MathBait™ Multiplication Part 6: Playing with Napier's Bones

MathBait
Apr 7, 2024
3 min read

Updated: Apr 14, 2024

Welcome to MathBait™ Multiplication Part 6, the penultimate unit in our multiplication series. In the previous unit, we introduced Napier's Bones, an exciting way to increase fluency and focus on structure to prepare students for multi-digit multiplication. Now that students have a set of bones and know how to use them, it's time to play! Below, we provide 10 hands-on activities to help you make the most out of Napier's Bones. We do not include any digital activities in this unit as we want students to play with physical bones to help build familiarity and fluency. Don't worry, the activities below are low-entry/high-ceiling, full of fun, and all contain our unique methods to ramp up engagement with students! Plus, we'll be back next week to wrap up MathBait™ Multiplication with some amazing activities and digital games focusing on prime numbers.

Prerequisites

Many of the games and activities below can be played with students of any age, so long as they can recognize digits. We focus here on playing with rods to increase exposure as well as look for and recognize patterns. However, for the biggest bang, we recommend students first complete MathBait™ Multiplication Parts 1-3 and have at least created their own set of bones (for step-by-step instructions see MathBait™ Multiplication Part 5).

Goal

The lessons and activities below allow students to informally play with numbers. Through exposure to Napier's Bones they are becoming more fluent and beginning to pick up on smaller nuances and patterns, as well as the structure of numbers. We recommend spiraling back to games in earlier units as you explore different ways to interact with your multiplication rods.

Each tab below contains a single game or activity that can be played with Napier's Bones. We go from simple explorations to strategy games that can be great for students through high school.

Do You Know My Rod?

This is a great activity for younger students just learning about multiplication.

Distribute a set of bones to each student. Teachers may also allow students to share a set at a table or in a small group and/or project a set of bones to the class.

Explain to students they will be going on a scavenger hunt. Teachers will provide two values. Students should try to find both values on one rod.

Tell students you are thinking of a rod that contains the numbers 4 and 8; can they find it? Allow students time to examine the bones in their set to find one that contains both numbers. Allow students to share what they found. Some students may have collected the 2 bone, others may have collected the 4 bone, other possible bones are 1,3, 6, 7, 8, or 9. Allow students with unique bones to share.

Ask students what bones do not contain the digits 4 and 8 (only 5) and prompt them to consider why this is. (All the multiples of 5 end in a 0 or 5, so there are no other one's place values. As the rods only go up to 9, and 9×5=45, there is a 4 but there is not enough values to also have an 8 in the ten's place.) Encourage students to determine how big their rod must be so that the 5 would have both a 4 and an 8. (The first 8 in the ten's place would be 80. Look for different ways to determine what multiple of 5 this must be. For instance, a student may notice 80=8×10=8×2×5, and thus 16×5 must equal 80. A 5-rod would need to go up to 16 to include both a 4 and an 8.)

Circle back to your rod. Announce the rod you are thinking of contains the number 4 (displayed on a rod as 04) and the number 8 (08) (rather than simply the digits 4 and 8), can students find your rod? Students should now be able to throw out all rods except 1, 2, and 4. Model a systemic logical approach by asking students to explain, in order, the rods we know it cannot be. For instance, we can throw out all rods above 5, as 5×1 is more than 4 (as is 6×1, 7×1, etc.) so none of these rods could possibly contain the value 4. We are left with 1, 2, 3, and 4. We can also throw out 3 because we know skip counting by 3's skips over 4. Checking the final rods 1, 2, and 4 we see that each contain the values 4 and 8.

In the next activity, students will build on this exploration to make combinations that are possible or impossible. These activities are helping students build number sense while also developing more familiarity with the multiplication table/rods.

Possible or Impossible?

This is recommended for younger students just learning multiplication or to build number sense in upper-elementary and middle school students.

Break students into pairs or small groups. Ideally, each student will have their own set of bones, however it is possible to play with one set per group.

Explain the game. Students should place the rods in front of them face down and secretly select a rod, not showing their group members. From their selected rod, students will pick two numbers (multiples of their number that are displayed on the rod) to announce to their group.

Next, each student has a turn to guess another member of their group's rod. They may only make one guess. If working in groups larger than two, a student can pick any other student in their group to guess their rod. Consider if you will or will not allow re-guessing. For example, in a group with students A, B, C, and D, if A selects to guess B's rod and guesses incorrectly, you may set the rules to allow C and D to also guess B's rod (or alternatively guess any other student's rod in their group), or set the rules such that once anyone has guessed, in this case student B, they have earned the point and no one else in their group can guess their rod. Both methods have merit and teachers may select to play without re-guessing as a first round and with re-guessing as a second round. Re-guessing forces students to carefully consider what they know to further narrow down the options.

To tally points, if any member of the group guesses a student's rod, that member earns a point. If no student can guess their rod, the student wins a point.

Begin with an example. Announce your rod contains 8 and 24 and allow students to guess. As 8 is a multiple of 4 and 8, and 24 is also a multiple of 4 and 8, the rod you have selected could be 4 or 8.

Allow students 10-15 minutes to play independently. Upon conclusion, announce the winners and go into a whole group discussion. Ask students for any strategies they found. Highlight that picking an "impossible" pair is the best strategy as there is no way to know for sure what rod you have picked. Here are some additional strategies to highlight:

Look for numbers that only appear once in the bones (or on a multiplication table). These numbers tell us right away the rod a student has picked.
Avoid perfect squares. While some perfect squares such as 4 and 36 appear on multiple rods, the second number can often give it away.
To create "impossible" combinations, try to use a number and its multiple (like 8 and 24).

Call out a few pairs and have students vote on if it is "possible" to know the rod or "impossible" as the pair of values appear on more than one rod. For example, 5 and 8 is possible as both numbers appear only on the 1 rod.

Patterns

This activity is recommended for younger students just learning multiplication or for older students to build number sense.

The goal of this activity is for students to examine their rods to find patterns. You may structure this activity more by directing students to examine certain areas, such as the diagonals or the one's place. It is fun to allow students to at least play detective on their own for 5-10 minutes before providing some guidance.

Here are some fun patterns to explore and discuss:

What do you notice about the one's place on all the even rods? (They all cycle between the even digits 2, 4, 6, 8, 0).
Can you find a rod that contains every digit in the one's place? (1, 3, 7, and 9 which are our odds, with 5 removed as multiples of 5 always end with a 0 or 5).
What is the pattern of the one's place on the 3, 7, and 9 rods? Will this continue if we extend the rods? (For 3's we have 3, 6, 9, 2, 5, 8, 1, 4, 7, 0 and for 7's we have 7, 4, 1, 8, 5, 2, 9, 6, 3. The 9's are fun to look at as they follow the pattern 9, 8, 7, 6, 5, 4, 3, 2, 1 which can lead to a great discussion on how adding 9 is the same as adding 10 then subtracting 1. This means the multiples of 9 will always have a ten's place that is 1 more and a one's place that is 1 less).
Examine the diagonals. What do you notice? (The diagonals also form fun patterns; the first diagonal is 0 followed by 010, 02020, 0304030, 040606040, 05181918150, and so on. The symmetry is the same symmetry we know from the table as a result of the commutative property.)
Add the digits of each multiple. What do you notice? (In this case, multiples of 3 and 9 always sum to their multiple. We do not recommend going deeply into this unless your students are mathematically advanced enough to understand why this happens. If so, you may wish to explore our video on Instagram or Facebook explaining this useful result).

Teachers may determine how deep to explore each pattern. We want students to see that numbers have a clear structure. The better we understand this structure, the easier it will be to multiply (as well as other more advanced topics). We also want to encourage making sense of patterns and identifying patterns we notice.

The Maze

This activity is great for students learning to multiply or practicing fluency.

To set up this activity, teachers will need to build a maze. Using our Digital Bones (see MathBait™ Multiplication Part 5), this can be done quickly and easily. Take a set of 8-10 bones and place them in any order you would like. Next, highlight a path from a value on the first bone to a value on the last bone. Finally, write down the series of digits you have crossed. For younger students, include spaces to help with processing. For older students, smash the numbers together, making it more difficult to identify where one begins and ends.

You may wish to provide students with a grid or blank bones. You can find a printable version in MathBait™ Multiplication Part 5.

To play, students follow the path, placing the numbers in the code as they go. To escape the maze, they must provide the exit code which is the order of the bones in the maze.

Bones 312845976 with a highlighted path through them — Show students a blank set of rods and a path. Provide the multiples crossed through: 18612104020243063495648

The basis of this activity is simply to recognize multiples. If we know the maze begins in the 6th row and the value is 18, we need only to determine what number times 6 equals 18 to quickly find the first bone must be a 3. This makes this activity great for students building fluency. Students may also consider common multiples.

Make the game less challenging by adding more movement, such as going up a few values on a rod. This will provide students with more multiples to work with and recognize.

To add more challenge to the game, show the shape of the path off the bone chart. This requires more problem solving to determine how these values could fit together. For a super challenge, only provide students with a few of the numbers without telling them where in the path they fall. This allows students to consider their multiples to determine, for instance, if 18 is on the path (somewhere) it must be in the 2nd, 3rd, 6th, or 9th rows. This can be an excellent introduction to factoring. To provide students with additional support, allow them to use their completed rods as they problem solve.

The Bone Collector

This activity is great for students new to multiplication or students practicing fluency.

Similar to Go Fish, place students in small groups and combine a half-set of bones (one for each digit) for each student. Place the bones upside down in the middle of the table and allow each student to grab 9 (or 10 if using the 0 bone).

The goal of the game is to gather a full set of bones (1-9 or 0-9). Players take turns selecting another player and asking about their hand. They cannot ask about the digit on the top of the bone (this also eliminates the first row).

On a player's turn, they first select a discard rod from their hand and lay it face down on the table. Next, they pick a player to converse with and ask the player a single yes/no question about their hand. The player must answer truthfully.

Suppose the game is played with four players, A, B, C, and D. Player A is in need of an 8-rod to complete their hand. On their turn, Player A lays out their discard rod and selects a player to converse with. They may select any player (B, C, or D). Player A asks Player C, "do you have any 16's?". If Player C has any bone with a 16, they must answer yes and give Player A their bone and pick up Player A's discard bone. If Player C doesn't possess any rod containing a 16, they answer no. Player A returns their discard bone to their hand and play passes to the next player.

Note that if Player C has more than one bone containing a 16, they may choose which bone to provide to Player A. This means, although Player A was in search of an 8-rod, by asking for a 16, they could receive a 4-rod instead.

The first player to have a full hand wins.

The Bone Collector requires students to look for and be familiar with multiples. If asking for a 16, students must either know, or find, that 16 is a multiple of 2, 4, and 8. The more students play, the more strategies they will develop. For instance, selecting a larger multiple can often narrow things down. In addition, students must work to become fluent as they do not have the bone they are looking for. If a player is in search of a 7-rod, they must recall the multiples of 7 to know what to ask for.

Secret Codes

This activity is great for students just learning multiplication or working on fluency practice. It can be a quick warm up or a long-form game over multiple rounds.

In this game, students make observations about numbers and multiples to help determine the secret code.

Boy holding a paper with Napier's Bones partially filled out.

Grab a set of empty rods and display them for the class to see. You can find a printable version in MathBait™ Multiplication Part 5.

Next, select a leader (this can be a teacher or student). Using a completed half-set of bones (one bone for each digit), in secret the leader will rearrange the bones in any order to create their Secret Code.

For instance, a code may be 5093627418. One of every bone is used exactly one time. For older or advanced students, the game can be played at a higher level of difficulty by allowing a full set of 20 bones (2 of each digit). This allows some digits to be used more than once and other digits not at all, such as the code 5613486325.

Each round, the leader will select a value to enter into the bones. They should display their value and its placement for all to see by adding to the empty set. If pressed for time, the leader may write out all ten clues at once.

For a super quick game, have the leader place one value on each bone, including tens and ones. Students use reasoning to determine the code. For instance, a 15 in the third row tells us 3× this number =15. Students will determine the value must be 5.

If you have more time, or more advanced learners, have the leader place only a single value (a ten or a one) in each column.

The remaining players have the chance to guess the Secret Code.

This game is great for both the leader and the players. In the image above, note the 7 in the one's place of the fourth rod. The only value on our table such that 9×? contains a 7 in the one's place is 9×3, so the fourth rod must be a 3. The leader must think critically about what clues they will give.

The criteria to win the game depends on your variation. It can simply be a fun activity without points to get the class started. If making Secret Code a regular activity, consider giving the leader a point if they stump their classmates. Have a special place in the classroom to display the current high score. Another alternative is simply the first person to guess the secret code wins.

Secret Codes not only helps students to become more familiar with multiples, it also encourages students to look for patterns within the bones or the multiplication table. Students develop reasoning and problem-solving skills as they play. In the board above, the 6 on the final bone tells students we are looking for a multiple of 2 with a 6 in the one's place. This could be 6 or 16. Both the leader and students must consider how to pick and interpret clues in order win.

Fast Adding

This activity is great for students developing fluency and ready for 2-digit multiplication. Fast adding is a great confidence builder. Consider using this activity as an introduction to multiplying large values with Napier's Bones.

Fast Adding is a simple activity that helps students practice their addition skills and fluency while building confidence in multiplication. To set up this activity, teachers should pick ~10 multiplication problems that do not involve carrying. As students advance, any value may be selected to also encourage students to consider place value.

We recommend starting with 12's. Some multiplication tables include 12's and it can be helpful for students to recognize multiples of 12 for tasks such as conversion between feet and inches.

Begin by asking students what the numbers on a bone represents (multiples). Next, ask them what happens when we put more than one bone together? When using multiple bones we create a multi-digit number. For instance, grabbing the 1 and 2 bone allows us to make the number 12.

Ask students to "read" the 12 bone. If needed, display how the bones work. We must sum the joints created, thus the first row of the 12 bone reads 0/10/2 we read this as 0 1+0 2 or 012 which is of course the number 12.

Allow students time to practice reading the bones. Create different combinations. When ready, begin Fast Adding.

Explain to students you will provide a multiplication fact. Students will use the bones to create the first value and then shout out the product. For example, we could ask students to find 87×5. Students must collect the 8 and 7 bone, navigate to the 5th row to find 4/03/5, and do the very simple sum to determine 87×5=435.

Continue to play a few rounds. While this seems like a very basic game you will be thrilled with the engagement. Because it is easy, students don't feel the anxiety and pressure that often comes with a new and complex topic. Students will go home and share with their families how they can multiply big numbers with ease. They will feel confident, so that when we try to find multi-digit products without the bones, they are prepared for the challenge.

stacking 40 and 35 to compute 87 times 5 mentally

Ask students if they might be able to determine 87×5 mentally without the bones. This should ideally take place after students have experience and understand how the bones work. They may come up with some great ideas. For instance, to quickly transition to multiplying 2-digit by 1-digit numbers without the bones, find each product and stack the result. If asking a student what is 92×4, their experience with the bones can help them to quickly calculate 2×4=8 and 9×4=36 to put together 368. Extended play with the bones will greatly increase student multiplication skills.

Guess Who

This game is ideal for students working on building fluency with multiplication.

Arrange students in groups of two and provide them with a half-set of bones (one bone for each digit) placed face down between them.

Each player selects a rod in secret, not showing it to their teammate.

Players take turns asking their partner questions. For instance, "Is 36 a multiple of your number?". For an added challenge, only allow students to ask questions about one digit at a time. Questions can be, "how many zeros are on your rod?", or "do you have any 5's in the one's place?"

Students playing one has a 3-rod, the other asks if 36 is a multiple of her number

The first student to guess the other player's rod wins.

Note: Be careful with terminology and setting the rules. For instance, in the image above 36 is a multiple of 3 although it is not shown on a standard bone. Students may ask "Is 36 a multiple on your bone?" or, for added challenge, allow students to guess questions that may not appear on the bone itself.

This game elicits thinking about common multiples as students consider what questions will have the most impact. Students are building familiarity with their "cards" as they must examine its values. Guess Who can also encourage vocabulary as teachers may require students to use words like "multiple", "factor", or even "divisible by" depending on student level.

Who Did It?

This is a great game both for elementary and middle school students. This game can be quite challenging. For younger students we suggest allowing them to work as a team (class against teacher), older students may work alone or in pairs. Who Did It? can be a great choice for a game day or a substitute. It may take up an entire class period but is well worth it! Students will be engaged, thinking, and building problem-solving and multiplication/factor skills.

Determine the number of bones based on the level of your students. The more bones, the more challenging the game. The activity leader will select and arrange their bones in secret. This number is the culprit. For added fun, provide a backstory such as this number robbed a bank or this number took a pencil from the teacher's lounge, etc. The activity leader should also pick a "crime", the crime is the row we will give clues for.

Each round, students will use clues to create a list of suspects, ultimately narrowing it down until someone determines who did "it". The clues will always be the same:

One clue for each middle joint forming a parallelogram, the clue is the sum of the joint without carrying. For instance, if the joint sums to 15, the clue is 15. Carrying from the previous joint is also not permitted.
One clue gives the sum of the first and last (triangular joint).

Note, clues can be given in any order. That is, the culprit may decide when to give players the information about the triangular joints.

Before play begins, announce the number of digits. The number of rounds (and clues) should be the number of digits. For instance, if the culprit is 312, we would allow for 3 rounds.

The activity leader will announce a clue each round. Clues are given as a joint appearing on the number.

For example, one clue may be "The third joint (hundred's place) of the culprit times 4 is 2." Using the bones and their understanding, students begin to build a suspect list.

A place value of 2 leaves us only with the options 2 and 0, 0 and 2, or 1 and 1. Since the clue told us we are multiplying by 4, this means 4 times the first rod must end in a 0, 1, or 2. Examining the 4th row of our rods leaves us with 3, 5, or 8. We can eliminate the 0 rod, as our first digit cannot be 0.

Narrowing down their suspect list, if their first rod is the 3, the second digit must have a 0 in the ten's place of the 4 column. Leaving us with 0, 1, or 2. If their first rod is the 5, the second digit must have a 2 in the ten's place, giving them 6 or 7 (as we are playing with a half-set of bones, the 5 cannot be repeated). Finally, if the first digit is 8, the second bone must have a 0 in the ten's place, again leading to 0, 1, or 2.

Encourage students to share their reasoning with their team. In the case above, as the multiples of 4 that end in a 0 are 12, 20, and 32, they can conclude that for 12 and 32 (ending in a 2) must be paired with a 1-digit multiple of 4, while 20 must be paired with a value whose multiple of 4 is in the twenties to find 20, 24, 28 or 5, 6, and 7 rods.

Provide students time to create their suspect list. After the first clue, they should have narrowed things down to:

Once the round is completed, the culprit will leave another clue. As we are working with a 3-digit number, the best clue left is "the second joint (ten's place) of the culprit times 4 sums to 4."

Time to play with the bones. Our second digit must be 0, 1, 2, 5, 6, or 7. We should consider the multiples of each of these values and 4 to find 0, 4, 8, 20, 24, and 28. Notice both 8 and 28 can be thrown out as they are too big (larger than 4). This means our second digit cannot be a 2 or a 7, and we eliminate these from our suspect list.

Continuing to narrow down their list, 30 would require a final bar with a 4 in the ten's place. The only such bar would be a 10-bar (or higher), which we do not have and can eliminate this option. 31 would require a 1-digit multiple of 4 for the final bar, narrowing it down to 0 or 2. 56 would also require a 1-digit multiple of 4 leaving 0, 1, or 2. 80 would require a 4 in the ten's place and can thus be eliminated. Finally, 81 would require a 1-digit multiple of 4 allowing for 0 or 2.

It's time for the culprit to provide the final clue. In this game, the only clue left is the sum of the triangular numbers, which the culprit tells us is 9. There are many problem-solving methods, as there are for the previous clues. We can consider the possible ways to make 9 (1+8, 2+7, ...), we could also consider the 3-digit multiples of 4. From our suspect list, a culprit in the 3-hundreds will have a thousand's place of 1, as 4×3=12, a culprit in the 5-hundreds will have a thousand's place of 2, as 5×4=20, and a culprit in the 8-hundreds will have a thousand's place of 3, as 8×4. That leads us to only a few options. If 310 or 312, the last digit must be an 8 (as 1+8=9). This is not the case with 310, so we can mark them from our list. If our culprit is in the 5-hundreds, the one's place must be a 7 (as 2+7=9). However, no multiple of 4 can end in a 7, which eliminates all the 500 values from our list. Finally, if our culprit is 810 or 812, the one's place must have a 6 (as 3+9=6). As 0×4=0 and 2×4=8, neither value will work. We have found the culprit! It must be 312.

There are many problem-solving approaches throughout this game. Not only are students working with the bones, considering possible values and multiples, they are also using logic skills. Students greatly enjoy a "whodunnit" so this is a fun game to get them thinking, playing with multiples and engaging in mathematics.

Spoons

This game is ideal for older students working on multi-digit multiplication and/or factoring.

Have you ever played Spoons? In this fast-paced card game, players attempt to gain a specific hand of cards through passing and discarding. Once a player has their hand, they grab a spoon. When one spoon is grabbed, any other player can also take a utensil. Here's the catch, there is always one less spoon than players. Thus, each round, one player is "out" and play continues until there is a single winner.

In this activity, we replicate the fun of Spoons with some fast-paced math knowledge. Group students into teams of 3-4 and place a full set of 20 rods on the table upside down. Have each student draw 3 rods. When the game begins, teachers will call out "discard" every 30 seconds (you may adjust the timing based on student level). On this call, students will discard one of their rods to the center pile as well as draw another.

Before beginning, teachers select a target 3-digit number. At the start of the game, announce the target number to all groups. Students will have the first 30 seconds to strategize before the teacher calls out "discard" and game-play begins. Students must discard one of the bones in their hand and draw a new bone.

To replicate spoons, consider adding the utensil element by allowing a player to grab a spoon when they find the target number amongst their bones. Like in the card game, players must be mindful of their surrounds as once one spoon has been grabbed, all players are eligible to try to snag their place in the next round. The player left without a spoon is out and play continues until only one player is left standing. Alternatively, once a player has the target number, they gain a point and the game continues with a new number as time allows.

In order to win, students must collect rods in their hand that contains the target number. For instance, if the target number is 120, students may notice the 0 rod along with any rod that contains a 12 will fit this requirement. As 12 is a multiple of 2, 3, 4, and 6, students could collect a 20 (as 20×6=120), a 30 (as 30×4=120), a 40 (as 40×3=120), or a 60 (as 60×2=120) to win, as any of these rods will contain the value 120. Note there are other rods they may collect too! For example, 24×5=120, thus a student could collect a 2 and a 4 rod as well.

This game has many benefits. At its core, it encourages students to consider larger multiples, or how to "build" a number. It also preps students for working with larger values and multiplying two-digit numbers. As they play, students are carefully examining the rods, gaining exposure and familiarity.

There are many strategies to find and discuss. Another to consider is the sum of the middle joint. Students may notice that to achieve a value of 2, the only options are 0+2, 1+1, and 2+0.

The depth of strategy here makes this game great for beginners as well as very advanced students. Spoons is especially excellent for students learning to factor. After each round, create a factor tree to show all the possibilities. The only option we have not yet mentioned for 120 is 8×15. In MathBait™ Multiplication Part 7, we will focus on factoring as another strategy to help students build strong multiplication skills.

Conclusion

In MathBait™ Multiplication Part 6, we have provided 10 fun games and activities for use with Napier's Bones. Using the rods, students are building familiarity and exposure to multiples and products. As they play detective, hunt for values, look for patterns, or try to create target numbers, students are focusing on the structure and applying reasoning and problem solving. Each of these activities is easy to play, super engaging for students, and will build both confidence and understanding. We hope you have a blast with these activities and share your experiences with us.

Next week sees MathBait™ Multiplication come to a close. With all new lessons, activities, and digital games, we will target factoring and prime numbers. Factoring is a key skill as it helps students to understand how to take apart and rearrange products, leading to improved fluency. It is also vital for fractions, GCF, LCM, and into algebra for working with polynomials and radicals. We will show you how teaching your students to factor early will not only help strengthen their multiplication, but also help students to see numbers as able to be manipulated, leading to enhanced problem solving skills and outcomes. And of course, we'll do it all through fun and engaging activities like only MathBait™ can!

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