Unit 5, Activity 1And 2, Vocabulary Self-Awareness Chart

Unit 5, Activity 1and 2, Vocabulary Self-Awareness Chart

Name ______Date ______Hour ______

Word / + /  / - / Example / Definition
function
linear equation
non linear equation
slope
geometric sequence
arithmetic sequence
y-intercept
independent variable
dependent variable

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 1, Function or Not

Name ______Date ______Hour ______

Determine whether each relationship below is a function. Justify your answer for each.

Number of hot dogs / Total Price
1 / $1.50
2 / $3.00
3 / $4.50
4 / $6.00
5 / $7.50

The table below shows the cost of hot dogs at the ball game.

The sale at Sale Street Hamburger stand reads, “Buy up to 3 hamburgers for $2.00, each additional hamburger costs $2.00.” Is the relationship a function? Prove your answer.

Determine whether each of the tables below represents a function. Explain your answers.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 1, Function or Not with Answers

Name ______Date ______Hour ______

Determine whether each relationship below is a function. Justify your answer for each.

Number of hot dogs / Total Price
1 / $1.50
2 / $3.00
3 / $4.50
4 / $6.00
5 / $7.50

The table below shows the cost of hot dogs at the ball game.

This is a function because with each hot dog that is purchased, the cost increases.

The sale at Sale Street Hamburger stand reads, “Buy up to 3 hamburgers for $2.00, each additional hamburger costs $2.00.” Is the relationship a function? Prove your answer.

This is not a function because if you buy 1, 2 or 3 hamburgers, it will cost $2.00 and this gives three x values, the same y value making it not a function.

Determine whether each of the tables below represents a function. Explain your answers.

A and D are not functions because there are two y values for the same x value

Not a functionfunctionfunctionNot a function

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 2, What’s a Function

Name ______Date ______Hour ______

Part I

These ordered pairs represent functions.

x / 7 / 4 / -3 / 5
y / 4 / 6 / 3 / 4

A) {(1, 2), (2, 3), (4, 8), (7, 5)} B)

C) D) Time it takes to travel to the mall

Speed (mph) / 50 / 55 / 60 / 65
Time (minutes) / 36 / 33 / 30 / 28

These ordered pairs do not represent functions.

A) {(4, 5), (7, 8), (4, 3), (2, 4)}B)

x / 7 / 4 / 5 / 5
y / 4 / 6 / 3 / 4

C) D) Does your age affect your GPA?

Age / 14 / 15 / 16 / 16
GPA / 3.6 / 3.3 / 3.0 / 2.8

What is a function?

Study the examples provided to determine the definition of a function. Discuss the possible answers with a partner. List the characteristics of each below.

FunctionNot a function

Part II A function is ______

______

Since a graph is a set of ordered pairs, it is also a relation.

These graphs represent functions.

These graphs represent ordered pairs that are not functions.

How can you determine if the graph represents a function?

Vertical Line Test - ______
______

Part III

Graph the function, y = 2x + 3 by making a table of values.

x / y
-2
-1
0

Use the vertical line test to determine if this is a function.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 2, What’s a Function? with Answers

Part I

These ordered pairs represent functions.

x / 7 / 4 / -3 / 5
y / 4 / 6 / 3 / 4

A) {(1, 2), (2, 3), (4, 8), (7, 5)} B)

C) D) Time it takes to travel to the mall

Speed (mph) / 50 / 55 / 60 / 65
Time (minutes) / 36 / 33 / 30 / 28

These ordered pairs do not represent functions.

A) {(4, 5), (7, 8), (4, 3), (2, 4)}B)

x / 7 / 4 / 5 / 5
y / 4 / 6 / 3 / 4

C) D) Does your age affect your GPA?

Age / 14 / 15 / 16 / 16
GPA / 3.6 / 3.3 / 3.0 / 2.8

What is a function?

Study the examples provided to determine the definition of a function. Discuss the possible answers with a partner. List the characteristics of each below.

FunctionNot a function

(Possible answers)(Possible answers)

No inputs repeatSome of the inputs repeat

No inputs are paired with two different outputsSame inputs have two different outputs

Part II A function is a set of ordered pairs in which every input has exactly one output

______

A graph is a set of ordered pairs.

These graphs represent functions.

These graphs represent ordered pairs that are not functions.

How can you determine if the graph of a relation is a function?

Vertical Line Test - If a vertical line drawn anywhere on a graph intersects the graph at more than one point, the graph is not a function.

Part III

If x = -2, y = -1

If x = -1, y = 1

If x = 0, y = 3

Graph the function, y = 2x + 3 by making a table of values.

x / y
-2 / -1
-1 / 1
0 / 3

Use the vertical line test to show that the graph is a function.

Students should draw a few vertical lines on the graph to show that the function passes the vertical line test.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 3, Stringy Situation

Name ______Date ______Hour ______

# cuts / 0 / 1 / 2 / 3 / 4 / 5
# pieces

Fold your string in half and make the number of cuts requested (diagram on table). Fill in the table with the number of pieces that result.

Explain what rate of change you observe from the table?

Use the information you have from the table and rate of change and determine the number of pieces for 6, 7, 8, 9 and 10 cuts. Describe how the data in the table and the rate of change helped you determine this.

Is it possible to make a prediction for the number of pieces from any number of cuts? Explain.

Record an equation that will represent the pattern in the table above.

What if you have 45 pieces of string resulting from an unknown number of cuts? Explain how you would determine the number of cuts.

Take another piece of string and fold it in the shape of an “S.” Predict the number of pieces if one cut is made as shown in the diagram at the right.

Cut the string and continue collecting the data to complete the table below:

# cuts / 0 / 1 / 2 / 3 / 4 / 5
# pieces

Record the equation that represents this new pattern.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 3, Stringy Situation with Answers

# cuts / 0 / 1 / 2 / 3 / 4 / 5
# pieces / 1 / 3 / 5 / 7 / 9 / 11

Fold your string in half and make the number of cuts requested. Fill in the table with the number of pieces that result.

Explain what rate of change you observe from the table?

The number of pieces ‘y’ increases by 2 each time an additional cut was made.

Use the information you have from the table and rate of change and determine the number of pieces for 6, 7, 8, 9 and 10 cuts. Describe how the data in the table and the rate of change helped you determine this.

Six cuts would result in 13 pieces, 7-15, 8-17, 9-19, 10-21 cuts, Answers will vary for second part but will probably have something to do with it increases 2 each time

Is it possible to make a prediction for the number of pieces from any number of cuts? Explain.

It is always 2 more than the one before so it is linear and yes, as long as the string is folded in half before the cut, it will remain the same.

Record an equation that will represent the pattern in the table above.

y=2x+1

What if you have 45 pieces of string resulting from an unknown number of cuts? Explain how you would determine the number of cuts.

45 = 2x + 1

45-1 = 2x

44 = 2x

22 = x

22 cuts

Take another piece of string and fold it in the shape of an “S.” Predict the number of pieces if one cut is made as shown in the diagram at the right.

Cut the string and continue collecting the data to complete the table below:

# cuts / 0 / 1 / 2 / 3 / 4 / 5
# pieces / 1 / 4 / 7 / 10 / 13 / 16

Record the equation that represents this new pattern.

y=3x+1

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 4, Find that Rule

Name ______Date ______Hour ______

Draw the 5th arrangement in each of these patterns and complete the table of values. Is the relationship a function?

x
(arrangement #) / y
(perimeter)
1
2
3
4
5
x
(arrangement #) / y
(perimeter)
1
2
3
4
5
x
(arrangement #) / y
(perimeter)
1
2
3
4
5

Name ______page 2 Find that Rule

x
(arrangement #) / y
(perimeter)
1
2
3
4
5
x
(arrangement #) / y
(# dots)
1
2
3
4
5

Name ______Date ______Hour _____

Complete the tables below using the patterns A, B, C, and E. Notice that y is the area of the arrangement in this section

Pattern D
x
(arrangement #) / y
(area)
1
2
3
4
5
Pattern C
x
(arrangement #) / y
(area)
1
2
3
4
5
Pattern A
x
(arrangement #) / y
(area)
1
2
3
4
5
Pattern B
x
(arrangement #) / y
(area)
1
2
3
4
5

Review the values in the tables, then write the rules for finding perimeter and/or area below.

Pattern / Rule for pattern for finding perimeter / Rule for pattern for finding area
A
B
C
D
E / Rule for finding the number of dots in pattern

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 4, Find that Rule with Answers

Pattern / Sketch the 5th arrangement / Rule for pattern for finding perimeter / Rule for pattern for finding the area
A / / y = 2x + 2 / y = x
B / / y = 2x + 8 / y = x + 3
C / / y = 4x + 8 / y = 2x + 1
D / / y = 4x + 2 / y = x2 + 1
E / / y = 2x + 3

Perimeter Tables

Pattern A
x
(arrang. #) / y
(P)
1 / 4
2 / 6
3 / 8
4 / 10
5 / 12
Pattern B
x
(arrang. #) / y
(P)
1 / 10
2 / 12
3 / 14
4 / 16
5 / 18
Pattern C
x
(arrang. #) / y
(P)
1 / 8
2 / 12
3 / 16
4 / 20
5 / 24
Pattern E
x
(arrang. #) / y
(#dots)
1 / 5
2 / 7
3 / 9
4 / 11
5 / 13
Pattern D
x
(arrang. #) / y
(P)
1 / 6
2 / 10
3 / 14
4 / 18
5 / 22
Pattern B
x
(arrang. #) / y
(area)
1 / 4
2 / 5
3 / 6
4 / 7
5 / 8
Pattern A
x
(arrang. #) / y
(area)
1 / 1
2 / 2
3 / 3
4 / 4
5 / 5

Area Tables

Pattern C
x
(arrang. #) / y
(area)
1 / 3
2 / 5
3 / 7
4 / 9
5 / 11
Pattern D
x
(arrang. #) / y
(area)
1 / 2
2 / 5
3 / 10
4 / 17
5 / 26

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 4, More Patterns and Rules

x (arr. #) / y (# tile)
1
2
3
4
5
x (arr. #) / y (# tile)
1
2
3
4
5

How many tiles will be in the 5th arrangement of pattern “A”? Explain.

Explain the rule for the number of tiles that will be in the nth arrangement of pattern “A”?

How many tiles will be in the 4th arrangement of pattern “B”?

Explain the rule for the number of tiles that will be in the nth arrangement of pattern “B”?

Make a graph of one of these patterns. Explain the pattern that the graph of the pattern creates (i.e., linear or not).

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 4, More Patterns and Rules with Answers

x (arr. #) / y (# tile)
1 / 2
2 / 4
3 / 8
4 / 16
5 / 32
x (arr. #) / y (# tile)
1 / 3
2 / 9
3 / 27
4 / 81
5 / 243

How many tiles will be in the 5th arrangement of pattern “A”? Explain.

There will be 25 tile in the 5th arrangement and the area will be 32 square units.

Explain the rule for the number of tiles that will be in the nth arrangement of pattern “A”?

The rule will be two raised to the power of the number of the arrangement or 2n.

How many tiles will be in the 4th arrangement of pattern “B”?

There will be 81 tile in the 4th arrangement or 34.

Explain the rule for the number of tiles that will be in the nth arrangement of pattern “B”?

The second pattern shows powers of three so the rule will be three raised to the power of the arrangement number. 3n

Make a graph of one of these patterns. Explain the pattern that the graph of the pattern creates (i.e., linear or not).

The graph is not linear, the growth is x2 each time.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 6, Use that Rule

Name ______Date ______Hour ______

Write the rule that represents each of the phrases below.
Sketch the first three figures in an arrangement that represents the rule.
Make a table of values to represent the first 10 arrangements in each pattern.
Identify and graph one linear and one exponential pattern.
Determine if each of the following is a function and explain why or why not..

Four times a number plus one.

A number squared minus one.

Two raised to the power of the figure number plus three.

A number plus five.

On the first day of a diet, she weighed 89 pounds, 92 pounds and 88 pounds on different scales.

Three times a number minus two

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 6, Use that Rule with Answers

Write the rule that represents each of the phrases below.
Sketch the first three figures in an arrangement that represents the rule.
Make a table of values to represent the first 10 arrangements in each pattern.
Graph one linear and one exponential pattern.
Explain why each is a function. They are all functions because for each x-value there is only one y-value.

Four times a number plus one.

Rule: 4x + 1

Answers for chart: (1,5); ( 2,9); (3,13); (4,17); (5,21); (6,25); (7,29); (8,33); (9,37); (10,41)

A number squared minus one.

Rule: x2 - 1

Answers for chart: (1,0); (2,3); (3,8); ( 4,15); (5,24); (6,35); (7,48); (8,63); (9,80); (10,99)

Two raised to the power of the figure number plus three.

Rule: 2x + 3

Answers for chart: (1,5); (2, 7); (3,12); ( 4, 19);( 5,28); ( 6, 39);( 7, 52);( 8, 76); (9, 84); (10, 103)

A number plus five.

Rule: x + 5

Answers for chart: ( 1, 6); (2, 7); (3,8); (4, 9); ( 5, 10); (6, 11); (7, 12); (8, 13); ( 9, 14); (10, 15)

On the first day of a diet, she weighed 89 pounds, 92 pounds and 88 pounds on different scales.

This will not be a function because on day one (x) there are three different values for y. (1, 89) (1, 92) and (1, 88)

Three times a number minus two

Rule: 3x -2

Answers for chart: (1, 1); ( 2, 4); ( 3, 7); ( 4, 10); (5, 13); (6,16); ( 7,19); (8,22); ( 9, 25); (10, 28)

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 7, Hexagon

Name ______Date ______Hour ______

Sketch the fourth hexagon in the pattern below.

Work with your group and find as many different ways as you can to compute (and justify) the perimeter of the 4th figure.

Based on your observations, determine the perimeter for the 10th figure number without sketching or constructing it. Write a description that could be used to compute the perimeter of any figure number in the pattern.

Use drawings, words, and symbols in your description. Make sure your description is clear enough so that another person can read and understand your thinking. Write your explanations using correct mathematics.

Write an equation that represents the description you wrote in #3..

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 7, Hexagon with Answers

Name ______Date ______Hour ______

Sketch the fourth hexagon in the pattern below.

The end on figure one is not part of perimeter of figure two, and this continues as the figure progresses. The two ends are always part of the figure, and this becomes the “y intercept”. Each time a hexagon is added, make sure the students see the “rooftop 2” and the “bottom 2” as the coefficient of the figure number.

Work with your group and find as many different ways as you can to compute (and justify) the perimeter of the 4th figure.

Make sure the students “see” each part of the equation in the figure. See notes on figures above.

Possible explanations: top and bottom are double the figure number and then two ends,

2(2x) + 2= 4(4) + 2 = 18

6 + 4(x – 1) six sides (counting the first hexagon and the last side on the last one in the pattern, added to four times one less than the number because only four sides are part of the perimeter.

6x – 2x + 2 hexagons have six sides and two of each are hidden (not part of perimeter) and then the two ends.

Based on your observations, determine the perimeter for the 10th figure number without sketching or constructing it. Write a description that could be used to compute the perimeter of any figure number in the pattern. Use drawings, words, and symbols in your description. Make sure your description is clear enough so that another person can read and understand your thinking.

Many possible answers here.

4(10) + 2 = 42

10 rooftops making 20 units on top and 20 on the bottom then the two ends.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 7, Polygon Extensions

Sketch the first three figures in a pattern using regular triangles in a row or train that increase by one each time.

Write a description that could be used to compute the perimeter of the regular triangles in your sketch. You may use sketches in your description but show your description mathematically, too.

Compare the description for the triangle pattern and the hexagon pattern. Write the triangle rule. Describe how the rule changes as it relates to the hexagons and the triangles.

Sketch the graphs for the regular polygon perimeters of the regular triangle, square, and heptagon (7 sides) and connect your graph to the rule that represents the perimeters .

Using what you know about slope and the perimeter relationships in these regular polygons, explain where the hexagon graph will be on the grid.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 7, Polygon Extensions with Answers

Extensions to the hexagon problem.

Write a description that could be used to compute the perimeter of the regular triangles in your sketch. You may use sketches in your description but show your description mathematically, too.

The pattern shows three sides in figure 1 for perimeter, then like the hexagon pattern, the side that the figures are joined by does not count at perimeter anymore. Figure 2 has 4 sides for perimeter, and figure 3 has 5 sides.

Compare the description for the triangle pattern and the hexagon pattern. Describe how the rule changes as it relates to the hexagons and the triangles.

The hexagon pattern increased by 4 each time because after the second figure, two sides were hidden as the pattern increased and the two ends stayed the same. The triangle pattern increased by 1 each time because two sides are hidden beginning with the 3rd figure and there are always the 2 ends. In both cases, although in figure 2 there is only one side hidden, in the other, there is an end that counts for each of the triangles. The hexagon rule was y = 4x + 2 and the triangle rule is y = 1x + 2 which you now write as y = x + 2.

Since two of the sides of any polygon will be “inside” and not counted as perimeter, these two must be subtracted from the number of sides of the polygon. There will also always be the two ends on the first and last polygon that only have one side that is inside the figure, so these two must be added.

(# sides of polygon – 2)(# of polygons) + 2 will give the perimeter.

Predict the graphs for the regular polygon perimeters and verify your prediction. Sketch the graphs for the regular polygon perimeters of the regular triangle, square, and heptagon (seven sides) and connect your graph to the rule that represents the perimeters.

Sides of polygon / Perimeter of figure equation
3 / y = x +2
4 / y = 2x +2
7 / y = 5x +2

Using what you know about slope and the perimeter relationships in these regular polygons, explain where the hexagon graph will be on the grid.

The hexagon graph will have a slope between the heptagon and the square because the slope is only 4.

Blackline Masters, Mathematics, Grade 8Page 5-1

Unit 5, Activity 8, Patterns and Slope

Name ______Date ______Hour ______

For this activity, you will need 15 square tiles.

Arrange three tiles in a rectangle as shown.