Today, I wanted to reintroduce the 8 Mathematical Practices to students using the following poster: Magnificent Mathematician Poster. So far this year, I have encouraged and defined each of the practices along the way; however, I hadn't placed them all on one poster. Prior to the lesson, I rewrote each practice in a student-friendly form. For example, Math Practice 1: Make sense of problems and persevere in solving them, I wrote, I can find a way to attack problems and I won't give up easily.
By placing these practices on a poster, I knew that I could continually refer to them for the rest of the year, keeping them at the forefront of learning. The goal is to make learning transparent with students. Teachers and students should be aware of best practices in mathematics.
Teaching the Practices
To introduce this poster to students, I said: Today, I want to explain what a Magnificent Mathematician does. Actually, I see Magnificent Mathematicians in each of you every day, but let's take a closer look and be more aware of what makes you magnificent!
I revealed each practice one-by-one, read the practice slowly, and then asked, What do you think this means? Often I would hear students say, "Oh, yeah, we're already doing this... It's when you..."
To make these practices even more meaningful, I asked students to choose the practice that they thought was the most important. Some students said that being able to defend your thinking is the most important. Then another student said, "No... not giving up is most important because..." The conversation was great! What was great about this conversation is that students were practicing Math Practice 3: Construct Viable Arguments!
Finally, we went through each of the practices and picked out the most important word(s). To make this a timely process, I had already prewritten what I thought were the most important words on colored paper before the lesson. Surprisingly, most of the words that I thought were important were also important to the students. One word that I left out that they insisted should be dubbed as "important" was the word 'attack'.
Reasoning for Teaching Multiple Strategies
During this Addition and Subtraction Unit, I truly wanted to focus on Math Practice 2: Reason abstractly and quantitatively. I knew that if students learned multiple strategies of adding and subtracting numbers, I wouldn’t only be providing them with multiple pathways to learning, but I would also be encouraging students to engage in “quantitative reasoning” by “making sense of quantities and their relationships in problem situations.” By teaching students how to use a variety of strategies, such as using number lines, bar diagrams, decomposing, compensating, transformation, and subtracting from nines, I hoped students would begin to see numbers as units and quantities that can be computed with flexibility.
I continued this lesson by introducing key vocabulary, necessary for students to know and understand when working with open number lines: Point.jpg, Starting Point & Ending Point, Landmark Numbers, Nearest 10, and Open Number Line. By teaching math vocabulary, students will have the tools to truly practice MP 3 (Constructing Viable Arguments).
Adding on an Open Number Line
Using the Open Number Line Poster, I explained and modeled on the board: When using an open number line to add, I first start by drawing a line with arrows on each side. These arrows mean that numbers go on indefinitely (or forever) in both directions. Whenever I take jumps to the right on a number line, the numbers increase. However, when I take jumps to the left, the numbers decrease. Can anyone remind me why this is an open number line? Students responded, "There aren't starting or ending points." "You can start wherever you would like." "You can start at 0 or another number."
Let's look at this addition problem: 2507 + 349. I wrote the problem on the board and drew a new number line. We have several options when adding on a number line. We can start with the larger addend or we can start with the smaller addend. We can take big jumps or we can take little jumps. Let's try a couple ways!
Student Number Lines
I passed out a Student Open Number Line to each student and asked team leaders to pass out dry erase markers and erasers. I created this student tool by taping two sheets of paper together and slipping a page protector over both sheets of paper on either end. Then, I taped the middle together. I created these number line models in place of using white boards because I wanted students to have an ample amount of space to show their thinking. Besides, students love to have new math tools! Providing students with a variety of math tools is an important part of Math Practice 5: Use appropriate tools strategically.
Starting with the Larger Addend
I continued and asked students to follow my steps on their white boards: I could start by making the larger addend, 2527, my starting point. Then, I could rewrite 349 in expanded form: 300 + 40 + 9. Next. I could take a jump of 300. What would I land on? "2,827!" Then I could take a jump of 40. What would I land on? "2,867!" Finally I could take a jump of 9, but I noticed that I could add part of the nine to get to a landmark number. What is a landmark number again? "A number that is easy to work with." Turn and talk: What could I add to 2,867 to get to a nicer number? Altogether, we agreed to take a jump of 3, land on 2,870 and then add the 6 (or the rest of the 9).
Starting with the Smaller Addend
I drew another open number line and explained: Another way we could add these two numbers is by starting with the smallest number, 349. We then followed the same procedures as above: decomposing 2,527 into 2000 + 500 + 20 + 7, starting on 349, taking jumps of 2000, 500, 20, 1, and 8 to get the same answer.
When finished, I asked students to turn and talk: Which strategy do you prefer... starting with the larger number or the smaller number? After providing students with time to discuss, one student said, "If you begin with the larger number, you take less jumps." Another student said, "Not always!" Then a student said, "When you begin with the larger number, you have to add less on." At this point, we agreed that making the larger number the starting point is often a more efficient strategy, but not always.
Taking Jumps to the Left to Subtract
Drawing another number line and modeling on the board, I said: Now it's time for subtraction! Let's look at 2876 - 349. One student reflected on the previous problem, "We already know that!" I always try to provide students with opportunities to practice Math Practice 8: Look for and express regularity in repeated reasoning. Plus, I wanted the focus to be on using an open number line instead of finding the answer. Again, following the same procedures as above, I modeled on the board while students took jumps on their own number lines. First, I modeled how to start on the right side of the number line and take jumps of 300, 40, 6, and 3 back to 2,527.
Taking Jumps to the Right to Subtract
Next, I showed students an even trickier way of subtracting.... by adding! On another open number line, we started with the subtrahend (349) and took jumps up to the minuend (2876). First, we took a jump of 51 to get to a landmark number, 400. Next, we took a jump of 2000, 400, and 76 to land on 2,876. We then added up the jumps to find the difference of 2,527.
In order to provide students with guided practice, I created a four column chart on the board with each of the column headings: Problem, Algorithm, Decompose one Number, and Number Line. As a class, we then solved, modeled, and discussed each of the following problems, one at at time:
I wanted to expose students to both addition and subtraction and numbers with zeros. I always want to start off with easier numbers to build math confidence and a solid understanding of the open number line model. Here's what the completed chart looked like when we were finished: Process Grid.
For each problem, students used the grid side of their white boards to solve the algorithm and their open number line boards to model their thinking. Other students showed all their work on the open number line board. When finished, students turned and shared their thinking with each other prior to discussing and modeling strategies as a class.
Problem: 75 + 11
For the first problem, students took jumps on the open number line in a variety of ways:
This was the perfect opportunity to get students thinking about which strategies are the most efficient: Comparing Strategies. As we celebrated students for using unique strategies, students naturally began trying new methods and learning from each other.
Problem: 575 + 128
Often students would finish earlier than others. I encouraged them to try as many strategies as possible: 575 + 50 + 50 + 20 + 8 = 575 + 25 + 25 + 25 + 25 +10 +10 + 4 + 4.
This provided me with the opportunity to support any struggling students: Providing Support.
Problem: 600 - 256
The most difficult part of transitioning to subtraction was honoring the fact that numbers increase when taking jumps to the right and numbers decrease when taking jumps to the left. Many students were unfamiliar with starting at the ending point to "take away." Here's an incorrect example: Oops! Subtracting to the Right.. I know it's blurry, so here's a recreation: Subtracting Right. Here's a correct example: 600 - 200 - 50 - 3 - 3 = 344.
I absolutely loved seeing this student using subtraction to verify his solution: 575 + 100 + 20 + 8 = 703 - 100 - 20 - 8..
Problem: 1,293 - 555
After students had time to solve, model, and discuss their work, I would ask students to complete the chart not the board. I would often ask students to model their thinking prior to sharing to maximize the learning time of others. Here's a Student Modeling on the Board. To encourage the student to construct a viable argument (Math Practice 3), I asked: Can you explain that more?
During this video, I also celebrate students who are successfully turning and talking. Immediately after this celebration, I immediately saw more students striving to meet this expectation. It goes to show: What you focus on in a classroom, you'll get more of!
Problem: 4,000 - 1,024
For the final problem, many students chose to subtract by moving to the left on the number line: 4000 - 1000 - 20 - 4. I was especially impressed with students who "added to the right" to subtract. I liked how this student in particular used landmark and compatible numbers. He first took a jump of 1 to get from 1024 to 1025 (landmark number). Then, he took a jump of 75 to get from 1025 to 1100 (compatible numbers). Next, he took a jump of 900 to get from 1100 to 2000: 1024 + 1 + 75 + 900 + 2000 = 4000. I was so proud of him!
Although students weren't provided with independent practice time during this lesson, I knew that the guided practice during this lesson would prepare students to practice independently in upcoming lessons.