Math: Grade 9 Forms Only
Everything in this unit appears ONLY on Grade 9 test forms: the Pythagorean theorem, cone and sphere volume, transformations, coordinates and scatter plots.
18 topics · 70 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
What the Grade 9 Forms Add
- How it worksHow does the Grade 8 standards list stack transformations, right triangles and round solids on top of Grade 7?
- Going furtherWhat happens to your study plan when you sit the test as a ninth grader instead of an eighth grader?
- Ties togetherHow do the circle and prism formulas from Grade 7 carry straight into cone and sphere questions?
Your turnhard1 of 3
New York's Grade 8 geometry standard on round solids says students will be GIVEN the formulas for the volume of cones, cylinders and spheres. The SHSAT guide says definitions and explanations for terms, symbols and formulas will NOT be given. A ninth grader meets 6 such questions.
What follows?The Pythagorean Theorem
- How it worksHow do the squares built on the two legs pour their area into the square on the hypotenuse?
- How it worksWalk me through a rearrangement proof of a squared plus b squared equals c squared using four copies of one triangle.
- Worked exampleFind the hypotenuse of a right triangle with legs of 9 and 12, then its perimeter and its area.
- Common mistakeWhy isn't the longest side automatically the one you label c when the right angle is drawn somewhere odd?
- Going furtherWhat happens to a squared plus b squared versus c squared when the triangle is obtuse instead of right?
Your turneasy1 of 3
A right triangle has legs measuring 12 units and b units. Write an expression for the hypotenuse, in terms of b.The Converse: Testing Whether a Triangle Is Right
- How it worksHow does comparing a squared plus b squared against c squared sort a triangle into right, acute or obtuse?
- Worked exampleFind whether the side trios 8, 15, 17 and 7, 8, 12 each hold a right angle.
- Common mistakeWhy doesn't 3, 4 and 5 making a right triangle mean 6, 7 and 8 makes one too?
Your turnmedium1 of 3
A triangle has sides 9, 15 and c, with c the longest. Write an expression, in terms of c, for c squared minus the sum of the squares of 9 and 15, a quantity that is zero exactly when the triangle is right.Pythagorean Triples Worth Memorizing
- How it worksHow does multiplying a trio like 3, 4 and 5 by any whole number keep it a right triangle?
- Worked exampleFind the missing leg of a right triangle with hypotenuse 26 and one leg 24, with no calculator.
- Going furtherWhat happens to your speed on a no-calculator test once 5-12-13 and 8-15-17 are in your head?
Your turneasy1 of 3
A right triangle has a hypotenuse of 16k units and one leg of 10k units. Write an expression for the other leg, in terms of k.Distance Between Two Points on a Grid
- How it worksHow do the horizontal gap and the vertical gap become the two legs of a hidden right triangle?
- Worked exampleFind the distance from the point at negative 3 and 2 to the point at 5 and 8 on a grid.
- Common mistakeWhy isn't the distance between two points just the sideways gap added to the up-and-down gap?
- Going furtherWhat happens to that distance when both points are shifted 4 units right and 7 units up together?
Your turnmedium1 of 3
Write an expression, in terms of y, for the distance between the point at negative 5 and 2 and the point at 8 and y on a coordinate grid.Right Triangles Inside a Solid
- How it worksHow does the diagonal across the base become one leg for the diagonal of the whole box?
- Worked exampleFind the longest straight rod that fits inside a box measuring 3 by 4 by 12 feet.
- Going furtherWhat changes about that hunt when the box is a perfect cube instead of three different edges?
Your turnmedium1 of 3
Write an expression, in terms of c, for the length of the longest straight rod that fits inside a box measuring 6 feet by 7 feet by c feet.Volume of Cones, Cylinders and Spheres
- How it worksHow does pouring water from a cone into its matching cylinder three times prove the one third?
- How it worksHow do the cone, cylinder and sphere volume rules line up when all three share a radius and a height?
- Worked exampleFind the volume of a cone of radius 6 cm and height 10 cm, left in terms of π.
- Worked exampleFind the radius of a sphere whose volume comes to 288 π cubic inches.
- Common mistakeWhy isn't the height of a cone the same thing as the slanted side running up to its tip?
Your turneasy1 of 3
A cone has a radius of r cm and a height of 15 cm. Write an expression for its volume, in cubic centimetres, in terms of r, leaving π in your answer.Translations
- How it worksHow does one arrow, written as a pair of numbers, move every point of a figure the same way?
- Worked exampleFind where a triangle with corners at 1 and 2, 4 and 2, and 4 and 6 lands after sliding 5 left and 3 down.
- Common mistakeWhy doesn't sliding a figure ever change its size or flip it into a mirror image?
Your turneasy1 of 3
A corner of a figure sits at (16, 15). The figure slides 5 units left and 3 units down. Find the sum of the two coordinates of the image of that corner.Reflections
- How it worksHow does the mirror line cut every point-to-image segment in half at a right angle?
- Worked exampleFind where the point at 6 and negative 2 lands after a flip across the x-axis and then the y-axis.
- Worked exampleFind the image of a triangle with corners at 2 and 1, 5 and 1, and 2 and 4 flipped across the line x equals 3.
- Going furtherWhat happens to a figure when you reflect it twice across the very same mirror line?
Your turneasy1 of 3
The point at (15, negative 2) is flipped across the x-axis and then across the y-axis. Find the x-coordinate of the final image.Rotations
- How it worksHow do the center of rotation and the turn angle together decide where each corner lands?
- Worked exampleFind where the point at 3 and 5 ends up after a 90 degree counterclockwise turn about the origin.
- Common mistakeWhy doesn't a half turn of 180 degrees give the same result as one flip across a single axis?
- Going furtherWhat happens to a figure after a quarter turn is repeated four times about the same pin?
Your turnmedium1 of 3
The point at 3 and 15 is turned 90 degrees counterclockwise about the origin. Find the x-coordinate of its image.Dilations
- How it worksHow do rays drawn from the center through each corner show exactly where the enlarged corners land?
- How it worksHow does the scale factor multiply both coordinates when the center of the stretch sits at the origin?
- Worked exampleFind the image of a rectangle with corners at 2 and 2, 6 and 2, 6 and 5, and 2 and 5 under a stretch of factor 3.
- Common mistakeWhy isn't a scale factor of one half a shrink toward the bottom left corner of the grid?
- Going furtherWhat happens to the area of a figure under a dilation with a scale factor of 5?
Your turneasy1 of 3
A rectangle measuring 8 by 6 units is stretched from the origin by a scale factor of k. Write an expression for the area of the image, in square units, in terms of k.Congruence Defined by Transformations
- How it worksHow do you describe a sequence of slides, flips and turns that lands one figure on its twin?
- Common mistakeWhy doesn't a dilation belong in any sequence meant to show two figures are congruent?
- Worked exampleFind a two-move sequence carrying corners 1 and 1, 3 and 1, 1 and 4 onto negative 1 and negative 1, negative 3 and negative 1, negative 1 and negative 4.
- Going furtherWhat happens to congruence when the two figures turn out to be mirror images of one another?
Your turnmedium1 of 3
A half turn about the origin carries a triangle onto its twin. Under that same half turn, write an expression for the sum of the two coordinates of the image of the point at (x, 15), in terms of x.Similarity Defined by Transformations
- How it worksHow does a dilation followed by a slide or a turn carry one figure onto a larger copy of itself?
- Worked exampleFind the scale factor and the moves taking a 3, 4, 5 triangle onto one with sides 9, 12 and 15.
- Common mistakeWhy isn't every pair of rectangles similar just because all eight of their corners are right angles?
- Ties togetherHow does the transformation view of similarity connect to solving for a missing side with a proportion?
Your turneasy1 of 3
A triangle with sides 4, 7 and 12 units is carried onto a similar triangle whose longest side is s units. Write an expression for the scale factor, in terms of s.Angle Relationships Built from Transformations
- How it worksHow does a half turn about the midpoint of a side build the parallel line that proves the angle sum?
- How it worksHow does the angle-angle test for similar triangles fall out of dilations and rigid moves?
- Worked exampleFind every angle in a row of three copies of a 30-70-80 triangle placed corner to corner along a line.
- Going furtherWhat happens to the angle-angle similarity test when the two triangles face opposite directions?
Your turneasy1 of 3
Copies of a triangle are set corner to corner along a straight line, so its three angles fill a straight angle. Two of them measure 48 degrees and 76 degrees. Find the third, in degrees.Reading and Plotting Points on the Coordinate Plane
- How it worksHow do the signs of the two coordinates sort every point into one of the four quadrants?
- Worked exampleFind the perimeter of the rectangle whose corners sit at negative 2 and 3, 6 and 3, 6 and negative 1, and negative 2 and negative 1.
- Common mistakeWhy isn't the point 4 across and 7 up the same place on the grid as the point 7 across and 4 up?
- Going furtherWhat happens to a point's quadrant when you flip the sign of only its second coordinate?
Your turneasy1 of 3
A rectangle has corners at negative 5 and 5, at x and 5, at x and negative 5, and at negative 5 and negative 5. Write an expression for its perimeter, in units, in terms of x.Point Graph and Shape Transformation Item Types
- How it worksHow do the point graph and shape transformation tools ask you to build an answer instead of choosing one?
- Worked exampleFind where to drag three corners after flipping a triangle at 2 and 1, 5 and 1, 5 and 3 across the y-axis.
- Going furtherWhat happens if you advance past a math item before checking where the point actually dropped?
- Common mistakeWhy doesn't a nearly-right placement earn partial credit on a drag-and-drop math item?
Your turnmedium1 of 3
A corner of a triangle sits at (11, 3). The triangle is flipped across the y-axis and then slid 4 units up. Find the sum of the two coordinates of that corner's image.Scatter Plots and Patterns of Association
- How it worksHow do positive, negative, linear, nonlinear and no association each look as a cloud of points?
- Worked exampleFind the outlier among study hours and scores of 1 and 60, 2 and 65, 3 and 71, 4 and 74, 9 and 52, 5 and 80.
- Common mistakeWhy doesn't a tight pattern between ice cream sales and drowning counts mean one is causing the other?
- Going furtherWhat happens to a scatter plot's pattern when one of the two variables switches units?
Your turneasy1 of 3
A scatter plot of study hours against score holds the points (1, 60), (2, 65), (3, 71), (4, 74), (5, 80) and (9, 62), where the last is the outlier. How many points below 80 does the outlier's score sit?Fitting a Line and Reading Its Slope
- How it worksHow do you place a line of fit so the dots end up balanced above it and below it?
- Worked exampleFind the predicted plant height at 7 hours of sunlight when the fitted line is 1.5 times hours plus 4.
- Common mistakeWhy isn't a line of fit supposed to touch as many of the plotted points as it possibly can?
- Going furtherWhat happens to a prediction when the fitted line is pushed far past the range of the collected data?
Your turneasy1 of 3
A fitted line predicts plant height as 3.5 times the hours of sunlight plus 6 centimetres. Write an expression for the predicted height, in centimetres, after h hours.