Pedagogical ideas

Activities sequence

• Number strips - even, odd, multiple of three
• Hint box
• Dot patterns - even, odd, & adding strips & patterns
• V & W patterns -
• More dot patterns - band & director model, name this model,
• Data tables and patterns -
• Adding patterns with number strips
• Arithmetic Sequences with tables and strips
• More arithmetic sequences - use an expression and make a table, strip, or dot pattern
• Pyramid sequences
• Prism sequences
• Tower sequences
• Expansion ideas

Focus questions

• What is a pattern?
• What objects, events, or systems can make a pattern?
• What are different types of patterns?
• What properties indicate patterns?
• Identify different ways to create patterns?
• Do patterns change in the same way?
• Do patterns alway change in a constant way?

Related resources

• Conservation & assessment tasks
• Blank graph sheet

Expansion resources

• Stairways and box stacking problems
• Growing squares & area perimeter activities
  • Pools & sidewalks problem or growing squares - An eccentric person wants to make containers, or pools, with sidewalks around them. The only requirement is the pool and sidewalk sections have to be square...
  • Toothpicks - Growing squares challenge worksheet. Use toothpicks to find formulas to determine the number of toothpicks added to make larger squares. Toothpicks in the perimeter, toothpicks to make the squares in the squares, and the number of squares in the growing squares.
  • Toothpicks - Growing squares challenge slides for the challenge above. Slides step through adding toothpicks to make growing squares, record data in a table, and ind the a pattern of increasing squares and increasing number of toothpicks.
• Body Proportions, height, body parts, bones, & Egyptian art
• Multiplication of whole numbers, multiplication of binomials (x+1)(x+1), & area patterns
• 8x8 Square patterns with three colors -visual geometric pattern

Dot patterns

Patterns can be made or represented with dots.

What does the pattern represent?

Pattern number	0	1	2	3	4	5
Dots for a pattern number
Change of the dots from previous pattern

 

• What is the pattern for how dots are added?
• What would be an expression to explain how many dots are in a pattern number?
• What numbers on the green red number strip do the dots represent.

 

Here is another dot pattern.

Pattern number	0	1	2	3	4	5
Dots
Change

 

• What is the pattern?
• What would be an expression to explain how many dots are in a pattern number?
• What numbers on the green red number strip do the dots represent.

 

Dot patterns and number strips together

Use ideas from both dots and number strips to think about patterns.

One of the dot patterns represents odd numbers and the other even numbers.

What color on the number strip represents odd?

What color represents even?

When you add even numbers, what kind of number do you get?

When you add odd numbers, what kind of number do you get?

If you visualize even numbers as dots, what happens when two are put together?

If you visualize odd numbers as dots, what happens when two are put together?

What if you add an odd and even number, what do you get?

Write three statements for adding odd and even numbers.

1.

 

2.

 

3.

 

Dot patterns with the red, light, and blue number strip.

 

Draw a pattern of dots for Red numbers in the pattern.

 

 

Draw a pattern of dots for Light numbers in the pattern.

 

 

Draw a pattern of dots for Blue numbers in the pattern.

 

 

Write a recursive formula and a direct formula for each colored pattern.

Recursive formulas

1. Red
2. Light
3. Blue

Direct formulas

1. Red
2. Light
3. Blue

 

What happens when different colored numbers are added? (R + L = ?) What color do you get?

Use the table to record the colors of the sums.

+	R	L	B
R
L
B

 

Dot patterns

Band and director pattern

Pattern number	0	1	2	3	4	5
Dots

 

Recursive formula

 

 

Direct formula

 

Name this pattern

Pattern number	0	1	2	3	4	5
Dots

 

Recursive formula

 

 

Direct formula

 

 

 

Adding Patterns with number strips

What pattern will there be an odd and even number strip are added?

What numbers will be on the new strip?

Make an even number strip (0, 2, 4, ...)

Make an odd number strip (1, 3, 5, ...)

Add the numbers on the even and odd number strips to make a third number strip (1, 5, 9 ...)

The pattern can be made with dots.

Write an expression for it.

 

How do the expressions for even and odd compare to it? (2n; 2n + 1; 4n + 1)

 

 

Use dot patterns and show how the dots can be rearranged from two patters in a pattern to make a third pattern.

 

 

More arithmetic sequences

Make a table, number strip, and dot pattern for the expressions.

Sequence D ; 9 + 3n

Sequence E ; 4 + 5n

Write your own

Sequence F ;

 

Pattern number	0	1	2	3	4	5	6	7	8
Sequence

 

 

Sequence

 

 

 

 

Prisms

Three dimensional figures have three common properties:

• Vertices
• Edges
• Faces

Use the prism diagrams, record the numbers of these properties in the table below, and find formulas for them and the relationships among them.

Prism	1	2	3	4
Vertices
Edges
Faces

 

Write a formula for the sequence increase of vertices.

 

Write a formula for the sequence increase of edges.

 

Write a formula for the sequence increase of faces.

 

Euler's formula; Vertices - Edges + Faces = 2 describes the relationship between the vertices, edges, and faces.

Explain how the formula fits with the prim sequences.

 

 

 

 

Show how the formulas for prisms relate to Euler's formula.

 

 

 

Triangular numbers

Here is a dot pattern and triangle shapes pattern. Make a table, find the pattern, write an expression

 

Pattern number

 

 

 

 

 

 

 

 

Log stacks

Write an expression to determine the number of logs in a stack or the number of logs in different rows of a stack.

Then make some log piles and use the expression to answer different problems.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hand shakes problem

Write the number of students in your class or other group.

If you were to talk to each person or shake their hand, how many talks would you have or how many hand shakes would you make?

The diagram below represents students, hand shakes, or talks.

Find a pattern and use it to answer some questions.

 

Number of students.

 

Number of shakes or talks.

 

How many shakes for _____ students?

 

How many students would there need to be to shake 200 different hands?

 

How does the total talk time increase if you talk to each person for 2 minutes?, 4 minutes, 6 minutes, ... 15 minutes?

 

Step pyramid problem

The diagram below shows a plan present to a pharoah to build a step pyramid.

If the center block is used as a cubic unit, how many blocks or cubes are needed to build it?

Step pyramid plan

 

Let's say that after the pharoah sees the pyramid, she thinks it needs to be higher. Of course the construct team needs to know how many more blocks will be needed. Find the pattern for the number of blocks needed to continue the pattern one cube higher, two cubes higher, and so on.

 

Trusses

Truss is a framework used to support a floor, bridge, roof, and other structures. They are often constructed with metal and lumber with triangular shapes.

Equilateral triangles can be used to make trusses. What is the pattern for adding triangles as trusses increase in size?

Truss sizes: 1, 2, 3, 4

Draw the truss, enter the length of the beam and the number of rods.

Truss
Length of beam	1	2	3	4
Number of rods

Describe the pattern.

 

Write a recursive formula.

 

How many rods are needed for a beam of ______ length?

 

Find a formula to find the number of rods (R) for a beam of any length (L).

 

 

Consider two truss lengths.

Six long

Three long

If we consider joining these two trusses at their sides, there are two places they can be joined.

Consider all the ways a six and three truss can be joined.

Combinations for joining six and three length trusses.

An outdoor shelter company makes shelters with two foot rods. They join three trusses, which they mount on two pilars.

Complete the table so it lists the sizes of the production trusses (short & long), their lengths, the length of the finished three part truss, and the rods used to make each.

	Rods	Length
Short truss
Long truss
Joined truss

 

The company will usually use three length trusses (6 foot) for the sides, but the length of the roof may vary from five truss lengths (10 foot) to ten truss lengths. Make a table for the number of rods needed for each truss.

 

	Side	Roof	Total
Five
Six
Seven

 

Creat a problem of your own and share it.

 

Staircases

Construction of stairs is an interesting process, which depends on two important values, known as the run and the rise.

Too investigate some stair patterns let's start by making a stair model.

Print the diagram below and make a staircase model.

Directions:

• Cut a slit along the line from A to C.
• Cut a slit along the line from B to D.
• Make nine accordion folds with the lines DC to AB.
• Square the stairs and fold where the wall would meet the floor.

 

 

Design and construct your own staircase on a piece of oaktag and use it for the following investigation.

Use a ruler, or T square, and triangle to draw straight lines.

Vocabulary

• The run is horizontal and the distance between the risers.
• The rise is the back of the stair and the height of each step.

Investigate patterns on a stair model

• Measure all the runs on your model.
• Measure all the rises on your model.
• Add all the runs on your model. and record the data in a table.
• Add all the rises on your model.
• What is the distance from the floor to the top step on your model?
• What is the distance from the wall to the front of the bottom step on your model?
• How do these distances compare to the sums of runs and rises.

Summarize patterns discovered in your model.

 

 

 

Investigate patterns on a staircase.

Background information:

On steps there is usually a stair tread that represents the run on the model. Usually the tread is wider than the run. Therefore, the measurement of a stair tread needs to be modified to accurately represent the stair's run.

Use what you know about stairs, select a staircase, collect data for it, and record it in a table.

Staircase data
Data collector
Stair location
Rise
Tread
Number of steps
Total rise (height)
Total run (depth = run + tread overhang)
Ease to climb

 

Write a formulas for the total rise and run of a staircase.

The higher the rise, the higher a person needs to step.

The lower the rise, the more steps needed in a staircase.

Make a graph or a side view diagram to show the rise and run for each stair in the measured staircase.

How does the step diagram compare with a line graph?