When he got to 4, he realized he already had crossed out all the multiples of 4! Ask students why this is. (Because 4 is made of 2's, every multiple of 4 is also a multiple of 2). Eventually he didn't have any more multiples to cross out and guess what was left? Almost every number on his multiplication table that wasn't crossed out was prime. Tell students there was one number left that wasn't prime - can they guess what it is?

Move directly into Activity 2 which will give students a refresher on primes before showing them the beautiful Sieve in action.

Activity 2: The Sieve

Before continuing to The Sieve, help students to solidify their understanding of primes. Provide each student with ten small items (such as paperclips or number cubes).

Start with 10. Ask students to make as many equal groups as they can with their ten items. Remind students this is how we find the factors of a number. As students have been practicing their factoring, they should be able to quickly make four groups: placing all ten items in one pile, creating two piles each with 5 items, creating 5 piles each with two items, and creating ten piles with 1 item each. Have students make a table representing the number of piles they can create; for 10 they should transcribe 5.

Allow students to continue (individually or in small groups or pairs), each round removing one item. Thus, next they will find the number of piles using 9 items and transcribe 3 (groups of 1, 3, or 9).

When we reach one, discuss what students found. They should notice the prime numbers always had exactly two groups. For instance, with 3 items we can create three groups placing a single item in each or we can make one big group with all three items. What happened with 1? Help students to see that with 1 item we can only make one group - place our 1 item in its own pile. Because of this, one is not prime. It is also not-not prime (what we call composite) because composite numbers are numbers with more than two groups. One is the only number with exactly one group. It is special because with a 1, we can build all the counting numbers - that is, every number has a factor of 1.

Conclude the activity by allowing students to play with the interactive applet below. Introduce the applet in what works best for your students. We recommend demonstrating first for the class before allowing each student to try it themselves and placing students in pairs or small groups to discuss the applet.

Play

©MathBait created with GeoGebra

Remind students a prime number is like a building block. From the primes we can build the counting numbers. Primes are special because they can only be built with one-blocks or one (sometimes very large) block of their size. Previously we factored a number like 18 into pairs (1 and 18, 2 and 9, and 3 and 6), the prime factorization breaks down a number completely so we can get to know it. The better we know the numbers we are working with, the more powerful we will be.

Now that students have an understanding of a sieve, it's time to build their own! In Activity 3, students will create their own Prime Quilt.

Activity 3: My Sieve Quilt

In this digital activity, students will work through the multiplication chart just like Eratosthenes. Unlike Activity 2, in My Sieve Quilt students will need to select all the multiples on their own. Encourage students to search for patterns as they work (for instance, 2's, 5's, and 10's make columns, while the multiples of 3 form a diagonal).

When students identify a prime number, they will select how they wish to color its multiples. This will create a beautiful and unique sieve for each student. In addition, as they find values such as 12 which have more than one prime factor less than 10, the color on their quilt will begin to blend. For example, selecting red for multiples of 2 and yellow for multiples of 3 will leave 12 as orange. Use these patterns to help students realize key findings such as all multiples of 6 are multiples of both 2 and 3.

At the end of the activity, students will have the chance to further design their quilt by adding textures as well as take a picture to memorialize their work.

Play

©MathBait created with GeoGebra

Activity 4: Prime Pop

The final activity of this lesson gives students the chance to practice identifying prime numbers. Not only is this directly strengthening their recognition, it is also further developing their multiplication skills as students focus on multiples. They should recognize two is the only even prime (as all other even numbers can be built with twos), any number with a zero or a five in the one's place cannot be prime, and use their multiplication knowledge to try to eliminate composite values.

As students play, numbers will fall from the top of the screen, the goal is to select only primes. It is not expected students will "catch" every prime. We will see repeated values as well as some very tricky values (91 isn't prime!) so encourage students to grab anything they think is prime. At the conclusion of the game they can win additional points which will change each time they play. The goal is to practice and aim for a high score! Every new game will be a new adventure.

Play

©MathBait created with GeoGebra

Benefit

In this lesson students dove into prime numbers. While prime numbers were introduced early in MathBait™ Multiplication, students now have the chance to gain a much deeper understanding of these special values. At this time we recommend returning to play Centipede as students recognize the game is encouraging prime factorization of values.

Understanding prime numbers will support student number sense and fact fluency. Ultimately, multiplication can be viewed as simply building with the same sized blocks and by gaining fluency with prime numbers, students now have the blocks they need to construct any positive integer. Combining this with the other strategies in MathBait™ Multiplication will skyrocket student number sense as they can break down, rearrange, and piece together values in a variety of methods. The more we play with numbers, the more comfortable and familiar we become.

Prime Factor It!

In our final lesson in MathBait™ Multiplication, students bring together their knowledge of primes, multiplication, and factoring to ultimately reflect on their multiplication journey, strategies learned, and progress made towards building multiplication fluency.

Activity 1: Can You Skip Count?

In Unit 1, we focused on skip counting. Students determined if a value was a multiple of a given number by asking "would we say this value when skip counting?". Now, we want to amp up this idea by introducing our understanding of prime numbers.

Remind students of the first activities they completed (such as The Whisper Game, Skip Count Pop, and BuZZ) if time allows. We recommend allowing students 10-15 to play BuZZ as a group or individually. Ask students to consider the difficulty as they play, is it easier than their first experience?

Multiplication is simply identifying the value we stop at when skip counting by a certain number. Ask students how we might be able to determine if 2 is a prime factor of a number. If needed, help guide students to realize that if 2 is a factor, this means the number is one we would say when counting by twos. As all numbers with 2 as a factor, or alternatively all numbers said when counting by twos, are even, all even numbers have 2 as a prime factor.

Later, students will learn about divisibility testing. They have essentially already learned this testing for prime factors of 2 and 5 (any even number is divisible by 2 and any number with a 0 or a 5 in the one's place is divisible by 5). Because we use a base-10 system, there is a nifty way to determine if a number is a multiple of 3 by using a remainder clock. You can learn more about this test on our Instagram and Facebook pages. Unfortunately, there is no good rule for 7's. While there is a divisibility test for 7 and 11, it is complex and not well-suited for young mathematicians. As students are not yet familiar with division, rather than divisibility testing, lean on what they do know - known multiplies and distribution.

Provide students with the following numbers and ask them to determine if each value is a multiple of 2, 3, 5, and/or 7.

• 48

• 54

• 99

Allow students to share their findings and explain their reasoning.

• Both 48 and 54 must have a prime factor of 2 as they are even and thus multiples of 2. 99 is odd and therefore doesn't have a 2 factor.

• All three numbers have a prime factor of 3. This may be difficult for students to identify. See the explanation below to add a new strategy to help identify prime factors.

• None of these values have a prime factor of 5 as all multiples of 5 have a zero or a five in the one's place.

• None of these values are multiples of 7 as 7×7=49 (skipping 48) and 7×8=56 (skipping 54). It may be more difficult to determine if 99 has a factor of 7. Starting at 7×10=70, we can skip count to find 77, 84, 91, and 98 to see counting by 7's will not stop at 99.
  

Identifying a factor of 3 above was challenging, next, students will be given larger values which may seem daunting. Discuss the following strategy before presenting students with their next round of values.

Announce to students you believe that if you add any two multiples of a number, the sum is also a multiple. Ask students to prove or disprove your belief. Place students in small groups and allow time to discuss. It can be beneficial to provide an example to solidify your statement. For instance, we know 15 is a multiple of 3 and 30 is a multiple of 3, so 15+30=45 must also be a multiple of 3.

Students may recall The Wizard activity played previously in MathBait™ Multiplication. By using their substitution skills, they can determine 15+30=3×5+10×3. This shows us a 3-wizard has cast a spell enlarging everyone on the field. If we reverse the spell we have 3×(5+10). Give students time to confirm 3×(5+10)=5×3+3×10. This tells us 3×15=15+30=45! Not only have we confirmed the sum is also a multiple of 3, we have also determined what times 3 will produce 45. While one example does not prove the rule, using an example can help students better see an abstract concept. While a formal proof is likely too advanced for students of this age, a more informal argument is simply to remember that multiplying can be thought of as repeated adding, computing something like 3×5 is the same as finding 3+3+3+3+3. Thus, when we have 3×5+3×10 (or adding any two multiples of the same number) we are adding together 5 threes and then adding 10 more which is the same as adding 15 threes. Rectangle diagrams can also be helpful in demonstrating this idea. Conclude by stating the statement once more - the sum of any two multiples of a number is again a multiple of that number.

Students can now use this strategy to help them determine the remaining values. Encourage students to decide if each of the primes under ten (2, 3, 5, and 7) can be skip counted to reach the number. Have students share their results and their reasoning.

• 138

• 140

• 148

• 210

• 287

Students should find the following:

• 138, 140, 148, and 210 all have a factor of 2 as they are even.

• 138 and 210 have a factor of 3. One way to determine this is by noticing 210=3×7×10. This is more challenging for 138. Students may determine since 60 is a multiple of 3, 120 is also (as 60+60=120). Adding another multiple of 3, 18, will land them at 138.

• 140 and 210 have factors of 5 as each is a multiple of 10 and 10=2×5.

• 140, 210, and 287 all have factors of 7. Students can note 140=14×10=2×7×10 and thus a 7 will appear in the factorization, similarly 210=3×7×10. While 287 may be more difficult to notice, students can determine 287=280+7 and as 280=7×40 and 7=7×1 both are multiples of 7 and therefore their sum is as well.

Conclude the activity by explaining that knowing if a number is a multiple can be very helpful! It is also powerful to be able to find the DNA of a number. In the remaining activities in this lesson, we will explore how to find all the prime factors of a given number. Breaking down numbers helps us to be able to build numbers as well and strengthens our multiplication skills. We want to now focus on what traits a number must have to help us to identify its factorization and its prime factorization. A good example to share with students is LEGO. If we are interested in building something new, it can help by taking apart existing sets and seeing how they work. This is what engineers often do. By studying how to build values, we are learning the inner-workings of numbers, which will allow us to build anything we can imagine!

Activity 2: Grow A Tree

In this activity, students will learn how to make a factor tree. This activity is intended to be an artistic display while also allowing students to practice and expand on their ability to factor values.

Previously, students used factoring and substitution to find tricky products. For instance, when attempting to factor 90, they may recognize 90 is a number said when skip counting by 5, but were unable to identify what times 5 would produce 90. By using a known fact such as 9×10=90, expanding into 3×3×2×5=90, and then combing the remaining factors to obtain 3×3=9 and 9×2=18, they could determine 18×5=90.

Now, our goal will be to find the prime factorization of any value. Students will use the same strategies to create their factor trees.

Begin by providing students with a blank sheet of paper (non-lined) and ask them to design a flower for each prime number under 10. Encourage students to determine the primes independently. If help is needed, students may refer to their prime quilt created in a previous lesson. This is a great activity to allow students to be creative! They may design any type of "flower" and specifically play to their interests. Students may design their prime flowers to match the number of petals, or write the prime number the flower represents in their design, or simply have a key that displays each flower and their corresponding prime.

As this activity can be time-consuming, here are a few ideas for teachers and parents to condense the activity.

• Print out basic flower shapes for class ahead of time. Rather than design each flower, students can simply pick the flowers they'd like to use from the existing supply.

• Use stickers! Grab a set of flower stickers from the dollar store and allow students to use stickers for their trees.

• Integrate technology. Many applications will allow students to cut and paste or duplicate. For instance, creating a Google Slide deck with "flowers" using polygons and sharing with students. Here is a very basic setup for using Google Slides.

Next, assign students a number. We recommend providing each student with a unique value, this will allow a various array of factor trees that can be used for a Gallery Walk, be added to student portfolios, and displayed around the room. Another alternative is to create small groups with the same value. This will allow students to compare their trees and see how different factorizations lead to the same result. We recommend selecting values with at least 4 prime factors, increase the number of prime factors as students become more proficient.

Begin with a tree trunk, students should write their assigned tree number on the trunk. Next, students begin branching. They should determine a multiplication fact they know about their given number and use this to branch their tree. If any of their factors are prime, they will use their flower; for any composite (not-prime) values, students will create a new branch.

Once each branch has terminated with a prime, their factor tree is complete! Complete the activity again with new values and even try some large values (like 128 or 164) and discuss the trees together. Some key takeaways are:

• Every counting number (positive integer) has a prime factorization.

• Every prime factorization is unique - no matter how we build our tree, the flowers will be the same.

• Knowing how to "build" a number or what a number is made of is powerful! For more on this, explore Marco the Great and the History of Numberville and The Kryptografima.

Looking for an alternative way to practice? Have students build their own number by first designing their tree. Students begin by placing flowers and multiplying to determine what "tree" they have created. Add in the final value to the stump upon completion. For example, if I want my tree to have two 7 flowers, three 5 flowers, one 3 flower, and two 2 flowers, we can build this tree first, then multiply to find 7×7×5×5×5×3×2×2=73,500! This is a great opportunity to build more multiplication skills as students will find pairing 5's and 2's will make for easier multiplication. Rather than computing 7×7×5×5×5×3×2×2, they may find (7×5)×(7×3)×(5×2)×(5×2) to lower the difficulty, as this becomes 35×21×10×10. Students can use Napier's Bones or skills previously discussed in MathBait™ Multiplication to find 35×21 (for instance, one good method is the distributive property, as 35×21=35×(20+1)=700+35=735. They are left with 735×100=73,500.

Activity 3: Music of the Primes

This activity is particularly fun as it allows us to view mathematics and numbers in a totally different light! In Marco the Great and the History of Numberville, we describe prime factors as number "traits". They are the basic DNA of a number. This analogy will take students far. It is particularly helpful for finding values such as the greatest common factor and least common multiple (The Kryptografima offers a plethora of games including the classroom favorite Super Sleuth which allows students to use Venn Diagrams and a visual representation to help organize factors).

In thinking about prime factors as "traits", we can begin to consider what would a number sound like? In Music of the Primes we do just that! Each of the primes have been assigned a note. In this activity, students begin by finding the prime factorization of a given value - be mindful! While we know each prime factorization is unique, the way in which we factor will give rise to new and interesting tunes for students to play. For example, factoring 90 as 2×5×3×3 will produce a different melody than factoring as 2×3×3×5 and so on.

Once their factorization is complete, they will be able to select the length of each note (whole, half, quarter) and "play" their number! What is particularly interesting about this activity is that each note was created using mathematics! We used sine curves to match a note's frequency, prime factorization to determine the notes, and the note length utilizes fractions - each working together to create the musical representation! Math is truly everywhere and Music of the Primes is not only a fun way to practice prime factorization, but also an interesting way to think about numbers and the mathematics all around us.

Music of the Primes is great for students just learning to factor a number as they need only to identify a single "trait". For example, if factoring 189 students can notice 3 is a factor. After entering 3, the activity will fill in the corresponding value of 63. Students then continue to factor 63 to complete their measure.

Make sure to have sound on! Unfortunately, the program we use to create our games, GeoGebra, is limited in its ability to "play" a note. The resulting sound is similar to a touchtone. Extend and connect to other areas by having students play their number on an accompanying instrument.

Play

©MathBait created with GeoGebra

Activity 4: What Times What?

In this activity, students will use prime factorization to determine different multiplication statements which reach a given product. This game is a great way to practice multiplication but also can include strategy, allowing students to play around with different values.

Begin by presenting students with a value. A great way to generate a value is to start with the prime factorization you are looking for. For example, if our goal is 2×2×3×3×5, multiply this out to find 180 and announce 180 to students.

Next, give students the factor you are searching for. Start with primes such as 2 or 3, then advance to composite values. The first student to find what times what wins a point.

Not all students thrive in competition-based activities or games. A great way to augment the game for differentiation is to create pairs. Allow students to play in teams. We recommend heterogeneous grouping for this activity (pairing students with different levels of understanding). Structure the game by allowing pairs to work together to find the prime factorization after announcing the number. Encourage students to work together and create a factor tree. For example, with 180, students can find the factorization in many ways. Consider pausing after this step to allow pairs to share their methods with the class. Here are few paths that are great to highlight:

• 180=18×10=(2×9)×(2×5) = 2×3×3×2×5

• 180=2×90=2×9×10=2×3×3×2×5

• 180=2×90=2×2×45=2×2×5×9=2×2×5×3×3

Now, announce the target factor. This can be any factor of your choice. You may ask the class "What times 2 equals 180?". Students who initially halved may find this value quickly, others may need to rearrange their factors and multiply back up. Set a time limit and provide points to all teams who find the correct value, or provide an extra point to the team who is first to solve. Increase the difficulty by asking for composites such as 6 times what is 180 or 15 times what is 180.

The goal here is to help students (1) work with factors, (2) practice multiplication, and (3) see one of the great benefits of prime factorization. Highlight that once we have the prime factorization, we can find what times any factor will produce the target value.

For example, if asking 15 times what is 180, students can use the prime factorization of 2×2×3×3×5 to first make 15 using 3×5; the product of the remaining factors will be the solution. In this case, we have 2×2×3×3×5=2×2×3×(15)=4×3×15=12×15=180. Thus, the solution to what times 15 is 180 is 12! Notice how this is also preparing students for division. A great approach to division is to prime factor the dividend and if taught correctly, this will also allow students to easily deal with remainders as well!

This game can be augmented in many ways. We've made a quick and basic digital activity to aid in random number generation and play. This can be used individually or in groups.

Play

©MathBait created with GeoGebra

Activity 5: Conclusion

In our final activity, students reflect on their journey through MathBait™ Multiplication. In reviewing the various activities, they will determine the strategies they liked (or disliked) and identify the multiplication facts they are confident in, and which ones they still find tricky. Reflection has many benefits. It helps students become more self-directed learners and aids in organizing the information they have gained as well as highlighting how far they have come and how much they have achieved!

Part 1: Table Dash

Begin by providing students with an empty multiplication chart. We have included one below for easy printing.

MathBait_Skip_Counting_Table.pdf

Download PDF • 103KB

Start by setting a timer for 1 minute. Provide students with colored pencils and ask them to start with purple. In this first minute, students should scan the table and shade any squares in purple of multiplication facts they do not know or they feel are difficult. Most commonly students struggle with their eights or values in the middle of the table such as 6×7.

Next, allow students 1 minute using an orange color to highlight any facts they feel are medium. Medium facts are facts they feel like they can figure out, but it might take more time.

Finally, allow students to color in the remainder of the chart with green. Green should represent facts they know right away. This will likely be multiplication by 1, 2, 5, and 10, among others.

Allow students about 5 minutes to reflect on their colored chart. What patterns do they notice? Ask students to complete the following sentence starters on a blank sheet of paper.

1. For any rows that are entirely green, complete the following: I can multiply by [row number] because. . . For example, "I can multiply by 1 because 1 of anything is the number I am multiplying by. Like 1×9=9."

2. For any rows that are mainly orange or purple, complete the following: Multiplying by [row number] is hard because. . . Encourage students to reflect on why this multiplication fact is hard, follow up by discussing a strategy. For instance, "Multiplying by 8 is hard because I am not good at skip counting by 8's". Have students then complete the sentence starter: A strategy I can use for multiplying by [row number] is. . . For 8's, students have many strategies they have learned in MathBait™ Multiplication. For one, they can skip-skip count by 4's. Alternatively, if a student is confident in multiplying by 4, they can remember an 8 is made up of two 4's, therefore, to find a value such as 8×6, they can find 4×6=24 and double it to compute 48.

Conclude this portion of the activity by providing students with a 1-minute timer to complete as many green squares as they can. Continue by allowing 2 minutes to fill in the orange spaces and 3 minutes for the purple. Encourage students to use the strategies they determined previously.

Part 2: My Plan

If students have followed along in MathBait™ Multiplication and followed the suggested pacing it is highly likely they are fluent in all if not the majority of basic multiplication facts. However, every student is unique and some students will require additional time and support for mastery. In this part of the activity, students will identify the games they enjoyed and the games they think will help support their fluency to create their own plan. It is important to remember that fluency (not memorization) comes from (1) understanding and (2) practice. MathBait™ games are excellent for practice. Students aren't completing worksheets or mindlessly solving problem after problem, but are instead engaged in interesting and entertaining novel activities to bolster fluency. Be careful to ensure your student has mastered understanding. If students are lacking understanding, practice will only do so much. In this case, consider revisiting some of the conceptual activities in MathBait™ Multiplication to ensure they have a solid foundation.

Allow students to review the games in MathBait™ Multiplication. We recommend providing time for students to replay any games that interest them. Have students think back on Part 1 and any multiplication facts they don't feel confident in; what game or games do they think will help them with these facts? Games in MathBait™ Multiplication Part 3 are great for building general fluency, but skip counting games in Parts 1 and 2 may be the most beneficial for dealing with tricky values as practicing skip counting will help students identify multiplies. In addition, strategy practice is a great way to build both fluency and conceptual understanding. Using the factoring techniques they have learned in this unit (Unit 7) can help by factoring one of the values and rearranging to make a more familiar product.

Have students pick 2-5 games they think will help them to grow their multiplication skills. Carve out time (as homework, as a warm up, or as weekend work) for students to spend 15-30 minutes simply playing. Reevaluate their skills each week and celebrate as they continue to build strong fluency skills. As students continue to new concepts, such as division and fractions, continuing to refresh their multiplication skills is highly recommended. In general, facts are not something that "stick" easily. Our memory works by building on patterns and connecting to existing knowledge. It is not uncommon for students to forget some of their multiplication facts. As with any mathematical topic, the less we use a skill, the harder it is to recall. MathBait™ games and activities are a great way to keep these concepts fresh and practicing over many months or even years will only strengthen recall. The great news is, by building a strong conceptual foundation, if a student's recall path becomes less traveled, their understanding will make it easy to refresh with little practice. Memorization does not work this way. Once a memorized fact is lost, it is lost. If you notice student skills waning, simply add in some MathBait™ games as warm up activities.

Benefit

In the final lesson of MathBait™ Multiplication, students worked on finding the prime factorization of a number. This helps students to understand how a number is built and identify multiplication statements that produce a given value. Not only do these activities allow students to practice multiplication backwards (by starting with a product and determining what values will multiply to reach it), helping to build a foundation for division, they are also learning the DNA of numbers. The more roads students have to build a number, the more multiplication understanding they will construct, helping not only with basic fact fluency, but also with many subsequent topics they will see in their future mathematical journey.

Students also reflected on their journey through MathBait™ Multiplication and constructed a plan to continue to strengthen their fluency. Reflection is a powerful activity. It allows us to celebrate accomplishment and identify pain points. It also allows students to assess their own understanding rather than have another assess it for them.

Conclusion

MathBait™ Multiplication has been a blast to create. We have experienced the joy on students' faces when a new game is introduced or a familiar game is replayed. We have watched students grow their multiplication skills drastically and even seen very young students build amazing fact fluency through fun and engaging activities. Next up we are releasing our MathBait™ Multiplication Database. An amazing site dedicated to making each activity easy to access, find, and share. If you have been playing along, please do share this labor of love with your colleagues, friends, and family so more students can master math through joy.

In our final unit, we focused on factorization. Factorization is a powerful tool. Understanding what a number is made of will help students not only with multiplication and fact fluency, but with many mathematical topics to come including fractions, GCF, LCM, and continuing all the way into algebra and working with quadratic equations. Developing this type of number sense early and fostering it throughout your students' mathematical journey will help build a solid foundation of understanding (not memorization!) as well as a love of numbers, as when students "get it", have those "aha" moments, and enjoy the process they are nourishing an interest in problem-solving, a confidence with manipulating values, and strengthening brain connections to last a lifetime.

Finally, we encourage you to continue to play! These games make for great warm up activities, review days, stations, and so much more. Many games allow for challenges well into middle and high school. Remember, what we learn with joy, we never forget so keep playing, keep laughing, keep learning through fun and engaging tasks and never encourage memorization - foster understanding and your students will go far!

The fun doesn't stop here!

For even more fun learning, make sure to check out our debut novel, Marco the Great and the History of Numberville.

For hundreds of games, missions, and loads of fun learning math, sign up for The Kryptografima by MathBait™ below.

Marco the Great

The Kryptografima

NOTE: The re-posting of materials (in part or whole) from this site is a copyright violation! We encourage you to use these activities with your students. You can not take any part of these activities and post them as your own or crediting MathBait™ without written permission. This includes making derivatives for Teacher Pay Teacher or other websites. The material here is not considered "fair use". Thank you for respecting the author's rights.