Alg1 Lesson 3.13: Finding a Formula for Triangular Numbers

Lesson 3.13: Finding a
Formula for
Triangular Numbers

Obj: 6C, 8A: I can formulate quadratic functions
given a sequence of shapes.

EQ: How do I find pentagonal numbers?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

Word Cloud

How will you cultivate relationships with your classmates today?

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Part 1: The Staircase

Revisited

Figure out how many total blocks they would need to
build staircases from 1 block tall to 10 blocks tall.
Continue the information in a table in the one shown
here, in which n is the staircase height and T(n) is the
total number of blocks needed.

Drag and Drop

What patterns do you see? Does the table of values represent a linear
relationship? You add the height of the last “stair” to the previous total blocks. For example, the total number of blocks needed for the 10th staircase is the amount of blocks needed in the

​

Placeholder

th staircase ​

Placeholder

​

Placeholder

In general, the pattern ​

Placeholder

linear
because the rate of change ​

Placeholder

constant.

Drag these tiles and drop them in the correct blank above

is not

is

9

plus

10

minus

times

Drag and Drop

Make an argument about whether the sequence of numbers is quadratic.

Because the ​

Placeholder

differences ​

Placeholder

constant, this ​

Placeholder

a quadratic sequence.

Drag these tiles and drop them in the correct blank above

first

are not

is not

second

are

is

This staircase pattern
sequence is called
“triangular numbers.” It is
possible to arrange the dots
in an equilateral triangle
shape, like bowling pins or
billiard balls.

Poll

Which game do you like to play more?

• Bowling

• Billiards

• I've never played either of these.

Part 2: Writing a Formula for

the Staircase

Use a quadratic function to model the triangular numbers.

The vertex form will not be helpful in this situation
because it is not clear from the data where the vertex of
the parabola will be. Likewise, the factored form will not
be helpful because the x-intercepts are not clear from
the data. This means that, as in the free-fall problems in
the previous lesson, the standard form of the quadratic
will be the best to use. Therefore, the function T will
have the form T(n)= an² + bn + c.

Multiple Choice

What does the parameter c correspond to in the graph of the function T(n)= an² + bn + c?

• Time of the maximum value

• Y-Intercept

• X-Intercept

• Maximum Value

Multiple Choice

Given the graph of the function T(n)= an² + bn + c, what is the value of c?

• 0

• 1

• cannot be determined

• 3

Multiple Choice

Given the graph of the function T(n)= an² + bn + c, should we expect a to be positive or negative?

• positive

• negative

• zero

Multiple Choice

Given the points, (0, 0), (1, 1), and (2, 3) find the quadratic equation.

• $$T(n)=\frac{1}{2}n^2+\frac{1}{2}n$$

• $$T(n)=2n^2+2n$$

• $$T(n)=\frac{n\left(n+1\right)\left(n+2\right)}{6}$$

• $$T(n)=n^2+n$$

Part 3: Using the Formula

T(n) = ½n² + ½n

Is equivalent to
the explicit
formula in the
image to the left.

Evaluate T of 8:

T(8) = ½(8)² + ½(8) = 32 + 4 = 36

∴ the 8th triangular number is 36.
This means it takes 36 blocks to make 8 stairs.

Find the 8th Triangular Number

The finite series
sum is the sum
of the current
triangle and all
triangles before
it in the
sequence.

Multiple Choice

What is the 15th triangular number?

• 120
• 100
• 110
• 105
• 55

Fill in the Blank

What is the 20th triangular number?

Fill in the Blank

What is the 100th triangular number?

,

Multiple Choice

Is 1000 a triangular number?

• No

• Yes

If you check the
numbers above and
below the value of
1000, you will get
values that are not
exactly 1000.

Multiple Choice

Is the triangular graph discrete or continuous?

• Discrete

• Continuous

Types of Data

Discrete: Something that is counted. The graph contains dots
and/or dashes only. There are gaps in between values.

Continuous: Something that is measured. The graph
contains a solid line. There can always be a number
identified in between two rounded values plotted
on these graphs.

Nominal: categorical data where there is no specific order.

Ordinal: categorical data with an order or rank.

Part 4: Summary and

Practice

Handout 3.13: Practice Writing Formulas for Quadratic Sequences

Practice Performance Task 3b:

Weaving a Rug

A rug weaver wants to create a large rectangular rug. He has 34 linear feet of a material that he plans to use on the entire outside border of the rug. He wants to use all this material. So, the rug must have a perimeter of 34 feet. The weaver already has a design for a rectangular rug that is 9 feet long and 8 feet wide.This design would produce a rug with the correct perimeter. However, he wants to be sure the rug will cover as much of the floor as possible, so he starts to experiment with changing the length and width of the rug design. To maintain the 34-foot border of the rug, he follows this rule: Whenever he decreases one dimension by x feet, he increases the other dimension by the same amount, x feet, as shown below.

a) The area of the rug the weaver will make can be
expressed as a function of x, the amount of material
removed from the 9-foot side and added to the 8-foot side. Write an equation for this function and explain your work.

b) Is it possible for the rug to have an area of 71.25 square feet? If it is possible, give approximate dimensions of the
rectangle and explain your work. If it is not possible,
explain why not.

c) The weaver made a chart of possible areas and determined that the largest possible area is 72 square feet using the original 8- by 9-foot design with no changes. Determine if he is correct and explain why. If he is not correct, provide the dimensions of the rug with the largest possible area and explain your work.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the
wheel as a class and spend a minute or so
discussing our answers.