Answers Investigation ACE Assignment Choices Problem. Core,,, Other Applications ; Connections, ; Etensions 7, ; unassigned choices from previous problems Problem. Core, Other Connections 7; Etensions ; unassigned choices from previous problems Problem. Core, Other Applications 7 ; Connections, ; Etensions, ; unassigned choices from previous problems Adapted For suggestions about adapting Eercise and other ACE eercises, see the CMP Special Needs Handbook. Connecting to Prior Units 7: Shapes and Designs;, : Filling and Wrapping Applications. a. and b. n c. The numbers seem to be getting bigger by a larger amount each time. The square number increases by consecutive odd integers, beginning with :,, 7,... X X= X 7 X= Y Y. a.,,,,, b. The rectangular number increases by consecutive even integers beginning with :,,,,,... c. The seventh number is greater than the sith number, or. The eighth number is greater than the seventh number, or 7. d. r = n(n + ), where r is the rectangular number. ( ) (). a. = = 7 b. yes; It is a triangular number because ( ) () = = (Note: students may substitute into the equation or continue the table to answer this.). a. Sam is correct because if you look at the triangles in Problem., each row of a triangle represents a counting number:,,...therefore the equation for n(n ) triangular numbers,, represents the sum of the numbers to n. b. c. d. n(n ) ACE ANSWERS Investigation Quadratic Patterns of Change 7
. Rectangular, because it satisfies the equation r = n(n + ), is the tenth rectangular number. = (). Triangular, because it satisfies the equation n(n ) t =, is the th triangular number: () =. 7. Square, because it satisfies the equation s = n, is the th square number: =.. None, because it does not satisfy any of the associated equations, is none of these.. a. The eight people on each side shake hands with others, so = handshakes will be echanged. 7() b.. 7 + + + + + + = =, or each of people shake hands with the other 7, but as this counts each (7) handshake twice, = handshakes will be echanged. c.. =. a.. = () b.. + + + = =, or each of 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 by cables, but as this counts two cables () between each two rooms, = cables will be needed. (7) b.. + + + + + = =. c. They are the same situation mathematically, where the cables are associated with high fives and rooms are associated with people.. Possible answer: P represents the area of a rectangle formed from a square of a side length n in which one dimension is decreased by unit, or P represents the number of handshakes between a team of n players and a team of n - players..possible answer: P represents the area of a rectangle with sides of length and n.. Possible answer: P represents the area of a rectangle formed from a square of a side Frogs, Fleas, and Painted Cubes length n in which one dimension is decreased by units.. Possible answer: P represents the area of a rectangle with perimeter units.. a. Graph ii; Possible eplanation: The n equation h = (n - ), which represents the relationship between the number of high fives and the number of team players, tells us that the graph will have -intercepts of and. b. Graph iii; Possible eplanation: The area of rectangles with a fied perimeter grows and then declines as the length of a side increases. That means the graph has a maimum point. c. Graph i; possible eplanation: Since the equation for the relationship described is y = ( + )( ), we know that y = when =- or. So, the -intercepts must be - and. d. Graph iv; Possible eplanation: The nth triangular number can be represented by n(n ) the equation T =. This equation that tells us the graph will have two -intercepts of and -. 7. Quadratic Relationship. Quadratic Relationship y y
. Quadratic Relationship y. a. + 7 + + + Connections. a. Epanded form: + + ; factored form: ( + )( + ) and epanded form: + + ; Factored form: ( + )( + ). We can t have negative values for sides given the area models in the student edition so factored forms like ( - )( - ) aren t possible even though they give you the terms and. b. No; epanded form: + + ; Factored form: ( + )( + ) (ecluding commutation of the factors of + and ( + ). Again there are more possibilities if we use non-whole-number factors of.. + + + + = + + or ( + )( + ) b. + 7 + = ( + )( + ); + + = ( + )( + ); + + is easier to do because has a coefficient of one.. ( - ) = -. ( + )( + ) = + +. ( - )( + ) = + -. ( + ) = + 7. ( + )( + ) = + 7 +. ( - )( + ) = + -. - + = ( - )( - ). - = ( - ). - - = ( - )( + ). + + = ( + )( + ). + = ( + ). - - = ( + )( - ). - - = ( + )( - ). a. Subdivide a rectangle into four parts. Label the area of one of the smaller rectangles as and the one diagonal to it as. Use these areas to find the dimensions of these two smaller rectangles. (Note that there are several ways to do this.) Once you have picked the dimensions, use them to find the area of the remaining two rectangles. If the sum of the area of these two rectangles is, then you picked the right ACE ANSWERS Investigation Quadratic Patterns of Change
dimensions. And the dimensions that you picked for the first two rectangles are the dimensions of the original rectangle. These two dimensions are the factors in the factored form of + +. Let the dimensions of the rectangle whose area is be and and the dimensions of the rectangle with area be and. The areas of the remaining two rectangles are and and their sum is. The dimensions of the original rectangle are + and +. Then write as +. Label the area of the two remaining rectangles as and. 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 property to check if the coefficient of is correct. 7. a. A pentagon has diagonals, a heagon has diagonals, a heptagon has diagonals and an octagon has diagonals. n(n ) b. An n-sided polygon has diagonals. This problem could be solved like the high fives problem. Each of n points echanges high fives with the other n - points (diagonals) ecluding the adjacent points and itself but as this counts each high five n(n ) twice, high fives will be echanged.. 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.) (Figure ) Figure nd train th train Frogs, Fleas, and Painted Cubes
b. Number of Rectangles in the First Ten Trains Train 7 Number of Rectangles 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.. a. Possible net: Bo A cm.7 cm cm cm cm.7 cm c. The number of rectangles increases by a greater amount each time. The pattern of increase is,,,,,...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. Bo B. cm Number of Rectangles in Trains Through Train Number of Rectangles 7 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 may use to solve this equation). ( ) () e. r = = = = rectangles. a. About 7. cm b. About 7. centimeter cubes c. layers d. < 7 cm ; this is the product of the number of cubes in one layer and the number of layers that fill the can.. cm. cm b. Bo A:.7 cm, Bo B:. cm ; To find the surface area of bo A you would have to add the area of the base which is = cm, the area of the two triangles each of which has an area of ( ) = cm and the areas for the side rectangles. The rectangles have dimensions and about.7. To find the.7 you use the Pythagorean theorem on the right triangle on the front of the bo with the side lengths of and to obtain the hypotenuse of ". =.7. So the surface area is + () + (.7) =.7 cm. To find the surface area of Bo B, you would have to add the area of the top and bottom two circles, which each have an area of p(.). cm, to the area of the ACE ANSWERS Investigation Quadratic Patterns of Change
rectangle which has an area of. p(.). cm. So the surface area is. +. =. cm. c. Bo A will require more cardboard to construct, since it has a larger surface area.. None of the above.. Quadratic. Eponential. None of the above. D. H; one way to eliminate certain choices is to notice that H is the only parabola that can have a minimum point since the other three parabolas open down. You can see this by looking at the coefficient of. st nd Etensions 7. a. ( + ) + ( + ) +... + ( + ) + ( + ) = () = =, b. This idea could be represented by the n equation s = (n + ), 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? bo. It just pairs the numbers in a number sentence rather than drawing arrows to make the pairings.. a.,, b.,, ; + = ; + = ; 7 7 + =. c. s = (n + ) + n(n ) For the Teacher: Students may not be able to find this equation. You might help them to write an equation by pointing out that each star has a center square and four triangular points. Each star number is thus composed of times the triangular number of the same number plus the net 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).. a..; + + 7 + + + + + + = () =, or each of the classmates shakes hands with others, but as this () counts each handshake twice, = handshakes echanged. Three diagrams that epress this are shown below. rd 7 Frogs, Fleas, and Painted Cubes
7 People: Handshakes: 7 A, B, C, D, E, F, G, H, I, J 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 b. Each of the friends shakes hands with others, but as one handshake is counted twice (when they shake hands with each other), there are () - = handshakes in all. Three diagrams that epress this are shown here.. a.,, b., c. h = n(n - ). For the teacher: Students may not be able to find this equation. You might help them write an equation by pointing out that there are four lines along which they can add + + +... + n dots. The equation for triangular numbers, n(n ), is multiplied by to get this sum. However, this counts the dots circled on the diagram an etra times for each heagon, for a total of n times for the whole figure. Thus, the complete equation for heagonal numbers is as n(n ) follows: h = ( ) - n = n + n - n = n - n = n(n - ). a. Of the remaining squares, are -by- squares, are -by- squares, and is a -by- square. ACE ANSWERS 7 Investigation Quadratic Patterns of Change
b. Possible answer: The squares found in a -by- grid, a -by- grid and a -by- 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-by-n grid is s = n + (n - ) +...+, where s is the number of squares. by grid by grid This shows that the pattern is a sum of squares. Possible Answers to Mathematical Reflections. a. The relationships in the handshake and triangular number problems have similar graphs and tables in which y increases at an increasing rate. The high-five situation and the triangular number pattern are applications of the same formula. However, one formula is n square: (n ). by grid squares: n (n ) squares: of of of of of of and the other is b. These functions are the same as in Investigations and, which also were epressed as the product of factors. The shape of the graphs are similar either a parabola that opens up or a parabola that opens down, with a minimum value or maimum value. Students might suggest things like this: In the tables a constant change in does not lead to a constant change in y as it did for linear relations, nor is there an inverse relation as there was for some of the investigations in Thinking with Mathematical Models.The graphs are all parabolas when viewed in the first and second quadrants. The equations that represent quadratic relations all have an (or n ) as the highest power of the input variable. However, students will not always see them in this form. They have seen epressions such as n(n + ) or n(n - ) or (n - ) n(n ) n(n ) or or. At this point they should not be epected to discuss the degree of the equation, but they should begin to notice some similarities in all of the epressions such as an epression with n is multiplied by another epression having n in it.. a. The rate of change for a sequence of numbers associated with a quadratic equation is not constant. However, the rate of change for each increment or increase seems to be constant. For eample, in the triangular numbers,,,,,,,..., the increases are +, +, +, +, +,... Each increase is one more than the last increase, and this one more is constant. For the handshake pattern, the increases are,, 7,,,...,and each increase is two more than the previous one. To predict the net entry you determine how the increments in the y-values are changing and continue the pattern. Thus, for the handshake pattern, if we know the sequence of y-values is..., 7,,, we can predict the net entry by adding to get. b. If a table of values indicates a symmetric increase/decrease or decrease/increase pattern, and if the change in y-values is increasing (or decreasing) by a fied amount, then the function is quadratic. Frogs, Fleas, and Painted Cubes