Our first Solution from North Cyprus came from Mr Atkinson English School of Kyrenia: My class tried this. Only about half of them could describe a rule to figure out the number of bricks in a row for any number. It seemed as if it took them a lot longer to complete the chart than I would have expected. It’s always good to start off with some test cases. even more serious falling hazard if the stairs are sloped. Because of the length of time used to fill in the chart, most groups did not have enough time to really do justice to answering the six questions posed in the worksheet. Justify why your formula works. This activity went very good. Since the staircase starts with a rise up to the first tread and there is one more rise from the last tread up to the next floor, I always have one less tread than the number of rises. No one used variables to describe it (even though we have done a lot of work with variables in this pre-algebra class.) All of the groups eventually came up with a plan that allowed them to get the chart filled in. Please help solve the below word problem, The Staircase Pattern - Math for Understanding - patterning for We are creating videos that Duration: 2:02 Posted: Mar 17, 2016 numWays (1) =1. How many blocks do you think would be in a 3-step up-and-down staircase? This answer also fulfills the alternative formula, because 7.06 inches times two is 14.12 inches, and 14.12 plus 10.44 is 24.56, which falls between 24 and 25 inches. There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. ... the number of blocks at each step in the three-step staircase and the total number of blocks for the entire staircase. Our first Solution from North Cyprus came from Mr Atkinson English School of Kyrenia: My class tried this. They again made tables like the following: Creating a formula was challenging for them. Mathematically speaking, a step function is a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in … I had students work in pairs on each activity for about 5- 10 minutes and then we discussed each part as a group. It's one of those problems where 2 steps is 3 blocks, 4 steps is 10 blocks, 5 steps is 15 blocks etc. The next time I have them do this activity I will have them work in pairs or in groups of three. Of course, this is really slow (O(2^N)) – we are doing a lot of repeated computations! ... the number of blocks at each step in the three-step staircase and the total number of blocks for the entire staircase. Groups: It is important to talk about Problem D2 as a whole group, as this question underscores the importance and convenience of the recursive formula. It was a short class, so students had about 20 minutes to work on it. formula 2 risers + 1 run = 23" to 24". Check out our newsletter, Daily Coding Problem, to get a question in your inbox every day. About ¼ of the students could figure out how to find the total number of bricks in a tower when they knew how many rows there were. Get a coding problem every day in your inbox! The step length must be between 56 and 67 cm. Staircase sequences Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Printable/supporting materials Printable version Fullscreen mode Teacher notes Similar reasoning tells us that if X = {1, 3, 5}, then our algorithm should be f(n) = f(n - 1) + f(n - 3) + f(n - 5). I showed them how to use variables for this particular problem. The extra challenges in this puzzle make a link to Task 18, Same or Different , and a template for … Exploring. And what we had to … Staircase sequences Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource Printable/supporting materials Printable version Fullscreen mode Teacher notes Give the students time to work out the number o… • 6. Towering numbers. So f(4) = f(3) + f(2). Let’s work through the following problem. I decided to try to do one of the “Algebraic Strategies” activities (Sum of Consecutive Numbers) with two of my calculus classes on a Friday afternoon after having taken a chapter test the previous day. When designing/building and fitting staircases formulas are used to ensure the treads and risers are the right size and comply with local building codes. For example, if X = {1, 3, 5}, you could climb 1, 3, or 5 steps at a time. Students were all able to come up with the pattern (nth table top has n2 blocks) very quickly. It's (stairs)* (stairs+1). I used The Staircase Problem / Towers / Fancy Staircases from the Algebraic Thinking class in my HOTS class. They are again asked to find a pattern. This made us realise how much could be done with algebra and how useful it is. I am looking for a formula that I can use in Excel to determine the correct radius of a handrail for a curved staircase. When we got together as a class during the last ten minutes to discuss any patterns they discovered, both classes made the comment that they could see patterns but that they had a difficult time putting the patterns down on paper as an algebraic expression of some type. This made us realise how much could be done with algebra and how useful it is. Act Three For me doing a vertical layout on a story pole helps me double check my math. Total Required Number of Tread = Total Stair Tread or Run/ width of one Tread = 90” / 10” = 9 Tread. It's one of those problems where 2 steps is 3 blocks, 4 steps is 10 blocks, 5 steps is 15 blocks etc. One of the groups immediately saw a pattern in the staircases and computed the answers. The problem was the stair problem where you can climb either 1 step or 2 steps at a time. Approach: For the generalization of … However, if the students prefer to use cubes, then they should record their results in a table. There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. They did the first one done by using the picture. Then, we’ll build up the array from zero using the same recurrence as before: This now takes O(N * |X|) time and O(N) space. One group did mention that they noticed that if they multiplied the middle number in a sequence by the number of numbers in the sequence that that would give them the sum. At this point, our meeting time was over, but we still wanted to see how close our predictions … Two of the groups concluded the formula for the nth tower as: 2n^2 - n. Question 118864: A set of staircases grows at a certain rate.If the rule to find out how many blocks are needed total to make a staircase with n number of steps is (n) (n)/2+1/2n=y then what is the rule to find out the number of steps in the staircase if y is given? Carpenters often use formulas and math when carrying out carpentry jobs such as roofing, to check a building is square and to calculate the length of rafters etc. They definitely had a hard time abstracting from the computation. For clear understanding, we are considering the below example of the doglegged staircase. Have struggling students create a table and track "stairs" and "steps on that trip" and "total steps after those trips." They continually evaluate the reasonableness of their intermediate results. For example, if N is 4, then there are 5 unique ways: What if, instead of being able to climb 1 or 2 steps at a time, you could climb any number from a set of positive integers X? numWays (N) = numWays (N-1)+numWays (N-2) This is same as the Fibonacci sequence formula. After students have an opportunity to draw and describe how they see the shape changing they are ready to engage in group work and further study. I need a rule that given y number of blocks you can tell how many steps are in the staircase. Handrails & Guards/Guardrails: A handrail is a railing that runs up a stair incline for users to … Some just took a number at random and tried to express it as different sums. Please make use of the below calculator (Input the Values in Inches) Over the next 2-3 days the students work in pairs or individually to solve the following problems. For my expansion of the staircase problem, I created a different pattern and set out to find an equation. Their goal will be to find a number rule that turns "stairs" into "total steps. Yes, it does. Begin the session by telling the students about up-and-down staircases: 2. That’s just the Fibonacci sequence, except shifted by one. ex.n(n)/2+1/2n=105 (solve for n) Answer by MathLover1(17568) (Show Source): We can do it a lot faster by just computing iteratively: Now, let’s try to generalize what we’ve learned so that it works if you can take a number of steps from the set X. The total run = the number of runs (treads) times the tread length. N = 3, 3 ways to climb: [1, 2], [1, 1, 1], [2, 1], N = 4, 5 ways to climb: [1, 1, 2], [2, 2], [1, 2, 1], [1, 1, 1, 1], [2, 1, 1]. The first part is entitled “Growing Squares” and uses table tops made out of square blocks. And only 2 or 3 of those could describe the rule in words. For the best results, aim between 62 and 64cm. Does this hold for N = 4? Steps in the calculation of the volume of concrete required for the staircase: Each component of the staircase is individually calculated. Common Core State Standards Math - Content Standards This formula is three times the formula for calculating triangular numbers – (n² + n)/2 I need a rule that given y number of blocks you can tell how many steps are in the staircase. The step length can be solved using Blondel's Formula: add the tread length to the height of two risers. From there we found that the formula would be 5 super stairs/ 16 super stairs = 21 super stairs / x seconds = 67.2 seconds. Staircase also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. I was somewhat surprised that a few of the groups started off filling in their charts in a quite disorganized fashion. While the students worked on the activity, I tried to walk around the classroom and listen to the discussions that were going on in the individual groups. Generalization of the Problem How to count the number of ways if the person can climb up to m stairs for a given value m. For example, if m is 4, the person can climb 1 stair or 2 stairs or 3 stairs or 4 stairs at a time. Approach: For the generalization of above approach the following recursive relation can be used. Generalization of the Problem How to count the number of ways if the person can climb up to m stairs for a given value m. For example, if m is 4, the person can climb 1 stair or 2 stairs or 3 stairs or 4 stairs at a time. Problem of the Month: Growing Staircases Overview: In the Problem of the Month Growing Staircases, students use algebraic thinking to solve problems involving patterns, sequences, generalizations, and linear and non-linear functions. The objective of the activity was to find all the possible ways to express each number from 1 to 35 as a sum of two or more consecutive counting numbers. Staircase Calculator. This was a pattern she recognized from Gauss and the Handshake problem. Squares To Stairs. To introduce this task ask students to think on their own about how they see the shape growing. So f(3) = f(2) + f(1). Everyone has different ways of working. To get a more comfortable staircase, it is best to have the treads around 30 cm long and the risers 15 to 20cm tall. How many combinations are there to get to the 10th step. The fancy stairs were very difficult to take to an abstract level, but seem to become easier if you break time into “odd fancies” and “even fancies”. Do you notice anything? The order of the steps matters. This formula will help you to design a staircase correctly. • Step stair riser openings: open stair risers are permitted provided the opening will not pass a 4" sphere (child safety). There must be a minimum of 36" of landing by the width of the stairs at both the top and bottom of each stairway. Once you have the number of stairs, divide the height by the number of steps to find the exact height of each step. Using these patterns, they were then asked to make predictions as to whether given numbers greater than 35 could be expressed as a sum of 2, 3, 4, or more consecutive counting numbers. Today we explore up-and-down staircases to find the pattern in the number of blocks they are made from. + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. Hopefully, some will be able to on the next exercise. Overall, I was disappointed in the results of the activity. For most stairways the landing is … Each entry cache[i] will contain the number of ways we can get to step i with the set X. 3. I began to manipulate this pattern by drawing it in a similar configuration to the staircase. When designing/building and fitting staircases formulas are used to ensure the treads and risers are … I found the data set that solves the answer to the question, but is there an equation that expresses the answer in terms of n? To keep track of the number of blocks/cubes in each staircase it might be useful to draw the staircases on graph paper. Step Function. I found the data set that solves the answer to the question, but is there an equation that expresses the answer in terms of n? Are you interviewing for programming jobs, or do you just enjoy fun programming questions? Most of the groups made a table of values similar to the following: The third part, “Towers”, was more challenging. Here is what they found: 25 blocks make an up-and-down staircase with 5 steps up and 5 steps down. The order of the steps matters. When we got to the third part to find a rule the faster students had it right away, but were so eager to tell the other students that they didn’t have the chance to think of it on their own. They then tried to use the previous formula from the staircases here in this problem as well. All the below-mentioned values in the calculation are considered from this image. I think things went well and I will do towering numbers next year. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. We can use dynamic programming to speed it up. How many blocks are in the staircase? Using the dissected figures, I was able to use my equation for the staircase as a foundation. They then tried to use the previous formula from the staircases here in this problem as well. Let’s work through the following problem. To solve this problem I decided to start with a low number of stairs, like $2$. I had them work in groups of two in one class and in groups of three students in the other. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. Staircase Calculation Formula is . It … Finally, using the first formula for run, subtract 7.06 from 17.5 to find that each stair measures 10.44 inches in depth. Let’s start with small cases and see if we can find some sort of pattern. Given N, write a function that returns the number of unique ways you can climb the staircase.The order of the steps matters. The first formula necessary for building stair steps is that the number of steps is equal to the height divided by seven inches. The problem was the stair problem where you can climb either 1 step or 2 steps at a time. However, calculations should always consider the specificities of each project, as well as local regulations in … I did not find any changes that I would make. I continue doing that and I noticed that the numbers of ways for a particular number of stairs was the sum of the numbers of ways to climb the stair for the previous two numbers of stairs. Staircase also appears on the Picture Puzzles Pattern & Algebra A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. The second group, while having less formal math training, actually attempted to create an algebraic formula. The activity actually has three main parts to it. For example, if N is 4, then there are 5 … It was cumbersome and ugly – but it worked. Please help solve the below word problem, The Staircase Pattern - Math for Understanding - patterning for We are creating videos that Duration: 2:02 Posted: Mar 17, 2016 numWays (1) =1. For example, if N is 4, then there are 5 unique ways: 1, 1, 1, 1; 2, 1, 1; I told the students that they had 40 minutes to look at the chart and the follow-up questions and then we would get together during the last 10 minutes of class to discuss the activity. Here is what they found: 25 blocks make an up-and-down staircase with 5 steps up and 5 steps down. Given N, write a function that returns the number of unique ways you can climb the staircase. Problem of the Month Growing Staircases ... of blocks needed to build a staircase with n stairs. At first, they had questions about whether they could use the number zero or negative numbers and had to be reminded what a “counting number” was. Stair/Rail Angle - the angle is most useful for determining the bevel cut on a stair rail post. What’s the relationship?The only ways to get to N = 3, is to first get to N = 1, and then go up by 2 steps, or get to N = 2 and go up by 1 step. Justify why your formula works. Example; Staircase has a run of 12.00" and a rise of 7.375" Staircase is against a wall that is curved with a consistent radius of 174' 11.75" Problem; What is the correct radius to bend the handrail to? So I took $2$ and worked out how many solutions there were. While they could describe the rule, they could not put it into an algebraic form. If I were to do this activity again, I would probably either spend a little more time at the beginning giving them more detailed directions or maybe go through a short, similar type of activity with them first. So the relationship looks like this: f(n) = f(n - 1) + f(n - 2), and f(1) = 1 and f(2) = 2. I have 5 students in the class this semester, which I divided into 2 groups. Consequently, the algebraic formula would be n². 1. Again, no one used variables. If you could please help out and explain how to find the answer, not necessarily just giving me the answer, than it would be much appreciated. If you could please help out and explain how to find the answer, not necessarily just giving me the answer, than it would be much appreciated. The extra challenges in this puzzle make a link to Task 18, Same or Different , and a template for creating new Triangle Numbers for smaller ones. Carpenters often use formulas and math when carrying out carpentry jobs such as roofing, to check a building is square and to calculate the length of rafters etc. Almost all could figure out the number of bricks in a row when they knew the actual row number. Have a look at it. Together count the steps so that the students understand why it is called a 2-step staircase. (HOTS stands for Higher Order Thinking Skills and is a non-mandatory mini math class that we offer opposite band where we play with math topics as well as puzzles and thinking games. Overall we spent anywhere from about 45 minutes for the fastest (least abstract thinking) group to 90 minutes for the group that really tried to go to the abstract. Maggie and Cynthia arrived at the same formula but came to it differently. The second part entitled “The Staircase Problem” uses pictures of staircases that have more and more steps. The first table top has one block, the second table top has four blocks, the third table top has nine blocks, and so on. Two of the groups concluded the formula for the nth tower as: 2n^2 - n. During the last few minutes of the class period we worked together as a class to see how this formula could be derived. Common Core State Standards Math - Content Standards I would probably also give them the entire period to work on it and then have them write something up and maybe spend the first 10-15 minutes of the next day’s class period discussing their results. Given N, write a function that returns the number of unique ways you can climb the staircase. How could you work it out? The mathematical topics that underlie this POM are finding and The same is true for the sea foam green and the fern green. Let’s work through the following problem. How many combinations are there to get to the 10th step. numWays (N) = numWays (N-1)+numWays (N-2) This is same as the Fibonacci sequence formula. If n < 0, then we should return 0 since we can’t start from a negative number of steps. The Staircase Problem / Towers / Fancy Staircases, The Staircase Problem -Towers (“Algebraic Strategies” activities). There exists a staircase with N steps, and you can climb up either 1 or 2 steps at a time. This used three-dimensional shapes. I then dissected it into two, smaller staircases. Total Required Number of Risers = Total Stair Rise / Height of One Rise = 60” / 6” = 10 Riser. This is again, very slow (O(|X|^N)) since we are repeating computations again. Most looked at each tower as a column surrounded by four staircases, when they calculated the number of blocks to be used. On the towers they developed strategies to compute the 1, 2, 3, 4, and 10th towers. They were somewhat frustrated with the what their results looked like after working the whole period on it so I sat down and we made it nicer looking together – but pointed out that it was the same thing that they created. For example – in the 4th figure above, the right side of the triangle has 4 black toothpicks, followed by 3, then 2, then 1. There are 21 stairs. I went over the problem the next class day and we talked about using variables. Consequently, the algebraic formula would be n². Problem of the Month Growing Staircases ... of blocks needed to build a staircase with n stairs. They were given a chart to fill in and then were to answer some questions about patterns they discovered while completing the chart. When we look at N = 3, the number of ways to get to 3 steps is 3, and they’re based off N = 1 and N = 2. Creating a formula was challenging for them. Solange multiplied that formula by 2 and came up with n (n + 1), or n² + n. She also represented this visually, by drawing the stairs and values of n² and n. Using her formula, Solange determined that the man would take 462 steps altogether. Since we can only get to the 4th step by getting to the 3rd step and going up by one, or by getting to the 2nd step and going up by two. If you know the number of stairs in the nth staircase, the number of stairs in the next staircase can be … I told them that once they had these done I had a story to tell them that might help them with the 100th (since they haven’t learned about arithmetic sequences yet) and then related the fable of Gauss and his teacher asking him to add all the numbers of 1 to 100 and how he arrived at the added the sum forward and backwards etc… It was a nice extension and eased some of the arithmetic while still concentrating on the patterns of the towers. This 4-inch opening dimension has the same basis as the rule that requires that the opening between stair balusters must be 4" or smaller. This is a great task. They again made tables like the following problems use my equation for the generalization of … problem the. Just took a number at random and tried to express it as different sums from Gauss the. We discussed each part as a foundation began to manipulate this pattern by drawing it a. 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