WORKBOOK 6: PAGE 20-24 Pascal s Triangle 2 nd Diagonal 2 Row 3 3 3 4 6 4 PA6-5 Finding Rules for Part I Draw on the board: Students will find simple additive, multiplicative or subtractive rules for. T-table input equation Ask your students: How many pentagons did I use? How many triangles? How many triangles and how many pentagons will I need for two such designs? For three designs? Remind your students that they previously used to solve this type of question. Invite volunteers to draw a T-table and to continue it to 5 rows. Ask: I want to make 20 such designs. Should I continue the table to check how many pentagons and triangles I need? Can you find a more efficient way to find the number of pentagons and triangles? How many triangles are needed for one pentagon in the design? What do you do to the number of pentagons to find the number of triangles in a set of designs? Ask volunteers how to derive the number of triangles in a particular row of the T-table from the number of pentagons. Students should see that the procedure they follow to find the number of triangles can be expressed as a general rule: The number of triangles is 5 times the number of pentagons. Mathematicians often use letters instead of numbers to express this type of rule. For example, they would use p for the number of pentagons and t for the number of triangles, and get the rule 5 p = t. This rule is called a or equation. Write these terms on the board beside the itself. Write another, such as 4 s = t. Explain that s represents the number of squares and t is the number of triangles as before. What rule does the express? What do you have to do to the number of squares to get the number of triangles? You can make a table of values that matches this. All you have to do is to write the actual number of squares instead of s and do the multiplication. The result of the multiplication is the number of triangles. Draw a T-table: 36 Copyright 2007, JUMP Math 6. TG Unit PA -2 p-48_r2.indd 36 28/07/207 :3:56 AM
WORKBOOK 6: PAGE 2-24 Number of Squares (s) Formula (4 s = t) Number of Triangles (t) 4 = 4 4 2 4 2 = 8 3 4 3 = Ask volunteers to fill in the missing numbers and to add two more lines to the table. Ask them to find the number of triangles for 25 figures (25 squares). Let your students practice drawing for more s, like: 3 t = s, 6 s = t (t number of triangles, s number of squares) Ask your students to create designs to go with the s above. Students should also learn to write s for patterns made with addition. Start with a simple problem: Rose invites some friends to a party. She needs one chair for each friend and one for herself. Can you provide Rose with a or equation for the number of chairs? Ask your students to suggest a letter to use for the number of friends and a letter for the number of chairs. Given the number of friends how do you find the number of chairs? Ask your students to write a for the number of chairs. Suggest that your students make a T-table similar to the one they used for multiplicative rules. Give your students several questions to practice writing rules and making, such as: a) Lily and Rose invite some friends to a party. How many chairs do they need? b) Rose, Lily and Pria invite some friends to a party. How many chairs do they need? They invited 20 friends. How many chairs will they need? c) A family invited several friends to a party. The number of chairs they need is 6 + f = c. How many people are in this family? If they invited 0 friends, how many chairs will they need? Ask your students to write a problem for the 4 + f = c. Explain to your students that the number that you put into a in place of a letter is often called the input. The result that the provides the number of chairs, for instance is called the. Write these terms on the board and ask volunteers to circle the input and underline the in the s you have written on the board. Draw several on the board, provide a rule for each and ask your students to fill in the tables. Start with simple inputs like, 2, 3 or 5, 6, 7 and continue to more complicated combinations like 6, 0, 4 and so on. 2 3 The rules you provide should include additive (Add 4 to the input), multiplicative (Multiply the input by 5) and subtractive rules (Subtract 3 from the input). WORKBOOK 6 Part Patterns & Algebra Copyright 2007, JUMP Math 37 6. TG Unit PA -2 p-48_r2.indd 37 28/07/207 :3:56 AM
WORKBOOK 6: PAGE 2-24 Suggest that your students try a more complicated task you write a table and they have to produce a rule for it. Ask them to think what was done to the input to get the. Give them several simple tables, like: 5 2 4 8 5 2 6 2 6 3 6 7 4 4 2 3 7 4 8 6 3 6 8 Assessment. Complete the tables: 2 0 2 2 20 24 53 7 37 Add 3 to the input Multiply the input by Subtract 5 from the input 2. Find the rules and the s (or equations) for the tables: 5 2 0 3 5 5 2 6 3 7 4 2 2 28 23 30 Bonus Find the rule, the and the for input = 7 and input = 0. 2 222 3 333 38 Copyright 2007, JUMP Math 6. TG Unit PA -2 p-48_r2.indd 38 28/07/207 :3:56 AM
WORKBOOK 6: PAGE 2-24 ACTIVITY Each student will need a spinner as shown and a die. Spin a spinner and roll the die. Write a for the rule given by the spinner and the die. For example, if the spinner shows Multiply and the die shows 3, the rule is Multiply by 3, and the is 3 A = B. Add Multiply Subtract ACTIVITY 2 A Game for Two Each pair of students will need a die and a spinner as in the previous activity. Player spins a spinner and rolls the die as before, but so that Player 2 does not see the result. Player writes the rule and the given by the spinner and the die and gives the other player 3 pairs of input and numbers. Player 2 has to guess the. ACTIVITY 3 Students could make a design using concrete materials (as in Questions 3 and 4 in the worksheets) and predict how many of each element in their design they would need to make 8 copies. Extensions. Find the rules by which the following were made: a) b) c) 40 2 80 3 20 50 2 00 3 50 250 2 500 3 750 2. (The Ontario Curriculum) Tell students that a family is having a party. This is the for the number of chairs they will need for the party: g + 4 = c. ASK: If g is the number of guests and c is the total number of chairs needed, how many people are in the family? (4) Point out that any change in the number of guests produces a change in the total number of chairs needed. For example, if there are two guests, g = 2 and the family will need 6 chairs; if there are three guests, g = 3 and the family will need 7 chairs; and so on. The number of family members is always 4, and it does not change. Next show a different for the number of chairs: g + f = c. Say that f represents the number of family members, and the represents a baby in the family who does not need a chair. This time, the number of family members can change, too. What other quantities can change? (the number of guests, the number of chairs) If the family has 0 chairs, how many guests and how many family members could be at this party? (There are different solutions to this problem. Students should find them systematically.) WORKBOOK 6 Part Patterns & Algebra Copyright 2007, JUMP Math 39 6. TG Unit PA -2 p-48_r2.indd 39 28/07/207 :3:56 AM
WORKBOOK 6: PAGE 25 PA6-6 Finding Rules for Patterns Part I Students will find simple additive rules for. Draw the following sequence of figures on the board and tell your students that the pictures show several stages in construction of a castle made of blocks. T-table input equation How will they keep track of the number of blocks needed for the castle? Invite volunteers to make a T-table with two columns the number of triangles and the number of squares. Ask your students to find a verbal rule and a that tell how to get the number of squares from the number of triangles. Could they predict the before building a T-table? There are three squares for every triangle. So there are three times more squares than triangles, and we can write that as Multiply the number of triangles by 3 or 3 t = s. Repeat with the next block pattern: Let your students practice with various patterns with multiplicative or additive rules. Assessment Make a T-table and write a and a rule for the patterns: a) (use s for shaded squares and u for the unshaded squares) b) (use r for rhombuses and t for trapezoids) 40 Copyright 2007, JUMP Math 6. TG Unit PA -2 p-48_r2.indd 40 28/07/207 :3:57 AM
WORKBOOK 6: PAGE 25-26 Extension A dragon has 44 teeth and 4 poisonous spikes on its tail. A dragon breeder uses 2 s: t = 44 d, s = 4 d. What does each describe? Can you write a that relates the quantities t and s, and does not involve d? Hint: Make a T-table with columns: dragons, spikes, teeth. Students should derive the t = s. They should also recognize that not all inputs make sense. A dragon can t have only one spike, for instance. Since a dragon has 4 spikes on its tail, only inputs that are multiples of 4 make sense. PA6-7 Direct Variation Show your students several sequences made of blocks with a multiplicative rule, such as: Students will find simple additive rules for. Invite volunteers to draw for the number of triangles and the number of squares: Figure Number (f) Number of Squares (s) Figure Number (f) Number of Triangles (t) T-table input equation Number of Squares (s) Number of Triangles (t) Ask your students to write a rule and a for each table. Remind your students the meaning of the terms input and. Explain to your students that when the rule is Multiply the input by, we say that the varies directly with the input. So in this pattern the Number of Squares varies directly with the Figure Number, and the Number of Triangles varies directly with both the Figure Number and the Number of Squares. Draw a sequence of squares with sides, 2, 3, etc. Ask your students to find the areas and the perimeters of the squares. Ask them to make a T-table for both and to check which quantity varies directly with the side length (perimeter). Encourage them to write a not only for the perimeter, but also for the area of the square. WORKBOOK 6 Part Patterns & Algebra Copyright 2007, JUMP Math 4 6. TG Unit PA -2 p-48_r2.indd 4 28/07/207 :3:57 AM
WORKBOOK 6: PAGE 26 Present a problem: The number of feet (f) varies directly with the number of people (p) (2 people 4 feet, 3 people 6 feet, 4 people 8 feet, f = 2 p). Does the number of paws vary directly with the number of cats? What is the? A cat has five claws on each front paw and four claws on each back paw. Make a T-table showing the number of cats and the number of claws and then another T-table showing the number of paws (add one paw at a time!) and the number of claws. Does the total number of claws vary directly with the number of paws or with the number of cats? Assessment Circle the tables where the input varies directly with the : 5 2 6 3 7 2 4 3 6 4 8 6 2 8 8 0 24 2 6 4 2 5 5 Ask your students to draw or build a sequence of figures in which each figure in the sequence is made by adding a fixed number of blocks to the previous figure. Students should try to create one sequence in which the number of blocks varies directly with the figure number and one sequence in which the number of blocks does not vary directly with the figure number. ACTIVITY EXAMPLE: Figure Figure 2 Figure 3 Figure Figure 2 Figure 3 The number of blocks varies directly with the figure number The number of blocks does not vary directly with the figure number. For sequences where the number of blocks does not vary directly, ask students if they can figure out a rule for determining the number of blocks in a figure from the figure number. (A method for finding such rules is taught in the next three sections, but you might challenge your students to figure out how to derive such rules themselves. As a hint, you might tell the students that, in the case where the number of blocks does not vary directly with the figure number, their rule will involve multiplication and addition.) Extension Find a sequence of rectangles so that the area will vary directly with length. Does the perimeter vary directly with the length? 42 Copyright 2007, JUMP Math 6. TG Unit PA -2 p-48_r2.indd 42 28/07/207 :3:57 AM
WORKBOOK 6: PAGE 27 PA6-8 Finding Rules for Patterns Part II Students will find the varying part of the pattern rule of type t = 2n + a, where n is the figure number. Draw the following sequence of figures on the board and tell your students that the pictures show several stages in construction of a castle made of blocks. Direct variation T-table input direct variation equation How could they find a to keep track of the number of blocks needed for the castle? The castle has two towers and a gate between them. The gate does not change from figure to figure, but the towers grow. Ask your students to find the rule for the number of blocks in the towers (2 Figure Number). They can use a T-table to find the rule if needed. After that ask them to find a T-table for the total number of blocks in the figures. In which T-table does the number of blocks in the column vary directly with the figure number the T-table showing the number of blocks in the towers or the one showing the total number of blocks? Give your students several more block patterns, shade the part that varies directly with the figure number, as in the first question of the worksheet, and ask them to find the rule for the number of shaded blocks. Ask your students to create a sequence of figures that goes with the table and to shade the part of the figures that varies directly with the figure number. They should leave the part that does not change from figure to figure unshaded. ACTIVITY Figure Number Number of Blocks 5 2 8 3 Extension A cab charges a $3.50 flat rate (that you pay just for using the cab) and $2 for every minute of the ride. Write a for the price of a cab ride. How much will you pay for a 4-minute cab ride? For a 5-minute ride? WORKBOOK 6 Part Patterns & Algebra Copyright 2007, JUMP Math 43 6. TG Unit PA -2 p-48_r2.indd 43 28/07/207 :3:57 AM
WORKBOOK 6: PAGE 28 PA6-9 Predicting the Gap Between Terms in a Pattern Students will understand the connection between the gap in the sequence and the part that varies directly. Difference in sequences Direct variations Give your students a pair of dice of different colors each and ask them to try the following game. Roll the dice, and write a sequence according to the rule: to find each term multiply the term number by the result on the red die and add the result on the blue die. Ask your students to find the difference between the terms of their sequences each time. After they have created several sequences, ask them what they have noticed. Likely, the students have noticed that the difference (gap) in the sequence equals the result of the red die, which is the multiplicative factor. Draw or make the following sequence: Figure Figure 2 Figure 3 Ask students to describe what part of the pattern changes and what part stays the same? Draw a T-table for the number of blocks in each figure of the sequence as shown. T-table input direct variation gap difference Figure Number 2 3 Number of Blocks gap Ask students to predict what the gap between the terms in the column (the Number of Blocks column) will be before you fill in the column. Students should see that the gap between terms in the T-table is simply the number of new blocks added to the pattern at each stage. Students should also see that to find the number of shaded blocks in a particular figure they simply multiply the figure number by the gap (since the gap is the number of new shaded blocks added each time). In the next lesson, students will learn how to use this insight to find rules for patterns made by multiplication and addition (or multiplication and subtraction). ACTIVITY A Game for Pairs The students will need pattern blocks and two dice of different colors. Player rolls the dice so that player 2 does not see the result. He builds a pattern of figures according to the rule: The number of blocks = Figure number Result on the red die + Result on the blue die. Player 2 has to guess what the results on the dice were. 44 Copyright 2007, JUMP Math 6. TG Unit PA -2 p-48_r2.indd 44 28/07/207 :3:58 AM