©2013 Judo Math Inc.

LINEAR EXPRESSIONS

©2013 Judo Math Inc.

7th grade

Algebra/Proportions Discipline: Orange Belt Training

Welcome to the Orange Belt – Linear Expressions If you’re like me, you regularly hear the word “expression” tossed around in math class and usually just brush it under the rug. But an expression is really not that difficult. It’s basically an equation, without the equals sign. So an equation would be 3x-2=5, and an expression would just be the 3x-2 part. Plugging things in for the x value would give possible “simplifications” of this expression.

An expression can be as simple as:

And as complex as:

YIKES!

There are lots of different types of expressions: quadratic expressions, rational expressions, and linear expressions to name a few. For 7th grade, we just look at linear ones. You can look forward to those others down the road in High School. Good Luck Grasshopper!

Order of Mastery: Linear expressions 1. Linear Relationships: 2 situations(7EE1) 2. Linear Expressions (7EE1) 3. Moving from sequences to Expressions 4. Combining like terms (7EE1) 5. Expanding Linear Expressions (7EE1) 6. Factoring Linear Expressions (7EE1) 7. Rewriting linear expressions (7EE2)

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1. Linear Relationships: 2 situations

Trick or Treating: Jenny got dressed up as a cheerleader on Halloween and went trick or

treating in her neighborhood. She left her house with three pieces of candy in her bag

that she had swiped from the big bowl her mom had prepared for trick or treaters.

After trick-or-treating at the first house, she had 5 pieces of candy total. After trick-or-

treating at the second house, she had 7 pieces of candy total. After the third house, she

had 9 pieces total! Jenny, being very good at math, began to notice a pattern unfolding

and decided to do some predicting.

1. Create a table (a t-chart would work well!) showing the house number and the amount of candy Jenny has after each house. If the same trend continues at all the houses she visits, how many pieces of candy would she have after five houses, six houses, and ten houses?

2. Assuming she doesn’t eat any while out on the road, how many pieces of candy will Jenny after going to 20 houses? 45 houses? x houses?

3. Jenny’s Mom bet her that she would not get more than 100 pieces of candy. How many houses will she have to visit to make sure she proves her mom wrong? Explain

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Dream Car Problem: Brett sets up a savings account to eventually put a down payment

on his dream car. The initial balance of the account is $5500 and $250 is added to the

account each month straight out of his paycheck (there are no other

transactions that occur on the account).

a. Write an expression that shows the balance of the account

after each month (m).

b. Make a table of the balance after m months for m=3, 6, 9, 12, 15, 18, 21.

c. Brett needs $10,000 for the down payment on his car. How long will it take

him to reach this balance?

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2. Linear Expressions

1. Both of the situations from the last two pages have examples of linear expressions. Look back over them and explain what the two situations have in common. “What was the linear expression” in each problem and how were they similar and/or different?

2. Write an expression for the sequence of operations. a. Add 3 to x, subtract the result from 1, then double what you have. b. Add 3 to x, double what you have, then subtract 1 from the result.

Explain in writing what is similar and what is different in each of your expressions.

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3. Laila tells Julius to pick a number between one and ten. “Add three to your number and multiply the sum by five,” she tells him. Next she says, “Now take that number and subtract seven from it and tell me the new number.” “Twenty-three,” Julius exclaims.

a. Write an expression that records the operations that Julius used.

b. What was Julius’ original number?

c. In the next round Leila is supposed to pick a number between 1 and 10 and follow the same instructions. She gives her final result as 108. Julius immediately replies: “Hey, you cheated!” What might he mean?

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3. Moving from sequences to expressions

PATTERNS AND SEQUENCES: One way to define a sequence of terms is to look at the term before it and find a “pattern”: 1, 6, 11, 16, 21, ___, ___, ___. The pattern is “add 5” so they next three terms would be 26, 31, and 36.

TABLES AND EXPRESSIONS: A more algebraic way to look at sequences is in a table. From a table, you can more easily create an expression that represents the sequence. This would come in handy as you try to figure out the 20th, 100th, and then nth term in the sequence!

Term 1 2 3 4 5 6 7 x

Value 1 6 11 16 21 26 31 5x-4

Continue the following sequences then complete the table with the expression:

a. 3, 4, 5, 6, 7, 8, 9, 10, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

b. 10, 20, 30, 40, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

+5 +5 +5 +5 +5 +5

What expression

represents this?

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c. 13, 15, 17, 19, 21, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

d. 5, 8, 11, 14, ___, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

e. 3, 9, 15, 21, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

f. 4, 7, 10, 13, 16, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

g. -7, -5, -3, -1, 1, 3, 5, 7 ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

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h. (Challenge) 1, 4, 9, 16, ___, ___, ___

Term 1 2 3 4 5 6 7 x

Value

How is part h different from the rest of the problems? Explain in writing.

i. Mrs. Strong is saving up to buy a new-er car. After 1 month, she had

$125 in her savings account. After 2 months, she had $250 in her

savings account. And after 3 months, she had $375 in her savings

account. Create a table to represent this information. Extend it through 6

months and then make a prediction about how much money she will have

after x months

Term

Value

Extra challenge: Try graphing the terms and values on the x, y axis below, using the

term for the x and the value for the y. Describe the shape of the graph and how it relates

to the table above.

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4. Combining like terms

The basic unit of algebra is a “term”. Here are some…

6a -xy 4b2 -0.7p -x2 9mn

A term has 3 parts:

1. a sign, being positive (+) or negative (-) 2. a number, called ‘the coefficient’ 3. a letter, called ‘the variable’

*Every term must have a sign. If there isn’t one, assume it is positive.

*Terms must have a coefficient. If there isn’t one, assume it is ‘one’. (n=1n)

Practice with the following terms:

Example Sign (+ or -) Coefficient Variable(s)

8f + 8 f

-10r

xy

½ m

-22ab2

-12a

105ab

-56yx

(write your own)

(write your own)

(write your own)

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You can only combine terms with the same variables. Here are some examples. CAREFULLY

read each one to understand the simplification!

B + B really means 1B + 1B,so the answer is 2B B + B = 2B

2a + 3a the answer is 5 variables called ‘a’ 2a + 3a = 5a

6y + 8ab since the letters are all different, no simplification! 6y + 8ab

9G2 + 1G since only one variable has a power, no simplification! 9G2 + 1G

Simplify the expressions by combining similar terms:

1. 7d +18d 2. 25x + (-14x) – (-10x)

3. 10 - 5k + 8k + 17 – 7 4. 9z2 + 19z2 + 13z + 11z + 18 - 2 + 3

5. 3m– (-17m) - 1 - 16 - 15m 6. 10 - 15v - 17v2 + 7 + 13v2 - 4

7. 14r - 4 + 12 8. p + 6 + 2p2 - 14p2 + 5 - 3p

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9. 6f + 14f + 11f2 + 17f3 - 19f + 13f4 10. 4g + 16 + 12g - 3 + 8

11. -4u - 6u3 - 11u2 + 10u 12. a – (-12a2)+ (-11a) - a + 6a4

13. 2w2 – (-19w2)+ 5w - 7w + 11w4 + 4w + 12a2 + 11a - 4a + 13a2 + 6a4

Extra integer challenge, just for fun

Arrange the integers from 1 to 15 in a line so that the sum of each adjacent pair is a prime

number. For example, 4, 1, 2, 3 would work for the numbers 1 to 4, since 4 + 1 = 5, 1 + 2 = 3,

and 2 + 3 = 5.

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5. Expanding Linear Expressions

In the previous section, we learned about combining like terms. In this section we will continue

working with expressions but we will be expanding them!

In the first discipline we learned about the commutative and associative properties. These can always be used to expand linear relationships as seen below.

Commutative -4x + (3 + x) = -4x + (x + 3) = (-4x + x) + 3 = -3x + 3

Associative -1/5 x + 3/5 x = (-1/5 + 3/5)x = 2/5 x

The most important property in expanding linear relationships, however, is the distributive property. Look at the diagram below and write your own “definition” for the distributive property.

My definition for the Distributive Property

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Use the distributive, associative and commutative properties to expand these expressions. PICK 2 to explain which properties you used and why.

1. 5(x+6) =

2. 1/3(2x+1) =

3. 2/3(2x-3) =

4. 6x(x-4) =

5. 8/3(2x-3) =

6. -3x(x2 -2x + 4)=

7. -1/2(3x - 1)=

8. -10(2x + 1/5 )=

9. -18(x2 + x - 2/3)=

10. 1/2(x2 +6x -4) =

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Another way to look

at the distributive

property is with a

picture. Examine

the diagram

below to see how the

distributive

property is modeled.

Now use a picture to expand 6(3x+2)= Explain how the picture method works and how it is related to your definition from above.

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6. Factoring Linear Expressions

Factoring uses the reverse of the distributive property: ab + ac = a(b+c). When factoring linear

expressions, it’s very handy to remember a trick you may have learned in 5th or 6th grade called greatest

common factor.

Then circle the factor that is the largest that they have in common. In this situation it is… 8!!!

Practice finding the GCF or these sets of numbers:

1. 6 and 8

2. 72 and 81

3. 90 and 100

4. 13 and 52

5. 50 and 95

6. 66 and 77

When we rewrite ab + ac as a(b + c), what we are actually doing is factoring. A factor in this case is

one of two or more expressions multiplied together. Factoring an expression means breaking the

expression down into bits we can multiply together to find the original expression. Why would we

want to break something down and then multiply it back together to get something equivalent to what

we started with in the first place? Oh, who knows. Those crazy mathematicians have a lot of time on

their hands. It actually will come in handy, trust us.

Greatest Common Factor

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 32: 1, 2, 4, 8, 16, 32

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Sometimes we can factor a term with a variable out so we need to see what these have in common:

22x and 77x2

22x= 2, 11, x

77x2 = 7, 11, x, x,

What these have in common is 11 and x so the common factor is 11x.

Factor the following expressions: Check to see that your answer is correct by multiplying them back

through…

1. 2x+4

2. 3x+9

3. -10x-5

4. -12x + 2

5. 3x2 - 27xy

6. 2x2 + 2x +2

7. 6xy+5x2y

8. 3x2 - 27xy +9x

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9. Factor the expression - 50x + 4y in two different ways.

10. 45x - 9y + 99z cannot be factored. Explain why.

Be Careful: Always check your answers to factorization problems. When you multiply factors

together, you should find the original expression. This step is especially important when negative

signs are involved, because they can be a tad tricky. In fact, you probably shouldn't trust them at all!

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7. Rewriting linear expressions

Rewriting expressions can show how things are related and make some problems easier to

solve... like percent problems which you may think are easy to solve any way, but let me show

you a new trick!

Read this…

Another name for the expression ‘100% of x’

Another name for the expression ‘300% of x’ is: 3x

If x increases by 20%, then the new amount is: x+0.2x=1x+0.2x=1.2x

If x decreases by 30%, then the new amount is: x−0.3x=1x−0.3x=0.7x

Now try completing this table

means expression simplify The same as

increase by 5% a+0.05a 1.05a multiply by 1.05

b+0.04b

1.2c

Decrease by 5% d-0.05d

Multiply by .85

f-0.75f

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Sample problem: Suppose you want to purchase something that costs $60 ant it increases in price by

25% and then decreases by 40%. What is the final price…

Solution: $60 (original amount)

$60 x 1.25 (100% + 25% increase) = $75

$75 x 0.6 (100% - 40% decrease) = $45 This is the final price!

You could simplify this by just doing: $60 x 1.25 x 0.6= _______ Did you get the same thing? Awesome!

Now you try:

1. Suppose an item costs $25.

If the price decreases by 30%, and then increases by 20%, the new price is

2. Suppose an item costs $10.

If the price decreases by 50%, and then increases by 50%, the new price is:

3. Suppose an item costs x.

If the price increases by 50%, and then increases by 50%, the new price is:

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4. Suppose an item costs $100.

If the price decreases by 50%, and then decreases by 50%, the new price is:

5. Suppose an item costs x.

If the price increases by 19%, and then decreases by 30%, the new price is:

6. Suppose an item costs x.

If the price increases by 20%, and then decreases by 30%, the new price is:

7. Does it matter which order the item is increased and decreased in? For example, if an item cost

$50 and it increased in price by 25% and then decreased in price by 50% WOULD IT BE THE

SAME as it decreasing by 50% then increasing by 25%? Solve this problem and then explain your

reasoning (you may need to use words like associative or commutative in your explanation!)

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TWO FINAL PUZZLES, just for fun;)

1. Jeremy had this problem on his first mathematics examination. He was having big problems with

answering the question. Replace each letter with a different number. Identical letters should be

replaced with the same number.

SPART

x 4

----------

TRAPS

Can you help him?

2.