The Amazing Houdini Introduction: After doing some math the Great Houdini will be able to tell you the magic number that is missing. How this is done, is to be found out later in this lesson. This lesson deals with place value, factoring, and manipulating equations. The end result will prove that a certain number will be a multiple of nine. This lesson is intended for a 7 th grade classroom. Standards: NYS Standards 7.A.4 Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation 7.A.5 Solve one-step inequalities (positive coefficients only) (See 7.G.10) 7.A.8 Create algebraic patterns using charts/tables, graphs, equations, and expressions Objectives: To figure out the secret to the trick on your own or in a group Break up a four-digit number into place value Factor a nine out of an equation and finish the equation Be able to perform the trick after the lesson is done. Say if a number is a divisible by nine with out actually doing the math to find out Instructional Protocol: This lesson is intended to be used in one 40-minute classes with time to start on their homework. Start the lesson with the great magic trick. Try the trick out many times with many students and do about 4 or 5 problems on the board to give the students a visual. Have students try to find the trick out after the trick has been performed many times. Students need to understand place value Students need to understand how to factor and then subtract what is left. Students need to find this trick fun. Do not let them get discouraged with the math. Keep up with positive encouragement. Students need to know the trick to know if a number is a divisible by 9. Add the digits up and see if they are multiple of 9. If so, then the number is divisible by 9.
The Amazing Houdini Teacher: Enter room dressed as a magician saying: Everyone write down a 4-digit number. Now rearrange those 4 digits to create another 4-digit number. Subtract the smaller number from the larger number. Pick one student. Ask him/her to circle on digit. Now tell me the other three digits. Tell the one student think hard of the circled number. Take awhile and then say the fourth number. (Hint: All four numbers need to add up to a multiple of 9. Thus, the number you guess needs to make all four digits added up a multiple of 9) Optional: Now that you have amazed the students; give some students 4 index cards. Have them write down each digit from the subtracted answer on to the index cards. Have them pick one card and keep it. (they could show it to the class if they wanted) You then look at the three cards and say what number is on the index card the student has (the circled number) After you are done impressing the class, or while you are doing the index cards you can choose five students and put their subtraction problems on the board. Before they will put their 3-digit answer down with out putting their circled digit yet. You can impress the class again by saying what the fourth digit is. Next, now that the class has visuals, have your students try to figure out how you came up with the magic circled digit. After a few minutes if your class has not come up with the answer, have your class get into groups (maybe 4) and see if combined they can come up with the solution. If not, you can tell the class that all that all digits from the answers add up to be a multiple of nine. If your students do not know the method to finding out if a number is divisible by 9, then either the day before or today mention it to them. Add the digits up and see if they are multiple of 9. If so, then the number is divisible by 9. After the students understand the concept, the next idea is to prove to them why this magic trick always works.
Proof: You could use an example from a student 4-digit number: 9567 Number rearranged: -6579 When subtracted: 2988 We can see that sum of 2988 is 27, a multiple of 9 We will prove that when a 4-digit number is subtracted from a 4-digit number with the digits rearranged the answer will always be a multiple of 9. We are going to write out the two numbers we are subtracting in expanded form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 9567 = 9 1000 + 5 100 + 6 10 + 7 1 6579 = 6 1000 + 5 100 + 7 10 + 9 1 We will now subtract the two expanded numbers but we are going to rearrange to bottom number to line up with the first number. ( 9 1000) + ( 5 100) + ( 6 10) + ( 7 1) ( 9 1) + ( 5 100) + ( 6 1000) + ( 7 10) ( ) = ( 9 999) + ( 5 0) + ( 6 9990) + ( 7 9) ( ) ( 9 1000 1 ) ( 5 ( 100 100) ) ( 6 ( 10 1000) ) ( 7 ( 1 10) ) = + + + We will now factor out a 9. (make sure your students understand this part) ( ) ( ) ( ) ( ) = 9 111 9 + 5 0 9 + 6 1110 9 + 7 1 9 We will now factor out a nine from the entire number ( ( ) ( ) ( ) ( )) = 9 111 + 5 0 6 1010 7 1 9 This just proved that the number 2988 is divisible or a multiple by 9 Q.E.D.
Guided practice: Prove that 2439 is divisible by 9 4928 2489 2439 Answer to proof ( 4 1000) + ( 9 100) + ( 2 10) + ( 8 1) ( 4 100) + ( 9 1) + ( 2 1000) + ( 8 10) ) = 4( 1000 100) + 9( 100 1) + 2( 10 1000) + 8( 1 10) = ( 4 900) + 9( 99) + 2( 990) + 8( 9) = 4( 100 9) + 9( 11 9) + 2( 110 9) + 8( 1 9) = ( 4 100 + 9 11 2 110 8) 9 Thus 2439 is a multiple of 9 Q.E.D. If there is time left in the lesson, you may do another example. Conclusion: Explain to students the process of guessing the magic number. If the number 2439 is the final number you get and the 3 is circled. Explain that you would know the numbers 2, 4, and 9. Those added up is 15. The first multiple of 9 is 9. This is too small, so we go to the next multiple of 9, which is 18. Thus, 15 + x = 18. x = 3. The magic number!! Time permitting: Have your students try it on each other and see if they can guess the magic number. More practice: Have your students try this on their parents or their friends and have them write down the process and how they did with the trick. Also, have them prove the number they got is a multiple of 9.
For extra practice: Have your students create an algebraic equation to find the magic number Answer: We know that all four digits must be a multiple of 9. Thus, a + b + c + d = 9 x. Let d be the magic number a + b + c = 9 x d If a + b + c 9then x = 1 If 9 < a + b + c 18then x = 2 If 18 < a + b + c 27 then x = 3 If 27 < a + b + c 36then x = 4 Knowing this we can find d, the magic number
Guided Notes for the Magic Houdini s Number Trick Prove that 2439 is divisible by 9 How do you guess the magic number?? (the magic number is the unknown number)
Calculator Trick This trick is where you tell your students to multiply numbers together to create a 5-digit number. Your students give you 4 of the digits and you add the digits up and the 5 th digit is the number that gives you the next multiple of 9. We need to have a multiple of 9 multiplied to in the series of numbers we are multiplying to have this trick work. Example: we need either two 3 s multiplied, a 3 and a 6, or a 9 for the trick to work. To make this trick almost full proof we can say that you cannot repeat a number when you multiply to get the five-digit number. There is only one case that does not work. The one case: If we multiply by a 1,2,4,5,6,7,8 we get 13440. This is the only combination that does not work. Any other combination that gives you a five-digit number guarantees you to have a number divisible by 9. Another option is to give the students a list of numbers they are allowed to use (can repeat) and make sure that a 3, 6, and 9 are included in all of the options. The key here is to keep out numbers like a 7 or 8 that will allow the students to get to a large number quickly. You definitely want to keep the 9 in the list. Example Lists: Student A s list Student B s List Student C s List 1 1 1 3 2 2 4 3 3 5 4 5 6 5 6 9 6 7 9 9 These are only a few ideas and there are many more.