Answers Investigation 3

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1 Answers Investigation Applications. a., b. s = n c. The numbers seem to be increasing b a greater amount each time. The square number increases b consecutive odd integers:,, 7,, c X X=. a.,,, b., X 7 X= Y Y c. The rectangular number increases b consecutive even integers:,,,, c d., 7; the 7th number is greater than the sith number and the th number is greater than the seventh number. e. r = n(n + ), where r is the rectangular number. ( + ). a. = () = 7 ( + ) b. Yes; = () = Note: Students ma also continue the table to answer this.. a. Sam is correct because if ou look at the triangles in Problem., each row of a triangle represents a counting number:,, c. Therefore, the equation for n(n + ) triangular numbers, represents the sum of the numbers to n. b. c. n(n + ) d.. Rectangular; it satisfies the equation r = n(n + ) where the th rectangular number () =.. Triangular; it satisfies the equation t = n(n + ) where the th triangular number: = (). 7. Square; it satisfies the equation s = n, where the th square number is =.. None; it does not satisf an of the associated equations.. a. The eight people on each side shake hands with others, so = handshakes will be echanged. b. ; = 7 * =, or each of people shake hands with the other 7, but as this counts each handshake twice, handshakes will be echanged. c. ; * =. a. ; * = b. ; = * =, or each of the people high five with the other, but as this counts each high five twice, high fives will be echanged.. a. ; = * =, or each of rooms connects with each other b cables, but as this counts two cables between each two rooms, cables will be needed. b. ; = * 7 =. Frogs, Fleas, and Painted Cubes Investigation

2 Answers Investigation c. The are the same situation mathematicall, where the cables are associated with high fives and rooms are associated with people.. Possible answers: P represents the area of a rectangle with dimensions n units b n - units. P could also represent the number of handshakes between a team of n plaers and another team of n - plaers.. Possible answer: P represents the area of a rectangle with dimensions units b n units.. Possible answer: P represents the area of a rectangle with dimensions n units b n - units.. Possible answer: P represents the area of a rectangle with a perimeter of units.. a. Graph II; Possible eplanation: The n(n - ) equation h =, which represents the relationship between the number of high fives and the number of team plaers, tells us that the graph will have -intercepts of and. b. Graph III; Possible eplanation: The area of rectangles with a fied perimeter increases and then decreases as the length of a side increases. c. Graph I; Possible eplanation: Since the equation for the relationship described is = ( + )( - ), the -intercepts must be - and. d. Graph IV; Possible eplanation: The nth triangular number can be represented n(n + ) b the equation T =. This equation that tells us the graph has two -intercepts, and This is not a quadratic function. Note: If the point (, -) were (, -), this would be a quadratic function.. This is a quadratic function with a minimum point.. This is a quadratic function with a minimum point.. This is not a quadratic function. Note: This has smmetr about the line =, but this has two linear segments; its equation is = +. Frogs, Fleas, and Painted Cubes Investigation

3 Answers Investigation. This is a quadratic function with a minimum point.. a. In each equation, second differences are constant, which means that all the equations are quadratic. The constant second differences for each equation are equal to a, where a is the coefficient of. = = 7 7 =... First First 7 First..... Second Second Second = First Second b. Since these are quadratic equations, second differences will be constant and will be equal to twice the number multiplied b. For = -, second differences will be -. For =, second differences will be. For = a, second differences will be a.. a. Table of (, ) Values b. = ( - ) = ( ) Because the maimum point is given, I can find the line of smmetr and complete the graph b plotting the corresponding point on the right side for each point on the left side. Frogs, Fleas, and Painted Cubes Investigation

4 Answers Investigation 7.. If ou etend the table, ou will get the following values: ( -, ), ( -, ), ( -, ), ( -, ), ( -, ). The second difference is. Connections. a.. a The epanded form is + + and the factored form is ( + )( + ). + + is easier to do because has a coefficient of one. b = ( + )( + ); + + = ( + )( + ) The epanded form is + + and the factored form is ( + )( + ). Note: The sides of area models cannot have negative values, so factored forms like ( - )( - ) aren t possible even though the give ou the terms and. b. No; the epanded form is + + and the factored form is ( + )( + ) or ( + )( + ) ( - )( - ). ( - ). ( + )( + ). ( + ). ( + )( - ). ( + )( - ) Frogs, Fleas, and Painted Cubes Investigation

5 Answers Investigation. a. Subdivide a rectangle into four parts. Label the area of one of the smaller rectangles as and the one diagonal to it as.. a. A pentagon has diagonals, a heagon has diagonals, a heptagon has diagonals and an octagon has diagonals. Use these areas to find the dimensions of these two smaller rectangles. The dimensions of the rectangle whose area is are and and the dimensions of the rectangle with area are and. The sum of the areas the other two rectangles should equal, so the are and. The dimensions of the original rectangle are + and +. Note: There are several was to do this. b. Look at the factors of the coefficient of and the factors of the constant term. Put these values in the factor pairs: (? +?)(? +?). Use the Distributive Propert to check if the coefficient of is correct. n(n - ) b. An n-sided polgon has diagonals. This problem could be solved like the high fives problem. Each of n points echanges high fives, or shares a diagonal, with the other n - points (ecluding the adjacent points and itself). Since this counts each high five twice, the epression is divided b.. a. The first train has rectangle, the square itself. The second train has rectangles, as shown below. The third train has rectangles, as in the problem. The fourth train has rectangles, as shown below. The fifth train has rectangles ( one-square rectangles, two-squares rectangles, three-square rectangles, four-square rectangles and five-square rectangle.) (See Figure.) Figure nd train th train Frogs, Fleas, and Painted Cubes Investigation

6 Answers Investigation b. Number of Rectangles in the First Ten Trains Train 7 Number of Rectangles 7. a. 7. cm a. b. 7. centimeter cubes c. laers d. 7 cm ; this is the product of the number of cubes in one laer and the number of laers that fill the can. e.. cm f. cm g. 7 cm ; this is the sum of twice the area of the base and the area of the paper label. Bo A cm.7 cm c. The number of rectangles increases b a greater amount each time. The pattern of increase is,,,,,c So, we could epect an increase of from the th train to the th train. The table below shows that there are rectangles in the th train. cm cm cm.7 cm Number of Rectangles in Trains Through Bo B Train Number of Rectangles. cm 7. cm. cm n(n + ) d. r = where r is the number of rectangles. Note: The numbers are the same as the triangular numbers, an observation that students ma use to solve this equation. ( + ) e. r = = () = rectangles Frogs, Fleas, and Painted Cubes Investigation

7 Answers Investigation b. The surface area of Bo A =.7 cm, and the surface area of Bo B =. cm. To find the surface area of Bo A, add the area of the base, * = cm, the total area of the two triangles, (. * ) = cm, and the total area for the two side rectangles, ( *.7) =.7. (B the Pthagorean Theorem, the hpotenuse of the triangle and the other dimension of the side rectangles is. +.7.) So, the surface area is =.7 cm. To find the surface area of Bo B, add the total area of the two circles, * p(.). cm, to the area of the rectangle,. * ( * p *.). cm. So the surface area is. +. =. cm. c. Bo A will require more cardboard to construct, since it has a greater surface area.. None of the above.. Quadratic. Eponential. None of the above. D. H; b looking at the coefficient of, H is the onl parabola that can have a minimum point because the coefficient is positive. The other three parabolas open down, because their coefficients are negative.. a. - + b. - Etensions. a. ( + ) + ( + ) + c( + ) + ( + ) = () =,. b. This idea could be represented b the n(n + ) equation s =, where s is the sum of the first s whole numbers. c. This method is the same as Gauss s method in the Did You Know? It pairs the numbers in a number sentence, which doubles the value. Then it divides b to get the sum. 7. a.,, b.,, c. s = (n + ) n(n + ) + Note: Students ma not be able to find this equation. You might help them to write an equation b pointing out that each star has a center square and four triangular points. Each star number n is thus composed of times the nth triangular number plus the (n + )th square number. For eample, the first star number has a center of (the second square number) and four points of (the first triangular number); the second star number has a center of (the third square number) and four points of (the second triangular number). st rd nd 7 Frogs, Fleas, and Painted Cubes Investigation

8 Answers Investigation. a. ; () =, or each of the classmates shakes hands with others, but as this counts each handshake twice, handshakes occur. This answer can also be epressed in different tpes of diagrams. b. Each of the classmates shakes hands with others, but as one handshake is counted twice (when the shake hands with each other), there are () - = handshakes in all. This answer can also be epressed in different tpes of diagrams a.,, b., People: A, B, C, D, E, F, G, H, I, J Handshakes: AB, AC, AD, AE, AF, AG, AH, AI, AJ BC, BD, BE, BF, BG, BH, BI, BJ CD, CE, CF, CG, CH, CI, CJ DE, DF, DG, DH, DI, DJ EF, EG, EH, EI, EJ FG, FH, FI, FJ GH, GI, GJ HI, HJ IJ Frogs, Fleas, and Painted Cubes Investigation

9 Answers Investigation c. h = n(n - ). Note: Students ma not be able to find this equation. You might help them write an equation b pointing out that there are four lines along which the can add c + n dots. The equation for the triangular numbers, n(n + ), is multiplied b to get this sum. However, this counts the dots circled on the diagram an etra times for each heagon, a total of n times for the whole figure. Thus, the complete equation for heagonal n(n + ) numbers is h = - n = n + n - n = n - n = n(n - ). b. Possible answer: The squares found in a -b- grid, a -b- grid and a -b- grid are shown below. Each grid contains n more squares than the previous grid, so an equation for the number of squares in an n-b-n grid is s = n + (n - ) + c +, where s is the number of squares. b grid b grid square: squares: of of of b grid. a. Of the remaining squares, are -b- squares, are -b- squares, and is a -b- square. squares: of of of This shows that the pattern is a sum of squares. Frogs, Fleas, and Painted Cubes Investigation

10 Answers Investigation. a. The graph of = + is a straight line with slope and -intercept (, ). The graph of = ( + )( + ) is a parabola with a minimum point at ( -., -.) and -intercepts at ( -, ) and ( -, ). The graph of = ( + )( + )( + ) increases as increases, then decreases, then increases again. It has three -intercepts at ( -, ), ( -, ), ( -, ). The graph of = ( + )( + )( + ) ( + ) is shaped like the letter W. It has two local minimum points, a local maimum point, and four -intercepts at ( -, ), ( -, ), ( -, ), ( -, ). Note: The terms local minimum and local maimum will be introduced in future mathematics courses. The refer to minimums and maimums over a given part of the graph, which are not necessaril the minimum or maimum for the entire graph. b. The equation = ( + ) has constant first differences. The equation = ( + )( + ) has constant second differences. The equation = ( + ) ( + )( + ) has constant third differences. The equation = ( + ) ( + )( + )( + ) has constant fourth differences. Frogs, Fleas, and Painted Cubes Investigation