Algebra Lesson Plan

 

Pattern is a unifying theme that weaves mathematical topics together. The study of patterns supports children in learning to see relationships, to find connections, and to make generalizations and predictions. Understanding patterns nurtures the kind of mathematical thinking that helps children become problem solvers and abstract thinkers. It is a problem-solving tool.

 

Understanding the concept of pattern means that the child recognizes in many forms the predictability and repetition that patterns imply.

 

 

Children first explore patterns with their own bodies, actions, and words. Before creating pictorial representations and patterns at symbolic levels, children need objects with which to make patterns.

 

Pre-K-Kindergarten

 

Children should focus on regularity and repetition in motion, color, sound, position, and quantity. They need to be involved in recognizing, describing, extending, transferring, translating, and creating patterns.

 

Begin with simple patterns. Have the students in a circle and have the first stand, then sit, then stand-sit-stand-sit…Introduce repeating pattern and use the letters  ABABABABAB

 

When the children are ready, expand the pattern to 3 components: stand up-stoop down-sit with legs crossed.

 

Students should name the positions because it adds an auditory cue to the visual pattern they are making.

 

Use the terms first, second, last, and next.

 

Begin a pattern and ask students to join in when they think they know the pattern: clap hands-slap knees, clap hands-slap knees, clap hands-slap knees, etc. Students should join in when they think they know the pattern and afterwards, be prepared to tell what the pattern is.

 

Move toward patterns with objects. Give the following clown pattern: red, blue, blue, yellow, red, blue, blue, yellow, red, blue, blue, yellow.

 

Students then need to transfer their patterns from one format to another. Can they represent the clowns with color tiles?

 

How can they show the stand-sit pattern with color tiles?

 

Patterns with pictures and symbols

Patterns created by using stamps, stickers or templates are pictorial patterns. Since young children need the opportunity to relate manipulatives to representations of the same ideas, making a pattern with objects, then drawing and coloring their patterns, will afford the children experiences in making written records of the physical world.

 

First Grade

 

There is continued emphasis on extending patterns and transferring them from one medium to another.

 

Students will create their own pattern by starting with a ‘design’ or unit to repeat and then repeating the design on long strips of paper (like adding machine tape).

 

Transfer patterns from one medium to another by representing the students’ patterns with Unifix cubes.

 

How could you name the designs using letters of the alphabet?

 

Predictable Literature- Children enjoy listening to stories with a predictable pattern that repeats over and over. Students can use the patterns found in stories to predict what will happen next in the story. Examples are: The Little Red Hen and The Gingerbread Man. The House That Jack Built is an example of a growing pattern.

 

Patterns with numbers- Students write 5 columns of numbers; 0-9, 10-19, 20-29, 30-39 and 40-49. Questions:

What kinds of patterns do you see? Do any patterns go across the paper? Down the paper?

Is there a pattern in the 10’s place?

What would the digits in the ones place be if I continued counting forward?

Put a marker on every 2. Does this make a pattern?

What do you think you will see if we put markers on every 4? On every 7?

 

 

Second Grade

 

Look at number patterns on the Hundred’s Board- Use copies I have and allow students to talk about the patterns they see.

 

This is the ideal time to begin making connections between concrete or pictorial patterns and numerical patterns.

 

Begin with an odd and even activity. Students will grab a handful of tiles and share them with a partner. Do they each have the same number of tiles (none left over)? Use the odd/even recording sheet to try and build a rectangle on the grid. If students have the same number of tiles with none left over, the number of tiles is an even number. Otherwise, it is an odd number.

 

 


1          2          3          4          5                6            7

 

Begin to discuss growing patterns. Growing patterns show an arithmetic change between pairs of elements in the pattern. The odd and even sequence is a growing pattern.

 

How Does it Grow?

 

To introduce the use of geometric figures in growing patterns, present the following drawing:

 

 

 

 


To help the students identify the pattern and the elements in the pattern, ask the following questions:

 

-         How are these figures alike? (They are made of rows of squares. Each row has 2 squares.)

-         How are these figures different? (There is a different number of rows in each figure. The number of small squares in each figure is different.)

-         How are these figures growing? (A row of 2 squares is added each time.)

-         How many rows of squares do you think will be in the next figure? (four) Why do you think so?

-         How many small squares will be in the next figure? (eight) How do you know? (Call on a student to draw the next figure and to verify the number of small squares.)

-         When the figure has ten rows, how many small squares will there be? (twenty) Have the students talk about how they decided on the number of squares.)

 

 

 

Make a table (third grade):

 

Figure

Number of Rows

Number of Squares

1

1

2

2

2

4

3

3

6

4

4

8

5

5

10

6

6

12

 

Can we tell the number of rows and the number of squares for any number?

 

Try 10, 15 and 20. Number of rows is nx1, number of squares is nx2.

 

To introduce students to a pattern composed of two different shapes, give the following pattern:

 

 

 


Ask some general questions:

-         Can you describe this pattern?

-         What will the next figure look like? How do you know?

 

To help students generalize the relationship between the number of small squares and the number of triangles in the figures and the number of triangles and squares and the position of the figure in the pattern, ask these questions:

 

-         The pattern continues. How many squares are there in the figure that has six triangles? (seven squares?) Seven triangles? (eight squares)

-         How can you find the number of squares in the figure when you know the number of triangles? (The number of squares is one more than the number of triangles.)

-         How many triangles are there in the figure that has 10 squares? (nine)

-         How can you find the number of triangles in the figure when you know the number of squares? (The number of triangles is one less than the number of squares.)

-         If there are nine shapes in all, what does the figure look like?(It has four triangles on top of five squares.)

-         If there are five triangles, how many shapes are there in all? (eleven)

 

Possibly give ‘How Does it Grow’ activity sheet for homework.

 

Third Grade

Students begin to create and use tables as a technique for analyzing and reporting patterns. Numbers are used to identify order in a sequence of data.

 

Use the Hundred’s Board and counters for the following activities:

  1. Have students place counters on the multiples of 3: (3, 6, 9,…). Ask students to describe the pattern. Explore patterns with other multiples.
  2. Ask students to describe the spatial pattern that results when they place counters on 11 and the multiples of 11. (An oblique or diagonal line of nine counters from 11 to 99)
  3. If you place a counter on all the squares where the digit five occurs, how many of the 100 squares will have counters on them? (19) What does the pattern look like? Repeat with other digits.
    1. Do you always get the same result? Why or why not? Can you predict what the pattern will look like if you cover all the eights?
  4. Have the students skip count by 2s and then by 3s, marking the multiples of 2 with one color counter and the multiples of 3 with a different color counter.
    1. Which numbers have 2 colors on them? (6, 12, 18, 25,…,96) Why? (They are multiples of both 2 and 3)
  5. Have students start at 3 and count on by 5s (3, 8, 13, 18,…)
    1. What is the pattern in the units digit?
    2. Can you use a table or an organized list to help you find out?
    3. Why does this pattern occur?
    4. Eighty-three is a number in this sequence. What number comes before 83 in the sequence? How do you know?
  6. For each of the following patterns, have the students tell by which number you are counting:

 

8, 14, 20, 26, 32, 44, 50,…

9, 13, 17, 21, 25, 29, 33,…

27, 41, 55, 69, 83,…

 

  1. Have the students find the following numbers in the chart: 7, 16, 25, 34, 43, 52, 61.
    1. Describe the visual pattern.(Each number is one square down, one square to the left)
    2. Discuss the numerical pattern (nine more, or counting by 9s)
    3. Add the digits and discuss their observations.

 

  1. Have the students cover the following numbers on their chart: 2, 13, 24, 35, 46, 57, 68, 79.
    1. Describe the pattern spatially and numerically. (Each number is one square down, one square to the right, 11 more)
    2. Is there a pattern in the sum or difference of the digits?
    3. Have students describe and explain their discoveries.

 

 

 

 

 

Describing Patterns Numerically

Show the following pattern and encourage students to describe it:

 

 

 

 

 

 

 

 

 

 

 


Ask students to describe the pattern.

Ask a student to draw an additional figure to extend the pattern. What would come next?

 

Next we are going to explore the pattern in a table and use the table to explore the patterns further.

 

Square Numbers

 

 

 

 

Term

1

2

3

4

Total Number of Circles

 

1

 

4

 

9

 

16

 

Use the numerical information in the table to extend your knowledge of the pattern. Can you tell how many circles would be needed in term 7?How do you know? How many circles would be needed in term 20? Could one of the circles have a total of 50 circles? Explain.

 

Grade Four

 

Children learn to represent patterns numerically, graphically, and symbolically, as well as verbally. They begin to look for relationships in numerical and geometric patterns and analyze how patterns grow or change. By using tables, charts, physical objects, and symbols, students make and explain generalizations about patterns and use relationships in patterns to make predictions.

 

Tiling a Patio

This activity enables students to observe patterns and relationships; make conjectures about patterns and test those conjectures; and discuss, verbalize, generalize, and represent patterns and relationships.

 

 

 

Present the problem:

            Alfredo Gomez is designing square patios. Each patio has a square garden area in the center. Alfredo uses brown tiles to represent the soil of the garden. Around each garden, he designs a border of white tiles. The pictures below show the three smallest square patios that he can design with brown tiles for the garden and white tiles for the border.

 

 

 

 

 

 


          Patio 1                 Patio 2                   Patio 3

 

 

 


Brown square 1     Brown square 2      Brown square 3

 

 

 

 

Patio Number

Number of Brown Tiles

Number of White Tiles

Total Number of Brown and White Tiles

 

 

 

 

 

 

 

 

 

Ask students to build the three patios using brown and white tiles or construction paper to show the garden and the border. Record the number of white and brown tiles for each patio in a table similar to the one below.

 

How many tiles will there be in the next largest patio?

 

Continue the table for the next two squares. Ask questions like the following that encourage the students to reason about the patterns. Focus on the number of brown and white tiles for a given patio, and help the students think about the border tiles given the number of brown tiles and the number of brown tiles given the number of border tiles.

 

1.      How do you know your answers are correct? (The number of white tiles along one side of the border is 2 more than the patio number.)

2.      Describe your methods for counting the different tiles. (Total of brown tiles = [patio number] x [patio number]. Total of white tiles = 4 x [patio number] + 4.

3.      What patterns do you see? What is your rule for the number of tiles on the patio and the number on the border? (Number of tiles on the patio = [patio number + 2] x [patio number + 2]. Number of tiles on the border = [total number of tiles on the patio] – [number of brown tiles].

4.      Test your rule with another example of a patio.

5.      If there are 36 brown tiles, how many white tiles are there? Explain how you got your answer. (If there are 36 brown tiles, then the center square has dimensions 6x6. Since there are two more white tiles along the side than the number of brown tiles on the side of the center square, there are 8 white tiles along the side of the patio. Therefore, the patio has a total of 8x8=64 tiles. Since 64-36=28, there are 28 white tiles..

6.      Can you make a square with forty-nine brown tiles? Explain why or why not. (Yes, 7x7 are the dimensions of the center [brown] square; 49 is a square number.)

 

Continue activity to Fifth Grade

Fifth-grade students can use new ways to represent the relationship between the number of tiles of each color and the number of the square patio in the sequence. You can begin to emphasize the idea of function more explicitly.

 

1.      Make a graph that shows the number of brown tiles in each square patio in your table. Make a graph that shows the number of white tiles in each square patio in your table.

 

 

Ask questions:

 

1.      As the patio gets larger, how does the number of white tiles change? How does the number of brown tiles change? What is your rule for these relationships? How are these rules represented in the table and in the graph? (As the patio gets larger, the number of white tiles increases by 4 each time to form a straight line on the graph. The number of brown tiles is squared each time; the line on the graph is not a straight line but a curved line. (In the higher grades, the students will learn that the curve is the graph of a quadratic function.)

2.      Use your graph to find the number of brown tiles in the seventh patio. (There are 49 tiles in the seventh patio.)

3.      Does the number of tiles ever exceed the number of white tiles in a patio? (Yes, beginning with patio 5, the number of brown tiles is greater than the number of white tiles.)

4.      Find the number of brown (white) tiles in the fifteenth patio; in the thirtieth patio; in the hundredth patio. (Patio 15 has 225 brown tiles and 64 white tiles; patio 30 has 900 brown tiles and 124 white tiles; patio 100 has 10,000 brown tiles and 404 white tiles.)

5.      If there is a total of 144 brown squares, what is the side length of the square patio including the border? How many white tiles are needed for the border? (The length of the side of the total patio is 14. The side of the brown square is 12. The patio border needs 52 white tiles.

6.      Can there ever be a border for a square patio with exactly twenty-five white tiles? Explain why or why not. (No. One rule for the number of white tiles is [(patio number)x4]+4. There is no patio number that will make [(patio number)x4]+4=25.)

 

This activity involves helping students make connections between the physical representations of the patterns and their verbal or graphical descriptions. The interaction between variables is the underlying mathematical idea addressed informally here. The natural extension of this notion is towards generalizing the pattern.

 

Patio Number

Number of Brown Tiles

Number of White Tiles

Total Number of Brown and White Tiles

1

2

3

4

5

.

.

.

n

1

4

9

16

25

.

.

.

n2

8

12

16

20

24

.

.

.

4(n+1)

9

16

25

36

49

.

.

.

(n+1)2

 

 

Grades 6-8

The middles grades experience continues to explore patterns and functions that provide opportunities for students to represent problems in a variety of ways and to translate among these representations as stories or problems, concrete objects, pictures, charts, tables, graphs, verbalizations, and symbols.

 

Moving from the arithmetic patterns of real numbers to a generalization using variables and real numbers to form expressions and sentences allows students to study the relationship between quantities. Thus, from the explorations of patterns a natural development of functions also emerges.

 


Staircase Problem

 

 

 

 

 

 

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