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Number and Operations
Properties of Operations
So far, you have seen a couple of different models for the operations: addition, subtraction, multiplication, and division. But we haven’t talked much about the operations themselves — how they relate to each other, what properties they have that make computing easier, and how some special numbers behave. There’s lots to think about!
The goal in this section is to use the models to understand why the operations behave according to the rules you learned back in elementary school. We’re going to keep asking ourselves “Why does it work this way?”
Think / Pair / Share
Each of these models lends itself to thinking about the operation in a slightly different way. Before we really dig in to thinking about the operations, discuss with a partner:
- Of the models we discussed so far, do you prefer one of them?
- How well do the models we discussed match up with how you usually think about whole numbers and their operations?
- Which models are useful for computing? Why?
- Which models do you think will be useful for explaining how the operations work? Why?
Connections Between the Operations
We defined addition as combining two quantities and subtraction as “taking away.” But in fact, these two operations are intimately tied together. These two questions are exactly the same:
27 – 13 = ____ 27 = 13 + _____.
More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you choose for a, b, and c!)
c – b = a c = a + b.
In other words, we can think of every subtraction problem as a “missing addend” addition problem. Try it out!
Here is a strange addition table. Use it to solve the following problems. Justify your answers. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know about the operations!
![\[ \begin{tabular}{| c | c | c | c |} \hline + & \textbf{A} & \textbf{B} & \textbf{C} \\ \hline \textbf{A} & C & A & B \\ \hline \textbf{B} & A & B & C\\ \hline \textbf{C} & B & C & A\\ \hline \end{tabular} \]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-dfc9537fd65b39b77b97c607062aa00c_l3.png "Rendered by QuickLaTeX.com")
A + C B + C A – C C – A A – A B – C
How does an addition table help you solve subtraction problems?
We defined multiplication as repeated addition and division as forming groups of equal size. But in fact, these two operations are also tied together. These two questions are exactly the same:
27 ÷ 3 = _____ 27 = _____ × 3.
c ÷ b = a c = a · b.
In other words, we can think of every division problem as a “missing factor” multiplication problem. Try it out!
Rewrite each of these division questions as a “missing factor” multiplication question. Which ones can you solve and which can you not solve? Explain your answers.
9 ÷ 3 100 ÷ 25 0 ÷ 3 9 ÷ 0 0 ÷ 0
Here’s a multiplication table.
![\[ \begin{tabular}{| c | c | c | c | c | c |} \hline \times & \textbf{A} & \textbf{B} & \textbf{C} & \textbf{D} & \textbf{E}\\ \hline \textbf{A} & A & A & A & A & A\\ \hline \textbf{B} & A & B & C & D & E\\ \hline \textbf{C} & A & C & E & B & D\\ \hline \textbf{D} & A & D & B & E & C\\ \hline \textbf{E} & A & E & D & C & B\\ \hline \end{tabular} \]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-cf6bafe9732254e04e9443de23ec9981_l3.png "Rendered by QuickLaTeX.com")
- Use the table to solve the problems below. Justify your answers. Important: Don’t try to assign numbers to the letters. Solve the problems just using what you know about the operations!
C × D C × A A × A C ÷ D D ÷ C D ÷ E
How does a multiplication table help you solve division (and exponentiation) problems?
Throughout this course, our focus is on explanation and justification. As teachers, you need to know what is true in mathematics, but you also need to know why it is true. And you will need lots of ways to explain why , since different explanations will make sense to different students.
Arithmetic Fact: a + b = c and c – b = a are the same mathematical fact.
Why is this not a good explanation?
“I can check that this is true! For example, 2+3 = 5 and 5 – 3 = 2. And 3 + 7 = 10 and 10 – 7 = 3. It works for whatever numbers you try.”
Addition and Subtraction: Explanation 1
Arithmetic fact:.
a + b = c and c – b = a are the same mathematical fact.
Why It’s True, Explanation 1:
First we’ll use the definition of the operations.
Suppose we know c – b = a is true. Subtraction means “take away.” So
means we start with quantity c and take away quantity b , and we end up with quantity a . Start with this equation, and imagine adding quantity b to both sides.
On the left, that mans we started with quantity c , took away b things, and then put those b things right back! Since we took away some quantity and then added back the exact same quantity, there’s no overall change. We’re left with quantity c .
On the right, we would be combining (adding) quantity a with quantity b . So we end up with: c = a + b.
On the other hand, suppose we know the equation a + b = c is true. Imagine taking away (subtracting) quantity b from both sides of this equation: a + b = c.
On the left, we started with a things and combined that with b things, but then we immediately take away those b things. So we’re left with just our original quantity of a .
On the right, we start with quantity c and take away b things. That’s the very definition of c – b . So we have the equation:
a = c – b .
Why It’s True, Explanation 2:
Let’s use the measurement model to come up with another explanation.
The equation a + b = c means Zed starts at 0, walks forward a steps, and then walks forward b steps, and he ends at c .
If Zed wants to compute c – b , he starts at 0, walks forward c steps, and then walks backwards b steps. But we know that to walk forward c steps, he can first walk forward a steps and then walk forward b steps. So Zed can compute c – b this way:
- Start at 0.
- Walk forward a steps.
- Walk forward b steps. (Now at c , since a + b = c .)
- Walk backwards b steps.
The last two sets of steps cancel each other out, so Zed lands back at a . That means c – b = a .
On the other hand, the equation c – b = a means that Zed starts at 0, walks forward c steps, then walks backwards b steps, and he ends up at a .
If Zed wants to compute a + b , he starts at 0, walks forward a steps, and then walks forwards b additional steps. But we know that to walk forward a steps, he can first walk forward c steps and then walk backwards b steps. So Zed can compute a + b this way:
- Walk forward c steps.
- Walk backwards b steps. (Now at a , since c – b = a .)
- Walk forward b steps.
The last two sets of steps cancel each other out, so Zed lands back at c . That means a + b = c .
- Read over the two explanations in the example above. Do you think either one is more clear than the other?
c ÷ b = a is the same fact as c = a × b.
Properties of Addition and Subtraction
You probably know several properties of addition, but you may never have stopped to wonder: Why is that true?! Now’s your chance! In this section, you’ll use the definition of the operations of addition and subtraction and the models you’ve learned to explain why these properties are always true.
Here are the three properties you’ll think about:
- Addition of whole numbers is commutative .
- Addition of whole numbers is associative .
- The number 0 is an identity for addition of whole numbers.
For each of the properties, we don’t want to confuse these three ideas:
- what the property is called and what it means (the definition),
- some examples that demonstrate the property, and
- an explanation for why the property holds.
Notice that examples and explanations are not the same! It’s also very important not to confuse the definition of a property with the reason it is true!
These properties are all universal statements — statements of the form “for all,” “every time,” “always,” etc. That means that to show they are true, you either have to check every case or find a reason why it must be so.
Since there are infinitely many whole numbers, it’s impossible to check every case. You’d never finish! Our only hope is to look for general explanations . We’ll work out the explanation for the first of these facts, and you will work on the others.
Addition is Commutative
Example: commutative law.
Addition of whole numbers is commutative.
What it Means (words):
When I add two whole numbers, the order I add them doesn’t affect the sum.
What it Means (symbols):
For any two whole numbers a and b ,
a + b = b + a .
Now we need a justification . Why is addition of whole numbers commutative?
Let’s think about addition as combining two quantities of dots.
- To add a + b , we take a dots and b dots, and we combine them in a box. To keep things straight, lets imagine the a dots are colored red and the b dots are colored blue. So in the box we have a red dots, b blue dots and a + b total dots.
- To add b + a , let’s take b blue dots and a red dots, and put them all together in a box. We have b blue dots, a red dots and b + a total dots.
- But the total number of dots are the same in the two boxes! How do we know that? Well, there are a red dots in each box, so we can match them up. There are b blue dots in each box, so we can match them up. That’s it! If we can match up the dots one-for-one, there must be the same number of them!
• That means a + b = b + a .
We can also use the measurement model to explain why a + b = b + a no matter what numbers we choose for a and b . Imagine taking a segment of length a and combining it linearly with a segment of length b . That’s how we get a length of a + b .
But if we just rotate that segment so it’s upside down, we see that we have a segment of length b combined with a segment of length a , which makes a length of b + a .
But of course it’s the same segment! We just turned it upside down! So the lengths must be the same. That is, a + b = b + a .
Addition is Associative
Your turn! You’ll answer the question, “Why is addition of whole numbers associative?”
Property: Addition of whole numbers is associative.
What it Means (words): When I add three whole numbers in a given order, the way I group them (to add two at a time) doesn’t affect the sum.
What it Means (symbols): For any three whole numbers a , b , and c ,
(a + b) + c = a + (b + c) .
- Come up with at least three examples to demonstrate associativity of addition.
- Use our models of addition to come up with an explanation . Why does associativity hold in every case ? Note: your explanation should not use specific numbers. It is not an example!
0 is an Identity for Addition
Property: The number 0 is an identity for addition of whole numbers.
What it Means (words): When I add any whole number to 0 (in either order), the sum is the very same whole number I added to 0.
What it Means (symbols): For any whole numbers n ,
n + 0 = n and 0 + n = n .
- Come up with at least three examples to demonstrate that 0 is an identity for addition.
- Use our models of addition to come up with an explanation . Why does this property of 0 hold in every possible case ?
Properties of Subtraction
Since addition and subtraction are so closely linked, it’s natural to wonder if subtraction has some of the same properties as addition, like commutativity and associativity.
Example: Is subtraction commutative?
Justin asked if the operation of subtraction is commutative. That would mean that the difference of two whole numbers doesn’t depend on the order in which you subtract them.
In symbols: for every choice of whole numbers a and b we would have a – b = b – a .
Jared says that subtraction is not commutative since 4 – 3 = 1, but 3 – 4 ≠ 1. (In fact, 3 – 4 = -1.)
Since the statement “subtraction is commutative” is a universal statement , one counterexample is enough to show it’s not true. So Jared’s counterexample lets us say with confidence:
Subtraction is not commutative.
Can you find any examples of whole numbers a and b where a – b = b – a is true? Explain your answer.
Lyle asked if the operation of subtraction is associative.
- State what it would mean for subtraction to be associative. You should use words and symbols.
- What would you say to Lyle? Decide if subtraction is associative or not. Carefully explain how you made your decision and how you know you’re right .
Jess asked if the number 0 is an identity for subtraction.
- State what it would mean for 0 to be an identity for subtraction. You should use words and symbols.
- What would you say to Jess? Decide if 0 is an identity for subtraction or not. Carefully explain how you made your decision and how you know you’re right
Properties of Multiplication and Division
Now we’re going to turn our attention to familiar properties of multiplication and division, with the focus still on explaining why these properties are always true.
Here are the four properties you’ll think about:
- Multiplication of whole numbers is commutative .
- Multiplication of whole numbers is associative .
- Multiplication of whole numbers distributes over addition
- The number 1 is an identity for multiplication of whole numbers
For each of the properties, remember to keep straight:
Once again, it’s important to distinguish between examples and explanations . They are not the same! Since there are infinitely many whole numbers, it’s impossible to check every case, so examples will never be enough to explain why these properties hold. You have to figure out reasons for these properties to hold, based on what you know about the operations.
1 is an Identity for Multiplication
We’ll work out the explanation for the last of these facts, and you will work on the others.
Example: 1 is an Identity for multiplication
The number 1 is an identity for multiplication of whole numbers.
When I multiply a number by 1 (in either order), the product is that number.
For any whole number m ,
m × 1 = m and 1 × m = m .
1 × 5 = 5, 19 × 1 = 19, and 1 × 1 = 1.
Why does the number 1 act this way with multiplication?
Why It’s True, Explanation 1:
Let’s think first about the definition of multiplication as repeated addition:
![\[ \underbrace{1 + 1 + \cdots + 1}_{m\text{ times}}\]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-d6347ad76b522b8b21970b28c00bffec_l3.png "Rendered by QuickLaTeX.com")
- On the other hand, 1 × m means to add the number m to itself just one time. So 1 × m = m also.
We can also use the number line model to create a justification. If Zed calculates 1× m , he will start at 0 and face the positive direction. He will then take m steps forward, and he will do it just one time. So he lands at m , which means 1 × m = m .
If Zed calculates m × 1, he starts at 0 and faces the positive direction. Then he takes one step forward, and he repeats that m times. So he lands at m . We see that m × 1 = m .
Why It’s True, Explanation 3:
In the area model, m × 1 represents m rows with one square in each row. That makes a total of m squares. So m × 1 = m .
Similarly, 1 × m represents one row of m squares. That’s also a total of m squares. So 1 × m = m .
The example presented several different explanations. Do you think one is more convincing than the others? Or more clear and easier to understand?
Multiplication is Commutative
Property : Multiplication whole numbers is commutative.
What it Means (words): When I multiply two whole numbers, switching the order in which I multiply them does not affect the product.
What it Means (symbols): For any two whole numbers a and b ,
a · b = b · a.
- Come up with at least three examples to demonstrate the commutativity of multiplication.
- Use our models of multiplication to come up with an explanation . Why does commutativity hold in every case ? Note: Your explanation should not use particular numbers. It is not an example!
Multiplication is Associative
Property : Multiplication of whole numbers is associative.
What it Means (words): When I multiply three whole numbers in a given order, the way I group them (to multiply two at a time) doesn’t affect the product.
What it Means (symbols): For any three whole numbers a , b , and c ,
(a · b) · c = a · (b · c).
- Come up with at least three examples to demonstrate the associativity of multiplication.
- Use our models of multiplication to come up with an explanation . Why does associativity hold in every case ?
Multiplication Distributes over Addition
Property: Multiplication distributes over addition.
What it means: The distributive law for multiplication over addition is a little hard to state in words, so we’ll jump straight to the symbols. For any three whole numbers x , y , and z :
x · (y + z) = x · y + x · z .
Examples : We actually did calculations very much like the examples above, when we looked at the area model for multiplication.
8 · (23) = 8 · (20 + 3) = 8 · 20 + 8 · 3 = 160 + 24 = 184
5 · (108) = 5 · (100 + 8) = 5 · 100 + 5 · 8 = 500 + 40 = 540
Which of the following pictures best represents the distributive law in the equation
![\[3 \cdot (2 + 4) = 3 \cdot 2 + 3 \cdot 4?\]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-51e1ae1c0cccb9c256005f84caf1271f_l3.png "Rendered by QuickLaTeX.com")
Explain your choice.
Use the distributive law to easily compute each of these in your head (no calculators!). Explain your solutions.
![\[45 \times 11 \qquad \qquad 63 \times 101 \qquad \qquad 172 \times 1001.\]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-3e45dd9c53c9e3e01df4590845eae867_l3.png "Rendered by QuickLaTeX.com")
Use one of our models for multiplication and addition to explain why the distributive rule works every time.
Properties of Division
It’s natural to wonder which, if any, of these properties also hold for division (since you know that the operations of multiplication and division are connected).
Example: Is Division Associative?
If division were associative, then for any choice of three whole numbers a , b , and c , we would have
a ÷ ( b ÷ c ) = ( a ÷ b ) ÷ c .
Remember, the parentheses tell you which two numbers to divide first.
Let’s try the example a = 9, b = 3, and c = 1. Then we have:
9 ÷ (3 ÷ 1) = 9 ÷ 3 = 3
(9 ÷ 3) ÷ 1 = 3 ÷ 1 = 3.
So is it true? Is division associative? Well, we can’t be sure. This is just one example. But “division is associative” is a universal statement . If it’s true, it has to work for every possible example . Maybe we just stumbled on a good choice of numbers, but it won’t always work.
Let’s keep looking. Try a = 16, b = 4, and c = 2.
16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8
(16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2.
That’s all we need! A single counterexample lets us conclude:
Division is not associative.
What about the other properties? It’s your turn to decide!
- State what it would mean for division to be commutative. You should use words and symbols.
- Decide if division is commutative or not. Carefully explain how you made your decision and how you know you’re right .
- State what it would mean for division to distribute over addition. You definitely want to use symbols!
- Decide if division distributes over addition or not. Carefully explain how you made your decision and how you know you’re right .
- State what it would mean for the number 1 to be an identity for division. You should use words and symbols.
- Decide if 1 is an identity for division or not. Carefully explain how you made your decision and how you know you’re right.
Zero Property for Multiplication and Division
You probably know another property of multiplication that hasn’t been mentioned yet:
If I multiply any number times 0 (in either order), the product is 0. This is sometimes called the zero property of multiplication. Notice that the zero property is very different from the property of being an identity!
1. Write what the zero property means using both words and symbols:
For every whole number n . . .
2. Give at least three examples of the zero property for multiplication.
3. Use one of our models of multiplication to explain why the zero property holds.
- For each division problem below, turn it into a multiplication problem. Solve those problems if you can. If you can’t, explain what is wrong.
5 ÷ 0 0 ÷ 5 7 ÷ 0 0 ÷ 7 0 ÷ 0
- Use your work to explain why we say that division by 0 is undefined .
- Use one of our models of division to explain why division by 0 is undefined .
Four Fact Families
In elementary school, students are often encouraged to memorize “four fact families,” for example:
2 + 3 = 5 5 – 3 = 2
3 + 2 = 5 5 – 2 = 3
Here’s a different “four fact family”:
2 · 3 = 6 6 ÷ 3 = 2
3 · 2 = 6 6 ÷ 2 = 3
- In what sense are these groups of equations “families”?
- Write down at least two more addition / subtraction four fact families.
- Use properties of addition and subtraction to explain why these four fact families are each really one fact.
- Write down at least two more multiplication / division four fact families.
- Use properties of multiplication and division to explain why these four fact families are each really one fact.
Going Deeper with Division
So far we’ve been thinking about division in what’s called the quotative model . In the quotative model, we want to make groups of equal size. We know the size of the group , and we ask how many groups . For example, we think of 20 ÷ 4 as:
How many groups of 4 are there in a group of 20?
Thinking about four fact families, however, we realize we can turn the question around a bit. We could think about the partitive model of division. In the partitive model, we want to make an equal number of groups. We know how many groups , and we ask the size of the group. In the partitive model, we think of 20 ÷ 4 as:
20 is 4 groups of what size?
When we know the original amount and the number of parts, we use partitive division to find the size of each part.
When we know the original amount and the size of each part, we use quotative division to find the number of parts.
Here are some examples in word problems:
![\[ \begin{tabular}{| c | c |} \hline \textbf{Partitive} & \textbf{Quotative} \\ \hline number of groups known & number in each group known\\ \ & \ \\ find the number in each group & find the number of groups\\ \hline A movie theater made \$6450 & A movie theater made \$6450\\ in one night of ticket sales. & in one night of ticket sales.\\ 430 people purchased a ticket. & Each ticket cost \$12.50.\\ How much does one ticket cost? & How many people purchased a ticket?\\ \hline \end{tabular} \]](https://pressbooks-dev.oer.hawaii.edu/math111/wp-content/ql-cache/quicklatex.com-acc42f202c4bb93aa33c895ff01f2069_l3.png "Rendered by QuickLaTeX.com")
For each word problem below:
- Draw a picture to show what the problem is asking.
- Use your picture to help you decide if it is a quotative or a partitive division problem.
- Solve the problem using any method you like.
- David made 36 cookies for the bake sale. He packaged the cookies in boxes of 9. How many boxes did he use?
- David made 36 cookies to share with his friends at lunch. There were 12 people at his lunch table (including David). How many cookies did each person get?
- Liz spent one summer hiking the Appalachin trail. She completed 1,380 miles of the trail and averaged 15 miles per day. How many days was she out hiking that summer?
- On April 1, 2012, Chase Norton became the first person to hike the entire Ko‘olau summit in a single trip. (True story!) It took him eight days to hike all 48 miles from start to finish. If he kept a steady pace, how many miles did he hike each day?
Write your own word problems: Write one partitive division problem and one quotative division problem. Choose your numbers carefully so that the answer works out nicely. Be sure to solve your problems!
Why think about these two models for division? You won’t be teaching the words partitive and quotative to your students. But recognizing the two kinds of division problems (and being able to come up with examples of each) will make you a better teacher.
It’s important that your students are exposed to both ways of thinking about division, and to problems of both types. Otherwise, they may think about division too narrowly and not really understand what’s going on. If you understand the two kinds of problems, you can more easily diagnose and remedy students’ difficulties.
Most of the division problems we’ve looked at so far have come out evenly, with no remainder. But of course, that doesn’t always happen! Sometimes, a whole number answer makes sense, and the context of the problem should tell you which whole number is the right one to choose.
What is 43 ÷ 4?
- Write a problem that uses the computation 43 ÷ 4 and gives 10 as the correct answer.
- Write a problem that uses the computation 43 ÷ 4 and gives 11 as the correct answer.
- Write a problem that uses the computation 43÷4 and gives 10.75 as the correct answer.
We can think about division with remainder in terms of some of our models for operations. For example, we can calculate that 23 ÷ 4 = 5 R3. We can picture it this way:
- Explain how the picture above illustrates 23 = 5 · 4 + 3. Where do you see the remainder of 3 in the picture?
- Explain the connection between these two equations.
23 ÷ 4 = 5 R3 and 23 = 5 · 4 + 3.
- How could you use the number line model to show the calculation 23 = 5 · 4 + 3? What does a “remainder” look like in this model?
- Draw area models for each of these division problems. Find the quotient and remainder.
40 ÷ 12 59 ÷ 10 91 ÷ 16
Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.
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Spectrum Math Grade 8 Chapter 3 Lesson 3 Answer Key Solving 1-Variable Equations
Students can use the Spectrum Math Grade 8 Answer Key Chapter 3 Lesson 3.3 Solving 1-Variable Equations as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 3 Lesson 3.3 Solving 1-Variable Equations Answers Key
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal: 4 + 17 = 21 4 + 17 + 5 = 21 + 5 (26 = 26) When the same number is subtracted from both sides of an equation, the two sides remain equal: 32 = 16 + 16 32 – 4 = 16 + 16 – 4 (28 = 28) Use these properties to determine the value of variables: x + 17 = 23 x + 17 – 17 = 23 – 17 x + 0 = 6 x = 6
40 – n = 19 40 – n – 40 = 19 – 40 0 – n = -29 n = 29
y – 14 = 3 y – 14 + 14 = 3 + 14 y + 0 = 17 y = 17
Find the value of the variable in each equation.
Question 1. a. a + 12 = 25 ____ Answer: a = 13 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: a + 12 = 25 Subracting 12 on both sides a + 12 – 12 = 25 – 12 a + 0 = 13 a = 13
b. 48 + d = 60 ____ Answer: d = 12 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 48 + d = 60 subtracting 48 on both sides 48 + d – 48 = 60 – 48 d + 0 = 12 d = 12
c. y – 19 = 18 ____ Answer: y = 37 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: y – 19 = 18 Adding 19 on both sides y – 19 + 19 = 18 + 19 y – 0 = 37 y = 37
Question 2. a. 31 – x = 16 ____ Answer: x = 15 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 31 – x = 16 subtracting 31 on both sides 31 – x + 31 = 16 – 31 0 – x = -15 -x = -15 x = 15
b. 11 + n = 25 ____ Answer: n = 14 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 11 + n = 25 subtractiong 11 on both sides 11 + n – 11 = 25 – 11 n = 14
c. m – 21 = 34 ____ Answer: m = 55 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: m – 21 = 34 Adding 21 on both sides m – 21 + 21 = 34 + 21 m – 0 = 55 m = 55
Question 3. a. 28 + b = 50 _____ Answer: b = 22 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 28 + b = 50 Subtracting 28 on both sides 28 + b – 28 = 50 – 28 b – 0 = 22 b = 22
b. p – 16 = 32 ____ Answer: p = 48 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: p – 16 = 32 Adding 16 on both sides p – 16 + 16 = 32 + 16 p – 0 = 48 p = 48
c. t + 22 = 57 ____ Answer: t = 35 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: t + 22 = 57 Subtracting 22 on both sides t + 22 – 22 = 57 – 22 t – 0 = 35 t = 35
Question 4. a. 33 + c = 514 ____ Answer: c = 481 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 33 + c = 514 Subtracting 33 on both sides 33 + c – 33 = 514 – 33 c – 0 = 481 c = 481
b. e + 19 = 37 ____ Answer: e = 18 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: e + 19 = 37 Subtracting 19 on both sides e + 19 – 19 = 37 -19 e – 0 = 18 e = 18
c. 16 + r = 0 ____ Answer: r = -16 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 16 + r = 0 Subtracting 16 on both sides 16 + r – 16 = 0 – 16 r – 0 = -16 r = -16
Question 5. a. 52 – n = 24 ____ Answer: n = 28 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 52 – n = 24 Subtracting 52 on both sides 52 – n – 52 = 24 – 52 0 – n = -24 n = 24
b. y – 15 = 18 ____ Answer: y = 33 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: y – 15 = 18 Adding 15 on both sides y – 15 + 15 = 18 + 15 y – 0 = 33 y = 33
c. 21 + n = 49 ____ Answer: n = 28 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 21 + n = 49 Subtracting 21 on both sides 21 + n – 21 = 49 – 21 n – 0 = 28 n = 28
Question 6. a. m – 5 = 18 ____ Answer: m = 23 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: m – 5 = 18 Adding 5 on both sides m – 5 + 5 = 18 + 5 m – 0 = 23 m = 23
b. 36 + s = 45 ____ Answer: s = 9 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 36 + s = 45 Subtracting 36 on both sides 36 + s – 36 = 45 – 36 s -0 = 9 s = 9
c. 21 – a = 7 ____ Answer: a = 14 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 21 – a = 7 Subtracting 7 on both sides 21 – a -7 = 7 – 7 14 – a = 0 14 = a
Question 7. a. 17 + d = 29 ____ Answer: d = 12 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 17 + d = 29 Subtracting 17 on both sides 17 + d – 17 = 29 – 17 d – 0 = 12 d = 12
b. x – 23 = 9 ___ Answer: x = 32 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: x – 23 = 9 Adding 23 on both sides x -23 + 23 = 9 + 23 x – 0 = 32 x = 32
c. 27 + f = 35 ____ Answer: f = 8 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 27 + f = 35 Subtracting 27 on both sides 27 + f -27 = 35 – 27 f – 0 = 8 f = 8
Question 8. a. r – 15 = 24 ____ Answer: r = 39 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: r – 15 = 24 Adding 15 on both sides r – 15 +15 = 24 + 15 r -0 = 39 r = 39
b. 27 – p = 3 ____ Answer: p = 24 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 27 – p = 3 Subtracting 27 on both sides 27 – p – 27 = 3 – 27 0 – p = -24 -p = -24 p = 24
c. 34 – x = 18 ____ Answer: x = 16 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 34 – x = 18 Subtracting 34 on both sides 34 – x – 34 = 18 – 34 0 – x = -16 -x = -16 x = 16
Question 9. a. y + 12 = 20 ____ Answer: y = 8 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: y + 12 = 20 Subtracting 12 on both sides y + 12 -12 = 20 – 12 y – 0 = 8 y = 8
b. n – 24 = 31 ___ Answer: n = 55 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: n – 24 + 24 = 31 + 24 n – 0 = 55 n = 55
c. 16 + p = 38 ___ Answer: p = 22 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables 16 + p = 38 Subtracting 16 on both sides 16 + p – 16 = 38 – 16 p – 0 = 22 p = 22
Question 10. a. 18 + q = 25 ____ Answer: q = 7 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 18 + q = 25 Subtracting 18 on both sides 18 + q – 18 = 25 – 18 q – 0 = 7 q = 7
b. m + 17 = 32 ___ Answer: m = 15 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: m + 17 = 32 Subtracting 17 on both sides m + 17 – 17 = 32 – 17 m + 0 = 15 m = 15
c. e + 29 = 36 ___ Answer: e = 7 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: e + 29 = 36 Subtracting 29 on both sides e + 29 – 29 = 36 – 29 e – 0 = 7 e = 7
Question 11. a. 39 – r = 34 ___ Answer: r = 5 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 39 – r = 34 r = 5
b. 42 + x = 56 ___ Answer: x = 14 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 42 + x = 56 Subtracting 42 on both sides 42 + x – 42 = 56 – 42 x + 0 = 14 x = 14
c. q – 21 = 35 ___ Answer: q = 56 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: q – 21 = 35 Adding 21 on both sides q – 21 + 21 = 35 + 21 q – 0 = 56 q = 56
Question 12. a. 18 + p = 22 ___ Answer: p = 4 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 18 + p = 22 Subtracting 18 on both sides 18 + p – 18 = 22 – 18 p – 0 = 4 p = 4
b. s – 32 = 9 ___ Answer: s = 41 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: s – 32 = 9 Adding 32 on both side s -32 + 32 = 9 + 32 s – 0 = 41 s = 41
c. 43 + n = 49 ____ Answer: n = 6 The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables: 43 + n = 49 Subtracting 43 on both sides 43 + n – 43 = 49 – 43 n – 0 = 6 n = 6
The Multiplication and Division Properties of Equality state that when each side of the equation is multiplied by the same number, the two sides remain equal: 3 + 4 = 7 (3 + 4) × 5 = 7 × 5 (35 = 35) When each side of the equation is divided by the same number, the two sides remain equal: 2 × 6 = 12 \(\frac{(2 \times 6)}{3}\) = \(\frac{12}{3}\) (4 = 4) Use these properties to determine the value of variables: n ÷ 5 = 4 n ÷ 5 × 5 = 4 × 5 n = 20
3n = 18 \(\frac{3 n}{3}\) = \(\frac{18}{3}\) n = 6
\(\frac{60}{n}\) = 4 \(\frac{60}{n}\) = 4n or 60 = 4n \(\frac{60}{n}\) = \(\frac{4 n}{4}\) 15 = n
Question 1. a. 5b = 35 _____ Answer: b = 7 \(\frac{5 b}{5}\) = \(\frac{35}{5}\) b = 7
b. \(\frac{60}{n}\) = 16 _______ Answer: n = 3.75 \(\frac{60 n}{60}\) = \(\frac{16}{60}\) n = \(\frac{16}{60}\) = 3.75
c. f × 12 = 72 _____ Answer: f = 6 \(\frac{(f \times 12)}{12}\) = \(\frac{72}{12}\) f = 6
Question 4. a. \(\frac{n}{20}\) = 4 _____ Answer: n = \(\frac{1}{5}\) \(\frac{n}{20}\) = \(\frac{4}{20}\) \(\frac{n}{20}\) = \(\frac{1}{5}\)
b. a × 12 = 60 _____ Answer: a = 5 \(\frac{(a \times 12)}{12}\) = \(\frac{60}{12}\) a = 5
c. 6p = 90 ____ Answer: p = 15 \(\frac{6 p}{6}\) = \(\frac{90}{6}\) p = 15
Question 5. a. x ÷ 7 = 11 ____ Answer: x = 77 x ÷ 7 = 11 x ÷ 7 x 7 = 11 x 7 x = 77
b. t ÷ 25 = 8 _______ Answer: t = 200 t ÷ 25 = 8 t ÷ 25 x 25 = 8 x 25 t = 200
c. \(\frac{x}{15}\) = 6 ____ Answer: x = 90 \(\frac{x}{15}\) = 6 \(\frac{x}{15}\) x 15 = 6 x 15 x = 90
Question 6. a. b × 16 = 64 ____ Answer: b = 4 b x 16 ÷ 16 = 64 ÷ 16 b = 4
b. 11d = 132 ____ Answer: d = 12 \(\frac{11 d}{11}\) = \(\frac{132}{11}\) d = 12
c. \(\frac{65}{m}\) = 5 _______ Answer: m = \(\frac{1}{13}\) \(\frac{65 m}{65}\) = \(\frac{5}{65}\) m = \(\frac{5}{65}\) = \(\frac{1}{13}\)
Question 7. a. \(\frac{n}{14}\) = 3 _____ Answer: n = 42 \(\frac{n}{14}\) x 14 = 3 x 14 n = 42
b. f × 9 = 99 ____ Answer: f = 11 \(\frac{f 9 }{9}\) = \(\frac{99}{9}\) f = 11
c. 4n = 60 ____ Answer: n =15 \(\frac{4 n }{4}\) = \(\frac{60}{4}\) n = 15
Question 8. a. e × 5 = 120 ___ Answer: e = 14 e x 5 ÷ 5 = 120 ÷ 5 e = 14
b. \(\frac{120}{m}\) = 10 ____ Answer: m = \(\frac{1}{12}\) \(\frac{120 m}{120}\) = \(\frac{10}{120}\)
c. b ÷ 9 = 7 ____ Answer: b = 63 b ÷ 9 x 9 = 7 x 9 b = 63
Question 9. a. 8t = 104 _____ Answer: t = 13 8t ÷ 8 = 104 ÷ 8 t = 13
b. \(\frac{b}{9}\) = 6 _____ Answer: b = 54 \(\frac{b}{9}\) x 9 = 6 x 9 b = 54
c. m × 18 = 54 ____ Answer: m = 3 m x 18 ÷ 18 = 54 ÷ 18 m = 3
Question 10. a. \(\frac{a}{6}\) = 12 ____ Answer: a = 72 \(\frac{a}{6}\) x 6 = 12 x 6 a = 72
b. 7m = 84 _____ Answer: m = 12 7m ÷ 7 = 84 ÷ 7 m = 12
c. a ÷ 4 = 18 ____ Answer: a = 72 a ÷ 4 x 4 = 18 x 4 a = 72
One-variable equations can be solved by isolating the variable on one side of the equation by performing inverse operations.
Addition t + 4 = 16 t + 4 – 4 = 16 – 4 t = 12 Subtraction 28 – r = 15 28 – r – 28 = 15 – 28 -r = -13 r = 13 Multiplication 5n = 65 5n ÷ 5 = 65 ÷ 5 n = 13 Division 72 ÷ r = 9 72 = r × r = 9 × r 72 = 9r 72 ÷ 9 = 9r ÷ 9 r = 8
Question 1. a. r × 13 = 13 _____ Answer: r = 1 r x 13 ÷ 13 = 13 ÷ 13 r = 1
b. w + 18 = 22 ____ Answer: w = 4 w + 18 – 18 = 22 -18 w = 4
c. 17 × v = 153 ____ Answer: v = 9 17 x v ÷ 17 = 153 ÷ 17 v = 9
Question 2. a. f ÷ 12 = 7 ____ Answer: f = 84 f ÷ 12 x 12 = 7 x 12 f = 84
b. y ÷ 8 = 17 ___ Answer: y = 136 y ÷ 8 x 8 = 17 x 8 y = 136
c. 24 – q = 13 ____ Answer: q = 11 24 – q – 24 = 13 -24 0 – q = – 11 q = 11
Question 3. a. d × 7 = 35 _____ Answer: d = 5 d x 7 ÷ 7 = 35 ÷ 7 d = 7
b. t ÷ 11 = 18 ____ Answer: t = 198 t ÷ 11 x 11 = 18 x 11 t = 198
c. v + 19 = 36 _______ Answer: v = 17 v + 19 -19 = 36 -19 v – 0 = 17 v = 17
Question 4. a. q + 8 = 16 ____ Answer: q = 8 q + 8 – 8 = 16 – 8 q – 0 = 8 q = 8
b. 66 ÷ w = 11 ____ Answer: w = 726 66 ÷ w x 66 = 11 x 66 w = 726
c. y ÷ 9 = 8 ____ Answer: y = 72 y ÷ 9 x 9 = 8 x 9 y = 72
Question 5. a. v – 8 = 9 ____ Answer: v = 17 v – 8 + 8 = 9 + 8 v = 17
b. 17 + d = 29 ___ Answer: d = 12 17 + d – 17 = 29 – 17 d = 12
c. 4 + s = 20 ____ Answer: s = 16 4 + s – 4 = 20 -4 s = 16
Question 6. a. 300 ÷ d = 20 ____ Answer: d = 6000 300 ÷ d x 300 = 20 x 300 d = 6000
b. 15u = 135 ____ Answer: u = 9 15u ÷ 15 = 135 ÷ 15 u = 9
c. \(\frac{x}{5}\) = 12 ____ Answer: x = 60 \(\frac{x}{5}\) x 5 = 12 x 5 x = 60
Question 7. a. q × 3 = 27 ___ Answer: q = 9 q x 3 ÷ 3 = 27 ÷ 3 q = 9
b. 28 ÷ r = 4 ___ Answer: r = 112 28 ÷ r x 28 = 4 x 28 r = 112
c. 11x = 77 ____ Answer: x = 7 11x ÷ 11 = 77 ÷ 7 x = 11
Question 8. a. w ÷ 4 = 13 ____ Answer: w = 42 w ÷ 4 x 4 = 13 x 4 w = 42
b. x – 16 = 20 ___ Answer: x =36 x – 16+ 16 = 20 + 16 x = 36
c. 20 – d = 13 ____ Answer: d = 7 20 – d -20 = 13 – 20 d = 7
Question 9. a. 29 – y = 19 ___ Answer: y = 10 29 – y -29 = 19 – 29 -y = -10 y = 10
b. d × 15 = 75 ___ Answer: d =5 d x 15 ÷ 15 = 75 ÷ 15 d = 5
c. 27 ÷ t = 9 ____ Answer: t = 243 27 ÷ t x 27 = 9 x 27 t = 243
Question 10. a. y ÷ 18 = 11 ___ Answer: y = 198 y ÷ 18 x 18 = 11 x 18 y = 198
b. w × 20 = 20 ____ Answer: w = 1 w x 20 ÷ 20 = 20 ÷ 20 w = 1
c. w + 14 = 22 ____ Answer: w = 8 w + 14 – 14 = 22 -14 w = 8
Question 11. a. j ÷ 20 = 3 ___ Answer: j = 60 j ÷ 20 x 20 = 3 x 20 j = 60
b. 12c = 156 ____ Answer: c = 13 12c ÷ 12 = 156 ÷ 12 c = 13
c. n + 16 = 31 ____ Answer: n = 15 n + 16 – 16 = 31 – 16 n = 15
Question 12. a. \(\frac{225}{r}\) = 15 ___ Answer: r = \(\frac{1}{15}\) \(\frac{225 r}{225}\) = \(\frac{15}{225}\) r = \(\frac{1}{15}\)
b. 12 – q = 8 ___ Answer: q = 4 12 – q – 12 = 8 – 12 -q = -4 q = 4
c. x – 19 = 1 ____ Answer: x = 20 x – 19 + 19 = 1 + 19 x = 20
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table below. Use mental math to find the total amount of rain. Explain your reasoning. Day Monday Tuesday Wednesday Thursday Rain (in.) 2.6 1.5 1.4 2.5 Lesson 3 Skills Practice Properties of Operations false; 8 ÷ 4 ≠ 4 ÷ 8 + (b + 2) = 5 + (2 + b) Commutative Property of Addition = (5 + 2) + b Associative Property of Addition = 7 + b Simplify.
Our resource for enVision Algebra 1 includes answers to chapter exercises, as well as detailed information to walk you through the process step by step. With Expert Solutions for thousands of practice problems, you can take the guesswork out of studying and move forward with confidence. Find step-by-step solutions and answers to enVision ...
Example 3 Simplify the expression. Justify each step. 9 + (3x + 4) 9 + ((3x)+ 4) = 9 + 4 + 3x Commutative Property of Addition = (9 + 4) + 3x Associative Property of Addition = 13 + 3x Simplify. Exercises Name the property shown by each statement. 1.7 · 1 = 7 2. 4+ (3y)+ 2) = (4 + 3y + 2 State whether the following conjectures are true or ...
Find step-by-step solutions and answers to Glencoe Algebra 1 - 9780079039897, as well as thousands of textbooks so you can move forward with confidence. ... Plan for Problem Solving. Section 0-2: Real Numbers. Section 0-3: Operations with Integers. Section 0-4: ... Chapter 3 Practice Test. Page 220: Chapter 3 Preparing for Assessment. Exercise ...
Chapter 5 Lesson 3: Properties of Operations. Property. Click the card to flip 👆. a statement that is true for any number. The following properties are true for any number. Click the card to flip 👆.
Why It's True, Explanation 1: First we'll use the definition of the operations. Suppose we know c - b = a is true. Subtraction means "take away.". So. c - b = a. means we start with quantity c and take away quantity b, and we end up with quantity a. Start with this equation, and imagine adding quantity b to both sides.
Properties of Operations. Name the property shown by each statement. State whether the following conjectures are true or false. If false, provide a counterexample. 7. The sum of an even whole number and an odd whole number is always odd. 8. Division of whole numbers is always commutative. Simplify each expression.
Writing Equivalent Expressions Using Properties. Worksheet. Combining Like Terms #2. Worksheet. Combining Like Terms #3. Worksheet. Equivalent Expressions: Matching Activity. Worksheet. Download and print 6.EE.A.3 worksheets to help kids develop this key sixth grade Common Core math skill.
Common core worksheets and activities for 3.OA.9 / Operations And Algebraic Thinking / Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic. / Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and ...
Use the Fundamental Counting Principle to find the total number of outcomes in each situation. rolling two number cubes and tossing one coin. choosing rye or Bermuda grass and 3 different mixtures of fertilizer. making a sandwich with ham, turkey, or roast beef; Swiss or provolone cheese; and mustard or mayonaise.
Choose one of the following to describe how you would plan to solve each problem. Do not solve the problems. A. Use only one operation, such as addition or multiplication. B. Use a combination of operations, such as division and addition. C. Use a different strategy. 5. MONEY Julia opened a savings account with a deposit of $36. She then ...
P.3 - L3 P.P. Key 7) If the area of a trapezoid is 48, and the bases 1 are 10 and 6, find the height of the trapezoid. c) 6 8) In triangle ABC, the measure of angle A is 60 degrees. If the measure of angle B is two times that of Angle C, we can conclude that: d) All of the angles are less than 90 degrees. 9) Solve for x: b) 40 12 1
001_009_CRM01_881031. Use the four-step plan to solve each problem. 1. GEOGRAPHY. The president is going on a campaign trip to California, first. flying about 2,840 miles from Washington, D.C., to San Francisco and. then another 390 to Los Angeles before returning the 2,650 miles back to.
Lesson 3 - Properties Of Operations. Lesson 3 Homework Practice Properties of Operations Name the property shown by each statement. 1. 1 · (a + 3) = a + 3 2. 2p + )(3q + 2) = (2p + 3q + 2 3. (ab)c)= c (ab 4. 2t · 0 = 0 5. m (nr) = (mn)r 6. 0 + 2s = 2s State whether the following conjectures are true or false. If false, provide a ...
Study with Quizlet and memorize flashcards containing terms like Commutative Property of Addition, Commutative Property of Addition, Commutative Property of Addition and more.
Georgia Milestones Assessment System Test Prep: Grade 3 English Language Arts Literacy (ELA) Practice Workbook and Full-length Online Assessments: GMAS Study Guide. Lumos Learning. 3. 2017. ACT Aspire Test Prep: 3rd Grade Math Practice Workbook and Full-length Online Assessments: ACT Aspire Study Guide. Lumos Learning.
Practice and Problem Solving Unit 3 Expressions and Equations Key B ... Lesson 17 Properties of Operations Study the example showing how to use properties of operations to write equivalent expressions with variables. Then solve problems 1Ð9. 1 What does ... Possible answer: (7 ¥ 3 x) 2 (7 ¥ 4) 21 x 2 28 Possible student work: 56 1 8 x 5 (8 ...
Good luck! Part 1: Order of Operations problems involving addition, subtraction, multiplication, and division. Problem 1:Simplify the numerical expression below. Answer. Problem 2:Simplify the numerical expression below. Answer. Problem 3:Simplify the numerical expression below. Answer.
Problem-Solving Investigation: Use a Venn Diagram. Use a Venn diagram to solve each problem. PHONE SERVICE Of the 5,750 residents of Homer, Alaska, 2,330 pay for landline phone service and 4,180 pay for cell phone service. One thousand seven hundred fifty pay for both landline and cell phone service.
Practice and Problem Solving: C 1. +10 2. −18 3. +104 4. −28 5. 8(−2 + 9 + 6) 6. gained $68 7. 4(−45) + 112 = −68; 68 ft lower 8. 17(5) + 5(−2) + 8 = 83; She got an 83. 9. 3(−20) + 2(−12) + (−42) + 57 − 15 = −84; $84 less 10. a. Positive, because there is an even number of negative factors. b. 2,880 Practice and Problem ...
Spectrum Math Grade 8 Chapter 3 Lesson 3.3 Solving 1-Variable Equations Answers Key. Find the value of the variable in each equation. Question 1. The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both ...
Lesson 3 Skills Practice. Similarity and Transformations. yes; Sample answer: A rotation and a dilation with a scale factor of 2 maps EFG onto XYW. yes; Sample answer: A rotation and a dilation with a scale factor of 2 maps HIJ onto QOP. no; Sample answer: The ratios of the side lengths are not equal for all the sides; −−. RQ. ON. = −1 ...
Lesson 19 Solve Equations. Solve. 6 Write the inverse of each operation. a. addition. b. multiplication 5 7 Solve each problem. Show your work. a. x 1 1 12 5 18 b. 5 x 1 2 7 3 8 Tim said that to solve the equation x 7 2 2 5 9, he would need to subtract 2 from both sides of the equation.