ࡱ> %` Jbjbj"x"x @@d<tZ.jjjjj999YYYYYYY$[h4^YQ96L999YjjZV;;;9 jjY;9Y;;6VXj^ ih9 W}Y VZHZW^9$^8X^X99;99999YY ; 999Z9999$ Chapter 2  TI Nspire"! CAS Activity  Expanding and Factoring Even though expanding is not part of the grade 11 curriculum, some investigation of expansion patterns lends itself to understanding of factoring patterns. To begin expanding expressions on the TI Nspire"! CAS, we will use the Expand command. From the Algebra menu, select Expand. Enter the expression and press . One of the actions that students enjoy is to reverse the process of expanding on the device. From the Algebra menu, select Factor. Move up using the key. When the result from the previous line is highlighted, press . The expression will be pasted into the entry line. Press to execute the factor command. Part 2 Factoring Trinomials Start a new Calculator page to explore a factoring pattern. A skill that benefits students is to look for patterns and communicate their findings. It is also good for them to see that not all trinomials factor. Using a CAS device such as TI Nspire"! CAS or the TI 89 allows students to explore these quickly. In this screen, the coefficient of the linear term has been increased in each line. The question that we could pose to students is  why do the last two trinomials factor while the rest do not? Hopefully, at least one student will come up with the idea that the sum of the constants equals the coefficient of the middle term. To confirm their hypothesis, continue the pattern until they are certain that there are no further combinations that work. Start a new Calculator page and enter the first few attempts at factoring a trinomial with a different constant. They should be able to explain why none of these trinomials work. In this screen, students will find that two of trinomials work while the rest do not. They should confirm that their hypothesis from the first example applies here. The pattern is continued here with negative coefficients for the middle term. As before, only a few of the trinomials are factorable. Students should be encouraged to explain why certain trinomials can be factored while others cannot be factored. Open a new calculator page to explore a different pattern. Keep the constant fixed at 6 for each of the examples. The initial value for the coefficient of x should be 7 and that can increase with each row. Once again, students should explain why one trinomial factored but the rest did not. This screen just continues the pattern for the remaining negative values of the coefficient of x. As before, students should explain why the one problem was factorable while the others were not. If students can explain the pattern, then they should also be able to predict which positive values of the coefficient will produce a trinomial that factors. The student worksheet contains two pages of these types of examples that groups of students can use to explore factoring. It is recommended that they work together for the first page and individually for the second page. Part 3 The Perfect Square Trinomial Start a new problem and open a Calculator page. This section begins with expansion patterns in the hope that seeing the pattern in expanding will assist in predicting the pattern for factoring. In this case, the perfect squares with constants from 1 through 5 have been expanded. Continuing the pattern, this screen shows the same problems using negative constants. Students should be prepared to explain the differences. Notice that the final expansion has not been completed but is set up for students to consider. If they have detected the pattern, then they should be able to predict the result. This screen simply displays the result that students were to predict from the previous screen. Open a new Calculator page and begin the factoring of perfect square trinomials. It is hoped that students will be able to predict the result before the command is executed. This is a bit different. If the class is confident of their abilities to predict the answer, ask them the value of k that will make this trinomial a perfect square trinomial. Hopefully, the students were able to provide the correct result and this can be tested by replacing k with the value and executing the command. The next one in sequence has been entered. At this point, we can show students that two different answers are possible. Part 4 The Difference of Squares Pattern Start a new problem and open a Calculator page. Notice the tab numbers at the top of the page. The difference of squares expansion pattern often causes more problems than it should. Students should be able to explain why the middle terms cancel out. Using a CAS allows students to generate several examples quickly. Each new line can be obtained by moving up to the previous problem and pressing , then editing the constants from the previous line. This can be abbreviated even further by using the variable k in the factors and then substituting for it at the end of the line. The * key is used between the expression and the substitution. 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