The activity looks at a sequence of growing patterns using sticks and blobs.

Transcription

1 Sticks and Blobs Key words: sequence, term Objectives Solve word problems and investigate in a range of contexts. Generate and describe sequences. Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence. Instructions for teacher: At the start of the lesson ask students to reset the TI-8 Plus as follows. Press 2 nd + to get Mem Choose 7 ( Reset) Choose 1 (All Ram) Choose 2 (Reset) KS Framework page ref:10, 11, 1, 1, activity looks at a sequence of growing patterns using sticks and blobs. If possible, allow students to make the patterns using cocktail/match sticks and modelling clay. Discuss the of sticks and balls of clay added on each time. Ask how they could decide on the of sticks/blobs needed to make a pattern with 100 squares. Have the students complete the table for pattern 1 up to the th term. Suggest that they could save time adding to the sequence by using the TI-8 Plus. Generate the first sequence (of sticks) Press ENTER + ENTER ENTER ENTER ENTER ENTER.. Ask why is entered first. Why do they add? Can they find the number of sticks for the 10 th diagram? 2 th? What is the difficulty in finding the number for the 100 th diagram? (Time-consuming and easy to lose count) Finding a rule would help. 1

2 Press STAT 1 ENTER Enter the of squares and of sticks into L1 and L2. Press 2 nd Y= for STATPLOT ENTER Choose these settings Press ZOOM and then 9 to set the axes automatically: Press TRACE to move the cursor to each point in turn and check the coordinates Discuss the finding of a rule with the students. Refer to the actual making or drawing of the diagrams. If they start with 1 stick, how many sets of sticks are added for the th diagram, the 9 th diagram etc Ask students to suggest a rule in words. Hopefully they will get Sticks = times the Squares plus 1. Explain that x will represent squares and y will represent sticks Press Y= Enter X,T,θ,n + 1 Press GRAPH An incorrect rule will give a line that does not go through all the points. It can be deleted by going to Y=, selecting the equation and pressing CLEAR 2

3 Answers to student activity sheet Pattern 1 10 squares = 1 sticks 2 squares = 76 sticks 0 squares = 11 sticks Three more sticks each time but we need four to start with so we have a complete square. For five squares there are x+ 1 sticks Sticks y = x + 1 Blobs y = 2x + 2 Pattern 2 Sticks y = x + 2 Blobs y = x + Pattern Sticks y = 6x + 2 Blobs y = x +

4 Pattern 1 Sticks and Blobs Draw the next two patterns of sticks and blobs Complete the table. How many sticks will be needed for 10 squares? 2 squares? 0 squares? You can use your TI-8 Plus to help. Start with a clear home screen of squares x of sticks Y 1 of blobs Y Press Why start with? ENTER + ENTER Why do we add? How many sticks are added on each time? ENTER ENTER ENTER ENTER Now let s graph the data Press STAT 1 ENTER Enter the from the first two columns of your table Press 2 nd Y= for STATPLOT Enter these settings: Press ZOOM and 9 to set the axes automatically

5 You should get this screen Press TRACE Use and to move the flashing cursor. Look at the x and y values at the bottom of the screen. What do you notice? Can you find a rule that connects the number of squares to the number of sticks? HINT: sticks are added each time, but there is also a stick on the end. For two squares, how many sets of sticks are there? For five squares, how many sets of sticks are there? For five squares there are x + sticks When you think you have the rule press Y= At Y1 enter your rule. If you think it is y = x + 2 you enter X,T,θ,n + 2 Press GRAPH This equation is WRONG because the line does not go through the points. correct line will join all the points. Keep trying until you have the correct rule. Now follow the commands to create the sequence, draw a graph and find the rule for the number of blobs.

6 For patterns 2 and : 1. Find the sequences. 2. Draw the graphs.. Find the rules.. Use the rules to find the of sticks and blobs for the 100 th diagram.. Write down all your observations. Pattern 2 of rectangles x of sticks Y 1 of blobs Y Pattern of rectangles of sticks of blobs 6