miin web viewunit 9: patterns, expressions ... word problems. pascal’s triangle. ... unit...

Unit 9 Math Mrs. ClementName: _________________

Teacher Check: ___________

Unit 9: Patterns, Expressions & EquationsWhat You’ll Learn...• To write a repeating decimal as a

fraction• To solve problems using patterns• To evaluate expressions to

represent a pattern• To evaluate an expression, given

the value• To create a table of values• To solve problems using the

graphs of a linear relation• To tell the difference between an

expression and an equation• To solve problems using equations

Why is it important?• We use patterns to translate between images and linear relations.• All graphs, images, maps, tables and expressions can be understood

as patterns. If you can crack the code, you are a genius!• Using equations is an effective problem solving tool• Using algebra to solve equations plays an important role in many

careers. For instance, urban planners use equations to estimate population growth.

• Have you ever run a red light? If you answered “no” it is because you have a basic understanding of algebra; Red equals STOP!

Unit 9 Math Mrs. Clement

+ I understand what this is and can do it very well√ I understand and can do most of this

− I have some trouble with this× I have a big problem with this

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Lesson Key WordsPractice

Questions/ Projects

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9.1 Describe Patterns What are patterns Increasing and decreasing

steps Variable and expressions

PatternVariablesExpressionsValueConstantNumerical coefficient

2325

9.2 Expressions and Equations Expressions vs. Equations Dividing by a constant Word problems

Pascal’s TriangleFibonacci SequenceAlgebraic expressionFlat feeSubstitution

43343945

9.3 Evaluate Expressions Model an expression Make a table of values Graph linear relations

EquationOperationCoefficientConstant termsconsecutive

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Book 7.277-101

9.4 Solving Equations One step equations Solve two-step equations

Unit Review and Test

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9.1 Describe PatternsInvestigate (textbook 348) How can we describe patterns?.

The numbers on a calendar form patterns. If you know the date of one Friday in January, you will know the next date is seven days later. Patterns can be made of shapes, colours, numbers, letters, words and more. Some patterns are easy to describe, while others are more difficult.

Name as many ways of representing patterns as possible:

Awesome video series on Simple Equations and Evaluations: https://www.youtube.com/watch?v=j_4FfSpFwOk

Not as awesome, but amusing and informative: https://www.youtube.com/watch?v=J2TYyUftI8k

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https://www.youtube.com/watch?v=J2TYyUftI8k

https://www.youtube.com/watch?v=j_4FfSpFwOk


It Started with Pizza: Patterns, Expressions and EquationsWorking in groups of 2 or 3, read through Dawn McMillan’s book “It Started with Pizza” to

complete the following questions:

Define in your own words:

Variables: _________________________________________________Equations: _________________________________________________

1. What does x represent in an equation?

2. What does x represent in the equation shown on page 5?

3. What is the solution to “How many hours” (page 8)? Show your work. You may use a diagram to demonstrate your understanding.

Define in your own words:Expression: _______________________________________________

4. What is the expression discussed on page 9?

5. Use the grid paper below to sketch your own bedroom. Calculate how much area is taken up in your room with furniture. How much area remains? Create a formula to solve the equation.

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6. Write your answers for the equations on page 15. Show your work.

7. Write an equation to show how many moons Jupiter has (page 24).

Describing PattersLet’s ExploreExample #1: Continue the sequence, then find the rule

a. 41, 38, 35, _____, _____, _____

Write a short reflection about your impressions of this book. What was useful?? What seemed straightforward?

What was confusing Do you have any remaining questions?

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The rule is: __________________________________b. 1, 4, 16, _____, _____, _____

The rule is: __________________________________c. 10, 15, 20, 10, 15, 20 , _____, _____

The rule is: __________________________________

Example #2: Find how each sequence was made. Then extend the pattern

Example #3: What type of language can we use to describe patterns?

A. Decreases by the same amount _______ 13, 20, 27, 34, 41B. Increases by different amounts _______ 9, 6, 9, 12, 9C. Increases by seven each time _______ 8, 12, 15, 19, 25

D. Increases and decreases by 3 each time _______ 45, 43, 41, 39, 37

Example #4: How can we use tables to organize patterns?

Allie planted an apple tree when she was four years old. Every year she records its height. Some of her data got lost. Can you figure out the missing heights?

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Allie’s Age Height of the tree

4 10056 1407 1608


Try it Out: 1. There are 14 apples in total. Write an

expression to show how you evenly divide all of the apples, except two, the three paper bags. Can you find the number of apples in each bag?

2. A bicycle company charges a flat rate of $6, plus $4 for every hour of rental.

a. Write an expression to show how much the bike will cost for each hour it is rented. Use h to represent hours.

b. How much will it cost to rent the bike for 3 hours?

3. Circle the equation that does not have the same answer as the other three:

4x + 5 = 21 21 = 5 + 4x 4x + 21 = 521 – 4x = 5

4. Underline the variables. Then write an expression for each cost. Use n for the variables.

Advanced

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5. Pick any number greater than one Pascal’s Triangle. Add the two numbers together directly above it. Do you see a pattern?

6. There are 8 dots in the figure on the right. Each pair of dots is joined by exactly one line segment. How can you find out how many line segments there are without counting every line?

Patterns… Everywhere!Another incredible (and famous) pattern is the Fibonacci Sequence. This pattern can be found throughout nature.

Describe the pattern in and trace the sequence in the following images.

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Cost ($)DVD’s are on sale for $5 eachA grocery store charges $0.39 for each lemon

A car rental company charges $50 flat fee plus $20 per day


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Due Date:______________Show You Know (textbook p. 354-56)

1. Describe the patterns of squares..

2. Describe the pattern of dots.

3. Olivia makes copies of the design below using one square and several triangles of the same size. Complete the chart and write the formula showing how to find the number of triangles form the number of squares.

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4. Diara makes an increasing pattern with blocks. After making the 3rd figure, she has only 14 blocks left. Does she have enough blocks to complete the 4th

figure?

Yes No

5. Haleigh makes an ornament using a hexagon (the white figure), rhombuses (the striped figure), triangles and squares.

a. How many triangles would Haleigh need to make 9 ornaments?

b. How many squares would she need to make 6 ornaments?

c. Haleigh used 4 hexagons to make ornaments. Hoe many rhombuses and how many triangles did she use?

9.2 Expressions and EquationsHomework Reflection:Which parts of the homework did you find pretty straightforward?

Which parts challenged you?

If you had difficulty with a certain question, what strategies did you use to solve the problem?

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Figure Number of Blocks

1 82 103 12


Investigate (textbook 348) How can we describe patterns?.

Let’s Explore

Example #1: Using a balance to find model equations.

Each weighs x kg and each weighs 1 kg. Scale A is balanced perfectly. Draw scale B so that there are only triangles on the first side.Draw scale C to show what balances one triangle.Under each scale, write an algebraic equation.

Example #2: Write 6x in different ways.

a. 6x = x + x + x + x + x + x

b. 6x = x + x + x + x + x + x

c. 6x = x + x + x + x + x + x

Example #3: Add expressions. The first is completed for you.a. 3x + x = ___4x___b. 5x + 2x = _______

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c. 7x – x = ______d. 4x + 3x + 1x = _______

Example #4: Fill in the blanks by crossing out numbers or variables that add to 0.

a. 4 + 3 – 3 = ____b. 8 + 6 – 6 = ____c. 4 + 3 – 3 + 7 – 7 + 6 =

____d. x + 12 – x = ____

e. 5 + 2 + 2 = ____f. x + x – x = ____g. 7 + x – x = ____h. x + x + x – x = ____

Example #5; Regroup the x’s together, then solve for x.a. 8x – 3x + x – 2 = 28 x = ________b. 5x + x – x – 2x + 4 = 19 x = ________c. 7x + 4 – 2x – 3 = 26 x = ________

Every time you see a number or variable subtracted by itself in an equation (3-3, 5-5, x-x), you can cross out both numbers or variables because they will add to 0. Crossing out parts of an equation that make zero is called cancelling out.

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Try It1. Circle the equations and underline the expressions

5n - 3 n + 6 = 7 7 + 3n 7 + 3n = 5 + 2na + b a + b = b + a 8 – 3n = 5 4 + 6n – 5n

2. What is the difference between an expression and an equation: __________________________________________________________________________________________________________

3. How are they the same? ___________________________________ _____________________________________________________

4. Write an equation that contains the expression 5n + 3. ____________

5. In the expression 5n + 3, what is changing, the 5n or the 3? ________

6. Write the coefficient in each expression:a. 2x – 7 ________b. m – 3 ________

c. 4n + 5 ________d. 6 + 7w ________

7. Write the coefficient of x in each expression. The first is completed for you.

BUZZWORD: in an expression, the number that is multiplied by the variable is called the coefficient

Some expressions have more than one variable. Each variable has its own coefficient like in the expression 3a + 7b + 7. Can you think of where this type of equation might be used?

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a. 3x + 4y + 7 ___3___b. 2u + 5x + 4w + 6

_____

c. 3 + 4x + w ______d. 3w + x ______

8. Write your own expression. Include one, two or even three variables.

9. Write the constant term in each expression. a. 4x + 7 = _____b. 8r _____c. 7u + 9 _____d. 5 + 3x _____

10. A car is travelling at a speed of 50km per hour. a. Write an expression for the distance the car travels in h

(hours): ______________b. What is the constant term? ___________c. What is the coefficient? ___________d. What does the variable in your expression represent?

i. The speed at which the car is travellingii. The number of hours the car travelsiii. The distance the car travels

Dividing by a Constant

In an expression, the quantity without the variable is called the constant term because it does not change.Example: 3x + 4 5x = 5x + 0

Constant term 4Constant term 0

Division is often written in fractional form:

12 ÷ 4 = 124 15 ÷ 5 = 155 w ÷ 7 = w715


Explore:Example #1: Solving division problems

a. 63 = 2❑b. 126 = ❑❑

c. 124 = ❑❑

d. 202 = ❑❑

Example #2: solve division problems with expressionsa. 3x = 12

x = 123x = 4

b. 2x = 10x = x =

c. 4x = 12x = x =

d. 2x = 14x = x =

Example #3: Solve each equation working backwards

a. x3 = 4

x = x =

b. x5 = 2x = x =

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Due Date:______________Show You Know (textbook p. 223)

1. Look at the algebraic expressions and equations below. Which are expression? Equations? How do you know?

a. 4w = 48

b. g – 11

c. 3d + 5

d. x12 = 3

e. i−510

f. 6 z+1=67

2. Hannah gives 10 CD’s to her sister. Now she owns 35 CD’s. Write an equation to find how many CD’s Hannah had to begin with.

3. Write an equation that would help you solve the following statements.

a. Seven more than a number is 18.

b. Six less than a number is 24.

c. Five times a number is 45.

d. A number divided by 6 is 7.

e. Three more than a number is 19.

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4. Find the value of n that makes the equation true.

a. 3n = 27

b. 2n + 3 = 27

c. 2n – 3 = 27

d. n3 = 27

5. Sketch balance scales to represent each equation. Solve the equation.

a. x + 12 = 19

b. x + 5 = 19

c. 4y = 12

d. 3k + 7 + 31

6. Write an algebraic description for each description.a. Four more than a number.

b. A number reduced by 4.

c. A number decreased by 10.

d. The sum of a number and 7.

e. Twice as much as a number.

BONUS ($5 ClementBucks): Marks’s dad is three times older than Mark. The sum of their ages is 48. How old is mark?

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Homework Reflection:

Which parts of the homework did you find pretty straightforward?

Which parts challenged you? If you had difficulty with a certain question, what strategies did you use to solve the problem?

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9.3 Graphing Linear Relations

Investigate (textbook 372) How can you determine a pattern on a coordinate grid?

Alexis makes a garden path using square and triangular tiles. Her pattern uses 6 triangular tiles for every 1 square tile. She writes a formula- an equation that shows how to calculate the number of triangles (t) from the number of squares (s).

6 x squares = triangles6 x s = t

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FormulasExample #1: Each chart represents a design for a path. Complete the chart.

Example #2: Find the number you must

add to the row number to get the number of chairs. Write a formula using r for the row number and c for the number of chairs.

Example #3: Write the Ordered Pair

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Squares (s)

4 x s = t Triangles (t)

1 4 x 1 = 4 42 4 x 2 =3 4 x 3 =

Squares (s) 7 x s = t Triangles (t)

1 7 x 1 = 7 7

2 7 x 2= 7

3 7 x 3 = 7

Squares (s)

4 x s = t Triangles (t)

5 4 x = _____

6 4 x = _____

7 4 x = _____

Row Chair1 52 63 7

r + 4 = c

Row Chair7 108 129 14

Row Chair1 92 103 11


How does the rule change when you switch the input and output numbers?

Write the ordered pair.

( 3,5)( )( )( )( )

An ordered pair consists of two numbers where the order they are written matters. We can treat input and output numbers as ordered pairs.

Example:

The ordered pair (2,1) is different than (1,2)

What have you noticed about patterns so far? What seems pretty straightforward? What is still too complicated?

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Graphing Linear RelationsExample #1: For each set of points, write a list of ordered pairs and complete the t-chart.

Example #2: Make a T-chart for each set of points on the coordinate grid.

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Example #3: How can we interpret graphs.

The graph shows the cost of making a telephone call to New York.

a. If you talked for two

minutes, how much would you have to pay?

b. How much does the cost rise every minute?

c. How much would you have to pay for 10 minutes?

The graph shows the distance Kathy travelled on a cycling trip.

a. How far had Kathy cycled after 2 hours?

b. Did Kathy rest at all on

her trip? How do you know?

c. When she was cycling, did she maintain the same speed at all times?

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The graph shows Olivia and Madi’s progress in a 120m race.

a. How far from the start was Olivia after 10

seconds?

b. How far was Madi after 15 seconds?

c. Who won the race? By how much?

Due Date:______________Show You Know (textbook p. 378)

1. Kyra makes a pattern with tiles and white tiles. Make a table of values showing the number of red tiles compared to the number of white tiles. Create a table for the first five patterns you could make.

2. This graph shows how many people visited a new web site every day.

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a. Imagine the pattern continues. Make a table of values for the first 7 values of d, starting at d+1.

b. Describe the pattern.

c. If the pattern continues, how many people will visit the site on day 12?

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3. Emily makes a stick pattern of triangles from stir sticks. How many triangles are shown in the diagram?

a. Graph your table of values.

b. What is the relationship between the sticks an the number of triangles?

c. How many sticks are needed to make 2007 triangles?

4. The diagram shows the number of guests that can be seated at a restaurant,

a. How many guests do you think can be seated at 5 small tables?

b. Make a table of values to show the number of guests that can be seated at 1, 3, 5, and 7 small tables.

c. Draw a graph using the values in the table.

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9.4 Solving One-step and Two-step EquationsInvestigate (Textbook 395) How do we solve more complicated equations?

When we think about algebra, we are really talking about using codes, patterns and images to help us imagine the answers to POSSIBILITIES.

A helpful image is the idea of scales.

For example, there is a one on the left side of the scale and a six and mystery number on the right. How can we balance the scale? If we move the mystery number to the right (five) we can balance the scales.

In the second example, you have a six and seven on the right. You have two choices: you can either place a 13 on the left side of the scale to balance 6 and 7 OR you can move the six over and pair it up with a 1. Either way, you have balanced the scale.

In the final example, you have more than one box, but the challenge remains the same. How to you balance the scale? What are all the possibilities?

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Solve One-Step Equations (x + a = b)Mrs. Clement and her mother spend $56 to take a ferry from Victoria to Vancouver. Mrs. Clement knows that the cost of one person is $11. So, the cost of two people is 2x $11 = $22. He decides to model the situation with the equation C + 22 = 56, where C is the cost of the car. How could she determine the cost of the car?

There are three way to solve an equation:

Example #1: Use mental math to solve the equation.

i + 4 = 12 --- i + 8

Step 1: Ask yourself “What number added to 4 makes 12?”Step 2: use mental math to solve.Step 3: input your answer to solve.

Example #2: Model and Solve the Problem

Jane and her sister cycle 4km to the shopping mall, then travel to their mom’s office. If they cycle 11km in total, how far is the shopping mall from the office?

Step 1: let d represent the distance from the shopping mall to the office. Sketch a diagram to represent the equation.

Step 2: balance the masses on each side of the scale.Step 3: Solve the equation.Example #3: Apply the opposite operation

Mrs. Clement needs to solve C + 22 = 56 to find the cost of C of taking the car to Vancouver. What is the cost? Check your answer…

Step 1: C + 22 = 56 can also be written as 56 – 22 = C

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Step 2. Solve. C= 34Step 3: Check your answer: 34 + 22 = __56__

Try it Out:

1. Write the number that makes each equation true.

a. 8 + 4 - = 8

b. 12 ÷ 4 x = 12

c. 8 x 3 ÷ = 8

d. 19 + 3 - = 19

2. Write the operation that makes each equation true.

a. 7 + 2 ___ 2 = 7b. 18 ÷ 3 ___ 3 = 18

c. 6 + 4 ___ 4 + 6d. 15 – 4 ___ 4 = 15

3. Write your own one-operation code. Have a friend try to solve it. Create one problem with a missing number and another with a missing operation.

4. Circle the expression that gets you back to m.7m – 7 7m ÷ 7 m ÷7 x 7

7 ÷ m x 7 7 + m -7 7 – m + 7

Solve Two-Step Equations (ax + b = c)

A clothing store has a sale. You can buy two T-shirt and sunglasses for only $19. How much is the cost of each T-shirt?

What could you use to model this equation?

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What information would make this equation easier to solve?

Write an equation that could represent this problem.

Example #1: Model equations

Step 1: Let’s say that t represents one t-shirt.

t x 2 (two T-shirts) OR t 2

Let’s say that s represents the sunglasses.

(t )2 + (s) = $19

Step 2: move the items so that the t-shirts are on one side of the equation. What information do you still need? Find out the cost of the sunglasses. Take this cost away from the total ($19 - $3). Now you can solve the problem.

(t) x 2 = 16t = 16 ÷ 2t = 8

Try it Out

Jason does the same operations to the secret number x. He gets 37 each time. Write an equation, then work backwards to find x.

Jason's operation

Start with x: ____________Multiply by 5: ____________Add 7: _____________The answer is 37: _________

Work Backwards…Write the equation again: ____________

Undo adding 7 by subtracting 7: ____________Write a new equation:____________________

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Undo multiplying by 5 by dividing by 5: _______Write the new equation. __________________ You solved x!

1. Solve for the variable by undoing each operation in the equation. The first question is solved for you.

a. 8x + 3 = 278x + 3 – 3 = 27 – 38x = 248x ÷ 8 = 24 ÷ 8x = 3 .

b. 4h – 3 = 37

4h - 3 _____= ______4h = ________4h÷ ____ = _______x =_____

c. 3s – 4 = 29

d. 2t + 3 = 11

e. 4z + 3 = 19

f. 8x + 3 = 3

2. A store charges you $3 per hour to rent a pair of figure skates. a. Write an expression for the cost of renting the skates. Use

h for hours.

b. Anastasia rented the skates for 4 hours. How much did she pay?

c. Kyra paid $15 to rent (hockey) skates. How many hours did she skate?

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3. Model and solve each equation. Check your answer with the “check back” method.

a. 3s + 1 – 7 b. 2 + 5n = 12

4. BONUS: Once again you were able to crack Jim’s code. He has now decided to make a new code that is as complicated as he can make it. Jim’s final message to his friends is:43 19 21 41 / 21 41 / 29 53 / 7 13 41 43 / 9 33 11 13 / 53 13 43

Jim’s new code uses an equation of the form ax + b = c to change the letters in the message to numbers in the coded version.Can you decode the message?

1=a 2=b

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miin web viewunit 9: patterns, expressions ... word problems. pascal’s triangle. ... unit...

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