This is not the type of test students are used to taking in math class. They are used to doing problems 1 through 20 and demonstrating one skill at a time. This assessment provides some information, asks students to figure something out and then use mathematical evidence to prove it. It is harder for students to know what they need to do to get a certain grade. On the skills practice they know they just need to get most of the answers right. But here there are no problem numbers and its harder for students to tell if their work is sufficient. It's also a beast to grade! This is why we spend time in the subsequent lesson examining some really good samples and use a rubric to help improve their responses before the actual test.
On an assessment day students are required to do the homework independently so I like to give them a little preview in the warm-up. I use box diagrams to model problems involving fractions and percents of numbers because my students have a wide range of prior knowledge gaps and misconceptions. The visual, more concrete, representation helps them develop the concept of finding fractions and percents of numbers.
The warm up Warm up fraction to percents asks what 3/4ths of 20 is when 1/4th is 5. The problem is modeled with a box diagram that is divided into 4 equal parts, one of which is shaded and labeled with a value of 5. A student might explain that if one of the parts equals 5, then all of the parts equals 5, so 3 of them equal 15. Others may set up a proportion and scale 1/4 up by 5.
The second problem tells students that 2/5ths of a number is 6 and asks what the number is. Some students might cut the 6 in half to find that all the equal parts are 3 and then multiply 3 by 5 to get 15. Others will add pairs of boxes and then add another half (6+6+3). One student explained it that the last one doesn't have its partner.
After students explain their methods I go back to problem one and say "so, we found that 3/4ths of 20 is 15. What percent of 20 is 15?" I may need to circle 3/4ths for them if they get stuck. Students sometimes get stuck when you ask them something they think they should already know as they try to remember it instead of try to figure it out. I don't want them searching their memory for an algorithm, I want them to make sense of it. I follow up by asking about the second question, "6 is what percent of 15" and I circle the two sections.
Before handing out the assessments Proportionality assessment I go over the directions with them. They are told that 60% of the lady bugs in my yard are spotted and are given a table that shows the numbers of spotted, non-spotted, and total lady bugs for my brother, my aunt, and my cousin. They are asked to figure out which of my family member's lady bugs are proportional to mine and which are not. They need to use ratio, percent, and graphing to show whether or not each is proportional. The same graph they have been using in previous lessons (My family's lady bugs series) is on the back of the assessment.
They are demonstrating several skills here. They should be able to write, simplify, and scale up ratios. They should be able to choose which ratios can be scaled up to a percent. They should be able to graph the ratios and percents. They should be able to write a percent as a fraction in simplest form. They should be able to determine that if the ratios when simplified or scaled up to common denominators are equal the quantities are proportional. They should be able to determine proportionality if the points on the graph lie in the same straight line through the origin.
I expect questions like "can I try...?", or "should I...?", or "do you want us to graph the percent or the simplified ratio?" I like to respond by telling them to try it and see if helps them figure out or explain or to try it out and see how it might help. This is also the first time they have to prove dis proportionality and they may get confused on the graph when all the points don't line up. I ask them why they think the points are supposed to line up. They will likely say, "because they have to if they are proportional" to which I respond, "then what does it mean if they don't" and remind them that the question is asking them 'which' are proportional and which NOT.
The hardest part of taking this kind of assessment for students is organizing their own work and clearly and logically explaining the connection between their conclusion and the evidence. This really speaks to the need to teach students how to present and articulate their findings. The rubric helped some, but in retrospect I think more sentence frames would have been helpful. Also, when I do this practice assessment again, I might have students present their findings to the class with a poster. The test results showed that most of the students were able to do the math, but had trouble explaining clearly how they knew. Discussion of student assessment samples.