Staircase Lesson Plan

 

 

 

 

 

 

 

 

 

 


On an overhead transparency, display the first 3 figures of a pattern. Ask a participant to describe the pattern. Then ask if anyone saw it any differently? Ask an additional person to describe how they see the pattern. The descriptions may include (but are not limited to):

  1. In the second figure, 2 blocks have been added to the right side of figure 1. In the third figure, 3 blocks have been added to the right side of figure 2.
  2. In the second figure, 2 blocks have been added underneath figure 1. In the third figure, 3 blocks have been added underneath figure 2.
  3. In the second figure, 2 blocks have been added like stairs to figure 1. One block was added to the left and one block was added on top. In the third figure, 3 blocks have been added like stairs to figure 2. One block was added to the left, one block was added in the middle and one block was added on top.

 

If no one has mentioned it, ask how many blocks are in each figure. (1, 3 and 6 respectively)

 

Ask: Can anyone describe what the 4th figure will look like? (Be sure to give participants time to think). Using one of the previous descriptions, a participant may say: You would add 4 blocks to the 3rd figure (to the right, below, or in stair steps). How many blocks does it have? (10) Can you build it? Wait for participants to build figure 4 with their blocks.

 

Ask: Can somebody tell me how many blocks figure 5 is going to have? (15) How did you know? (added 5 to the 4th figure and it had 10). Can you build it? Wait for participants to build figure 4 with their blocks.

 

If this pattern were to continue, what do you think the 10th figure would look like? Who can describe it? (A participant may say it will have 10 columns of heights 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10) Ask if others agree. Can somebody describe the 20th figure? (A participant may say it will have 20 columns of heights 1, 2, 3, 4, 5, 6, 7, 8, 9… 20)

 

Who thinks they could describe the 50th figure? (a ‘triangle’ figure with 50 blocks on the bottom and 50 blocks in the right column. Each column decreases from 50 by 1 down to the last column on the far left side with 1 block in it). How many cubes are in the 50th figure? How can we figure it out…If the pattern is to continue, how many blocks are needed to make the 50th staircase?

 

We don’t have all day to count each block one at a time and it would take almost as long to add each column together (50 total). So without using one of those methods, think about (and share) ways you could go about determining the total number of cubes in the 50th pattern.

 

Have students share some ideas. The purpose of sharing ideas is to give everyone a place to ‘start’. Groups with no clue may get an idea of where to begin. Groups with the wrong idea may be able to correct their thinking and head in the right direction. You don’t want to throw a problem out there and wait for a miracle to happen. It’s okay to give some help to get started.

 

Tell groups that their task is to determine the total number of cubes in the 50th arrangement without having to count the blocks one by one.

 

Here it is important for the instructor to step back and let the groups attempt the problem without interference. In other words, Don’t spy. The instructor must determine when it is appropriate to start questioning methods the groups are using to solve the problem. It is NOT the instructor’s job to ‘fix’ each group’s mistakes by telling them what they are doing wrong and what they should be doing. It IS the instructor’s job to ask appropriate questions in order to determine what the groups are thinking and to help THEM establish whether they are on the correct path to solve the problem. Ideally, the instructor will relate what the students know/prior knowledge (based on their efforts with the problem), in order to offer guidance and suggestions. The questions might include:

 

-         Explain to me how this group is solving the problem. (Depending on whether or not the students are on the right track or not will determine the follow-up question).

-         Wrong idea but don’t realize they are wrong: Will your method work for figure 8? Let’s check and see.

-         Students who still don’t know where to begin: Let’s look back at figure 6. What is a way other than counting each block that you could determine the total number of blocks in the figure?

-         If a group knows a ‘formula’ for determining the answer, make sure they can explain why it works by using the blocks.

-         Students who solve the problem: Can everyone in your group explain the method you have used? Will this method work for any number of staircases? Can you write a formula based on your method that would work for any number of staircases?

 

If groups begin to finish before others, be prepared with the following extra questions:

 

-         Will there be a staircase that is made of exactly 100 blocks?

-         Which staircase in the sequence will be the first one to use more than 100 blocks to build?

-         An additional pattern will be available if needed (attached).

 

When students have solved the problem and each member of their group can explain it, they can record their results on a piece of butcher paper. The challenge is for them to come up with a way to describe their method in a written format that someone else can understand.

 

Each group will then be invited to share their results with the rest of the class. This is a difficult time for the instructor as he/she is responsible for making sure the mathematics is correct and that the students’ reasoning matches their equations.

 

*See following 2pages for possible student solutions/explanations.

 

When all the groups have shared, pass out the lesson plan to the participants and talk about the pieces to the lesson.

 

The discussion following the problem is the time when it would be appropriate to include the dimensions of powerful learning and how they played out during the lesson.

 

  1. How did the instructor introduce the problem? (The instructor didn’t give them the problem, she ‘set up’ the problem by walking the participants through several examples. She also had them build and describe certain figures in the pattern).
  2. How is this problem different from traditional problem-solving tasks? (Students are often given a problem and expected to solve it. Could you use clue words to solve this problem?)
  3. How did you feel when you heard what the task/problem was?
  4. Once you were given the problem, what happened? (The instructor asked groups to think about and share ways they might go about solving the problem). Why do you think the instructor did this? (It gave participants an idea or place to begin. Often, if students have no idea how to even approach a problem, they will give up or wait for the teacher to come tell them what to do.)
  5. What was your role during problem solving? (To try, to work together, to talk ideas out, to not give up)
  6. What did you expect of the other members of your group? (To participate, to help, to try)
  7. What did you expect from the instructor? (To give help, not answers; to allow time and not invade). How long did the instructor wait before asking questions of individual groups? Why was that important? What kinds of questions did the instructor ask? How did you feel when you were interrupted while working on the problem? Did the instructor help too much, or not enough?
  8. How did you feel during this activity? (Frustrated? Comfortable? Anxious? Dumb?)
  9. What kind of environment did we have? How did you feel when you were sharing? Did anyone NOT want to share? What could we have done to make you feel better?

 

 

Possible equations and explanations to the staircase problem:

 

Example 1

 

(B+1)H    This group used ‘B’ to represent the base and ‘H’ to represent the height

2                              of the figure. They created the same exact figure, flipped it upside down and aligned it with their original figure, thus producing a rectangle.

Although the height stayed the same, the linear measure of the base increased by one. They multiplied the base x height to determine the area (or number of cubes). They divided the amount by 2 because they had 2 of

1     +
 

B
 

n
 
the same figures. Dividing the total number of cubes by 2 gives the number of cubes in one of the figures.

H
 

N
 
 


Same reasoning different equations:  N(N+1)   ;    (n + 1) x (n ÷ 2)

                                                                 2

Example 2

 

 

N x N  +  N     This group took one staircase and ‘filled in the square’. Meaning,

    2          2         they did not increase either the base or the height, they just added

enough blocks to make it a perfect square. They multiplied side times side to determine the total number of blocks and divided by 2 which gave them exactly half the square (2 triangles). They had to add half pieces of ‘steps’ back to the figure to account for the stair steps, which the group discovered was 1/2 the side.

 

Same reasoning different equations:  n2 + 1(n)  ;   s2 + 1/2s 

2           2         2

Example 3

   

N
 
n(1/2n) + 1/2n   This group made columns of all the same size (n) by adding 1

 1/2n
 

 n(1/2)
 
    cube to the column missing one, 2 cubes to the column missing         two, 3 cubes to the column missing three, etc. They found that by doing this the figure becomes half the length of the base, but the height stays the same [n(1/2n)]. They always had 1/2 of a column (n) left, so they added back 1/2 n.

 

Same reasoning different equations:  (1/2b•h) + 1/2b   ;

x(x÷2) + (x÷2)

 

Example 4

 

X + X (X-1)    This group took away the first column ‘X’ and left it alone. The

2                       
X-1

  2
 
number of columns they now have is one less than X written (X-1). They make as many columns of ‘X’ as they can which results in the number of remaining columns being divided by 2. They now have (x-1)/2 columns of ‘X’. They add the original column of ‘X’ back into the equation. This picture looks similar to the picture in the previous example, but the reasoning is different.

 


X
 
The equation for this picture is: 6 + 6 (6-1)

                                                                2

        

 

X-1
 
 

 


The different formulas (equations) are expressed in ways that reflect how the students were thinking about the problem. Do the participants recognize that each formula can be simplified and result in the same formula?