Lesson 1.4Additive Patterns and Arithmetic Sequences

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1. For each square, determine the perimeter P without measuring. Then complete the table on the next slide.

Part 1Handout 1.4.A: Perimeters of Squares

4 cm 8 cm

12 cm 16 cm

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1. Finish determining the perimeter P without measuring. Then complete the table.

2. ANSWER THE QUESTIONS ON THE NEXT SLIDE USING THE INFORMATION FROM SLIDES 2 & 3

Part 1Handout 1.4.A: Perimeters of Squares

Side Lengths of Squares (s) in centimeters

Perimeter (P) in centimeters

1 42 83 124 165 20

20 cm

1. What patterns do you see?Each perimeter was a multiple of 4

2. Given the value s, how do you determine the value of P ?We multiplied the side length (s) by 4 to determine the Perimeter (P).

3. Looking at the Table you created, which values represent the “input” and which values represent the “output” in the situation?Input is the “sides” (x-column on a table)Output is the “Perimeter” (y-column on a table)

4. Is this an example of Direct Variation? Why or why not?Yes it is, because the inputs were all multiplied by the same value to get the outputs.

Stop here!

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1. Look closely at each visual pattern. After a few moments try to draw what the next for each Pattern.

2. Answers explores questions on Slides 7-9

Handout 1.4.B, p. 15

Part 2Handout 1.4.B: Visual Patterns

a. Can you describe the rule to determine the value of C given the value of n using only multiplication? Why or why not?There is no rule using only multiplication to determine the value of “C”. It is an addition of 2. There is no number you can multiply by all the “n” to get all the values for “C”.

b. Which quantity is the input, and which is the output?The input values are “n” (the first column) and the output values are “C” (the second column).

c. Does the table represent a Direct Variation?No, direct variation means ALL of the input values are being multiplied by the same number to get all of the output values, this table does not follow that pattern.

d. Graph this table on WWW.DESMOS.COM what do you notice?When graphed we notice it does create a straight line, and has a constant rate of change.

e. If you connect the points and extend the line, would the line pass through the origin?No it doesn’t go through the origin.

Handout 1.4.B Visual Patterns

Stage Number (n)

Number of Circles (C)

1 1

2 3

3 5

4 7

5 9

6 11

a. Can you describe the rule to determine the value of T given the value of n using only multiplication? Why or why not?There is no rule using only multiplication to determine the value of “T”. It is again an addition of 2. There is no number you can multiply by all the “n” to get all the values for “T”.

b. Which quantity is the input, and which is the output?The input values are “n” (the first column) and the output values are “C” (the second column).

c. Does the table represent a Direct Variation?No, direct variation means ALL of the input values are being multiplied by the same number to get all of the output values, this table does not follow that pattern.

d. Graph this table on WWW.DESMOS.COM what do you notice?When graphed we notice it is exactly the same table as the previous slide, it does create a straight line, and has a constant rate of change.

e. If you connect the points and extend the line, would the line pass through the origin?No it does not.

Handout 1.4.B Visual Patterns

Stage Number (n)

Number of Triangles (T)

1 1

2 3

3 5

4 7

5 9

6 11

a. Can you describe the rule to determine the value of S given the value of n using only multiplication? Why or why not?There is no rule using only multiplication to determine the value of “T”. It is again an addition of 2. There is no number you can multiply by all the “n” to get all the values for “T”.

b. Which quantity is the input, and which is the output?The input values are “n” (the first column) and the output values are “S” (the second column).

c. Does the table represent a Direct Variation?No, direct variation means ALL of the input values are being multiplied by the same number to get all of the output values, this table does not follow that pattern.

d. Graph this table on WWW.DESMOS.COM what do you notice?When graphed we notice it is exactly the same table as the previous slide, it does create a straight line, and has a constant rate of change.

e. If you connect the points and extend the line, would the line pass through the origin?No it does not.

Handout 1.4.B Visual Patterns

Stage Number (n)

Number of Squares (S)

1 5

2 9

3 13

4 17

5 21

6 24

Stop here!

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Handout 1.4.B Visual Patterns Debrief

1● It isn’t possible to use a multiplication rule to

determine the value of C because the relationship between n and C is NOT an example of direct variation.

● n would be the input and C is the output. ● There is an addition pattern! If I add one to the n

column, I have to add two to the C column. ● The points will lie on a line because there is a

constant increase in both sides of the table, but the line will NOT pass through the origin.

● The other points on the line do not represent the sequence of circle figures because we can only use whole numbers.

This pattern is discrete because only whole-number values of n and their corresponding values of C apply to this situation.

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1. Next, try to develop a formula (also known as a function/equation) to determine the number of circles in any stage of the pattern.

2. Navigate to desmos.

3. Once you are there, adjust the sliders in the formula to get the line to align with the plotted points.

a. Note: C=mn+b *replace “m” and “b” with the

values that make the line go

through the points.

1. READ THIS ENTIRE SLIDE FIRST BEFORE NAVIGATING TO DIFFERENT APPLICATIONS.

2. Click on the Desmos link.

3. Adjust the sliders to make the “line” connect all the points.

Formula: C=mn+b m=2 b=-1C=2n+(-1) C=2n-1

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Another way to explain the formula is to describe the calculations in each stage.

1. Let’s look at the pattern another way.

2. Study the calculation pattern.

3. I notice the the first stage has no +2s, the second stage has one +2, the third stage has two +2s and so on…

4. The number of +2s is one less than the stage number.

5. Reread steps 4 and 5 to try to make sense of the formula.

Notice that there is an addition +2 in every stage.

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Stage numbers: 1, 2, 3, 4, etc.Circles in each stage: 1, 3, 5, 7Triangles in each stage: 1, 3, 5Squares in each X stage: 5, 9, 13Circles in each arrow stage: 7, 10, 13, 16

1. Look at another way we can display number patterns.

2. These lists of numbers each represent a sequence.

3. These particular sets of numbers are both examples of arithmetic sequences.

DEFINITIONS: Sequence: An ordered list of numbers. The list could be finite, or it could be infinitely long.

Arithmetic Sequence: A sequence in which the numbers increase by a common difference.

Part 3

Mark as “Turned in” AFTER all slides are completed

Examples of Direct Variation VS Not Direction VariationYes-Direction Variation:

Table: If you CAN multiply all of the “input” values by the SAME number and get all the “output” values. The EXAMPLE → all of the input values are multiplied by 2, to get the output values.

Graph: The LINE crosses through the ORIGIN.

Formula/Equation: Just multiplication - EXAMPLE: y = 2x

No-Not Direction Variation

Table: If you CANNOT multiply ALL of the “input” values by the SAME number and get all the “output” values. The EXAMPLE → all of the input values cannot be multiplied by the same number, to get the correct output values.

Graph: The line DOES NOT crosses through the ORIGIN.

Formula/Equation: If there is some number being added or subtracted at the END of the formula/equation EXAMPLE: y = 2x + 1

X Y

1 1

2 3

3 5

4 7

5 9

6 11

X Y

1 2

2 4

3 6

4 8

5 10

6 12

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