Today we are reviewing for the final assessment of this unit. Students are asked to find the cos(75). As I move around the room I make sure that I have at least one group using sum and difference identities and at least one group using half angle identities.
After about 4 min., I have students put the 2 different methods on the board. (review student work slide 1).
When both answer are shown I will comment that the answer do not look the same. "Are they the same?" Some students will begin to check with their calculators while others will start discussing that both process involved an angle referenced by a 30 degree angle. Once the students have shown that the answers are equivalent using a calculator I ask what can we learn from this situation. Students forget that we can write numbers in many different ways but it looks a lot more confusing when the number involves radicals and fractions.
"We think these are true, how can we show they are true?"
I want to challenge some of the advanced students so I give the class the challenge as extra credit. The students are told they can use any resources but they must cite any resources used. Students are given 2 days to find a mathematical verification.
We are going to review today. I will do this by giving student problems to work then share with the class. Students are given a few minutes to solve the problems. We then put the problems on the board and discuss any questions or concerns.
On slide 2 the students verify an identity.
Slide 3 reviews sum and difference identities. The students struggle with this problem so I spend more time with this question. I remind students to write out each step so they can do the substitutions correctly.
Even with the strategy suggested above some students will not substitute correctly. I am still working on understanding why this is such a struggle for students.
On slide 4, students solve using different methods to find cos (2u). Both methods are shared. We discuss when one method might be more appropriate to use. Most students say if you know cosine then use the cosine version of the identity, if you know sine use the sine form. Some students say that it is easier for them to find both fractions and not think about which form of the cos 2u to use.
The biggest mistake on the double angle identities us not computing the numbers correctly. I remind students that squaring a negative results in a positive answer, and that a whole number such as 2 is 2/1 and you can make 1 a fraction by putting a number over itself.
The biggest mistake in using the half-angle (slide 5) is deciding whether the answer is forgetting to determine if the result is positive or negative. I review how to make this decision. Some students find the quadrant of u and think that is what determines the sign for the half angle. I remind them that the equation is for u/2 so we want the quadrant u/2 is going to be in.
After reviewing as a whole class, I want to work with some students individually on prior skills such as fractions. I ask the students what were the 2 things we did in this unit. They will say verify identities and find trigonometric values for functions using the identities. I explain that this will be what is tested.
Student work on any old assignments and I help those that need help on previous skills. Students also make sure they have their reference sheet prepared for the assessment.