I asked students to make a big circle around the room. To practice our decimal understanding in the hundredths place, we played math "Buzz." We counted by hundredths starting with 1.42. Every time we got to a 5 in the hundredths place or a new whole number, we would say "Buzz". It was a tedious process, but I think it was good for students to hear the sounds of the counting and the change to the next tenths place. I think I undervalued that experience, not thinking about the fact that for some students, the auditory process was important. They hesitated. We counted up to 2.15 and stopped. I am not sure they thought it was much fun! A few students had to sit down when they missed, but if someone missed, they could jump in their spot. That part was fun, because then there was a chance again to stay in the circle.
I started today's lesson with my Decimals Strategies SB file. We reflected how to compare decimals and then I told them I expected that they would master proving it on a number line. This proof really satisfies the standard fully and deepens the meaning of decimals for them, just like it did when they were in the beginning stages of counting.
We reviewed the first page of the SB file together and came up ideas for remembering the difference between the place values. Some students remembered that when comparing a tenth and hundredth place value, it was helpful to create the hundredths place by placing a zero in that place value. The two of them explained together that "it was easier to look at" and "that way we can see that it is more." We compared the concept to whole numbers at first and then I circled back to reading and understanding the difference and meaning of the decimal place values Comparing Decimals Classroom Notes shows how we moved along in the lesson.
What is the difference between value and place value? applied now to decimal numbers as it did with whole numbers in the past. Students were able to connect this meaning well to decimals. This transitioned beautifully as we practiced plotting numbers and comparing with a number line, learning to draw, placing numbers correctly and talking about the "how" to do it.
Students took turns as we first practiced tenths. As soon as I thought they were ready, we began hundredths ( page 3). We talked about how the metric ruler is connected to understanding our decimals and the meaning of the numbers. We talked about how and why 1.3 and 1.30 are the same value. It was a rich discussion that flowed simply through drawing and number comparison.
As we practiced between tenths and hundredths on the SB file, I asked students to volunteer plotting. We discovered together what benchmark number the decimals fell between. i.e. 3.65 compared to 3.45 would be between 3 and 4 on the number line. Please refer to the SB classroom notes to help you see how we played on the last two pages.
I closed the SB Lesson: Do you understand that there are one hundred hundredths in a whole and that the whole changes from 0 to 1, 1 to 2 each time?
We counted by tens to one hundred on the number line to reinforce this concept.
I asked: Do you understand we are comparing the same size wholes when we compare decimals? I explained that when we compare .32 and .45, we are comparing it against the number line.
From their answers, I understood that they were just starting to understand the concept of the whole as applied to decimals. I plan to return to the concept later, as they develop more comfort and skill with comparing decimal numbers.
I assigned the Number line /Number Sentences from the worksheet resource for my above grade level achieving students. This sheet forced them to decide which number sentence was true and then after they chose the number sentence, they needed to plot the sentence on the number line. This way, they could see if their choice was correct, truly showing them how they can prove their answer on the number line. This fully satisfies the standard and supports the CCSS common thread of proving the "why" of your answers.
Students who needed more concrete understanding and practice needed to use the Comparing Decimals (Hundredeths) sheet. They needed to compare the values, and then plot the number line. That way, they were only working with one comparison at a time and then could prove their answer on the number line. As they worked, I paid special attention to their drawing, accuracy and understanding. The other group seemed to be fairly well versed in plotting the number line. There were some questions and I needed to guide some.Connecting the ruler to the numberline shows how a student is using a metric ruler as her guide. Students are standing around us, watching to help them understand too.We worked in class for about 20 minutes on our assignment. I didn't stop until I felt sure they could do it independently at home. Another Assignment was given to this student and shows how they plotted their work. This all was very rigorous work and time consuming, but the value of the learning in this assignment is worth the effort. They enjoyed it.
To close the lesson, asked students:
What does a number line show you that the symbols of greater than and less than don't?
My one student who always responds quickly and also had a little trouble plotting, said, "When I finally did it right, I could see exactly where the dots were."
I asked if she could see that the one dot was farther to the right, or showed us a larger value?
She said yes.
I told them that that was what I had hoped everyone would see, but that they needed to prove it independently to me now. I assigned 10 problems total. They had completed about 5 or 6 in class. This gave them enough to practice at home.