MathBait™ Mastering Multiplication Part 3
- MathBait
- Mar 18, 2024
- 4 min read
Updated: Apr 14, 2024
Welcome to Part 3 of MathBait™ Multiplication, our final installment for students learning and building fluency with one digit products. Don't be too sad! What lies ahead is a treasure trove of 20 activities and 9 digital games! We also have loads of great content on the way as we advance to multi-digit multiplication and prime numbers which continue to aid in fluency as well as support a solid foundation for higher math. Get ready to have an absolute blast learning!
Thus far, students became confident in skip counting before advancing to processing two pieces of information (the number of groups and how many items in each group). Now, we shift our focus to formally multiplying. In the following lessons, students will be introduced to the multiplication symbol and continue to build on their previous understanding to master their multiplication skills.
Prerequisites
Students should complete MathBait™ Multiplication Parts 1 and 2. They should be able to fluently skip count by 2, 3, and 5 and are beginning to strengthen their ability to count by 4, 6, 8, 9, and 10. Students should also be able to answer simple questions such as "if I count by 3's five times what number do I stop on?". It is okay if students are still using their fingers. Finally, students should be familiar with the word multiple and be able to determine if small values are multiples. For instance, is 14 a multiple of 2? Students can confirm 14 is a number said when we skip count by 2's and is thus a multiple of 2.
If students are not ready for Part 3, that is okay! It is better to take it slow than rush things. Spend more time on Part 2 until students feel confident in their skip counting, their ability to recognize multiples, and to determine the total number of items.
Goal
In MathBait™ Multiplication Part 3, students will be introduced to the multiplication symbol. They will continue to build fluency through novel activities. While in some lessons students will multiply directly, many activities build fluency through creating multiple pathways to access products. This not only strengthens student recall of basic facts, but also helps prepare them for algebra and other activities in which they will utilize their multiplication knowledge for problem solving.
It is highly recommended that students play all the games provided here multiple times over a few weeks. Spiraling back to earlier games will help develop a strong fluency and multiple methods of accessing their multiplication facts.
Select a lesson to view details. 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.
Symbols and Scale
In this lesson, students will practice finding the missing value. The purpose of these activities is to strengthen fluency by providing different perspectives of multiplication facts.
Warm Up: Using Symbols
Begin by telling students that we often use symbols to avoid having to write so much. Allow students time to think of a symbol they are familiar with. If needed, explain to students that a symbol is a small picture, drawing, or shape that has a meaning. Students may instantly think of emojis which are a great example. Allow students to share a symbol and what it means. For instance a heart can convey love while a knife and fork might indicate a food court or meals nearby. Signs on the road often have symbols. This can be a quick or in-depth conversation depending on where students take it and time available.
Draw out a large X and tell students this is the symbol for multiples. Wait to use the term "multiplication". Explain that when we see something like 3×4, it is asking us the fourth number or the fourth multiple when counting by 3's. Order matters here. Draw back to the Hide and Seek game from MathBait™ Multiplication Part 2. When we move to row 3 we are counting by 3's. In other words, 3×4 is the same as going down to row 3 and counting over 4 spaces. At this time, we recommend calling the × symbol "ex". This avoids confusion and having to process too much information. Some students may already be familiar with the notation, but for students new to multiplying, allow them time first to practice with the notation before introducing terminology.
Ask students to write 3-4 statements like 3×4 and trade with a partner. Students should find the total for each statement. For instance, in 3×4 students will identify the fourth number said when counting by 3's (12). The goal here is to make multiplication easy and approachable by leaning on existing understanding. We are not doing anything new, but rather finding an easier way to communicate. Writing out "what is the 5th number we say when counting by 4's" is a lot. Now, we can simply write 4×5 to mean the same thing.
If time permits, return to MathBait™ Multiplication Part 2 and play a group round of Hide and Seek. Replace each clue the dragon gives with a multiplication statement. For example, if the dragon says, "I start at 5 and hop 4 spaces", have students identify this as 5×4 and find the value of 20.
Activity 1: What's That Called?
In this activity, we will provide students with the terminology for the × symbol. This is an investigation to allow students to feel like they have discovered the name on their own and to help make sense of the new terminology.
Draw out the large X on the board and ask students to share what this means. Drawing on the Warm Up, students should recall this is the symbol we use for skip counting. A statement such as 4×5 tells us to find the 5th number when skip counting by 4's.
In the Warm Up, we called this symbol "ex", because it looks like an X! But, later we will see there are many symbols we can use. Some people use a dot. Write out 4·5. Explain that in middle school, we no longer use the × and instead use a dot. Some people use parentheses like 4(5). While we don't care much about the different ways to write ×, we shouldn't call it "ex" because it doesn't always make an ex!
Provide students with their skip counting chart. Have each student write out a statement like they did in the warm up (such as 3×4). Remind students of the activity in MathBait™ Multiplication Part 2 where they made squares on their chart. This time, we will make rectangles.
With a colored pencil or crayon, ask students to draw out the rectangle that matches their statement. It can be helpful to model for students. For example, if we picked 3×4, I'll go down to row 3 and trace out 4 over. Next, complete the rectangle starting in the top left corner.
Ask students to count the number of small 1-by-1 squares in their rectangle. What do they notice? Here are a few key points to highlight:
There are 12 squares in a 3×4 rectangle and 12 is the 4th number said when skip counting by 3's.
The rectangle is 4 tall and 3 wide, which are the numbers we used.
Explain to students that we often describe a rectangle by how big it is. We would call this rectangle 3-by-4 because it is 3 wide and 4 tall. For this reason, we often call the × symbol "by". Allow students to each say the statement they selected using the word "by".
Another word we can use for the × symbol is "times". If possible, place students in small groups and allow them 5-7 minutes to come up with an idea why we might call this symbol times. After their discussion, ask students what something like 5×6 means. Students should recall this meant the 6th number we say when counting by 5's. It's like we count 5 how many times? "6!".
Conclude by explaining we can say 5×6 as "5 times 6" because it is telling us to count by fives six times. We call this multiplication. Allow students to guess at why we might call this multiplication. Explain or bounce off student comments to note that when we multiply we are finding multiples. Five times six is asking us to find the 6th multiple of 5, so we are multiplying.
If time permits, allow students to practice with a partner. Each student writes a multiplication statement such as 3×6 and they practice saying 3×6 in multiple ways: "3 times 6", "the 6th number counting by 3's", "counting by 3, 6 times", "3 by 6 (like a rectangle)". Explain it is not necessary that students remember all the different ways to say the same thing, our goal is to recognize these words and understand the meaning behind the words. They can pick the way they like best to communicate.
Activity 2: What is Multiplication?
Now that students have been formally introduced to the multiplication symbol as well as the term "multiplication", this is a great time to dive into what multiplication means. Notice we have intentionally held back on this previously. Our program waits to introduce the meaning of multiplication and ways to interpret products until students already feel confident in their skills from building on skip counting. This helps reduce cognitive dissonance and misunderstandings and promotes a positive feeling around fact fluency and multiples.
The best way to explain multiplication is for students to "discover" the meaning on their own. This activity is designed as an exploration; allow students to drive the conversation and share their thoughts.
Provide students with graph paper. This activity can be completed with coloring, cutting, or folding. You may allow students to select their medium or choose a medium that works best for your classroom.
We will be counting by 3's. If coloring, have students color bars of length three to create a graph up to about 12. For cutting, cut out the bars, and if folding, provide students with four bars of length 12 and allow them to fold to create the multiplies of three. (Note: if cutting or folding, little hands may have difficulty with small pieces. You can create wider three-strips to help accommodate developing fine motor skills).
Ask students what they notice and wonder about the bars they have created. They created the bars using skip counting, is there another way they know how to make 3, 6, 9, and 12?
The goal is to lean into repeated addition and help students to see that skip counting is simply adding the same amount each time. When we count by 1's, we are adding 1 (demonstrate if needed, starting at 1, 1+1=2, 2+1=3, ...). Counting by 2's can also be accomplished by adding 2, and so on and so forth. In this way, multiplication can save us a lot of time. Is it easier to write 3+3+3+3+3 or 3×5?
Continue the conversation and ask students to compare their 3-bar and their 6-bar, what is similar and what is different? The goal of this conversation is to help students see the 6-bar is constructed of two 3-bars and is thus twice the size. If students are not familiar with words like "twice", this is a good time to introduce the terminology. You might ask how many of the 3-bars can fit into one 6-bar. Explain that multiplication has the power to "scale" items or change their size. We love to use a Honey, I Shrunk the Kids or similar example to imagine taking something and drastically increasing its size.
The purpose of this activity is simply to allow students to play with the idea of multiplication. They have already mastered skip counting, now it's time to think about how this power is useful. If time permits and students would benefit from additional exploration, encourage them to be the wizard and cast enlargement spells on items. Students can begin using graph paper to draw an item (like an apple, ant, flower, etc.) and then use their wizardry to enlarge the item. Use the markings on the graph paper to help guide them. For instance, to turn an ant that is 3 squares long 5 times its size we would need to make the length that of five 3-bars (or 15).
Once students have a good feel for how their skip counting skills can be used as repeated addition and scaling, move on to Activity 3.
Activity 3: Emoji Mystery
In this activity, students practice using symbols to convey meaning while also building their multiplication facts. Players are given three emojis, each emoji is a symbol standing in for a number from 1 to 10. They must use the clues to decode the number hiding behind each symbol.
For example, in the instance 😄×😍=30 and 😍×😔=48, students can deduce that both statements contain the symbol 😍. They can "hunt" through the chart to identify a common row or column that contains both 30 and 48 to find both are multiples of 6. From there, they determine what times 6 leads to 30, finding 😄=5 and what times 6 leads to 48 to determine 😔=8.
For graduated fluency, allow students to begin with both "Show Table" and "Show Hints" turned on to use their detective skills with the multiplication grid. As they improve, challenge them to turn each off. They may continue to use skip counting or other strategies. In gaining exposure and playing with the chart, students are continuously building more and more familiarity without learning anything new.
Play
©MathBait created with GeoGebra
Activity 4: Growth Laser
The purpose of this activity is to encourage students to consider multiplication as scaling. Until this point, multiplication has been viewed through the lens of skip counting with a brief introduction to repeated addition and scaling. Now, students have the chance to experiment with multiplying in a new and fun way.
Start by providing students with graphing paper. Have them draw a segment that is 2-3 boxes or units long. Explain they will create a growth laser. This is a machine that can increase the size of something. For younger students, stick to whole numbers, however, for older or more advanced students they will learn this machine can create any size by using rational values. For instance, taking a 3-unit segment and multiplying it by 5/3 can turn it into a segment of length 5. If students are not working with fractions, explain this laser will increase the size of items by making copies. To create a segment twice as long, we must set the dial to 2 and it will multiply the segment by 2.
Allow students to draw their new segment. This might come intuitively to some students and others may have more of a struggle with spacial reasoning. Remind students that to "double" means to make two of. Next, ask students to set their growth laser to 3 and draw the result. Continue upwards to 10 (maximum power) and have students make a table of their results. What do they notice? Help students to see this is a skip counting table, the length of the new segment is exactly how many 2's (or 3's depending on their initial segment) they have counted.
In our digital game, students have a growth laser and must help the Dino grow to save their babies. Level one is direct multiplication, in level two students begin to refine their skills by thinking about the closest multiple.
Play
©MathBait created with GeoGebra
Benefits
In this lesson, students were introduced to the multiplication symbol. This should be an easy transition from their previous knowledge, as in Part 2 students practiced determining facts such as "6 groups of 2 gives us 12". Now we will represent this same idea as 2×6=12. It is much easier to gain student buy-in from the position that we are saving time and making something easier, rather than introducing something new and making things more complicated. By using the perspective of 2×6 is much easier to write than "the number we get by counting 2's six times", the notation is an extension of their existing knowledge that makes things more pleasant for them.
In Emoji Mystery, students continue to use a completed multiplication table to support their growth. This is akin to Hide and Seek from MathBait™ Multiplication Part 2 as they are exploring and gaining additional exposure to the multiplication table without any expectation of memorization. Students build problem solving skills by seeing two multiples of a number and determining what the number is before identifying the missing factor.
Finally, in Growth Laser Rescue, students will use multiplication facts directly. They may continue to use their fingers or count up. For instance, if the Dino is 4ft and the building is 28 ft, students may count up by 4's to determine how to set the laser. We want to continue to expose students to the topic in a fun and non-threatening way. If needed, allow students to continue to use a completed multiplication chart and encourage them to call on it less and less. In level 2 of Growth Laser, students build a higher level of fluency as they consider values close to multiples.
Missing Numbers
In this lesson, we continue to build on problem solving skills. We are not focusing on memorizing the table, but instead working on building patterns and identifying relationships between numbers. In this way, students are connecting new tasks to previous understanding for long lasting retention and success.
Warm Up: A Riddle
Begin by drawing a 2 by 2 grid with row and column headers for students to see. Place a 6 and 9 in the first row and an 18 in the second blank of the second row. Explain this is just like the multiplication table. The numbers to the left tell us what we are counting by and the numbers on the top tell us how many we have counted.
To solve this riddle students may notice that to achieve 9 they must be counting by 1, 3, or 9. They can eliminate 9 from the first row entry as they cannot count by 9's and reach 6. Therefore, the first row entry must be a 1 or a 3. From there, they should be able to fill in the rest of the table.
An alternative strategy is to consider what 9 and 18 have in common. We can reach both 9 and 18 by counting by 9's or by counting by 3's, so the top of this column must be a 3 or a 9. Similarly, 6 and 9 can be found by counting by 1's or by 3's.
Provide students with time to try to find the missing values and allow them to share their thoughts and strategies.
Activity 1: Missing Numbers
In this activity, students will continue to play the warm up exercise with new values. The goal is to recognize the relationship between values as well as gain practice using a multiplication table to reference the columns and rows.
Play
©MathBait created with GeoGebra
Benefit
The theme here is that in order for students to develop strong fact fluency they need:
Scaffolded instruction that connects multiplication to previous knowledge
Exposure to multiplication in various formats
Practice that requires them to recall information in novel ways
By learning multiplication in this way (rather than memorizing), students are strengthening their number sense and building connections for easier recall. In the previous lesson, Growth Laser is particularly helpful as students aren't asked to simply calculate 9×6, but instead must find the multiple of 9 that is closest to 56, without going over. This requires them to think in new ways rather than directly recalling a memorized fact.
In this lesson, students are not only recalling products, but also using their problem solving skills to find what the values in a given row or column have in common. Not only is this helping solidify their knowledge of multiplication, it is further reinforcing the patterns and relationships between numbers they previously explored.
A Quick Note
Before we dive into the next lesson, we wanted to take a moment to circle back on the foundation of MathBait™ Multiplication with, what else, a story.
A world leader was once asked in an interview to calculate 8×7. He fumbled around, turning a silly ice breaker into international news. How is it possible that our leaders cannot complete basic multiplication?
Well, it's the same idea we have been emphasizing throughout this series. He probably learned through memorization. You stare at the table, you repeat it again and again, you take timed tests in order to cement these facts in your head.
And then, you lose them.
You lose them because memorization doesn't stick, it doesn't have anything to hold on to. In order to build lasting knowledge we must scaffold learning into manageable pieces and connect each piece to prior understanding. At MathBait™ we use storytelling because it has the ability to connect daily life, things we understand from simply being human, to mathematical concepts, automatically setting up students to better understand (and thus remember) as it is connected, anchored, to existing knowledge.
As everyone rushed to gossip on the leader's blunder, a particular commenter stuck out among the noise. He explained that rather than guessing or trying to rattle through his memory on the spot to pull out that one little square from this sacred table, it would have been much more impressive if he said, "Ooh, that's a tough one, but I know 7×4=28 and so I can double that to get 56."
As teachers, we often forget the long term. Our job is to make sure our students know the content in front of us, so that becomes the focus. However, if we expand our focus to consider why students need to know this (and it's not because the district or state or leadership have told us so!) we can not only help our students with this year's content, but set them up for long term success.
Multiplication is the foundation for many advanced topics. In the second book of our Marco the Great series, students will be introduced to factoring quadratics and simplifying radicals. These tasks rely on a solid understanding of numbers and multiplication. However, we often work backwards. That is, to determine the roots of a quadratic we undo multiplication, we factor, to identify what values could multiply to 24.
It is a much stronger skill for students to have the ability to problem-solve rather than spout out facts. Our Multiplication series started with skip counting and showing students we only need a small handful of facts to derive the remainder of the table. This is a great time to circle back to this idea and not allow students to get too caught up in remembering each product. In fact, in higher math they will need to decompose values and it is often counterproductive to multiply everything out as we end up needing to rip it back apart to solve.
The activities in this lesson fit well into an analogy:
We have built a multiplication store in our brain. There, we store all the multiplication facts we can. Note, if we memorize these facts, the store will quickly go out of business as soon as we stop shopping there regularly!
Next, we have carved out many different ways to get to our store. Just like in real life, a store needs a lot of traffic to be successful, it needs multiple ways to access it. We haven't directly practiced products much at this point, because this is only one route and we want to have many.
Finally, until our store is popular, people will need a map to get there. In MathBait™ Multiplication Part 3, we are working on traveling to the store (lots and lots!) and taking many different paths to get there. The more we visit the store on different routes, the easier it will be for us to quickly run to the store and grab a product without much thinking. Just like driving to familiar places becomes almost automatic.
The moral of our tale is that although students will practice their facts a lot in Part 3, we still want to avoid memorization. We want to practice facts from many different routes and perspectives. This will not only solidify fact fluency and faster recall, but will also support students much more in the long term and help build the skills they will need through high school and even college math! Figuring out 8×7 is much more impressive and impactful than trying to keep it memorized throughout life.
Multiplication BINGO!
Thus far, students have indirectly strengthened their multiplication skills. They have identified missing numbers, found patterns within the multiplication table, and developed an understanding of the relationship between values. Now it is time to practice multiplication in action.
It is important that students have already developed a conceptual understanding through the previous activities. A lack of understanding of multiplication, especially combined with a push to memorize or quickly recall tables, is a set up for failure. Students will continue to struggle and possibly lose interest in math. Many students enjoy puzzles, mysteries, and unknowns. They like to solve problems and feel a great sense of accomplishment when they are successful. By first building each step using their previous understanding, they have a strong foundation and are now ready to strengthen their connections through recall.
Our first game provides students with time to think and problem solve. There is no rush to compute a product quickly, which is why it is ideal at this stage of learning.
Warm Up: Find It
To get students ready to multiply, write out a few numbers from the multiplication table on a board where everyone can see (for example 24, 36, 48, 56, and 72). Provide students with a multiplication expression that matches a number on the board (like 8×3). When students have found the product, instruct them to put their thumb up. Have a student share their solution and strategy before moving on to the next.
An alternative method of play is to provide students with a similar matching activity individually to reduce time pressure. The general goal is simply to get students thinking about multiplication in preparation for the activity. Ideally, students will have started to move away from skip counting at this time. Although, skip counting is still an excellent tool for tricky values. For instance, if they are unsure what 8×6 is, they can notice that 8×5=40 and they need one more 8 to find 48. If students are still heavily learning on skip counting at this time, we recommend spiraling through the previous activities (including the skip counting activities in MathBait™ Multiplication Part 1) to build more comfort before proceeding forward. It is not necessary for students to know all their products or be able to quickly recall them at this time.
Activity: Bingo
Our digital Bingo can be played individually (against a clever computer player) or in a group/with others. The goal of this activity is to strengthen multiplication fluency and continue to help students utilize multiplication notation. Speed is not the goal here, so if playing in a group, be mindful of how long it may take a student to determine the value. For younger students, playing in pairs is an excellent support system. Students may share a card/device in order to work together on the products.
Remembering that this should be a fun, low stress experience is vital!
Against the Computer (1-player)
Select the "against the computer" option from the drop-down menu. Next, slide the red, green, and blue values to select your marker color. Click Play to begin.
Click on "New Board" to find a board you'd like to play. Rules are the same as traditional Bingo, only one cell may be marked each round - even if you have a repeated value on your board. When happy with your board, lock it in.
Select "Call" for the host to call out a product. It will be shown on the top right of the screen. Search your board for the solution. For instance, if the call is 3×9, search for 27. Selecting another value will display an error, "That isn't a valid space!". If 27 is clicked, it will be marked with your color and you may proceed to the next call. If your board does not contain a 27, select "Not on my board" and play will continue to the next call.
The computer will also play! They have been trained to look for the best option making for a competitive game. The first to earn 5 in a row (vertically, horizontally, or along the two diagonals) wins.
Note, the game will not check your board for you. If the product is on your board and you select "Not on my board" the game will continue without warning.
Playing with Others
Have all players or teams open the game on their own device and select "with others" from the drop-down menu.
One (and only one) player will be the caller. We recommend a teacher is the caller for the first few rounds. The caller should check the "I am the caller" box on the home screen. All other players should leave this checkbox unchecked.
All players may select their marker color and pick play to begin. Click "New Board" to generate boards. Once each player is happy with their board, lock it in.
Caller
The Caller will see a "Call" button at the top of their board. The Caller may play along on their board or simply call out the products. When the expression appears, the caller calls out the multiplication statement to all players. If playing, they may search their board for the product or select "Not on my board". After giving players time to mark their boards, select "Call" to pull the next product and continue until someone has a Bingo.
Players
For all other players, rather than a "Call" button, they will see a drop-down to record what was called. Use the drop-down to enter the value when called out. This is a great way to ensure everyone has correctly heard the value. The call will appear in the top right after selecting the check. Continue play as normal, searching your board for the value or selecting "not on my board".
Have fun!
Play
©MathBait created with GeoGebra
Benefit
Multiplication Bingo is low stress. Rather than a worksheet, students are motivated to win and also may consider where they mark off a value to increase their odds, introducing elements of gamification. Students playing together or in pairs can discuss their products and strategies allowing them both time to share their ideas but also consider alternative viewpoints. This game contains traditional fact fluency practice helping to strengthen recall-ability as well as help identify some of the trickier products students are not yet familiar with. Allowing a student to share their solution and strategy after each round also increases confidence.
Table Mash Up
Many of our previous games practice multiplication from different directions. Rather than simply calculate 5×6, we ask students to play with numbers in various ways. This helps build connections between products and factors, as well as helps to develop relationships between values.
In this lesson, students will work more directly with products. In MathBait™ Multiplication Part 1, we discussed why completing a times table is not beneficial for fact fluency. Students may complete a multiplication table through skip counting, memorization, or other methods that do not directly correlate to understanding. Thus, our multiplication is a mash up! It allows students to practice with products without using the format of the table as a crutch. In addition, students will utilize their logic and problem solving skills while also continuing to develop an intuitive understanding of how values relate to each other.
Warm Up: All Mixed Up
If you have already completed Napier's Bones from MathBait™ Multiplication Part 4, this is a great time to pull them out. If not, for this warm up students will begin by creating multiplication rods.
Provide students with 10 strips of paper (about 1 by 6 inches). On the top of each strip, write out the numbers from 1 to 10. Next, write the multiples of the value listed on strip. In essence, students are creating a multiplication table they can rearrange into different orders.
Allow students to play with their strips. What order would they place the table in? Ordering our strips from smallest to greatest has some benefits, but are there other arrangements that are interesting? For instance, placing the even numbers next to each other allows us to more easily see our skip and skip-skip counting patterns. A similar instance helps us to see the skip and skip-skip counting patterns with 3, 6, and 9. Encourage students to stagger their strips; what if we don't line everything up?
The goal of this activity is simply for students to play with multiplication strips and look for patterns. In creating the strips they are practicing their multiplication and skip counting. In playing, they find patterns, consider the table from a new perspective, and gain familiarity with multiples through exposure. There is no right or wrong answer or way to play.
Activity 1: Table Mash Up
In this digital game, students will practice with a mixed up multiplication table. They will have the chance to directly practice their facts, utilize their problem-solving skills, and continue to hunt for number relationships.
Game play includes 4 levels. Level 0 has no timer. This can lower pressure and is a great place to start. However, with no timer, students must complete the full table which can be time consuming. Levels 1-3 include timers. Students have until the timer runs out to fill out their table.
This is not a traditional timer game or timed test!
The purpose of the timer is a gamification element. It adds an urgency which helps students to strengthen their connections and also can result in positive feelings of accomplishment. However, the goal of the timer is not to cause pressure. Reiterate to students they do not need to complete the table before time runs out. Instead, do their best and try to beat their own high score. Teachers may wish to play first as a demonstration and might find it difficult to complete the table with the timer themselves!
After selecting a level, players will pick 10 cells to unveil. Discuss strategies with students. There can be luck involved, but we can also pick cells that will better help us to decode the grid. Remind students of their Missing Numbers activity previously and the strategies they found helpful.
Next, students will see a drop down for each row and column header. They will update these with a number from 1-10. Each number only appears once across the columns and once across the rows, just like a standard multiplication table. In this game, however, they are not in order from least to greatest. Students will use the clues from the cells they have revealed to make their best guess. Once they have selected each header, the game begins. Note, players can change the headers at anytime in the game. If they find their headers do not work, they can update them using the new information they have gained.
If playing on Levels 1-3, every 10 seconds a new cell will be revealed. This will provide vital clues on the value of the headers. Students will select a cell and enter the product. They can check their value (each check costs 5 points) to either ensure their multiplication is correct, or to gain more information about the headers. If students are confident, they can enter their value without checking to maximize their points.
Once the time is up, or the table is filled if playing on Level 0, students will have the chance to check their values. Each play will provide a new mash up!
We recommend students start on Level 0. This gives them the chance to understand the game and practice. As they feel more comfortable, challenge them with additional levels. Our games in MathBait™ Multiplication are designed to be played many times. As students continue with the games and build stronger fluency, they will be able to return to Table Mash Up and achieve more and more success.
Play
©MathBait created with GeoGebra
Benefit
This activity is a fun way for students to directly practice multiplication while also enhancing their problem solving skills and recognizing the relationships between numbers, such as common factors.
The mix-up of the times table forces students to get away from blindly filling out a table and instead strengthening their fact fluency and recall. The timed element can make students feel proud and confident (especially the first time they complete the full table). Graduated levels allows students to continue to push their understanding, fluency, and recall to "beat the game".
An Ode
It's finally time to increase the speed of recall. This is not memorization. Throughout our program, students have built a strong conceptual understanding of multiplication. The more they work with products and the more they work on building different routes for retrieval, the easier it will become to quickly identify a fact. This game is a blast and can be a challenge even for adults! Players are racing against themselves trying to increase their personal best. This makes An Ode a great game for students to play at home (they can screenshot their results to share with teachers), with a substitute, during a short class, game day, or at stations.
There are a lot of feelings about timed activities. Truthfully, timed activities have no correlation to math anxiety (for more on this, check our our article Timed Activities: Stressful or Impactful). In fact, timed activities are an excellent way to strengthen the connection between understanding and recall, but they must be done right! It is important that timed activities:
Do not occur until mastery or have different levels for varying abilities.
Are fun or engaging. When timing is an element of a game, the human brain releases chemicals that can bring on good feelings and accomplishment. When timing is an element of a test or assessment, the human brain releases chemicals like cortisol that bring about stress and even inflammation.
There is a valid reason for the timer. This can be pedagogical or a game element, but we shouldn't time for time's sake. Our goal here is to help students build a strong understanding of multiplication and improve their ability for recall, not to produce anxiety or encourage memorization.
Thus, if your students aren't ready for this activity, wait. These are meant for students who have done well on the previous activities and feel ready and excited to test their skills in a fast-paced environment. If your students aren't quite ready, return to Part 2 (Rainbow Multiples is a class favorite and a low stress way to build strong foundational skills) and the earlier games here in Part 3. Emoji Mystery, Growth Laser Rescue, Missing Numbers, Table Mash Up and Multiplication Bingo are all games with high replay-ability that will continue to build mastery even for older students who have been multiplying for a long time.
Warm Up: Strategy Session
The purpose of this warm up is to discuss with students the different activities they have completed and strategies they have learned. This is a great pre-game reminder to help students add speed to their toolbox.
Remind students of Part 1. Here, they practiced skip counting. They learned that we can count by 1's, but we can also count by any other number as well. After practicing counting by 2's, 3's, and 5's they learned how to use this knowledge along with skip-skip counting, to count by other values as well. Because 4 is made of two 2's, they could skip count their 2's to count by 4's. Ask students to share their experience. What did they like about skip counting? What was their favorite activity? What strategies helped them to count by larger values?
Next, move to Part 2. Here, they learned how to work with two pieces of information. They were still skip counting, but also keeping track of how many numbers they counted. They learned the term multiple and played games like Hide and Seek and Rainbow Multiples. Ask how they knew if a number in Rainbow Multiples was a multiple of 4 or 6. If time permits, playing Rainbow Multiples again (possibly on a higher level with larger values) can help students to think of and share the strategies they learned.
If students are ready, they might be interested in learning how to identify larger multiples of 3 and 9. We have not included this in MathBait™ Multiplication as younger students often see this as a "trick" and don't learn the reasoning behind why it works. You can find our video explanation on our Instagram or Facebook page.
Finally, discuss the activities they have done in Part 3. Explain that throughout these activities they have built a "Multiplication Store" in their brain. This store holds all the answers they are looking for. The different activities built roads to the store. This allows them many ways to access their Multiplication Store, but they still need a map. The more often they visit the store, the easier it is to find it. Soon, they won't need a map at all! This next activity will help them get used to finding their store.
Activity: An Ode
This game is based off a great app called CMYK created by S. Scott. In CMYK, players are given four buttons at the base of the screen (cyan, magenta, yellow, and black). Colored bars begin falling from the top of the screen and players eliminate a bar by launching a ball of the matching color.
In An Ode, bars will fall from the top like in CMYK but each bar contains a number. At the base of the screen, players will find buttons labeled 1-9 as well as a black button with a white square. To eliminate the bar, players must send out two balls that equal the value on the bar. For example, a bar with the number 8 can be eliminated with a 1 and an 8 or with a 2 and a 4.
Use the colors to help you! Each bar has a color which corresponds to the colors of the numbers at the base of the screen. The bar color is one of the factors of the number. That means a bar labeled 8 could appear as red (1), orange (2), green (4), or purple (8). While this might not be very helpful for smaller values, for a large value like 48, this can help students by tying back to the previous activities such as Emoji Mystery and Missing Numbers. A purple 48 tells players that it can be eliminated with an 8 (as the 8 is colored purple) and some other value. They need only to find the second value (6) in this case.
However, for numbers with multiple factors, any two values whose product matches will eliminate the bar. A red 8 does not mean players must use 1×8. The colors are simply a helpful tool. The red 8 could also be eliminated with the orange 2 and the green 4.
Squares have special powers. Using the black square on the far right correctly will clear the entire board! To use the Square Power for a square number (such as 1, 4, 9, 16, ... ( you can review MathBait™ Multiplication Part 2 for activities on squares)) click on the black square first. Before the black ball hits the bar, select the number you wish to square. For example, to use Square Power on 36, select the black square followed by the 6.
Finally, you can get ahead by launching many balls at once. But be careful! You must launch in order so that the ball hits the intended bar. Each number also contains only one ball to launch. When the ball is launched, a red x will appear over the number. When the red x is gone, you know you may launch a new ball from that number. For example, if the first bar is 48 and the second bar is 14, you may launch 6, 8, 2, and 7 to quickly eliminate both bars as soon as the balls hit. But, if the bars read 48 followed by 24, you could launch 6, 8, 3, and wait for the 8 ball to return to launch the second 8. Alternatively, you could launch 6, 8, 4, and wait for the 6 to return to launch another 6.
This is a fun fast-paced game. Play for a high score! You can watch a demo game on our Instagram or Facebook page. Players have 3 hearts, each mistake will remove a heart. The more bars you eliminate correctly in a row will increase your multiplier and your points.
Desktop Version
The following is our desktop version. This is great on a tablet, iPad, Chromebook, or computer. As always, we recommend playing on full screen mode.
Play
©MathBait created with GeoGebra
Mobile Version
An Ode is a fun game to play anywhere! We have created a mobile version which boasts a taller aspect ratio and larger buttons. This can also be played on a tablet with vertical orientation and can be helpful for younger children.
Play
©MathBait created with GeoGebra
Benefit
Throughout MathBait™ Multiplication, each lesson provided novel activities helping students to grow their understanding. Rather than simply memorizing products, they found relationships between values, looked for what numbers have in common, and approached multiplication from every angle. This strengthened students' foundation.
Long term fluency requires two key factors: (1) Understanding and (2) Practice. Understanding stores the information in long-term memory while practice helps students to quickly retrieve this knowledge. The more students work with multiplication, the more easily they will be able to recall. But be careful! Direct practice of facts will weaken the underlying structure. Our program built many roads to access this information, practicing by only taking one path will make the others less traveled and harder to find when students need them.
This lesson provides students a fun and challenging way to practice recall. To master multiplication, spiral the activities in MathBait™ Multiplication. These games can be played again and again and will continue to help students strengthen their understanding of numbers and products.
Centipede
Prime numbers are the building blocks of Natural. Understanding prime numbers is a game changer for students and can benefit not only multiplication, but factoring, fractions, and many other topics students will face along their mathematical journey. In fact, prime numbers are so important, a key finding of primes is called The Fundamental Theorem of Arithmetic - that's right, it's fundamental.
We spend a good deal of time on primes in Marco the Great and the History of Numberville and dedicated an entire area (The Castle) to primes and prime games in The Kryptografima. In MathBait™ Multiplication, students were briefly introduced to the idea of prime numbers in Part 2, but we will not fully tackle prime numbers until the final installment. This gives students time to build more maturity and familiarity with products. That being said, this is a great time to continue to build an intuitive understanding of prime 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. Notice in both this game and An Ode, we are working on multiplication backwards. Rather than giving an expression such as 3×4, we provide the product and allow students to build the expression. This builds multi-directional fluency and will greatly support students as they encounter fractions as well as in higher levels such as Algebra.
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.
©MathBait created with GeoGebra
Benefit
This lesson continues to help students build fluency in multiplication. Students continue to recognize products such as 5×6, but also can now build problem-solving skills to help them tackle any values they have found difficult. For instance, if a student is having trouble with 5×6, they can now envision this as 5×2×3, making quick work to find 10×3=30.
Not only does this help students with fact fluency, it is also building towards important skills such as prime numbers and factorization. Students are further continuing to create multiple pathways to access their multiplication knowledge. They are building more resilience as forgetting a fact isn't a game over, they have now developed skills to look at the problem in a new way and lean on what they do know rather than being pulled down by what they don't.
Distribution
In this lesson, we focus on the distributive property. Understanding this mathematical property will help students use problem solving to determine some of the more difficult facts as well as supporting their long term retention and recall.
Warm Up: Let's Eat
Start with a story. This can be very informal so that students do not even think we are talking about math or have started the lesson.
"Last weekend I was having a small party with my friends. I didn't have time to cook and gather snacks so I went to Taco Bell [feel free to enter your own restaurant]. They offered a special which contained 1 drink, 3 tacos, and 2 burritos. I ordered 5 specials but when I got to the window I was sure they didn't give me everything. They handed me only 1 drink, so I knew something was wrong. While the cashier was giving me an attitude asking why I even needed 5 drinks as there was only one of me in the car, I was trying to do the math to figure out how many tacos and burritos I should have received."
At this point ask students how many of each item you should have been given. They will probably be quick to figure it out. You ordered 5 combo meals, so you should receive 5 of each item. This is 5 drinks, 5 of the taco plates, and 5 of the burrito plates. Breaking it down further, since each taco plate contains 3 tacos, 5 of these totals 15 tacos and similarly 5 burrito plates total 10 burritos.
Explain to students this is called the Distributive Property. Invite them to share their own thoughts and examples of where they have seen this used in their life. Connecting to existing understanding is a sure-fire path to mastery.
If needed and time permits, using rectangles can further help support understanding. Mimic the passage below from Marco the Great and the History of Numberville. Allow students to make bars to figure out some of the more difficult products (for instance, to find 8×7, allow students to draw 8 bars each 7 tall and discuss how they might split them up to more easily determine the product. Some examples would be to split each bar into 3 and 4 so they have eight 3-bars and eight 4-bars or 5 and 2 which is a particularly good strategy for students who have developed confidence in counting by 2's and 5's).
©MathBait
Activity 1: Gold Rush City
In this activity, students complete a role-playing game. This is a great game as students have a big reward at the end - they can design their own city!
Remind students of the distributive property. In Gold Rush City students begin by directing their miners to different areas on the map. If they select an area without any gold, an X will appear on the map and they can continue searching. If they find gold, they will first be asked to collect all the bars by dragging the slider. Next, they must compute how much gold they have found. Each bar of gold contains some number of nuggets, students will compute by multiplying the number of nuggets by the number of bars.
Now it is time to distribute the wealth. Students move a saw to decide where to split the gold. They can decide to give more (or even much more) to themselves, more to their workers, or split it evenly when possible. But be careful! If you are too unfair, the miners will revolt!
Some students may be interested in understanding when the miners revolt. This is a complex algorithm created for this game. It takes into account how much they have mined (for instance, lots of X's on the map means they mined many areas without any prize which increases their irritability). It's also taken into account how you have split the wealth. Sometimes an uneven splitting may anger them, but not cause them to revolt. Players will see "tensions are high!" and should try to make their miners happy on the next split to calm things down.
If the miners revolt, players will need to click on the little dots. Each dot is a miner attempting to steal their money! The faster the player clicks the less money the miners will get away with.
After splitting the gold, students use the distributive property to determine how much they keep and how much the miners receive. For example, if we collected 6 bars of gold, each containing 5 nuggets, the total is 30 nuggets. Splitting it so that the miners keep 2 nuggets in each bar mean the miners keep 2×6=12 and the player keeps 3×6=18. Notice 12+18=30. Correctly finding the products on the first try will also give players hints as to where to search on the map.
There are 10 gold mines on the map. Once players have found all 10 gold mines they advance to a screen that allows them to spend their wealth and create a city. They can select which items they would like to purchase and construct their city. Students can return to the purchase page as often as they would like to buy more items (so long as they have money remaining). When finished, they can select to take a picture of the city they have created!
©MathBait
Students complete 30 multiplication problems as they progress through the game. Problems are random, with the exception of student choice in how they distribute the wealth. At the end of it all, they are rewarded with the ability to play and create something of their own!
Play
©MathBait created with GeoGebra
Activity 2: Big Numbers
So far, we have focused on 1-digit by 1-digit multiplication and multiplication by 10. As we continue in the series, we will soon turn our focus into multi-digit multiplication. This is a great place to see the distributive property in action.
Provide students with a multiplication question such as 5×46. Allow students a moment to consider how they might compute this value. Remind students of the distributive property. Can splitting this up be helpful?
Rewrite the question as 5×(40+6). Ask students how we would utilize the distributive property. Allow them to guide you to 5×40+5×6. Announce we know how to find 5×6, what is it? But how can we find 5×40?
Remind students that 40 is simply saying we have 4 tens. This means 5×40 is asking us how many tens do we have when we skip count by fives, 4 times. Allow students to compute 5×4=20 to conclude the solution is 20 tens. Ask students how we can write the value "20 tens". Help as needed to see that 20 tens is simply placing the 20 in the ten's place. That is 20 tens which equals 200. If students have not yet worked with place value, we recommend waiting to complete this activity and the next until after a unit on place value for the most success.
Avoid encouraging students to "add a zero". This creates a misunderstanding. We are not simply adding a zero, in fact we can't just add values to the end of a number. Instead, we are focusing on placing our value in the ten's place. To do this, we must also place a zero in the one's place as otherwise we wouldn't know if our 20 meant 20 tens or 20 ones.
Now ask students if anyone can think of a different way to approach 4×50. Encourage students to use the distributive property again. Since 50=5×10, this means 4×50=4×5×10 or 20×10. We can count by 20's just like we count by 2's. Have the class count by 20's together: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200. Now we can see that 10 groups of 20 is 200.
Allow students to practice with a partner. Each student will ask their partner a basic fact they have been working on, but with one value changed to tens. For example, rather than asking "what is 3×4?" they might ask what is 3 times 40 or what is 30 times 4. Students can use either "tens" or the distributive property to support them.
Activity 3: The Wizard
In Marco the Great as well as in The Kryptografima, we introduce the distributive property as a wizard which has many benefits in Prealgebra tasks. Remind students of the Growth Laser activity. Multiplication can be thought of as a growth spell cast by a wizard. However, like most spells it hits everyone on the field. When we point our laser and pull, everyone gets increased, whether we like it or not!
In this activity, students will have their first look at multi-digit multiplication. Students should be familiar with multiplication through the previous activities. This game is based on Numberville and thus has some terminology students may not be familiar with. In our experience, they catch on quite quickly.
To play, students will see a multiplication problem such as 6×78. It is framed as a 6-wizard casting a spell on the house of 78. Their job is to take the house and write it as two terms, in this case 70+8. The game will help students to organize information as they click on each occupant and determine the impact of the spell. Finally, they sum the values together.
Play
©MathBait created with GeoGebra
Benefit
In this final lesson of MathBait™ Multiplication Part 3, students focus on the distributive property. They first develop an intuitive understanding of what it is and why it makes sense before testing their skills by distributing the wealth.
The distributive property is powerful throughout mathematics but especially in learning multiplication and in introducing multi-digit multiplication. Just as students can use their problem solving skills to determine 8×7=2×4×7=2×28, they can utilize the distributive property to make more complicated products approachable. If a student is unsure of 6×8 they can write this as 6×(4+4)=6×4+6×4=24+24=48. While it may take more time at first, over time students will also develop increased speed and fluency in using the distributive property and it can be an important skill to mentally compute larger facts.
We also introduced 2-digit by 1-digit multiplication, focusing on tens. This is a great starter to the next part in our series where we will advance to multi-digit multiplication. It is also a sturdy bridge as larger numbers can be intimidating. Here, we are reducing these products to the multiplication we already know and are comfortable with.
Gold Rush City is a great game for students who might find math boring or unattractive. It not only makes multiplying fun, it can also help students to tackle unfamiliar products as they are encouraged to maximizing their wealth. While splitting the gold into easier to compute products is beneficial for multiplying, this may not give them as much money as they would like to build their city.
Conclusion
MathBait™ Multiplication Part 3 is a beast! Designed for students who have completed Parts 1 and 2, or for students in need of additional fluency practice, Part 3 helps encourage students to learn, explore, and play! Throughout MathBait™ Multiplication, each lesson provides novel activities helping students to grow their understanding. Rather than simply memorizing products, they find relationships between values, look for what numbers have in common, and approach multiplication from every angle. This strengthens students' foundations through engaging and meaningful tasks.
What makes MathBait™ Multiplication different from memorization? If you have been following along with the series you might have thought "we are still just practicing the skills, what makes this any different?". In a nutshell, memorization is like steroids. It's a quick fix that helps builds muscle and improve performance. There are also side effects including irreversible damage and stopping steroids causes a withdrawal effect. For math, the memorization steroid is just the same. If students are not constantly practicing, the information isn't connected to anything tangible and is lost, which is why we often forget some of the more tricky products. With MathBait™ Multiplication, we are continuously building in small doses on previous knowledge and providing scaffolding and exposure. Not only does this build muscle and improve performance, it is helping develop relationships and connections that are both stored in long-term memory and more likely to stick because of said connections.
Our games are designed for replay-ability. We recommend students play and play often! With many choices for group and individual games, students can both build fluency and have fun in the process.
We hope you enjoyed Part 3 of our MathBait™ Multiplication series. There is so much to come!
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.
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.
Comments