Solve It! Friday- Task #9

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek9: Gomez is 8 years old; Pugsley is 10 years old; and Frankenstein is 16 years old. 

Thursday Tool School: Computational Fluency Math Tools

Last week, I talked about using Base 10 blocks to develop a conceptual understanding of multiplication of larger numbers. Today, I would like to talk about how to use Base 10 blocks to do the same with division. 

When students first begin to multiply and divide with larger numbers, we often jump to using the algorithm too quickly. However, students really need time to develop these skills so that they have a solid foundation for the algorithms later. Today's post will offer some ways a Base 10 model can be used to connect to the algorithm. 

In the model below, the illustration shows a step-by-step model of how an understanding of the division process is developed through using Base 10 blocks. 

• Notice the regrouping of the hundreds in the first step. Instead of allowing students to think that "7 doesn't go into 3," the model helps them understand that because we cannot divide 3 hundreds into 7 groups, we must regroup the hundreds into tens, add them to the two original tens for a total of 32 tens, and then separate them into 7 groups. 
• From here, students see that the remaining 4 tens are then regrouped into units and added to the four original units for a total of 44 units. 
• Finally, students are able to see that once the 44 units are divided into 7 equal groups, there are two leftover units. 
• When the value of each group is counted, we have 4 tens and 6 units, or 46. There are two remaining units that will not be separated into the 7 groups.   

Sound Off! The traditional algorithm is shown below. Compare this process to the one above. What connections do you notice? What language is necessary to better support our students' understanding of the division process. 

What I'm Reading Wednesday: Making Number Talks Matter- Chapters 1 & 2

I recently began participating in a nationwide book study for the book, Making Number Talks Matter, by Cathy Humphreys and Ruth Parker. Specifically, this book is geared toward grades 4 - 10. 

Since I am reading the book and learning more about how to use number talks and emphasize strategic thinking in the classroom, I decided to pass that knowledge along to you. Each Wednesday, I plan to share some observations, reflections, and next steps with you. 

Throughout my career as both an elementary and middle school math teacher and now as math coach, I see the struggles that our students have with basic computations. To see a student write down and solve 17 - 9 or 12 x 11 is disheartening. After many years of this, I started to wonder why our students struggle with basic computational thinking and why they do not have the flexibility necessary to manipulate these facts. Making Number Talks Matter addresses these questions and gives a framework for using successful number talks in the classroom. 

What are number talks? 

Number talks are daily routines that require students to demonstrate flexibility in working with numbers and solving basic problems without using paper and pencil to find the solution. 

Where's the value in doing number talks?

Besides building more confident math students, number talks require students to be flexible in their thinking about numbers and operations. In addition, students increase their ability to articulate their thinking and refine their mathematical communication skills through the use of number talks. 

The chart below illustrates the basic flow of a number talk. 

The authors suggest a few other key ideas for successful number talks: 

1. Utilize wait time. 
2. Ask "why." Encourage students to use clear language to explain their thinking. 
3. Encourage creativity-- try to highlight a variety of strategies and probe students to think of alternative methods when few have been offered.
4. Listen to students' responses asking for clarity when needed but careful not to reveal personal thoughts or opinions. 
5. Use number talks regularly. 
6. Encourage students to use content-specific vocabulary.
7. Record what students say. Exactly. Careful not to interpret the meaning of their words.
8. Encourage students to communicate with each other when questions arise or clarity is needed. 
9. Encourage multiple answers to enhance the learning opportunities.
10. Encourage the use of clear communication skills. 
11. Have an alternative problem just in case the chosen one goes awry. 
12. Use caution when deciding to interject your thoughts.
13. Encourage students to use non-standard methods. 

Note: In order to honor the authors' work, I will only share my own personal experiences, thoughts, and reflections as related to the book's content. If something really strikes me, I will share a quote from time to time with the appropriate citations. 
Reference: Humphreys, C and Parker, R. (2016). Making Number Talks Matter. Portland, Maine: Stenhouse Publishers

Transformation Tuesday: Cooperative Learning- Flexible Grouping Strategies

One of the most important aspects of cooperative learning is how students are grouped together. Depending on the activity, it may be necessary to divide into groups of 2, 3, 4, or more. Using a flexible tool that will create a variety of groups instantaneously is essential. With this in mind, today's strategy is how to create instant flexible groups. I have several effective strategies that I want to share with you today. 

The first strategy is a basic grouping technique that I have seen in many classrooms and will provide three groups: same-number groups, letter groups, and different-number groups. It is extremely useful when you use table groups (with students numbered 1-4) and partner groups (with A and B) a lot. This strategy offers an additional grouping of students with the same number. See the example below. 

Free Resource Alert! Click here to download a free copy of this resource. 

To use this resource, copy enough mats for each group of four, laminate the cards, cut-out each individual square and tape them to the corner of each desk. Note: You can also just copy them, cut them out, and tape them down from the back and spread clear packing tape across the front. This will laminate them instantly!

The second strategy offers the opportunity to create more of a variety of groups. I created it at a time when I needed to have quick access to multiple grouping opportunities so that my students did not get bored with their same group. I originally created them with 3.5 inch x 5 inch index cards and stickers; however, the example below was created with clipart from my computer and can easily be manipulated to achieve the number and type of groups that you want. 

As you can see, there are six grouping symbols here. First, there is a group number, then, moving clockwise, there is a colored star, an animal, a pencil, a shape, and a letter. Depending on your grouping needs, the number, star, animal, pencil, and shape can be used to create groups of different sizes. For example, if you want groups of three to be the star, then divide the number of students you have by three and use that number of different stars to create the card. The letter can be used for partners when the letters A and B are used. 

Similar to the first strategy, the card can be attached to individual student desks or each student can receive their own card. Passing the cards out randomly each time grouping is necessary adds addition combinations of student groups because students will most likely receive a card that is different than the one received the previous time around. 

The last strategy is a hybrid of the two strategies shared above. It offers the most flexible grouping techniques: table, number, shape, shape color, facing students, adjacent students, letter, and number/ letter combination. See the example below. 

This cooperative learning mat offers eight flexible grouping strategies. The group sizes range from 2 to 16 when used with a class of 32 students. It includes same-number student groups, different-number students groups, and the letter student groups like the first strategy; however, this grouping tool also offers a number/ letter combination student group (the A and B are in a different location on every other mat). Additionally, shape and shape color (8 colors included with the set) student groups can be created as well. Partner groups can be created with "face" or "shoulder" student groups.

To Use this Mat: Consider the students' everyday seating arrangement to be home base. Place a mat in the center of each table. Each student then uses the symbols on the portion of the table mat nearest to them to determine his/ her group for the next activity. From here, after completing the activity, students can return to home base or use the table mat at their current location for a new group if needed. 

Grab a copy of this resource at The Routty Math Teacher Store! 

Click here to view the resource. 

One additional tidbit: I love to use the table mats/ desk tags to assign jobs and task roles as well. For example, I may ask the students at desk number 3 to be the materials manager, while I may ask students at desk number 1 to be the group leader. There are endless possibilities!

Solve It! Friday- Task #8

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek8: Left Side- Lizard Tails; Middle Left- Frog Juice; Middle Right- Rabbit Feet; Right Side- Snake Eyes

Like this Halloween-themed problem? 

Click here or the picture below to check out 

Thursday Tool School: Computational Fluency Math Tools

Last week, I talked about using Base 10 blocks to reinforce addition and subtraction concepts. Today, I would like to talk about how to use Base 10 blocks to reinforce multiplication skills. 

When students first begin to multiply and divide with larger numbers, we often jump to using the algorithm too quickly. However, students really need time to develop these skills so that they have a solid foundation for the algorithms later. Today's post will offer some ways a Base 10 model can be used to connect to the algorithm. 

Like I mentioned last week, using Base 10 blocks to model multiplication is a great way to "see" a model and connect it to the algorithm at the same time. In the picture below, I model how Base 10 blocks can be used side-by-side with a multiplication area model to illustrate the connection between an area model for multiplication and its often confusing and hard-to-understand algorithm. 

Specifically, the model shows how using Base 10 blocks to model two-digit by two-digit multiplication relates to the area model and the traditional algorithm. 

Sound Off! What models do you use to teach multiplication? 

Transformation Tuesday: Cooperative Learning Strategies- Teach 'n' Tip

One of the most powerful learning tools we have in the classroom is our students. As many of us have witnessed, students seem to learn better from each other than they do from us. Our students really do speak the same language! It's amazing to watch a pair of students really supporting each other and learning together through supportive collaboration. Today's strategy is a general one that can be used during any part of the lesson cycle. It's called "Teach 'n' Tip" and encourages students to support their classmates during partner work with a series of questions or tips to help advance their thinking about a skill or concept. 

An important caveat, however, is that most students do not know how to do this. They think that they are helping by giving the other student the answer. In order to help students develop the supportive collaboration skill, use purposeful questioning to support student thinking in classroom discussion and in your individual interactions with students. As students begin to experience this model, they will start responding to each other in the same way. 

Teaching Tip: You may want to create an anchor chart with sentence starters or question stems that students can refer to during their partner work. In addition, you may also want to probe students to determine what questions they could ask their partner if he/she were stuck on a specific part. Either way, brainstorming responses in advance will help all students be good supportive collaborators.

This strategy is a great way to get students talking and learning cooperatively; however, you will want to be intentional about the way you group the students. Pairing students with a large difference in ability levels, i.e. students who do not have similar achievement levels, or different learning styles is usually not effective and can be a burden for the more able student. 

The example below is a very basic conversation that occurred while two students were playing a fraction comparison game. Imagine the possibilities if all of our students interacted and collaborated this way. Wow!

Sound Off! What are your thoughts about today's strategy? 

Solve It! Friday- Task #7

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek7: Pumpkin = 3; Hat = 2; Ghost = 2; Cat = 5; Bat = 1

Thursday Tool School: Math Tools for Computational Fluency

Today's tools are Base 10 blocks. I can remember when I first started teaching, the tool we had the most of was Base 10 blocks. We had an abundance of them! Each teacher had a large classroom set or a tub of random ones; however, I did not know what to do with them as an upper elementary school teacher. Over the years though, I have discovered some essential uses that I will share with you over the next few weeks.

Both as a teacher and as a math coach, I see students struggling with understanding basic operations. Specifically, I see students struggle to demonstrate true understanding of subtraction with zeros. Typically, I see students slashing zeros and adding numbers above them with no meaning whatsoever-- a meaningless process because it is not tied to a conceptual understanding. I feel like that is one of the best uses for Base 10 blocks, to truly illustrate this concept in a more meaningful and concrete way.

Modeling addition and subtraction with Base 10 blocks is a great way to get students to "see" a model and connect it to the algorithm at the same time. In the picture below, I have modeled how Base 10 blocks can be used side-by-side with the subtraction algorithm to illustrate the connection between real-life "borrowing" (regrouping) and the slashing and changing quantities students do.

When connected to a model, students are more easily able to see why the zero in the tens place eventually becomes nine tens and the zero in the ones place becomes ten ones. This is often hard for students to see as they do not understand that the zero in the tens place first became ten tens after regrouping the hundred into tens and that the second zero became ten after regrouping one of the tens into ten ones.

Below, I have included some games to practice using a model and accompanying algorithm to understand regrouping with addition and subtraction.

• Race to a Flat- Students roll two dice (6- or 10-sided), add the numbers on the dice together, take that many units, or rods and units, to add to their total, and regroup their quantity until they have achieved a flat (100). (Variation-- Play "Race to a Cube (1,000)" following the same directions but using one die as the tens place and the other die as the ones place. (Two different colored die work great here.)
• Race to Zero- Beginning with a flat, students roll two dice (6- or 10-sided), add the numbers on the dice together, subtract that many units from their collection, and  regroup as needed until they reach zero. (Variation-- Play "Race to Zero" from a cube (1,000) following the same directions but using one die as the tens place and the other die as the ones place. (Two different colored die work great here.)

Sound Off! How do you use Base 10 blocks in the classroom?

Transformation Tuesday: Cooperative Learning- Question Me

Today's cooperative learning activity is a general strategy that can be used to get all students involved in content-specific conversations. Often times, complete participation is a limiting factor of cooperative learning experiences. The "Question Me" strategy gives each student a role and can be used with any concept or skill. 

Here's how it works: 

1. Create teams of four. Assign each team member a number (1 – 4).

2. Give each team a set of questions.

3. Direct students to complete the assigned tasks below:

a. Student #1 shuffles the cards and organizes them on the table. Student #1 then asks Student #2 to choose a card.

b. Student #2 chooses a card and reads the question aloud.

c. Student #3 answers the question with a few moments of think time if needed.

d. Student #4 agrees or disagrees with Student #3’s answer and provides support for their opinion. (Teams should resolve any differences in opinion at this time.)

e. Student #1 praises Student #3 and #4 for their responses with one positive comment.

4. Students rotate roles.

a. Student #1 becomes Student #2.b. Student #2 becomes Student #3.c. Student #3 becomes Student #4.d. Student #4 becomes Student #1.

5. Students repeat steps 3 and 4 until all questions have been answered.

See the example below for a more detailed explanation. 

The question modeled below is part of a lesson on comparing fractions using the missing piece strategy. The question is "Which fraction's missing piece is half the size of a fourth? Explain your answer." Students then use a graphic of several fractions that are each one-piece away from being one whole to support their responses. (CCSSM 4.NF.A.2)

Sound Off! How might you use this strategy in your classroom? 

Solve It! Friday- Task #6

Here's how Solve It! Friday works:
1. Each Friday morning (at 12:00 AM Central Time), I will post one problem-solving task. Note: In some cases, I may post more than one version of the task to reach a wider variety of grades. 
2. Before the next Friday, use the task with your students. 
3. Have students solve the problems individually or with a group. 
4. Individual students or student groups create posters using numbers, pictures, and words to illustrate the solutions. Note: The blank backs of old book covers make great poster paper! 
5. Either via a math talk session or a gallery walk, be sure to have students share their responses with other students. 

I would love to see your students' responses and showcase them on social media. Please post your students' responses to Twitter using the hashtag #RMTSolveIt(week number). For privacy, please be sure that students' names and other identifying information is located on the back of the poster. Be sure to check out other classes' solutions using the same hashtag to filter the Twitter results. 

I look forward to seeing your students' work! Thanks for sharing! 

Solution: The real beauty of this task is in the process. Please emphasize that with your students. It may take some time to solve this problem. Validate their efforts and ask questions to move them in a different direction if needed. For your convenience, I have provided the solution below:

#RMTSolveItWeek6: The Redden family will see 12 spiders on the 11th day. 

Like this Halloween-themed problem? 

Click here or the picture below to check out 

Thursday Tool School: Math Tools for Computational Fluency

This month will be a little different than last month. Instead of featuring a specific math tool, I will feature math tools that can be used to practice computational fluency skills. Today's math tools are dominoes. Dominoes are one of those versatile tools that have many purposes. I realize that dominoes are not a standard math tool, but they are great to have laying around. If you don't have any in your classroom, you can pick some up at the dollar store (usually one dollar per set). 

The activity I am sharing with you today is called "Boneyard Numbers." What I love about this activity is how it can be adapted to meet the needs of all students. Whether a student just adds everything together or goes out on a limb and tries to use all of the operations, the activity can meet his or her needs. For this reason, "Boneyard Numbers" makes a great station and fast finisher activity. 

Here's how it works: 

1. With a partner, spread-out one set of face-down dominoes. This is called the boneyard.

2. The tallest player goes first.

3. In turn, each player grabs two dominoes from the boneyard. 

4. Using the four numbers indicated on the dominoes, create a number sentence using addition, subtraction, multiplication, and/ or division with a final result of any number from 0 to 12.

5. Once a target number is reached, cover the number on the game board with your marker. 

6. Continue playing until someone covers four numbers in a row, column, or diagonal. 

Note: Some students will ask if they can use exponents. This is completely up to you. I sometimes tell students that they can only square numbers. Squaring a number does not affect the four digits on the dominoes and can be used even if the student's set of dominoes does not include a two. 

Here's an example: 

Since my numbers are 6, 5, 5, and 3, I can write: 

* 6 x 5 ÷ 3 - 5 = 5 OR

* 6 x 3 x (5 - 5) = 0 OR

* 6 + 5 + 3 - 5 = 9 OR

* 3^2 (3 squared) + 6 - 5 - 5 = 5

There are many other ways to use the numbers. The number sentences above are just a few combinations. 

Note: Students who have experience following the order of operations should do so here as well. Younger students may not be aware of the order of operations and will usually just write the numbers in the order in which they plan to complete the operations. This is okay as long as they can describe the order correctly. 

Free Resource Alert! Click here for a free download of the "Boneyard Numbers" game. 

Transformation Tuesday: Cooperative Learning- Fact or Fib

This month, I will be focus on engaging instructional strategies that incorporate cooperative learning structures. For this first full week of October, I want to feature a strategy I just recently learned about while creating an activity for some teachers with whom I work. The activity is called "Fact or Fib" and is an adapted Lead4ward instructional strategy. 

What I love about "Fact or Fib" is how the structure of the activity focuses on students' understanding and justification of a standard or skill. Students must have an example, or even better, a counterexample, to illustrate their answers. 

Here's how it works: 

1. Group students in group of 3 or 4.

2. Give each student the "Fact or Fib" desk tent or have students write "fact" and "fib" on individual index cards. 

3. Give the students a statement based on the current standard or skill. For example, "When you multiply two numbers together, the product is greater than the two factors that were multiplied together*." 

4. Give each student think time to determine whether the statement is a fact or a fib. Then have students create examples or counterexamples to justify their response on a whiteboard. 

5. Once all students have written justifications, each student displays his/her desk tent to show whether they believe the statement to be a fact or a fib. 

6. In turn, each student states whether the statement is a fact or a fib and gives his/her justification.

7. After all students have shared, the group should come to a consensus and have several examples or counterexamples to support the group's decision. 

8. Each group shares their consensus with the class. 

9. After all groups have shared, verify and/ or clarify student responses addressing misconceptions where needed. 

10. Repeat the process with another statement. 

Click here to download a copy of the desk tent.

The picture below is an example of how "Fact or Fib" can be used as an assessment tool. It was a bonus to the September "Transformation Tuesday" blog series included in the October edition of "Teaching Tidbits," my monthly newsletter. Click the banner at the bottom of this post to get your copy of "Teaching Tidbits."

Sound Off! How will you adapt this activity in your classroom? Write about it in the comments below!

* CCSSM- 5.NF.B.5.B

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