[NEW] 2026 9th-Grade DIGITAL SHSAT Math Review (Official Problems)

Shsat review

13 chapters7 takeaways25 key terms7 questions

Overview

This video provides a review of key math concepts for the 9th-grade digital SHSAT, focusing on problem-solving strategies. It covers topics such as estimating values of square roots, understanding angle relationships with parallel lines, identifying irrational numbers, distinguishing between linear and non-linear functions, applying geometric transformations, using the vertical line test for functions, finding the line of best fit, calculating points of intersection for graphs, comparing rates of change, solving equations with absolute values, modeling linear functions for real-world scenarios like taxi fares, and solving proportions. The presenter emphasizes understanding the underlying principles and offers practical methods for tackling these problems.

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Chapters

To determine the value of a square root of a non-perfect square without a calculator, square the numbers bounding the target range (7 and 8 become 49 and 64).
Square the answer choices (e.g., sqrt(37) becomes 37).
Identify which squared answer choice falls within the squared range (e.g., 57 is between 49 and 64).
The number whose square falls within the range is the number located between the original bounds on the number line.

This strategy allows you to accurately place numbers involving square roots on a number line without needing a calculator, a common constraint on standardized tests.

To find which number is between 7 and 8, square 7 and 8 to get 49 and 64. Then, square the answer choices. If sqrt(57) is an option, squaring it gives 57, which falls between 49 and 64, making sqrt(57) the correct answer.

When a transversal line intersects two parallel lines, corresponding angles are equal.
Angles on a straight line (a linear pair) sum to 180 degrees.
To find an unknown angle, use the properties of parallel lines to find an adjacent or related angle, then use the linear pair property if necessary.

Understanding angle relationships formed by parallel lines and transversals is crucial for solving geometry problems involving unknown angles.

If angle T is 47 degrees and line K is parallel to line M, then the corresponding angle to T is also 47 degrees. If angle J and this 47-degree angle form a linear pair on a straight line, then J = 180 - 47 = 133 degrees.

An irrational number has a non-terminating and non-repeating decimal expansion.
Irrational numbers cannot be expressed as a simple fraction (a/b).
Common examples of irrational numbers include pi and the square roots of non-perfect squares.
Repeating decimals (like 0.777...) and terminating decimals (like 3.0) are rational.

Distinguishing between rational and irrational numbers is fundamental in number theory and is frequently tested on math exams.

The square root of 7 is irrational because 7 is not a perfect square (1^2=1, 2^2=4, 3^2=9), meaning its decimal representation goes on forever without repeating. In contrast, the square root of 9 is 3, which is rational.

A linear function can be written in the form y = mx + b, where x has an exponent of 1 and is in the numerator.
Functions are non-linear if the variable 'x' has an exponent other than 1 (e.g., x^2) or if 'x' is in the denominator (e.g., 1/x).
Functions like y = x/4 are linear because x is in the numerator and has an exponent of 1.

Recognizing whether a function is linear or non-linear determines the shape of its graph and the methods used to analyze it.

The function y = x/4 is linear because 'x' is in the numerator and its exponent is 1. The function y = 2/x is non-linear because 'x' is in the denominator.

Transformations (translation, rotation, reflection) change a shape's position or orientation but not its size or shape.
Reflection across an axis mirrors the shape, flipping it over the axis.
Rotation turns the shape around a fixed point (the origin or center).
A 180-degree rotation results in an upside-down orientation, while 90-degree rotations change orientation significantly.

Understanding how transformations affect geometric figures is key to solving problems involving congruent shapes and coordinate geometry.

To transform triangle JKL into J'K'L', a reflection across the y-axis followed by a 90-degree counterclockwise rotation about its center correctly aligns the triangle's orientation and position.

A graph represents a function if and only if no vertical line intersects the graph more than once.
This test ensures that for every input (x-value), there is only one output (y-value).
Graphs like circles or vertical lines fail the vertical line test because a single vertical line can intersect them at multiple points.

The vertical line test is a quick and visual method to determine if a given graph satisfies the definition of a function.

A circle fails the vertical line test because a vertical line drawn through its center intersects the circle at two points. A straight line with a positive slope passes the test, as any vertical line intersects it only once.

The line of best fit is a straight line that best represents the trend in a scatter plot of data.
A good line of best fit has an equal number of data points above and below it.
The slope of the line of best fit should generally match the overall trend of the data (positive slope for increasing data, negative for decreasing).
The y-intercept should be reasonable given the data points.

The line of best fit helps in visualizing and predicting trends in data sets, which is essential in statistics and data analysis.

For data points trending upwards, a line with a positive slope, like y = x, is a better line of best fit than y = -x + 2 if it has roughly an equal number of points above and below it, and its slope aligns with the data's upward trend.

The point of intersection of two graphs is the coordinate (x, y) where both equations are true simultaneously.
To find it, set the two equations equal to each other (since y = y).
Solve the resulting equation for 'x'.
Substitute the found 'x' value back into either original equation to solve for 'y'.

Calculating the point of intersection is fundamental for solving systems of equations and understanding where different mathematical models meet.

To find the intersection of y = 2x + 5 and y = -4x + 3, set them equal: 2x + 5 = -4x + 3. Solving for x gives x = -1/3. Plugging this back into y = 2x + 5 gives y = 2(-1/3) + 5 = -2/3 + 15/3 = 13/3. The intersection point is (-1/3, 13/3).

In a linear function y = mx + b, 'm' represents the slope, which is also known as the rate of change.
The rate of change indicates how much 'y' changes for each one-unit increase in 'x'.
For a table of values, the rate of change (slope) can be calculated using the formula (y2 - y1) / (x2 - x1) using any two points.
A larger 'm' value signifies a steeper slope and a faster rate of change.

Comparing rates of change helps in understanding which linear relationship grows or decreases more quickly.

Comparing y = 4x + 3 (rate of change = 4) with data from a table that yields a rate of change of 5/6, the function y = 4x + 3 has a greater rate of change because 4 is significantly larger than 5/6.

To find the value of 'y' for a given 'x' in a function, substitute the 'x' value into the function's equation.
Perform the arithmetic operations to calculate the resulting 'y' value.
This process is equivalent to finding the y-coordinate of a point on the graph of the function when the x-coordinate is known.

This skill is fundamental for understanding function notation and how to find specific points on a graph.

Given the function y = 2x - 4 and the point (12, y), substitute x = 12 into the equation: y = 2(12) - 4 = 24 - 4 = 20. Thus, y = 20.

The absolute value of any expression is always non-negative (zero or positive).
An equation where the absolute value of an expression is set equal to a negative number has no solution.
This is because the absolute value can never produce a negative result.

Understanding the properties of absolute value is crucial for solving equations and inequalities correctly, especially avoiding impossible scenarios.

The equation |4x + 6| = -2 has no solution because the absolute value of 4x + 6 can never be a negative number like -2.

Linear functions model situations where there's a constant rate of change (slope) and a fixed initial or flat fee (y-intercept).
Identify the rate of change (often associated with 'per' or changing quantities) and the y-intercept (fixed charges, initial fees).
Surcharges that apply only once (e.g., initial charge, end-of-trip fee) contribute to the y-intercept (B).
The slope (M) is typically associated with costs that vary based on usage (e.g., per mile, per minute).

This skill allows you to translate real-world pricing or cost structures into mathematical equations for analysis and prediction.

For a taxi fare with a $3 initial charge, $0.70 per 1/5 mile, a 50-cent surcharge, and a $1 surcharge, the equation is y = 3.5x + 4.50. The slope (3.5) comes from converting $0.70 per 1/5 mile to $3.50 per mile, and the y-intercept (4.50) is the sum of the one-time charges ($3 + $0.50 + $1).

A proportion is an equation stating that two ratios are equal.
Cross-multiplication is a reliable method to solve proportions: if a/b = c/d, then ad = bc.
Alternatively, proportions can be solved by finding a common factor or by simplifying one ratio to match the other.
Ensure units are consistent or accounted for when setting up proportions.

Proportions are used to solve problems involving ratios and scaling, common in geometry, rates, and measurement conversions.

To solve 0.44/1 = x/25, cross-multiply: 0.44 * 25 = x * 1. This gives x = 11.

Key takeaways

1Mastering estimation techniques, like squaring numbers to compare square roots, is essential for problems without calculators.
2Geometric principles, such as angle relationships with parallel lines and the properties of transformations, are fundamental for spatial reasoning.
3Understanding the definitions of number types (rational vs. irrational) and function types (linear vs. non-linear) is critical for classification problems.
4The vertical line test provides a simple visual check to determine if a graph represents a function.
5Lines of best fit and points of intersection help analyze data trends and solve systems of equations.
6Real-world scenarios can often be modeled using linear functions, requiring careful identification of the slope and y-intercept.
7Absolute value equations have specific constraints; setting an absolute value equal to a negative number results in no solution.

Key terms

Square rootNumber lineParallel linesTransversalCorresponding anglesLinear pairIrrational numberNon-terminating decimalNon-repeating decimalLinear functionNon-linear functionGeometric transformationReflectionRotationTranslationVertical line testFunctionLine of best fitSlopeY-interceptPoint of intersectionRate of changeAbsolute valueProportionCross-multiplication

Test your understanding

1How can you determine if a square root is between two integers without a calculator?
2What is the relationship between angles formed when a transversal intersects parallel lines, and how can this be used to find unknown angles?
3What are the defining characteristics of an irrational number, and how do they differ from rational numbers?
4What are the key conditions for a function to be classified as linear?
5Explain the vertical line test and why it is used to identify functions.
6How do you set up and solve a system of two linear equations to find their point of intersection?
7Describe the process of modeling a real-world cost scenario, like a taxi fare, using a linear function, identifying the slope and y-intercept.