Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model. For each task today, students shared their strategies with peers (sometimes within their group, sometimes with someone across the room). It was great to see students inspiring others to try new methods and it was equally as great to see students examining each other work for possible mistakes!
Task 1: 15/5
While we have completed several division number talks in the past, it has been a while! Still in the multiplication mode, many students created 15 x 5 arrays.
Before discussing the above task, I modeled a simpler array on the board, 6/2. I asked, What is 6 divided by 2? Students responded, "3." I then drew three arrays: 2x6, 6x2, and 2x3 and asked students: Which array looks right? After discussing this question, I then drew another 2 x 3 and explained: If I draw 6 squares on the inside of this array, does it make sense that 6 squares divided into 2 rows equals 3 in each row? Soon, I heard students say, "Oh, now I get it!" and "That makes sense now."
At this point, students went back to the original task, 15/5 without any prompting! Many students drew a 3x5 array: 15:5 =3. This student experimented with decomposing the 15 into multiples of 4+1, 15:5 = (8+4):4 + (2+1):1..
Task 2: 30/5
During the next task, most students doubled the array from the first task: 30:5 = 2(15:5).
Task 3: 45/5
Then, students solved 45/5 by adding on another 15/5: 45:5 = (15:5) x 3.. Some students found other ways to creatively divide 45: 45:5 = (6+10+2):2 + (9+15+3):3. Others took a less efficient approach to solving this problem by making a 9 x 5 and drawing all the squares: 45 square array!
Task 4: 450/5
For the final task, students guessed that the quotient would be 90. They used the last task and said, "If we multiply the dividend by 10, then we'll have to multiply the quotient by 10." I wrote 90 with a question mark after the equation and said, "Let's see if it really does equal 90! Prove it!" Here, a student used the previous task and multiplied: 10 x 45/5, 450:5 = (150x3):5. I also liked watching this student decompose the 5 and the 450: 450:5 = 450:(2+3).
Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Listed Tasks.
Sequence of Division Lessons
When developing a learning progression for multi-digit division, I decided to teach the standard algorithm (long division) prior to developing a conceptual understanding of this process using hands-on tools and visual representations (such as base 10 blocks, money, grid paper, arrays, and ratio tables).
Here's why: In order for students to truly gain a deep understanding of the algorithm, they have to be provided with multiple opportunities to practice the algorithm. If I waited until the end of this division unit to teach this abstract process, students would only have a couple days of practice, leading to very few students achieving proficiency with the standard algorithm.
Also, in order for students to truly see the relationship between division models (such as the money model) and the algorithm, I want to provide them with the opportunity to use the algorithm and models side-by-side in upcoming lessons.
Lesson Introduction & Goal
To begin the lesson, I then shared today's goal: I can solve division problems with 2-digit dividends using the standard algorithm. I explained with great enthusiasm: Today, you get to learn how to complete simple division problems using the standard algorithm, which is also called long division! Students seemed both excited and nervous at the same time.
Song & Poster
To hopefully get all students equally excited, I invited students to sit on the carpet at the front of the room. Pointing to the Division Poster, I said: Whenever you use the standard algorithm to solve multi-digit division problems, you always follow the same Long Division Steps:
4. Brother-Bring Down
I then showed the following song about long division and pointed to the lyrics on the Division Poster when appropriate. Students immediately recognized the tune of the song. Some said, "I've heard this before!" Others began singing along: Students Singing Along Smiles and giggles filled the air.
I played the song through three times to help students learn the lyrics and steps of long division:
1. First you must DIVIDE.
2. Then you MULTIPLY.
3. And SUB-TRA-CT!
4. Then you BRING IT DOWN.
5. The last think you do is CHECK!
Reviewing Division Terms
Before moving on, I pulled the Division Terms poster from a past lesson off our math wall. I reminded students: With the equation, 18 divided by 3 = 6, the 18 is the dividend (the number being divided). The 3 is the divisor, which also sounds like the divider! This number represents the number of groups we are dividing 18 into. In this case, we are dividing 18 into three groups. And what's the answer to a division problem called again? "Quotient!"
I then showed students two other common ways of writing division problems (see bottom of poster). The ordering of the divisor and dividend can be different, depending on how the division problem is written. Since this can be confusing for students, I wanted to begin addressing the ways to write division problems right away.
To provide students with a real world application (Math Practice 4), I introduced today's Skylanders Problem: David has ______ shelves in his room. He has ______ Skylander figures. If he places... (at this point I asked students to help me write the rest of the problem to encourage active engagement) ...an equal number of figures on each shelf, how many Skylanders will be on each shelf? (I chose to include Skylanders in math today because several boys in my classroom love collecting these figures!)
For guided practice today, I wanted to provide my visual students with a fun, yet structured way to learn the steps. I passed out a page protector, an 2-Digit Dividend Template, and a paper clip to each student. I then modeled how to outline boxes within the algorithm grid, one color at a time. Singing the song, I said: First you must divide... I placed an red D on the side of the paper. We then outlined the top two boxes red: Building the Algorithm Mat 1.
We then moved on to then you multiply (orange boxes) and sub-tra-ct (yellow boxes): Building the Algorithm Mat 2. Then you bring it down (blue boxes): Building the Algorithm Mat 3. These colors also coordinated with the Dad, Mom, Sister, and Brother Long Division Steps from earlier. Whenever possible, I organize information by color to help the brain categorize new content and skills.
When finished, I asked students to place the algorithm inside the page protector and grab a dry erase marker and eraser for today's lesson.
I filled in the blanks to the Skylander problem above: Let's say that David has 2 shelves in his room. And let's say he has 22 Skylanders. If he divided the figures equally, how many would be on each shelf? Students knew the answer right away, "11!" Good! Now, let's see if we can get the correct answer by following the steps for the standard algorithm.
I explained: Remember... 22 is called the dividend. We always place the dividend, or number you are dividing inside the division sign. How many groups are we dividing 22 Skylanders into? "2 groups!" How do you know that we are dividing into 2 groups? "Because there are 2 shelves!" Good! Now the 2 is called the divisor, or number that we are dividing by. We always place the 2 on the outside of the division sign like so...
When we are finished solving this problem, the quotient, or answer to a division problem, will be found at the top of the division sign. We then labeled: dividend, divisor, and quotient. Before solving the next problem, a student suggested that we write these terms on the actual paper inside the page protector. I thought this was a great idea!
Then I very slowly modeled the standard algorithm step-by-step while students completed the problem alongside of me: Algorithm Mat 22:2. We placed the paper clip on the side of the page and moved it from D to M to S to B and back to D... to track the long division steps. We also sang along to the tune of the song!
I explained (and sang):
1. We sang together: First you must divide! We placed the paper clip on "D." I then asked: How many 2s are there in 2? Students said, "One!"
2. Write the 1 above the 2 in the tens place, in the first red box.
3. We sang together: First you must divide! Then you multiply! I moved the paper clip to "M." What is 1 x 2? "Two!"
4. Write the 2 below the 2 in the tens place, in the orange box.
5. We sang together: First you must divide! Then you multiply! And sub-tra-ct! We moved the paper clip to "S." I asked: What is 2 minus 2? "Zero!"
6. Write the 0 below the 2, in the yellow box.
7. We sang together: First you must divide! Then you multiply! And sub-tra-ct! Then you bring it down! We moved the paper clip to "B."
8. Now we bring down the 2 in the ones place to the blue box.
(Later this week, we will discuss how this 2 is really 2 tens, but for now, I wanted students to focus on learning the process of long division.)
9. At this point, we start over at division. We moved our paper clips back to the "D." I asked and pointed: How many 2s are in 2? "One!"
We then continued this process until we achieved a remainder of 0. We then discussed: How many Skylanders will be on each shelf? How many skylanders are left over? How do you know there will be 0 Skylanders remaining?
We continued on, solving division problems with 2-digit dividends and no remainders, following the same procedures as above:
To build the staircase of complexity, I included division problems involving a divisor greater than 2. With each problem, I provided a little less guidance and I also encouraged students to try solving the problem ahead of me. Here are student examples of the above problems:
Here, a student explains the procedure on her own: Student Explaining Steps.
Picking math partners is always easy as I already have students placed in desk groups based upon behavior, abilities, and communication skills. Before students began working, I asked them to discuss how they would like to support each other today. I gave them many examples: Do you want to take turns talking out loud? Do you want to solve quietly and then check with each other? Or do you want to turn and talk anytime you get stuck? Students always love being able to develop a "game plan" with their partners!
Goal & Modeling
We then wrote D, M, S, & B alongside the edge of our papers to help us remember the steps. Next, I modeled the first row of problems for the class, asking for feedback along the way: What do I do first? Then what? What does this represent? Where do I put the 2? What is this number called again? I wanted to be sure all students knew and understood the assignment.
Monitoring Student Understanding
Once students began working, I conferenced with every group. My goal was to support students by providing them with the opportunity to explain their thinking and by asking guiding questions. I also wanted to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3).
Here are a couple students conferences. This student explained how she solved 93/3, Student Solving 93:3
Here's a student working on Learning Steps. Sometimes it can be discouraging for students who want to pick the steps up quicker, but this is a perfect opportunity to encourage Math Practice 1: Persevere! Throughout this lesson, we discussed the importance of having a great attitude and not giving up easily!
Most students were able to complete this practice page. Here's a Student Example.