This week, my students and I prepared for my first assessment over Common Core material. Through different formative assessments this past week and a half, I felt my students were ready. Last week, after a one-question check-up, I found I needed to spend an extra day teaching how to change a repeating decimal into a fraction. I gave the check-up again and felt students were ready. The students worked on a review together that was similar to my interim assessment and we went over it towards the end of class. Again, I felt comfortable with how the students were doing. The assessment I gave can be found here and here are some observations I made during each question.
Overall, students did a really good job telling my a number is rational or irrational. But, sorting the numbers proved to be a little more difficult with some numbers being common mistakes. The fraction 1/3 was probably the most missed number of any. When I returned their quiz, students told me it was because they weren’t sure what 1/3 was equal to as a decimal. Most of them were using the idea that a rational number was a decimal that either terminated or repeated. If students had remembered that rational numbers could be written as the quotient of two integers, maybe they would have done better. But, I don’t think that definition is as student-friendly as “if the decimal repeats or terminates, it is a rational number.” I felt that with Common Core, it was important for students to come up with their own definitions and criteria as to what was and was not a rational number. Next time I teach this, I think I will be sure to repeat every day that, if a number can be written as the quotient or ratio of two integers, it is rational.
Question 2: Given the following numbers, graph them on a number line.
This question gave students more problems that any of the other questions. Many students graphed 1/3 and 1/4 farther to the right than any of the other numbers. Although students will manipulate fractions sooner and more often in Common Core, it was apparent that students’ knowledge of fractions at this point was very fragile. As I questioned students when we were going over their quiz, some of them said that 1/4 was 1.4. Some said it was 4. Some said they had no idea, so they just guessed where the fractions would go. Next year I will spend much more time working on ordering fractions. Another problem students had was where to put the square root of 5 and 2.236. Before students started their quiz, I told them and wrote on the board that the square root of 5 was approximately 2.2360679775. Even with this knowledge, there were students that had the two numbers switched. We talked about how 2.236 was equivalent to 2.236000000…. and that the square root of 5 would be larger in the hundred-thousandths place, making it the larger of the two. Next time I am going to spend more time ordering numbers that are very similar and point out the differences to help students determine which is larger.
Question 3: Write the repeating decimal 0.282828… as a fraction. Then explain what you did.
This question was a heartbreaker for me. Almost every student that took the quiz did this perfectly. They knew the exact procedure to end up with 28/99. However, there were many students who did not write what they did. A few said they forgot to or didn’t read that part, but many students said they didn’t know how to put into words what they did. This is going to be one of the interesting things as the year progresses because it is such an important part of the Common Core. My strategy for “fixing” this is, when I give homework, assigning one question and then having students explain what they did. Hopefully this practice will become habit as the year progresses.
Question 4: Lylah believes that 2/3 is an irrational number because it is a decimal that goes on forever. Logan disagrees and says that 2/3 is a rational number because, although it goes on forever, it does repeat, making it a rational number. Determine who is correct and why.
This question came about after we had the same disagreement in one of my classes. The two students had the same reasoning and we had a great discussion in class about whether 2/3 was rational or irrational. The students did great debating the answer and the entire class joined in. Many students did well on this question, but again, the students the did not know what 2/3 was as a decimal struggled to get this answer correct. Some students also correctly said that 2/3 was a repeating decimal, therefore a rational number, but said that 2/3 = 66.6666…. Again, I think students’ knowledge of fractions is very muddy and hopefully I can do some different things this year to clear things up, at least a little bit.
Overall, I thought the first assessment went really well. I gained some great insight into what students think and what they can do. On Monday we are doing our retake and I’m interested to see how well students improve, not that I pointed out their misconceptions.