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.
Task 1: 15 x 17
For the first task, 15 x 17, some student decomposed both the 15 and the 17 one time: 15 x 17 = 5(7+10) + 10(7+10). Other students decomposed both numbers repeated times: 15 x 17. Either way, both students were developing number sense in their own ways. Here, I conference with a student who is Confused on 15 x 17.
Task 2: 15 x 34
For the next task, students figured out that 34 is double 17 so 15 x 34 = 2(15 x 17). We established the answer ahead of time using mental math (510). Sometimes knowing the answer helps guide and direct students as they experiment with different strategies. Again, students came up with multiple strategies: 15 x 34 = 10(30+4) + 5(30 +4) and 15 x 34. However, I was most impressed with this student: Doubling 15 x 34. I loved how she took 15 x 17 (the previous task) and showed how to use this information to find 15 x 34!
To begin today's lesson, I introduced students to the goal: I can round multi-digit numbers.
Pointing to the Rounding Anchor Chart, I explained key concepts while writing on the anchor chart. This is because students learn more when they can experience the construction of an anchor chart (instead of the teacher constructing the chart before the lesson).
I explained: Rounding is when you find a number that is close to a given number. You might ask, What is the nearest 10? ...or What is the nearest hundred? ...or What is the nearest million?
Whenever we round numbers, it is helpful to locate the benchmark numbers on the number line. We can use a straight number line or a bent number line like this. If we are rounding to the nearest 10, the benchmark numbers are the nearest multiples of 10, such as 10, 20, and 30.
The midpoint always comes directly in-between two benchmark numbers. What would the midpoint be between 10 and 20? Students quickly caught on, "15!" What would the midpoint be between 20 and 30? "25!"
Going to the Nearest Gas Station
I proceeded: Let's say that you are in a car and you are traveling along this number line. I showed students a picture of a car (I placed wall putty on the back for easy placement on the number line). Let's say that there is a gas station located at each benchmark. If you ran out of gas at mile marker 11, would you go to the gas station at mile marker 10 or the gas station at mile marker 20 to get gas? Which gas station is closest? We discussed how we could "roll back down the hill" to the gas station at mile marker 10. Continuing on, we discussed how we would round 12, 13, 14, and then 15 using the gas station concept. Students decided that we could roll down to the next gas station (at the mile marker 20) once we got to the midpoint. Then, we moved on to 16, 17, 18, 19, 20.... 32. Students loved the car and gas station model. The students who struggle with math the most especially latched on to this idea!
Once students understood the reasoning behind rounding based on midpoints and benchmarks, I wrote a helpful rhyme on the anchor chart: "If it's 4 or less, give it a rest! If it's 5 or more, raise the score!"
This was a great point to ask students, "What patterns are you noticing?" Students came up with many observations: "Both midpoints end with 5." "The benchmarks end with 0." The longer students discussed patterns, the more in-depth they became. As a class, we finally came up with, "Any number at or above the midpoint, round up" and "Any number below the midpoint, round down."
For guided practice, I wanted students to focus on rounding numbers (2-digit to 6-digit) to the nearest ten. I created a sequence of tasks that gradually increased with complexity:
For each task, students drew a bent number line on their white boards, labeled the benchmark and midpoint numbers, and turned and talked to explain which benchmark they would round to using the model. Turning and talking was an important part of this activity as it supports Math Practice 3: Construct viable arguments and critique the reasoning of others. In addition, by asking students to model their thinking using a number line, they were taking an abstract task and representing it symbolically (Math Practice 2: Reason abstractly and quantitatively).
Here, a student is Rounding 63 to the nearest ten. I provided guidance by referring to the vocabulary (benchmark numbers, midpoint, number line), asking the student to place the number being rounded on the bent number, and by singing our rounding song. In this video, I'll told students that we won't sing the song every time, but as you will see, I have a hard time following through with this promise! In the following videos, you'll see that we follow the same procedures with larger numbers:
Rounding the larger numbers to the nearest ten were more challenging. However, by providing students with the number line model and peer conversation, all students were successful.
During independent practice time, students worked on two practice pages on their own. I loved how the first practice page required students to identify the benchmark numbers (which required much more of students that remembering the procedure of rounding). On the next page, students practiced rounding numbers 2-digit and 3-digit numbers to the nearest ten without having to identify the benchmark numbers.