I begin this class with the 2012 SAT mean math score on the board (mean = 514, standard deviation = 117). I challenge my students to decide how well a sample score (I just make one up like 585) compares to the given mean and give them a minute or two to think about it individually, then have them pair-share their ideas. (MP2, MP7) I ask whether anyone wants to share what they've discussed and accept all input without comment. I'm hoping someone asks for additional information or says they can't make a good comparison with just the mean, but if not, I ask leading questions to get my students to think about using the standard deviation as a measure of how far from the mean my sample score is. For example I might say "You've calculated that the sample score is 71 points higher than the mean. Is that a lot higher? Is it a good score?" If there are still students struggling to make the connection I might give an example like "If you compare your height to someone else you might just say you're taller or shorter, but if you want to talk about your height in general, you might want to say what percentage of the population you are taller or shorter than." I continue with these kinds of questions and examples until my students are clear on the idea that we want to compare using a standardized measure about the mean. When everyone indicates understanding by fist-to five I explain that the measurement of standard deviations from the mean is called the Z-score. I then walk them through finding the Z-score for the given sample (You can see an educreations video of this in my resources) My students are often surprised to find that a score 71 points above the mean is only 0.61 standard deviations from the mean. At this point I remind them that the normal curve has approximately 68% of the values within 1 standard deviation from the mean.
You will need copies of the ACT SAT worksheet.docx and the Z-Scores.docx handout for this part of the lesson. I copy the ACT SAT on the front and the Z-Score on the back to reduce the amount of paper I have to handle.
Independent Work 10 minutes: see my video Z-score.mp4 for further explanation of this section I tell my students that we will be using Z-scores to get a better understanding of both ACT and SAT test scores and also to compare their scores for the two tests. (MP1) (If your students have not yet taken either of these tests, you can use data from the collegeboard website or the sample scores I've included in my resources.) I explain that they will be working independently for this part of the lesson and may use their calculators if they want or may do the calculations by hand. (MP5) This gives those who are less comfortable with the graphing calculator a chance to see how the mathematics works manually, at least for their initial experience with z-scores. It may not be the quickest option, but part of learning to use tools appropriately is to use what works best for yourself in any given situation. I also tell them that they may work with their own ACT/SAT scores in place of one of the sets of sample scores on their worksheet.
Team work 15 minutes: Now that my students have had some practice working with Z-scores, I tell them that there are many uses for Z-scores besides tests. In fact almost any measured quantity that might be compared between two or more objects has a mean and standard deviation and thus could use Z-scores. I give some easily measured examples (height, hair length, armspan), ask for additional suggestions and have my students vote on three measurements to use as a class. I then tell them they will be working in partners to collect those measurements and post them on the board. When everyone has posted their data, I tell them to work with their partner to complete the second page of their worksheet (Z-Scores), reminding them that because there are multiple steps it will take diligence to avoid making mistakes. (MP2, MP6) While they're working I walk around giving encouragement and assistance as needed.
When all my students have completed the Z-score assignment, I tell them they have the remainder of class to write a response to the following statement: "Z-scores should be used in place of standardized test scores like ACT and SAT." I assure them that they may turn in this assignment at the beginning of class tomorrow if they need additional time and give the following criteria to clarify my expectations.
This closure assignment gives my students an opportunity to reflect about Z-scores beyond just the mathematics of the calculations, it let's them consider applications and even the end-users of these calculations. (MP3)