About these worksheets
These worksheets introduce students to the concept of variables and algebraic thinking. Activities include using substitution to evaluate expressions, writing and graphing inequalities on a number line, identifying numerical coefficients, solving for unknown values using all four operations, and exploring powers, bases, and slope. Aligned with sixth through eighth grade standards, these resources build the foundation for pre-algebra and algebra.
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- Replace a variable with a given number and simplify the expression.
- Use substitution to find the value of an unknown in an equation.
- Check whether a number makes an equation true by substituting it in.
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- Choose the correct inequality symbol (>, <, ≥, ≤) based on what the sentence says.
- Write an inequality from a short word problem using a variable to represent the unknown number.
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- Read an inequality sign (>, <, ≥, ≤) and say what values it allows.
- Graph an inequality on a number line by choosing an open or closed circle and shading the correct direction.
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- Decide whether an endpoint should be written with < or > versus ≤ or ≥ by noticing open and closed circles.
- Write the correct inequality (like x > 3 or x ≤ -2) to match the direction of the shading.
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- Match an inequality to the correct number line graph.
- Use open and closed circles to decide whether the endpoint number is included.
- Use the shaded direction on a number line to tell if the solutions are greater than or less than a value.
- Pick out the number that multiplies the variable in a term (like the 7 in 7x).
- Recognize when a coefficient is negative and include the sign (like -3 in -3y).
- Know that a variable by itself has a coefficient of 1 or -1 (like x or -x).
- Tell the coefficient apart from the variable and any exponent (like the 4 in 4m^2).
- Identify the coefficient even when it is written as a fraction or decimal (like 0.5 in 0.5t).
- Solve simple equations to find the missing number or letter.
- Work with equations written in a horizontal format (like x + 7 = 15).
- Check that the value you found makes the equation true when you plug it back in.
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- Identify the base and the exponent in an expression written with powers.
- Write the full multiplication that a power represents (like 5^3 means 5×5×5).
- Evaluate simple powers of integers to find the value.
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- Find the slope of a line by comparing the rise and run shown with two triangles on a graph.
- Compare two slopes to tell which line is steeper and explain why using rise/run.
- Write slope as a ratio or fraction and simplify it when possible.
About these worksheets
Students practice expressing proportional relationships as equations and solving circle equations in standard form. These worksheets develop the ability to identify constants of proportionality, use variables to model real-world relationships, and apply algebraic techniques to geometric formulas. Suitable for seventh and eighth grade math.
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- Turn a proportional relationship into an equation using a variable.
- Identify the constant of proportionality (unit rate) and use it as the number that multiplies the variable.
- Translate a word statement about “times as much” or “per” into an equation with an equals sign.
- Match a situation to the correct equation form (like y = kx) and label what the variables represent.
- Identify the center and radius from a circle equation written in standard form.
- Rewrite a circle equation into standard form by completing the square.
- Solve for missing values in a circle equation by substituting given points or coordinates.
About these worksheets
These worksheets cover a wide range of algebraic expression skills, from simplifying and expanding to factoring. Students practice combining like terms, using the distributive property, rewriting expressions as multiples of a sum, solving linear equations with variables on both sides, and expanding polynomials using the box method. Topics also include perfect square trinomials and matching equivalent expressions, making these resources ideal for sixth through eighth grade algebra.
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- Rewrite an expression like 3x + 6 as a multiplication problem with parentheses, like 3(x + 2).
- Find the greatest common factor of the terms so you know what number or variable can be factored out.
- Keep expressions equivalent while working with coefficients and variables.
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- Practice combining like terms to make an expression simpler.
- Practice using the distributive property to remove parentheses.
- Practice following the order of operations when simplifying expressions with addition and subtraction.
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- Rewrite algebra expressions into an equivalent form that’s simpler to read and work with.
- Use the distributive property to expand expressions with parentheses.
- Combine like terms to simplify expressions with variables.
- Rewrite expressions that include fractions by simplifying and combining terms correctly.
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- Practice using the distributive property to remove parentheses in algebra expressions.
- Multiply a number or variable across terms inside parentheses, including with fractions.
- Keep track of positive and negative signs while expanding and simplifying.
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- Practice pulling out the greatest common factor when the coefficients are fractions.
- Rewrite an expression as a product using the distributive property in reverse.
- Simplify the factored form so the numbers and fractions are in lowest terms.
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- Turn a percent in a story problem into a decimal you can use in a calculation.
- Match a word problem to the correct decimal expression that represents “percent of” an amount.
- Decide when to multiply by a decimal to find a percent of a number and when to add or subtract to find the new total.
- Combine like terms to make an expression shorter and easier to read.
- Keep track of positive and negative signs while adding and subtracting terms.
- Simplify expressions that include up to three different variables.
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- Recognize when two algebra expressions have the same value even if they look different.
- Simplify expressions by combining like terms.
- Rewrite expressions by factoring out a common number or variable.
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- Solve linear equations where the variable shows up on both sides of the equals sign.
- Combine like terms to simplify each side before solving.
- Use the distributive property to clear parentheses in an equation.
- Move terms across the equals sign using addition, subtraction, and multiplication to isolate the variable.
- Handle equations that need more than one step to get the variable alone.
- Multiply two polynomials by breaking them into smaller parts in a box (grid).
- Distribute each term to fill the boxes and keep track of positive and negative signs.
- Combine like terms to write the expanded polynomial in simplest form.
- Use a box (grid) to multiply a binomial by itself, like (x + 5)(x + 5).
- Combine like terms from the grid to write the final expanded quadratic expression.
- Connect the box method to the distributive property so you can see where each term in the answer comes from.
- Practice expanding squared binomials like (x + 5)^2 into a trinomial.
- Use the perfect square patterns (a + b)^2 and (a - b)^2 to expand quickly without doing full FOIL every time.
- Recognize when a trinomial matches the pattern of a perfect square.
- Factor perfect square trinomials into a squared binomial like (a+b)^2 or (a-b)^2.
- Use the middle term to decide whether the binomial uses addition or subtraction.
About these worksheets
These worksheets bring together coordinate geometry and algebraic reasoning. Students use similar triangles to find missing coordinates and rise values, rotate shapes around the origin, and identify points of intersection by solving systems of equations. Aligned with eighth grade standards, these activities strengthen graphing skills and spatial reasoning on the coordinate plane.
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- Use similar triangles on a graph to figure out a missing rise between two points on a line.
- Set up and solve a proportion using rise and run to keep the slope the same.
- Check that your answer makes sense by comparing slopes from different triangles on the same line.
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- Find a missing x- or y-coordinate on a graph by using similar triangles.
- Set up and solve a proportion from matching side lengths in two similar triangles.
- Practice rotating points and shapes around the origin on a coordinate plane.
- Tell the difference between clockwise and counterclockwise rotations.
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- Find the exact point where two lines cross by solving both equations together.
- Use substitution to plug one equation into the other and solve for x and y.
- Use elimination to combine equations and solve for the ordered pair.
- Check your solution by plugging the x and y values back into both equations.
About these worksheets
Students apply mathematical formulas to solve geometry problems involving right and non-right triangles. These worksheets cover the Pythagorean theorem for finding missing side lengths and the Law of Cosines for calculating unknown angles. Designed for eighth grade and above, these resources connect algebraic computation with geometric understanding.
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- Identify the hypotenuse and the two legs in a right triangle.
- Use the Pythagorean theorem (a² + b² = c²) to find the missing side c.
- Square the leg lengths and add them correctly before solving for c.
- Take a square root to get the final length of c.
- Use the Law of Cosines formula to find a missing angle in a non-right triangle when you know all three side lengths.
- Set up the equation by matching each side length to the correct angle across from it.
- Solve for the angle by isolating the cosine value and using an inverse cosine (arccos) on a calculator.
About these worksheets
These worksheets explore number concepts essential for middle school math, including square roots, cube roots, rational and irrational numbers, laws of exponents, scientific notation, radicals, and powers of ten. Students practice estimating radical values, simplifying expressions with exponents, and performing operations in scientific notation. Aligned with eighth grade Common Core standards, these resources build a strong number sense foundation for high school math.
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- Recognize the perfect squares that are closest to a given square root.
- Decide which two whole numbers a square root falls between.
- Use nearby perfect squares to make a quick estimate of a square root’s size.
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- Decide whether a number is rational or irrational.
- Recognize that fractions, integers, and whole numbers are rational because they can be written as a ratio of integers.
- Tell that terminating decimals and repeating decimals are rational numbers.
- Identify common irrational numbers like pi and square roots that do not simplify to a fraction.
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- Decide which two whole numbers a square root falls between.
- Use nearby perfect squares to judge whether a square root is closer to the lower or higher whole number.
- Compare the sizes of different square roots by thinking about their values.
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- Place square roots on a number line by figuring out which two whole numbers they fall between.
- Estimate the decimal value of a square root well enough to plot it in the right spot.
- Use nearby perfect squares (like 16 and 25) to judge how close a square root is to a whole number.
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- Rewrite expressions with exponents using the product and quotient rules.
- Simplify powers raised to powers by multiplying exponents.
- Rewrite expressions with zero and negative exponents using reciprocals.
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- Practice rewriting negative exponents as fractions (e.g., 3⁻² becomes 1/3²)
- Multiply powers with the same base by adding their exponents, even when some exponents are negative
- Raise fractions to a power by applying the exponent to both the numerator and denominator
- Simplify expressions step by step to reach a final whole number or fraction
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- Evaluate expressions with squared and cubed numbers.
- Solve simple equations where a number is squared or cubed to find the missing value.
- Recognize perfect squares and perfect cubes and match them to their roots.
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- Rewrite negative exponents as reciprocals so the exponent becomes positive.
- Evaluate expressions with negative powers to get the correct fraction or decimal value.
- Work with negative exponents on whole-number bases, fractions, and powers of 10.
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- Recognize perfect squares and perfect cubes so you can solve quickly without a calculator.
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- Break a number into factors and spot pairs that make a perfect square.
- Use exponents to write repeated factors more simply, like 3×3 as 3^2.
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- Practice rewriting square roots by pulling out perfect-square factors.
- Use factor pairs to break a number under a radical into simpler parts.
- Simplify radicals all the way to a number times a square root (like 3√2).
- Recognize which radicals are already in simplest form.
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- Rewrite whole numbers and decimals as one digit times a power of 10.
- Use place value to decide how many places the decimal point moves when converting.
- Work with both positive and negative exponents to show very large and very small numbers.
- Read and write numbers in a scientific-notation style and convert back to standard form.
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- Compare two powers of ten and decide which one is larger or smaller.
- Figure out how many times bigger one power of ten is than another.
- Connect powers of ten to place value shifts, like moving the decimal left or right.
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- Multiply two numbers written in scientific notation.
- Multiply the coefficients and combine the powers of 10 using exponent rules.
- Rewrite answers in proper scientific notation by shifting the decimal and adjusting the exponent.
- Understand what n! (factorial) means as multiplying whole numbers from 1 up to n.
- Fill in missing factorial values by continuing a multiplication pattern.
- Notice how factorials grow fast and compare which factorial is larger.
