Mathematics: Algebraic Reasoning – Grade 6

Algebraic reasoning begins with understanding how to represent mathematical ideas using symbols. This involves converting written descriptions into algebraic expressions and vice-versa. This skill is crucial for communicating mathematical ideas clearly and for solving problems in various contexts.

An algebraic expression is a combination of numbers, variables (letters), and operation symbols ($+$, $-$, $\times$, $\div$) that represents a mathematical relationship. Unlike an equation, an expression does not contain an equality sign ($=$). For example, the expression $7.20x - \$20$ could represent the daily profit of a company, where $x$ is the number of units sold and $\$20$ represents fixed costs.

Variables are letters that represent unknown numbers or quantities that can change. We typically use lowercase letters for variables, such as $a, b, c, x, y, z$. It's generally a good practice to avoid using 'o', 'i', and 'l' as variables because they can easily be confused with numbers (0, 1, 1).

A coefficient is the numerical factor multiplied by a variable. In the term $6n$, $6$ is the coefficient, meaning $6$ times $n$. If a variable appears by itself, like $x$, its coefficient is implicitly $1$ (i.e., $1x$).

A constant is a number that stands alone in an expression; its value does not change. In the expression $7.20x - \$20$, $\$20$ is the constant representing fixed costs. Similarly, in $3a + 5$, $5$ is the constant.

Translating written descriptions into algebraic expressions requires recognizing keywords that indicate specific mathematical operations. Here's a breakdown:

• Addition ($+$): sum, total, more than, increased by, added to, plus
• Subtraction ($-$): difference, less than, decreased by, subtracted from, minus, fewer than
• Multiplication ($\times$ or juxtaposition): product, times, multiplied by, of, twice, triple
  • For example, 'six times a number $n$' can be written as $6 \times n$, $6 \cdot n$, or most commonly, $6n$.
  • 'A third of a number $n$' can be written as $\frac{1}{3}n$ or $\frac{n}{3}$.
• Division ($\div$ or fraction bar): quotient, divided by, per, ratio of

It's important to pay close attention to the order of operations, especially with subtraction and division. For instance, '$3$ less than $x$' translates to $x - 3$, not $3 - x$. While addition and multiplication are commutative (meaning the order of the numbers does not affect the result, e.g., $x + 3 = 3 + x$), subtraction and division are not. However, $x - 3$ can be rewritten as $-3 + x$ using the commutative property of addition for negative numbers.

Consider a practical example: A parking garage charges $\$5$ per hour plus a $\$3$ entrance fee. If $h$ represents the number of hours parked, the total cost can be expressed as $5h + \$3$. This expression helps calculate costs for any parking duration.

To better understand and construct expressions, visual tools can be very helpful:

• Pictorial Representations: Drawing diagrams to represent quantities and relationships.
• Tape Diagrams: Rectangular bars that represent quantities, useful for showing parts of a whole or comparisons.
• Algebra Tiles: Concrete manipulatives that represent variables (e.g., $x$) and constants (e.g., $1$), allowing for a hands-on approach to building expressions.

When faced with a real-world problem, avoid jumping straight to the expression. Instead, ask yourself:

1. What do I know? Identify the given information.
2. What am I trying to find? Determine the unknown quantity that needs to be represented by a variable.
3. Am I combining or separating groups? This helps determine if addition/multiplication or subtraction/division is appropriate.
4. Are the groups the same size? This is key for deciding between addition/subtraction and multiplication/division.
5. Can I draw a picture or model to help? Visualizing the problem often clarifies the relationships.

Maintaining a graphic organizer or a list of keywords for each operation can serve as a valuable reference as you encounter more complex scenarios.

Inequalities are mathematical statements that compare two expressions using symbols like $>$, $<$, $\geq$, and $\leq$. Unlike equations that show exact equality, inequalities describe relationships where one side is greater than, less than, greater than or equal to, or less than or equal to the other side.

Greater than ($>$): The left side is strictly larger than the right side.

• Example: $x > 5$ means "$x$ is greater than 5"
• This includes values like 6, 7, 10, but NOT 5

Less than ($<$): The left side is strictly smaller than the right side.

• Example: $y < 10$ means "$y$ is less than 10"
• This includes values like 9, 5, 0, but NOT 10

Greater than or equal to ($\geq$): The left side is larger than or exactly equal to the right side.

• Example: $a \geq 3$ means "$a$ is greater than or equal to 3"
• This includes 3, 4, 5, and all numbers greater than 3

Less than or equal to ($\leq$): The left side is smaller than or exactly equal to the right side.

• Example: $b \leq 7$ means "$b$ is less than or equal to 7"
• This includes 7, 6, 5, and all numbers less than 7

Variables can appear on either side of an inequality symbol. These statements are equivalent:

• $x > 8$ ("$x$ is greater than 8")
• $8 < x$ ("8 is less than $x$")

Both expressions describe the same relationship – $x$ represents values larger than 8.

Number lines provide a visual way to represent inequality solutions:

Open Circle (○): Used for strict inequalities ($>$ or $<$)

• The value at the circle is NOT included in the solution
• Example: For $x > 4$, place an open circle at 4

Closed Circle (●): Used for inclusive inequalities ($\geq$ or $\leq$)

• The value at the circle IS included in the solution
• Example: For $x \geq 4$, place a closed circle at 4

Shading Direction:

• For $>$ or $\geq$: Shade to the RIGHT of the circle
• For $<$ or $\leq$: Shade to the LEFT of the circle

Consider these practical situations:

Height Requirements: "You must be at least 48 inches tall to ride this roller coaster."

• This translates to $h \geq 48$, where $h$ is height in inches
• A person who is exactly 48 inches tall CAN ride (inclusive)

Speed Limits: "The speed limit is 35 mph."

• This means $s \leq 35$, where $s$ is speed in mph
• Driving exactly 35 mph is legal (inclusive)

Temperature Ranges: "The temperature must be below 32°F for water to freeze."

• This translates to $t < 32$, where $t$ is temperature in Fahrenheit
• At exactly 32°F, water starts to freeze but hasn't completely frozen yet

To verify if a value satisfies an inequality, substitute it into the original statement:

For $x > 3$:

• Test $x = 5$: Is $5 > 3$? Yes ✓
• Test $x = 2$: Is $2 > 3$? No ✗
• Test $x = 3$: Is $3 > 3$? No ✗ (not strictly greater)

Inequalities appear frequently in everyday situations:

• Budget constraints: "I can spend no more than $\$50$" → $c \leq \$50$
• Capacity limits: "The elevator can hold at most 15 people" → $p \leq 15$
• Minimum requirements: "You need more than 80% to pass" → $s > 80$
• Age restrictions: "You must be 13 or older to use this app" → $a \geq 13$

Understanding inequalities helps you model and solve real-world problems where exact values aren't required, but ranges or limits are important.

Substitution is the process of replacing variables in an algebraic expression with specific numerical values, then using the order of operations to find the final result. This fundamental skill allows you to find the value of expressions in real-world contexts and verify solutions to problems.

Think of substitution as replacing a placeholder with its actual value. If you have the expression $3x + 5$ and you know that $x = 4$, substitution means replacing every $x$ with $4$:

$3x + 5 = 3(4) + 5 = 12 + 5 = 17$

The parentheses around the $4$ help clarify that we're multiplying $3$ by $4$.

After substitution, always follow the order of operations (PEMDAS/BODMAS):

1. Parentheses/Brackets first
2. Exponents/Orders (powers and roots)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Example 1: Evaluate $2x^2 - 5$ when $x = 3$

Step 1: Substitute $x = 3$ $2(3)^2 - 5$

Step 2: Apply order of operations $2(3)^2 - 5 = 2(9) - 5 = 18 - 5 = 13$

When expressions contain multiple variables, substitute each variable with its corresponding value:

Example 2: Evaluate $4a + 3b - c$ when $a = 2$, $b = -1$, and $c = 5$

Step 1: Substitute all variables $4(2) + 3(-1) - (5)$

Step 2: Apply order of operations $4(2) + 3(-1) - (5) = 8 + (-3) - 5 = 8 - 3 - 5 = 0$

To avoid confusion with multiple variables, use different colors:

• Use red for all $a$ values
• Use blue for all $b$ values
• Use green for all $c$ values

This visual strategy helps ensure you substitute the correct value for each variable.

Be especially careful when substituting negative numbers:

Example 3: Evaluate $-2x^2 + y$ when $x = -3$ and $y = 4$

Step 1: Substitute carefully with parentheses $-2(-3)^2 + (4)$

Step 2: Apply order of operations $-2(-3)^2 + (4) = -2(9) + 4 = -18 + 4 = -14$

Important: $(-3)^2 = 9$, not $-9$, because the negative sign is inside the parentheses.

Perimeter of a Rectangle: The perimeter formula is $P = 2l + 2w$ If length $l = 12$ feet and width $w = 8$ feet: $P = 2(12) + 2(8) = 24 + 16 = 40$ feet

Distance Traveled: Using $d = st$ (distance = speed × time) If speed $s = 65$ mph and time $t = 3$ hours: $d = (65)(3) = 195$ miles

Company Profit: Using $\text{Profit} = 7.20x - \$20$ If $x = 15$ products sold: $\text{Profit} = 7.20(15) - \$20 = \$108 - \$20 = \$88$

Always verify your substitution by:

1. Double-checking that you substituted the correct value for each variable
2. Reviewing your order of operations steps
3. Asking yourself if the answer makes sense in context

• Forgetting parentheses around negative substituted values
• Mixing up which value goes with which variable
• Applying operations in the wrong order
• Combining unlike terms before substitution

Mastering substitution gives you the power to evaluate any algebraic expression when you know the values of its variables. This skill is essential for solving equations, checking solutions, and working with formulas in real-world contexts.

Properties of operations are mathematical rules that allow you to rearrange and simplify expressions while keeping their value unchanged. Understanding these properties gives you flexibility in how you work with algebraic expressions and helps you recognize when different-looking expressions are actually equivalent.

Two expressions are equivalent if they have the same value for any value of the variable(s). For example:

• $2x + 3x$ and $5x$ are equivalent
• $4(y + 2)$ and $4y + 8$ are equivalent

Think of equivalent expressions like different ways to describe the same amount of money: four quarters, ten dimes, and one dollar bill all represent $\$1.00$.

Addition: $a + b = b + a$

• $x + 7 = 7 + x$
• $3m + 2n = 2n + 3m$

Multiplication: $a \times b = b \times a$

• $5 \times y = y \times 5 = 5y$
• $x \times 3 = 3 \times x = 3x$

Important: Subtraction and division are NOT commutative:

• $5 - 3 \neq 3 - 5$ ($2 \neq -2$)
• $12 \div 4 \neq 4 \div 12$ ($3 \neq \frac{1}{3}$)

Addition: $(a + b) + c = a + (b + c)$

• $(x + 3) + 7 = x + (3 + 7) = x + 10$

Multiplication: $(a \times b) \times c = a \times (b \times c)$

• $(2 \times x) \times 5 = 2 \times (x \times 5) = 10x$

The associative property shows that grouping doesn't matter for addition and multiplication.

Basic Form: $a(b + c) = ab + ac$

This property allows you to "distribute" multiplication over addition or subtraction:

Example 1: $3(x + 4) = 3x + 12$ Example 2: $5(2y - 3) = 10y - 15$ Example 3: $-2(m + 6) = -2m - 12$

Like terms have the same variable raised to the same power:

• $3x$ and $7x$ are like terms
• $4y^2$ and $-2y^2$ are like terms
• $5a$ and $3b$ are NOT like terms

Combining like terms:

• $3x + 7x = 10x$
• $5y - 2y = 3y$
• $4a + 2b - a + 5b = 3a + 7b$

Example 1: Simplify $2(3x + 1) + 4x$

Step 1: Apply distributive property $2(3x + 1) + 4x = 6x + 2 + 4x$

Step 2: Combine like terms $6x + 2 + 4x = 10x + 2$

Example 2: Simplify $3(x - 2) - 2(x + 1)$

Step 1: Apply distributive property $3(x - 2) - 2(x + 1) = 3x - 6 - 2x - 2$

Step 2: Combine like terms $3x - 6 - 2x - 2 = x - 8$

Sometimes expressions have multiple layers of grouping:

Example: Simplify $-2[3(x - 3)]$

Step 1: Work from inside out $-2[3(x - 3)] = -2[3x - 9]$

Step 2: Apply distributive property again $-2[3x - 9] = -6x + 18$

Store Sales: Tamika sells candy bars for $\$1$ and popcorn for $\$3$. If she sells $y$ candy bars and $(y - 7)$ bags of popcorn:

Original expression: $1y + 3(y - 7)$ Simplified: $y + 3y - 21 = 4y - 21$

This shows her total sales depend on the number of candy bars sold.

Algebra tiles provide a visual way to understand equivalent expressions:

• Variable tiles represent $x$
• Unit tiles represent constants like $1$
• Grouping tiles together shows addition
• Removing pairs shows subtraction

To check if two expressions are equivalent, substitute the same value for the variable in both expressions:

For $2(x + 5)$ and $2x + 10$, let $x = 3$:

• $2(3 + 5) = 2(8) = 16$
• $2(3) + 10 = 6 + 10 = 16$ ✓

Since both give the same result, they're equivalent.

Understanding properties of operations gives you powerful tools to simplify expressions, solve problems efficiently, and recognize patterns in algebraic thinking.

When you're given an equation or inequality along with a set of possible values, your task is to determine which values make the mathematical statement true. This process is fundamental to understanding what it means to "solve" an equation or inequality.

A solution to an equation is a value that, when substituted for the variable, makes the equation true (both sides equal). A solution to an inequality is a value that, when substituted for the variable, makes the inequality statement true.

Example: For the equation $3x + 8 = 14$, let's test $x = 2$: $3(2) + 8 = 6 + 8 = 14$ ✓

Since both sides equal 14, $x = 2$ is a solution.

When given multiple values to test, work through each one methodically:

Example: Which values make $4x + 5 > 25$ true: $\{-3, 4, 5, 6, 12\}$?

Test $x = -3$: $4(-3) + 5 = -12 + 5 = -7$. Is $-7 > 25$? No ✗ Test $x = 4$: $4(4) + 5 = 16 + 5 = 21$. Is $21 > 25$? No ✗ Test $x = 5$: $4(5) + 5 = 20 + 5 = 25$. Is $25 > 25$? No ✗ Test $x = 6$: $4(6) + 5 = 24 + 5 = 29$. Is $29 > 25$? Yes ✓ Test $x = 12$: $4(12) + 5 = 48 + 5 = 53$. Is $53 > 25$? Yes ✓

Solutions: $x = 6$ and $x = 12$

Equations typically have one specific solution (or sometimes no solution or infinitely many solutions). Inequalities often have multiple solutions or ranges of solutions.

For the equation $2x - 1 = 9$, only $x = 5$ works. For the inequality $2x - 1 > 9$, any value greater than 5 works (like 6, 7, 10, etc.).

Some equations and inequalities have variables on both sides:

Example: Test $x = 3$ in $2x + 1 = x + 4$ Left side: $2(3) + 1 = 6 + 1 = 7$ Right side: $3 + 4 = 7$ Since $7 = 7$, $x = 3$ is a solution ✓

Number Line Testing: For inequalities, you can use a number line to visualize which values work:

For $x > 3$, test values:

• $x = 1$: Is $1 > 3$? No
• $x = 3$: Is $3 > 3$? No
• $x = 5$: Is $5 > 3$? Yes

Algebra Tiles: Use physical or virtual tiles to represent equations and test values by replacing variable tiles with unit tiles.

Budget Problem: Maria has $\$75$ and buys shirts for $\$12.75$ each and dresses for $\$9.50$ each. The expression $75 - 12.75s - 9.50d$ represents her remaining money.

If she buys 2 shirts and 1 dress, does she have money left? $75 - 12.75(2) - 9.50(1) = 75 - 25.50 - 9.50 = \$40$

Yes, she has $\$40$ remaining! 💰

• Order of Operations: Always follow PEMDAS when substituting values
• Sign Errors: Be careful with negative numbers and subtraction
• Inequality Direction: Remember that $>$ means "greater than," not "less than"
• Multiple Solutions: Don't stop after finding one solution to an inequality

When testing values:

1. Organize your work by showing each substitution clearly
2. Double-check your arithmetic
3. Think about whether your answers make sense
4. Look for patterns in which values work and which don't

As you test values, you'll develop intuition about:

• Which types of values are likely to work
• How changing the variable affects the expression
• The relationship between equations and their solutions

This foundational skill prepares you for more advanced equation-solving techniques and helps you verify solutions to ensure they're correct.

One-step equations involving addition and subtraction are the foundation of algebraic problem-solving. Understanding how to solve these equations gives you the tools to find unknown values in many real-world situations.

Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. The equal sign ($=$) is like the fulcrum in the center.

For the equation $x + 3 = 9$:

• The left side has $x + 3$
• The right side has $9$
• These must be equal for any solution

Addition and subtraction are inverse operations – they "undo" each other:

• If you add 5, you can undo it by subtracting 5
• If you subtract 8, you can undo it by adding 8

Type: $x + a = b$ (where $a$ and $b$ are integers) Strategy: Subtract $a$ from both sides

Example 1: Solve $x + 6 = 10$

Step 1: Identify what's been added to $x$ $x + 6 = 10$ (6 has been added)

Step 2: Subtract 6 from both sides $x + 6 - 6 = 10 - 6$ $x = 4$

Step 3: Check your solution $4 + 6 = 10$ ✓

Type: $x - a = b$ (where $a$ and $b$ are integers) Strategy: Add $a$ to both sides

Example 2: Solve $y - 8 = -3$

Step 1: Identify what's been subtracted from $y$ $y - 8 = -3$ (8 has been subtracted)

Step 2: Add 8 to both sides $y - 8 + 8 = -3 + 8$ $y = 5$

Step 3: Check your solution $5 - 8 = -3$ ✓

Sometimes the variable appears on the right side of the equation:

Example 3: Solve $7 = z + 2$

Step 1: Subtract 2 from both sides $7 - 2 = z + 2 - 2$ $5 = z$

This is the same as $z = 5$.

Example 4: Solve $x + (-5) = 12$

This is the same as $x - 5 = 12$

Step 1: Add 5 to both sides $x - 5 + 5 = 12 + 5$ $x = 17$

Example 5: Solve $m - (-3) = 8$

This is the same as $m + 3 = 8$

Step 1: Subtract 3 from both sides $m + 3 - 3 = 8 - 3$ $m = 5$

Money Problem: Alex has some money in his wallet. His grandmother gives him $\$10$ for his birthday. He now has $\$28$. How much did he originally have?

Let $x$ = original amount Equation: $x + 10 = 28$ Solution: $x = 28 - 10 = \$18$

Temperature Change: The temperature dropped by 15°F during the night. If the morning temperature is 23°F, what was the evening temperature?

Let $t$ = evening temperature Equation: $t - 15 = 23$ Solution: $t = 23 + 15 = 38°F$

Bar Diagrams: Draw rectangles to represent the equation parts

For $x - 4 = -13$:

[  x  ] - [4] = [-13]

Number Lines: Show the movement from one number to another

For $x + 6 = -7$, start at -7 and move 6 units left to find $x = -13$.

Algebra Tiles: Use physical manipulatives where:

• Large rectangles represent variables
• Small squares represent +1
• Shaded squares represent -1

1. Identify what operation is being performed on the variable
2. Apply the inverse operation to both sides
3. Simplify both sides
4. Check your solution by substituting back into the original equation

• Only working on one side: Always perform the same operation on both sides
• Wrong inverse operation: Remember addition undoes subtraction and vice versa
• Sign errors: Be careful with negative numbers
• Forgetting to check: Always verify your solution works

Key phrases that indicate addition/subtraction equations:

• "more than," "increased by," "plus" → addition
• "less than," "decreased by," "minus" → subtraction
• "total," "sum" → addition
• "difference," "remaining" → subtraction

Learning to solve addition and subtraction equations builds the foundation for solving more complex equations and gives you powerful tools for solving real-world problems involving unknown quantities.

Multiplication and division equations require a different approach than addition and subtraction equations, but the same principle applies: use inverse operations to isolate the variable while keeping the equation balanced.

Multiplication and division are inverse operations:

• If you multiply by 4, you can undo it by dividing by 4
• If you divide by 7, you can undo it by multiplying by 7
• $4 \times 3 \div 4 = 3$ (back to the original)
• $20 \div 5 \times 5 = 20$ (back to the original)

Type: $ax = b$ (where $a$ and $b$ are integers, $a \neq 0$) Strategy: Divide both sides by $a$

Example 1: Solve $3x = 21$

Step 1: Identify the coefficient of $x$ $3x = 21$ ($x$ is multiplied by 3)

Step 2: Divide both sides by 3 $\frac{3x}{3} = \frac{21}{3}$ $x = 7$

Step 3: Check your solution $3(7) = 21$ ✓

Example 2: Solve $-5y = 30$

Step 1: Divide both sides by -5 $\frac{-5y}{-5} = \frac{30}{-5}$ $y = -6$

Step 3: Check your solution $-5(-6) = 30$ ✓

Type: $\frac{x}{a} = b$ (where $a$ and $b$ are integers, $a \neq 0$) Strategy: Multiply both sides by $a$

Example 3: Solve $\frac{x}{4} = 9$

Step 1: Identify what $x$ is divided by $\frac{x}{4} = 9$ ($x$ is divided by 4)

Step 2: Multiply both sides by 4 $4 \cdot \frac{x}{4} = 4 \cdot 9$ $x = 36$

Step 3: Check your solution $\frac{36}{4} = 9$ ✓

Example 4: Solve $\frac{m}{-3} = 8$

Step 1: Multiply both sides by -3 $(-3) \cdot \frac{m}{-3} = (-3) \cdot 8$ $m = -24$

Step 3: Check your solution $\frac{-24}{-3} = 8$ ✓

Example 5: Solve $42 = 6z$

Step 1: Divide both sides by 6 $\frac{42}{6} = \frac{6z}{6}$ $7 = z$

This is the same as $z = 7$.

When the coefficient is a fraction, multiply by its reciprocal:

Example 6: Solve $\frac{2}{3}x = 10$

Step 1: Multiply both sides by $\frac{3}{2}$ (reciprocal of $\frac{2}{3}$) $\frac{3}{2} \cdot \frac{2}{3}x = \frac{3}{2} \cdot 10$ $x = 15$

Step 3: Check your solution $\frac{2}{3}(15) = \frac{30}{3} = 10$ ✓

Solar Panel Problem: A warehouse wants to generate 34,000 watts of power. Each solar panel generates 200 watts. How many panels are needed?

Let $p$ = number of panels Equation: $200p = 34,000$ Solution: $p = \frac{34,000}{200} = 170$ panels

Parking Lot Problem: An outlet mall has 4 identical lots that hold a total of 1,388 cars. How many cars fit in each lot?

Let $c$ = cars per lot Equation: $4c = 1,388$ Solution: $c = \frac{1,388}{4} = 347$ cars per lot

Rate Problem: A delivery truck travels 392 miles in 7 hours. What is its average speed?

Let $s$ = speed in mph Equation: $7s = 392$ Solution: $s = \frac{392}{7} = 56$ mph

Bar Diagrams: Show the relationship between parts and whole

For $4x = 20$:

[x][x][x][x] = [20]

Each $x$ represents $\frac{20}{4} = 5$

Algebra Tiles: Use grouping to show multiplication

For $3x = 12$, arrange tiles in 3 equal groups, each containing $x$, totaling 12 unit tiles.

Multiplication can be written several ways:

• $5x$ (coefficient notation)
• $5 \cdot x$ (dot notation)
• $5(x)$ (parentheses notation)
• $(5)(x)$ (double parentheses)

All represent "5 times $x$."

Multiplication and division equations often arise from:

• Unit rates: miles per hour, cost per item
• Scaling: recipes, maps, proportions
• Area and volume: length × width, etc.

1. Identify whether the variable is multiplied or divided
2. Apply the appropriate inverse operation to both sides
3. Simplify the result
4. Check by substituting your answer back into the original equation

• Forgetting to apply operations to both sides
• Confusing multiplication and division inverses
• Sign errors with negative coefficients
• Not simplifying fractions in the final answer

Mastering multiplication and division equations expands your problem-solving toolkit significantly, allowing you to solve problems involving rates, scaling, and many real-world situations where quantities are related through multiplication or division.

When equations involve decimals and fractions, you can use algebraic reasoning and mental math strategies rather than complex procedures. Understanding number relationships and fact families helps you find unknown values efficiently.

Fact families show the relationship between operations:

• If $3 \times 4 = 12$, then $12 \div 3 = 4$ and $12 \div 4 = 3$
• If $7 + 5 = 12$, then $12 - 7 = 5$ and $12 - 5 = 7$

You can extend this thinking to algebraic equations with decimals and fractions.

Example 1: Find $m$ in $0.15m = 0.60$

Mental Math Approach:

• Think: "0.15 times what number equals 0.60?"
• Since $0.15 \times 4 = 0.60$, we have $m = 4$
• Check: $0.15 \times 4 = 0.60$ ✓

Example 2: Solve $\frac{c}{0.15} = 0.5$

Mental Math Approach:

• Think: "What number divided by 0.15 equals 0.5?"
• This means $c = 0.5 \times 0.15$
• Calculate: $0.5 \times 0.15 = 0.075$
• Check: $\frac{0.075}{0.15} = 0.5$ ✓

Example 3: Find $g$ in $\frac{2}{9} + g = \frac{8}{9}$

Mental Math Approach:

• Think: "$\frac{2}{9}$ plus what equals $\frac{8}{9}$?"
• Since the denominators are the same, focus on numerators: $2 + ? = 8$
• Therefore: $g = \frac{8-2}{9} = \frac{6}{9} = \frac{2}{3}$
• Check: $\frac{2}{9} + \frac{6}{9} = \frac{8}{9}$ ✓

Example 4: Solve $p - \frac{3}{5} = \frac{7}{10}$

Mental Math Approach:

• Think: "What minus $\frac{3}{5}$ equals $\frac{7}{10}$?"
• Convert to common denominators: $\frac{3}{5} = \frac{6}{10}$
• So: $p - \frac{6}{10} = \frac{7}{10}$
• Therefore: $p = \frac{7}{10} + \frac{6}{10} = \frac{13}{10}$
• Check: $\frac{13}{10} - \frac{6}{10} = \frac{7}{10}$ ✓

Example 5: Find $x$ in $\frac{1}{3} \times x = \frac{1}{15}$

Pattern Recognition:

• Think about the relationship: $\frac{1}{3} \times ? = \frac{1}{15}$
• Notice that $15 = 3 \times 5$
• So $\frac{1}{15} = \frac{1}{3 \times 5} = \frac{1}{3} \times \frac{1}{5}$
• Therefore: $x = \frac{1}{5}$
• Check: $\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}$ ✓

Example 6: Solve $2.5 + y = 4.8$

Mental Math:

• Think: "2.5 plus what equals 4.8?"
• $y = 4.8 - 2.5 = 2.3$
• Check: $2.5 + 2.3 = 4.8$ ✓

Example 7: Find $k$ in $k \div \frac{1}{4} = 12$

Understanding Division by Fractions:

• $k \div \frac{1}{4}$ means "how many fourths are in $k$?"
• If the answer is 12, then $k = 12 \times \frac{1}{4} = 3$
• Check: $3 \div \frac{1}{4} = 3 \times 4 = 12$ ✓

Recipe Problem: A recipe calls for $\frac{8}{25}$ kilowatt of power per solar panel. The average store needs 30 kilowatts. How many solar panels does the store need?

Let $n$ = number of panels Equation: $\frac{8}{25} \times n = 30$

Mental Math Solution:

• $\frac{8}{25} = 0.32$
• Think: "0.32 times what equals 30?"
• $n = 30 \div 0.32 = 93.75$
• Since you can't have partial panels, round up to 94 panels

Number Lines: Show decimal and fraction relationships visually

For $x + 0.3 = 0.8$, start at 0.8, move 0.3 units left to find $x = 0.5$

Fraction Bars: Use visual models to understand part-whole relationships

For $\frac{3}{4} - y = \frac{1}{4}$, visualize removing $\frac{1}{4}$ from $\frac{3}{4}$ to get $y = \frac{2}{4} = \frac{1}{2}$

1. Convert to familiar forms: Change decimals to fractions or vice versa if it helps
2. Use benchmark numbers: Relate to halves, quarters, tenths that you know well
3. Think multiplicatively: Use "groups of" thinking for multiplication and division
4. Check with estimation: Does your answer make sense?

Memorizing these helps with mental math:

• $0.5 = \frac{1}{2}$
• $0.25 = \frac{1}{4}$
• $0.75 = \frac{3}{4}$
• $0.1 = \frac{1}{10}$
• $0.2 = \frac{1}{5}$

As you work with decimal and fraction equations:

• Look for patterns in number relationships
• Use estimation to check if answers are reasonable
• Connect to previous knowledge about operations
• Think flexibly about different ways to represent the same number

This approach to decimal and fraction equations emphasizes understanding over memorizing procedures, building stronger number sense and algebraic reasoning skills.

A ratio is a mathematical way to compare two or more quantities by showing their relative sizes. Unlike subtraction, which shows the difference between quantities, ratios show multiplicative relationships – how many times larger or smaller one quantity is compared to another.

A ratio describes a multiplicative comparison that relates quantities within a given situation. For example, if there are 12 girls and 18 boys in a class, the ratio of girls to boys is 12:18, which tells us that for every 12 girls, there are 18 boys.

Ratios can compare:

• Part to part: Girls to boys (12:18)
• Part to whole: Girls to total students (12:30)
• Different units: Miles to gallons, dollars to hours

Every ratio can be expressed in three equivalent forms:

1. Fraction notation: $\frac{12}{18}$ or $\frac{2}{3}$
2. Word notation: "12 to 18" or "2 to 3"
3. Colon notation: "12:18" or "2:3"

All three forms represent the same relationship! Choose the form that makes most sense for your context.

The context of a problem determines what the ratio represents and sometimes the order in which it should be written.

Example 1: In a parking lot, there are 24 cars 🚗 and 8 trucks 🚚.

• Cars to trucks: 24:8 or 3:1 (for every 3 cars, there's 1 truck)
• Trucks to cars: 8:24 or 1:3 (for every 1 truck, there are 3 cars)
• Cars to total vehicles: 24:32 or 3:4 (3 out of every 4 vehicles are cars)
• Trucks to total vehicles: 8:32 or 1:4 (1 out of every 4 vehicles is a truck)

Ratios can be written in different forms depending on what information is most useful:

Original numbers: 24:8 tells us the actual count of cars and trucks Simplified form: 3:1 shows the basic relationship pattern

Both are correct! Sometimes the context determines which form is more helpful.

Example 2: A recipe calls for 6 cups of flour and 4 cups of sugar.

• For cooking: Use 6:4 (shows actual amounts needed)
• For understanding proportions: Use 3:2 (shows basic ratio pattern)

Part-to-Part Ratios compare different parts of a whole:

• Red marbles to blue marbles: 5:3
• Boys to girls in a class: 12:15

Part-to-Whole Ratios compare one part to the total:

• Red marbles to all marbles: 5:8 (if there are 5 red and 3 blue)
• Boys to all students: 12:27 (if there are 12 boys and 15 girls)

Bar Models: Use rectangular bars to show ratio relationships

For a ratio of toy cars to airplanes of 3:5:

Cars:     [■][■][■]
Airplanes:[■][■][■][■][■]

Number Lines: Show ratios as positions on a line

Ratio Tables: Organize equivalent ratios systematically

Sports Statistics: A basketball player makes 15 free throws out of 20 attempts

• Success ratio: 15:20 or 3:4
• Miss ratio: 5:20 or 1:4

Cooking Recipes: To make purple paint, mix 2 parts red with 1 part blue

• Red to blue ratio: 2:1
• Red to total paint: 2:3

School Surveys: In a class of 28 students, 12 prefer math and 16 prefer science

• Math to science preference: 12:16 or 3:4
• Math preference to total: 12:28 or 3:7

When a problem specifies the order, you must follow it:

Question: "What is the ratio of red marbles to blue marbles?" If there are 4 red and 6 blue: Answer must be 4:6 or 2:3

Question: "Write a ratio describing the marble collection." With 4 red and 6 blue: Many answers are possible: 4:6, 6:4, 4:10, 6:10, etc.

• Reversing the order when the problem specifies sequence
• Adding instead of comparing (ratios show multiplicative, not additive relationships)
• Confusing part-to-part with part-to-whole ratios
• Forgetting to label what each number represents

To develop strong ratio understanding:

1. Always identify what quantities are being compared
2. Use color coding to track different parts
3. Draw visual models when possible
4. Check that your ratio makes sense in context
5. Practice expressing the same ratio in different forms

Ratios provide a powerful way to describe relationships between quantities and form the foundation for understanding rates, proportions, and percentages.

A rate compares two quantities with different units, while a unit rate tells you how much of one quantity corresponds to exactly one unit of another quantity. Understanding unit rates helps you make comparisons, solve problems, and make informed decisions in everyday situations.

A rate is a special type of ratio that compares quantities with different units. Common examples include:

• Speed: 60 miles per hour (distance per time)
• Price: $\$3.50$ per pound (cost per weight)
• Density: 15 students per classroom (people per space)
• Fuel efficiency: 30 miles per gallon (distance per fuel)

A unit rate is a rate where the second quantity is 1 unit. It answers the question "How much per one?"

Example: If Tamika reads 500 words in 4 minutes:

• Rate: $\frac{500 \text{ words}}{4 \text{ minutes}}$
• Unit rate: $\frac{500}{4} = 125$ words per minute

The unit rate tells us that Tamika reads 125 words every single minute.

Method: Divide the first quantity by the second quantity

Example 1: A car travels 240 miles using 8 gallons of gas. What is the fuel efficiency?

Step 1: Write the rate $\frac{240 \text{ miles}}{8 \text{ gallons}}$

Step 2: Divide to find the unit rate $\frac{240}{8} = 30$ miles per gallon

Interpretation: The car travels 30 miles for every 1 gallon of gas.

Example 2: You pay $\$9.87$ for 3.3 pounds of apples. What is the unit price?

Step 1: Write the rate $\frac{\$9.87}{3.3 \text{ pounds}}$

Step 2: Calculate the unit rate $\frac{9.87}{3.3} = \$2.99$ per pound

Interpretation: Each pound of apples costs $\$2.99$.

Unit rates make it easy to compare different options and find the "best deal."

Example: Which cereal is the better buy? 🥣

Option A: 16 ounces for $\$3.50$

• Unit rate: $\frac{\$3.50}{16} = \$0.219$ per ounce

Option B: 12.4 ounces for $\$2.42$

• Unit rate: $\frac{\$2.42}{12.4} = \$0.195$ per ounce

Option C: 11.5 ounces for $\$2.35$

• Unit rate: $\frac{\$2.35}{11.5} = \$0.204$ per ounce

Best buy: Option B (lowest cost per ounce)

Driving Speed: If you travel 180 miles in 3 hours:

• Unit rate: $\frac{180 \text{ miles}}{3 \text{ hours}} = 60$ mph
• This helps you estimate travel times for other distances

Work Rate: If you earn $\$120$ for 8 hours of work:

• Unit rate: $\frac{\$120}{8 \text{ hours}} = \$15$ per hour
• This helps you calculate earnings for different work periods

Population Density: If 2,400 people live in a 4-square-mile area:

• Unit rate: $\frac{2,400 \text{ people}}{4 \text{ square miles}} = 600$ people per square mile
• This helps compare crowding between different areas

Bar Models: Show the relationship between quantities

For $\$9$ for 3 pounds of apples:

$9  → [■][■][■] (3 pounds)
$3  → [■]       (1 pound)

Number Lines: Show scaling relationships

Rate Tables: Organize equivalent rates systematically

Pay attention to units when expressing rates:

• "Miles per hour" means $\frac{\text{miles}}{\text{hour}}$
• "Dollars per pound" means $\frac{\text{dollars}}{\text{pound}}$
• "Students per classroom" means $\frac{\text{students}}{\text{classroom}}$

The word "per" indicates division and tells you which unit should be 1 in the unit rate.

Example: A recipe calls for $4\frac{1}{4}$ cups of flour for $8\frac{3}{4}$ servings. What is the unit rate of flour per serving?

Step 1: Convert to improper fractions

• $4\frac{1}{4} = \frac{17}{4}$ cups
• $8\frac{3}{4} = \frac{35}{4}$ servings

Step 2: Calculate the unit rate $\frac{\frac{17}{4}}{\frac{35}{4}} = \frac{17}{4} \times \frac{4}{35} = \frac{17}{35}$ cups per serving

Step 3: Simplify $\frac{17}{35} ≈ 0.49$ cups per serving

Calculator Use: When working with decimal division, use a calculator for accuracy Estimation: Round numbers to check if your answer is reasonable

For $\frac{\$9.87}{3.3}$, estimate: $\frac{\$10}{3} ≈ \$3.33$ Actual answer: $\$2.99$ (reasonable! ✓)

When to use rates:

• Describing relationships: "500 words in 4 minutes"
• Showing actual quantities involved

When to use unit rates:

• Making comparisons: "Which is faster?"
• Predicting: "How far in 5 hours?"
• Finding best deals: "Lowest price per pound?"

Mastering unit rates gives you a powerful tool for making mathematical comparisons and solving real-world problems involving different quantities and units.

Equivalent ratio tables are powerful tools that organize related ratios in a systematic way, helping you see patterns and solve complex problems involving part-to-part and part-to-whole relationships. These tables are especially useful for scaling recipes, solving mixture problems, and finding missing values.

Equivalent ratios represent the same relationship between quantities, just scaled up or down. They're like equivalent fractions – different numbers that express the same proportional relationship.

Example: The ratio 2:3 is equivalent to 4:6, 6:9, 8:12, etc. All represent the same relationship: "for every 2 of the first quantity, there are 3 of the second."

Two-column tables show equivalent part-to-part ratios:

Example: A punch recipe uses 2 cups of juice for every 3 cups of water.

Pattern: Each column is multiplied by the same number

• Row 2: $2 \times 2 = 4$, $3 \times 2 = 6$
• Row 3: $2 \times 3 = 6$, $3 \times 3 = 9$
• Row 4: $2 \times 4 = 8$, $3 \times 4 = 12$

Three-column tables show part-to-part-to-whole relationships:

Example: Making paint by mixing 1.5 ounces yellow with 2.4 ounces blue.

Key insight: Total = Yellow + Blue for each row

Use patterns in the table to find unknown quantities:

Example: Jeremy's punch recipe

Finding the missing values for 30 cups total:

• Pattern: Each row is multiplied by 2 compared to the row above
• For 30 cups: This is 3 times the original recipe ($10 \times 3 = 30$)
• Syrup: $2 \times 3 = 6$ cups
• Water: $8 \times 3 = 24$ cups
• Check: $6 + 24 = 30$ ✓

Tables reveal both within-row and between-row patterns:

Within-row patterns:

• How do the numbers in each row relate to each other?
• Example: Water is always 4 times the syrup amount

Between-row patterns:

• How does each row relate to other rows?
• Example: Each row doubles the previous row's values

Example: A field trip has a ratio of 15 students to 3 adults.

Problem: If 42 people go on the trip, how many are students and how many are adults?

Solution approach:

• Student-to-adult ratio: 15:3 or 5:1
• Student-to-total ratio: 15:18 or 5:6
• Adult-to-total ratio: 3:18 or 1:6

For 42 people total:

• Students: $\frac{5}{6} \times 42 = 35$
• Adults: $\frac{1}{6} \times 42 = 7$
• Check: $35 + 7 = 42$ ✓ and $35:7 = 5:1$ ✓

Example: A pie recipe serves 8 people and needs:

• 2 cups sugar
• 6 cups apples

Question: How much sugar and how many apples for 5 pies? Answer: 10 cups sugar and 30 cups apples

Bar Models: Represent each row as grouped bars

Number Lines: Show scaling relationships

Concrete Materials: Use counters or blocks to build table relationships physically

• Adding instead of multiplying to find equivalent ratios
• Forgetting to scale all parts equally
• Mixing up part-to-part and part-to-whole relationships
• Not checking that totals equal the sum of parts

Example: Hiking rate problem

Drew hikes at a steady rate. In 30 minutes, he hikes 2 miles.

Unit rate: $\frac{2 \text{ miles}}{30 \text{ minutes}} = \frac{1}{15}$ mile per minute

For 2 hours (120 minutes): 8 miles

Equivalent ratio tables prepare you for:

• Proportional relationships in Grade 7
• Linear equations and graphing
• Scale factors in geometry
• Percentage calculations and interest problems

Mastering ratio tables gives you a systematic way to organize proportional thinking and solve complex scaling problems efficiently.

Percentages are everywhere – from test scores and sales tax to discounts and tips! Understanding that percent means "per hundred" and connecting it to ratio relationships gives you powerful tools to solve percentage problems without complex procedures.

Percent literally means "per hundred" or "out of 100." Every percentage can be written as a ratio with 100 as the denominator.

• 25% means $\frac{25}{100}$ or "25 out of every 100"
• 70% means $\frac{70}{100}$ or "70 out of every 100"
• 125% means $\frac{125}{100}$ or "125 out of every 100" (more than the whole!)

Every percentage problem involves three components:

• Part: The portion you're finding or given
• Whole: The total amount (represents 100%)
• Percentage: The rate or proportion

Basic relationship: $\frac{\text{Part}}{\text{Whole}} = \frac{\text{Percentage}}{100}$

Question type: "What is 40% of 120?"

Ratio approach: $\frac{\text{Part}}{120} = \frac{40}{100}$

Visual with bar model: 📊

100% → [■][■][■][■][■][■][■][■][■][■] (120)
 40% → [■][■][■][■]                     ( ? )

Solution using equivalent ratios:

• If 100% = 120, then 10% = 12
• So 40% = 4 × 12 = 48

Answer: 40% of 120 is 48

Question type: "15% of what number is 12?"

Ratio approach: $\frac{12}{\text{Whole}} = \frac{15}{100}$

Reasoning: If 15 out of 100 equals 12, then 100 out of 100 equals what?

Solution using unit rate:

• If 15% = 12, then 1% = $\frac{12}{15} = 0.8$
• So 100% = 100 × 0.8 = 80

Answer: 15% of 80 is 12

Question type: "What percentage is 60 out of 115?"

Ratio approach: $\frac{60}{115} = \frac{\text{Percentage}}{100}$

Solution using division:

• $\frac{60}{115} ≈ 0.522$
• Convert to percentage: $0.522 × 100 = 52.2\%$

Answer: 60 is approximately 52.2% of 115

Bar Models: Show part-to-whole relationships clearly

For "70% of 120 = ?":

100% → [■■■■■■■■■■] = 120
 70% → [■■■■■■■]   = ?

Number Lines: Show percentage as positions

Ratio Tables: Organize equivalent ratios systematically

Sales Tax: You buy a $\$45$ shirt with 8% sales tax. What's the total cost?

Solution:

• Tax amount: 8% of $\$45$
• Using mental math: 8% = 8 × 1% = 8 × $\$0.45$ = $\$3.60$
• Total cost: $\$45$ + $\$3.60$ = $\$48.60$

Test Scores: You got 34 questions correct out of 40 total. What's your percentage? 📝

Solution:

• $\frac{34}{40} = \frac{85}{100} = 85\%$
• You can find this by noticing that 40 × 2.5 = 100, so 34 × 2.5 = 85

Discounts: A $\$80$ jacket is 25% off. What's the sale price?

Solution:

• Discount amount: 25% of $\$80$ = $\$20$
• Sale price: $\$80$ - $\$20$ = $\$60$
• Alternative: 75% of $\$80$ = $\$60$ (since you pay 100% - 25% = 75%)

Percentages less than 1%: 0.5% of 200

• Think: 0.5% = $\frac{0.5}{100} = \frac{1}{200}$
• So 0.5% of 200 = 1

Percentages greater than 100%: 150% of 80

• Think: 150% = 100% + 50%
• 100% of 80 = 80, and 50% of 80 = 40
• So 150% of 80 = 80 + 40 = 120

Using benchmark percentages:

• 10%: Move decimal point one place left
• 50%: Divide by 2
• 25%: Divide by 4
• 1%: Divide by 100

Example: Find 35% of 80

• 10% of 80 = 8
• 30% of 80 = 3 × 8 = 24
• 5% of 80 = 0.5 × 8 = 4
• 35% of 80 = 24 + 4 = 28

Increase problems: "The price increased by 20%"

• New price = Original + (20% of original) = 120% of original

Decrease problems: "The population decreased by 15%"

• New population = Original - (15% of original) = 85% of original

Comparison problems: "How much greater is 45 than 36?"

• Difference: 45 - 36 = 9
• Percentage increase: $\frac{9}{36} = \frac{25}{100} = 25\%$

• Confusing part and whole in the ratio setup
• Forgetting to convert between decimal and percentage form
• Using the wrong base for percentage increase/decrease problems
• Rounding too early in multi-step calculations

Understanding percentages through ratio relationships gives you flexibility to solve problems using visual models, mental math, and logical reasoning rather than memorizing complex procedures.