expressionsandequations7thgrade20120705.notebook 1 October 26, 2012 www.njctl.org This material is made freely available at www.njctl.org and is intended for the noncommercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative www.njctl.org 7th Grade Math Expressions & Equations www.njctl.org 20120705 table of contents Table of Contents Inverse Operations One Step Equations Two Step Equations MultiStep Equations Variables on Both Sides More Equations Graphing & Writing Inequalities with One Variable Click on a topic to go to that section. The Distributive Property Combining Like Terms Simple Inequalities involving Addition & Subtraction Simple Inequalities involving Multiplication & Division Common Core Standards: 7.EE.1, 7.EE.4 Apr 258:15 PM The Distributive Property Return to Table of Contents Apr 258:19 PM An Area Model Find the area of a rectangle whose width is 4 and whose length is x + 2 4 x 2 Area of two rectangles: 4(x) + 4(2) = 4x + 8 4 x+ 2 Area of One Rectangle: 4(x+2) = 4x + 8 Apr 258:41 PM The Distributive Property Finding the area of the rectangles demonstrates the distributive property 4(x + 2) 4(x) + 4(2) 4x + 8 The 4 is distributed to each term of the sum (x + 2)
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This material is made freely available at www.njctl.org and is intended for the noncommercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.
Click to go to website:www.njctl.org
New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
www.njctl.org
7th Grade Math
Expressions & Equations
www.njctl.org
20120705
table of contents
Table of Contents
Inverse OperationsOne Step EquationsTwo Step EquationsMultiStep EquationsVariables on Both SidesMore EquationsGraphing & Writing Inequalities with One Variable
The Distributive Property is often used to eliminate the parentheses in expressions like 4(x + 2). This makes it possible to combine like terms in more complicated expressions.
30 The lengths of the sides of home plate in a baseball field are represented by the expressions in the accompanying figure.
A 5xyzB x2 + y3zC 2x + 3yzD 2x + 2y + yz
yz
yy
xx
From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Which expression represents the perimeter of the figure?
variables
Inverse Operations
Return to Table of Contents
equation
What is an equation?
An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in
2+3=5
92=7
equation
Equations can also be used to state the equality of two expressions containing one or more variables.
In real numbers we can say, for example, that for any given value of x it is true that
4x + 1 = 14 1
If x = 3, then
4(3) + 1 = 14 1 12 + 1 = 13
13 = 13
equation
When defining your variables, remember...
Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand.
While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes.
Try It!
Write an equation with a variable and have a classmate identify the variable and its value.
Both sides need to contain the same quantity in order for it to be "balanced".
inverse
For example, 20 + 30 = 50 represents an equation because both sides simplify to 50. 20 + 30 = 50
50 = 50
Any of the numerical values in the equation can be represented by a variable.
Examples:
20 + c = 50 x + 30 = 50
20 + 30 = y
Aug 3110:50 PM
Why are we Solving Equations?
First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to.
Now, we are told what it simplifies to and we need to find the value of the variable.
When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).
inverse
In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.
Let's review the inverses of each operation:
Addition Subtraction
Multiplication Division
Oct 309:46 PM
There are four properties of equality that we will use to solve equations. They are as follows:
Addition PropertyIf a=b, then a+c=b+c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation.
Subtraction PropertyIf a=b, then ac=bc for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation.
Multiplication PropertyIf a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation.
Division PropertyIf a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.
inverse
To solve for "x" in the following equation... x + 7 = 32
Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides.
x + 7 = 32 7 7
x = 25
In the original equation, replace x with 25 and see if it makes the equation true.
For each equation, write the inverse operation needed to solve for the variable.
a.) y +7 = 14 subtract 7 b.) a 21 = 10 add 21
c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 12
move move
move move
inverse
Think about this...
To solve c 3 = 12
Which method is better? Why?
Kendra
Added 3 to each side of the equation
c 3 = 12 +3 +3 c = 15
Ted
Subtracted 12 from each side, then added 15.
c 3 = 12 12 12c 15 = 0 +15 +15 c = 15
inverse
Think about this...
In the expression
To which does the "" belong?
Does it belong to the x? The 5? Both?
The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable.
So:
question
31 What is the inverse operation needed to solve this equation?
7x = 49
A Addition
B Subtraction
C Multiplication
D Division
question
32 What is the inverse operation needed to solve this equation?
Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.
This means that you undo in the opposite order (PEMDAS):1st: Addition & Subtraction2nd: Multiplication & Division3rd: Exponents4th: Parentheses
Whatever you do to one side of an equation, you MUST do to the other side!
Apr 1111:19 PM
Examples:
3x + 4 = 10 4 4 Undo addition first 3x = 6 3 3 Undo multiplication second
x = 2
4y 11 = 23 + 11 +11 Undo subtraction first 4y = 12 4 4 Undo multiplication second
y = 3
Remember whatever you do to one side of an equation, you MUST do to the other!!!
two step practice
67x = 836 6 7x = 77 7 7 x = 11
3x + 10 = 46 10 10 3x = 36 3 3 x = 12
4x 3 = 25 +3 +3 4x = 28 4 4 x = 7
2x + 3 = 1 3 3 2x = 4 2 2 x = 2
9 + 2x = 239 9 2x = 14 2 2 x = 7
8 2x = 88 8 2x = 16 2 2 x = 8
Two Step Equations
Solve each equation then click the box to see work & solution.
What if you did it a little differently?4x + 8 = 2x + 264x 4x Subtract 4x from both sides 8 = 2x + 26 26 26 Undo Addition 18 = 2x 2 2 Undo Multiplication
9 = x
Recommendation: Cancel the smaller amount of the variable!
Apr 1111:19 PM
Example:
6r 5 = 7r + 7 2r 6r 5 = 5r + 7 Simplify Each Side of Equation5r 5r Subtract 5r from both sides (smaller than 6r) r 5 = 7 + 5 +5 Undo Subtraction r = 12
Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Define t = Mary's time in hourst + 0.25 = Jocelyn's time in hours
Relate
Write 12t = 9(t+0.25)
example
12t = 9(t + 0.25)
12t = 9t + 2.259t 9t
3t = 2.253 3
t = 0.75
It took Mary 0.75h, or 45 min, to get home.
Step 1 distribute the 9 inside the parenthesis(pull)
90 You are selling tshirts for $15 each as a fundraiser. You sold 17 less today then you did yesterday. Altogether you have raised $675.
Write and solve an equation to determine the number of tshirts you sold today.
Be prepared to show me your equation!
Aug 1012:59 PM
91 The length of a rectangle is 9 cm greater than its width and its perimeter is 82 cm.
Write and solve an equation to determine the width of the rectangle.
Be prepared to show me your equation!
Aug 1012:59 PM
92 The product of 4 and the sum of 7 more than a number is 96.
Write and solve an equation to determine the number.
Be prepared to show me your equation!
Aug 1012:59 PM
93 A magazine company has 2,100 more subscribers this year than last year. Their magazine sells for $182 per year. Their combined income from last year and this year is $2,566,200.
Write and solve an equation to determine the number of subscribers they had each year.
Be prepared to show me your equation!
Aug 1012:59 PM
94 The perimeter of a hexagon is 13.2 cm.
Write and solve an equation to determine the length of a side of the hexagon.
Three less than a number is less than three times that number.
The sum of two consecutive numbers is at least thirteen.
Three times a number plus one is at least ten.
x 3 < 3x
x + (x + 1) ≥ 13
3x + 1 > 10
Answer
Answer
Answer
Solution Sets
A solution to an inequality is NOT a single number. It will have more than one value.
10 2 3 4 5 6 7 8 9 1012345678910
This would be read as the solution set is all numbers greater than or equal to negative 5.
Solution Sets
Solution Sets
Let's name the numbers that are solutions of the given inequality.
r > 10 Which of the following are solutions? 5, 10, 15, 20
5 > 10 is not trueSo, not a solution
10 > 10 is not trueSo, not a solution
15 > 10 is trueSo, 15 is a solution
20 > 10 is trueSo, 20 is a solution
Answer:15, 20 are solutions of the inequality r > 10
Solution Sets
Let's try another one.
30 ≥ 5d; 4,5,6,7,8
30 ≥ 5d30 ≥ 5(4)30 ≥ 20
30 ≥ 5d30 ≥ 5(5)30 ≥ 25
30 ≥ 5d30 ≥ 5(6)30 ≥ 30
30 ≥ 5d30 ≥ 5(7)30 ≥ 35
30 ≥ 5d30 ≥ 5(8)30 ≥ 40
Answer: 4,5,6
Graphing Inequalities
Graphing Inequalities with Greater/Less Than or Equal To
An open circle on a number shows that the number is not part of the solution.It is used with "greater than" and "less than".The word equal is not included.< >
A closed circle on a number shows that the number is part of the solution.It is used with "greater than or equal to" and "less than or equal to".< >
Graphing Inequalities
Remember!
Open circle means that number is not included in the solution set and is used to represent < or >.
Closed circle means the solution set includes that number and is used to represent ≤ or ≥.
Step 1: Figure out what the inequality solution requires. For example, rewrite x is less than one as x < 1.
Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, an open circle since it represents the starting point for the inequality solution but is not part of the solution.
1 02345 1 2 3 4 5
Graphing Inequalities
Graphing Inequalities
Step 4: Draw a line, thicker than the horizontal line, from the dot to the arrow. This represents all of the numbers that fulfill the inequality.
Step 3: Draw an arrow on the number line showing all possible solutions. For numbers greater than the variable, shade to the right of the boundary point. For numbers less than the variable, shade to the left of the boundary point.
1 02345 1 2 3 4 5
1 02345 1 2 3 4 5
x < 1
Graphing Inequalities
10 2 3 4 5 6 7 8 9 1012345678910
Step 1: Figure out what the inequality solution requires. For example, rewrite x is greater than or equal to one as x > 1.
Step 2: Draw a circle on the number line where the number being graphed is represented. In this case, a closed circle since it represents the starting point for the inequality solution and is a part of the solution.
A store's employees earn at least $7.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.
Let e represent an employee's wages.
Graphing Inequlities
Try this:
The speed limit on a road is 55 miles per hour. Define a variable, write an inequality and graph the solution.
• Solving onestep inequalities is much like solving onestep equations. • To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations.
Simple Inequalities
12 > x + 6
To find the solution, isolate the variable x.
Remember, it is isolated when it appears by itself on one side of the equation.
Simple Inequalities
Step 1: Since 6 is added to x and subtraction is the inverse of addition, subtract 6 from both sides to undo the addition.
12 > x + 6 6 6
6 > x
Simple Inequalities
Step 2: Check the computation. Substitute the end point of 6 for x. The end point is not included (open circle) since x < 6.
12 > x + 612 > 6 + 612 > 12
0 1 2 3 4 5 6 7 8 9 10
Simple Inequalities
Step 3: Check the direction of the inequality. Choose a number from your line (such as 4) and check that it fits the inequality.
0 1 2 3 4 5 6 7 8 9 10
6 > x6 > 4
Simple Inequalities
10 2 3 4 5 6 7 8 9 1012345678910
k > 5
3 3k + 3 > 2
A. k + 3 > 2
Solve and graph.
5 is not included in solution set; therefore we graph with an open circle.