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Math: Grade 9 Forms Only (NYS Grade 8)

GRADE 9 FORMS ONLY. Everything New York adds in Grade 8: exponents, scientific notation, roots, irrationals, slope, functions, systems and scatter plots.

19 topics · 81 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.

Properties of Integer Exponents

  1. How it worksHow does writing out the factors prove that xa times xb equals x to the a plus b?
  2. How it worksHow does a power raised to another power multiply the exponents rather than adding them?
  3. Worked exampleFind the simplest form of (3x2)3 times 2x4 divided by 6x5.
  4. Common mistakeWhy isn't 23 times 24 equal to 47 when both bases are 2 and both exponents appear?
  5. Going furtherWhat happens to the exponent rules when the base itself is a fraction?
Your turneasy1 of 3
Simplify x5 times x5, writing your answer as a single power of x.

Zero and Negative Exponents

  1. How it worksHow does dividing 23 by 23 force the value of 20 to be exactly 1?
  2. How it worksHow does a negative exponent flip its base down into the denominator of a fraction?
  3. Worked exampleFind the value of 32 times 3 to the negative 5 as a single fraction.
  4. Common mistakeWhy isn't 5 to the negative 2 a negative number when the exponent is negative?
Your turnmedium1 of 3
Find the value of 32 times 3 raised to the power negative 6.

Scientific Notation

  1. How it worksHow does the exponent count the places the decimal point has to walk?
  2. How it worksHow does keeping exactly one digit in front of the decimal make the form standard?
  3. Worked exampleFind 6320000 and then 0.00019 rewritten in scientific notation.
  4. Common mistakeWhy isn't 25 times 104 proper scientific notation when it equals the right amount?
Your turnmedium1 of 3
A number is written as 72 times 10 raised to the power 6, where 72 is a two-digit number. Rewritten in proper scientific notation, what is the new exponent?

Multiplying and Dividing in Scientific Notation

  1. How it worksHow do you repair a product such as 45 times 106 that drifted out of standard form?
  2. Worked exampleFind the product of 3.2 times 105 and 4 times 10−2 in scientific notation.
  3. Worked exampleFind the quotient of 8.4 times 107 divided by 2.1 times 103.
  4. Going furtherWhat happens when you try to add two scientific-notation numbers with different exponents?
Your turneasy1 of 3
Find the product of 3.2 times 105 and 4 times 10 raised to the power negative 3.

Estimating with Powers of Ten

  1. How it worksHow does rounding to one digit times a power of ten turn a hard ratio into a subtraction?
  2. Worked exampleFind roughly how many seconds fit in a century using powers of ten alone.
  3. Common mistakeWhy doesn't a larger leading digit make a number larger once the powers of ten differ?
  4. Worked exampleFind which is greater, 4 times 1012 or 9 times 1011, without writing either one out.
Your turneasy1 of 3
Using 3 times 107 seconds per year, about how many seconds are there in 154 years?

Square Roots and Cube Roots

  1. How it worksHow does the square root symbol undo squaring the way subtraction undoes addition?
  2. How it worksHow does a cube root ask for the edge of a cube whose volume you already know?
  3. Worked exampleFind the solutions of x2 = 144 and of x3 = 125, and say why only one has two answers.
  4. Common mistakeWhy isn't the square root symbol in front of 16 asking for both 4 and -4?
  5. Going furtherWhat happens to the square root of a number sitting between 0 and 1?
Your turneasy1 of 3
Find the square root of 294. Give your answer to two decimal places.

Irrational Numbers and Decimal Expansions

  1. How it worksHow does a decimal that eventually repeats prove a number is rational?
  2. How it worksHow does knowing 225 is a perfect square tell you the square root of 224 is irrational?
  3. Worked exampleFind which one is irrational among 0.75, the square root of 49, the square root of 20 and 22/7.
  4. Common mistakeWhy isn't π equal to 22/7 when the two agree for the first two decimal places?
Your turnmedium1 of 3
The number 22 is not a perfect square. Find the square root of 22 to three decimal places.

Approximating Irrational Numbers on a Number Line

  1. How it worksHow do the two nearest perfect squares trap an irrational root between two whole numbers?
  2. How it worksHow does testing 3.1 and then 3.2 pin the square root of 10 down to a single tenth?
  3. Worked exampleFind which two consecutive tenths the square root of 40 falls between.
  4. Worked exampleFind the order of 3.5, the square root of 12, 11/3 and π from least to greatest.
  5. Going furtherWhat happens to your estimate when you square each guess and compare instead of guessing again?
Your turnmedium1 of 3
A student guesses that the square root of 42 is 6.3. Find the value of 6.3 squared minus 42, which measures how far the guess misses.

Linear Equations with Variables on Both Sides

  1. How it worksHow does gathering every x term on one side turn the problem back into a two-step equation?
  2. Worked exampleFind x in 3(x - 1/2) = x + 2.
  3. Common mistakeWhy doesn't 4x + 3 = 4x + 7 have a solution when it looks like any other equation?
  4. Going furtherWhat happens when both sides of an equation collapse into exactly the same expression?
Your turneasy1 of 3
Solve 7(x - 1/2) = x + c for x, writing your answer in terms of c.

One Solution, No Solution, or Infinitely Many

  1. How it worksHow does simplifying until the variable disappears reveal which of the three cases you have?
  2. Worked exampleFind which of 5x - 2 = 5x + 1, 5x - 2 = 3x + 8 and 5x - 2 = 5x - 2 has no solution at all.
  3. Common mistakeWhy isn't an equation that ends up reading 0 = 0 an equation with no solution?
Your turnmedium1 of 3
Write an expression, in terms of c, for the value of b that makes 7x + b = 7(x + c) true for every x.

Slope as a Rate of Change

  1. How it worksHow does rise over run compress a picture of steepness into one number?
  2. How it worksHow does the slope triangle keep its shape wherever you slide it along the same line?
  3. Worked exampleFind the slope of the line through (2, -1) and (6, 7), and draw the triangle you used.
  4. Going furtherWhat happens to the slope of a line that runs perfectly flat or perfectly upright?
  5. Common mistakeWhy isn't a line of slope -3 gentler than a line of slope 2?
Your turneasy1 of 3
Write an expression for the slope of the line through the points (2, -1) and (15, y), in terms of y.

Finding Slope from a Table, a Graph, or an Equation

  1. How it worksHow does a table give up its slope when you divide the change in y by the change in x?
  2. How it worksHow does the number sitting in front of x in y = mx + b announce the slope before you plot?
  3. Worked exampleFind the slope from a table where x runs 2, 5, 8 while y runs 11, 20, 29.
  4. Worked exampleFind the slope and the starting value of 4x + 2y = 10 by rewriting the equation.
Your turneasy1 of 3
A table for a linear relationship shows y = 11 when x = 2, and y = c when x = 15. Write an expression for the slope, in terms of c.

The y-Intercept and Slope-Intercept Form

  1. How it worksHow does setting x to zero locate the exact spot a line crosses the vertical axis?
  2. Worked exampleFind the equation of the line through (0, -4) with slope 3, then find y when x is 5.
  3. Common mistakeWhy isn't the y-intercept simply the first number written down in an equation?
  4. Worked exampleFind the equation of the line passing through (1, 5) and (3, 11).
Your turneasy1 of 3
A line passes through (0, -4) with slope 7. Write the expression that y equals, in terms of x.

Linear Functions and What Makes a Function

  1. How it worksHow does the vertical line test catch a graph that fails to be a function?
  2. How it worksHow does y = mx + b guarantee a straight line while area equals s squared bends away?
  3. Worked exampleFind whether a rule pairing 1 with 3, 2 with 5 and 1 with 7 counts as a function.
  4. Common mistakeWhy isn't every straight line drawn on a grid the graph of a function?
Your turneasy1 of 3
A linear function has f(0) = 14 and climbs by 7 every time x increases by 1. Write an expression for f(x), in terms of x.

Comparing Functions Across Representations

  1. How it worksHow do you pull the rate of change from a graph, a table and an equation to line them up?
  2. Worked exampleFind the faster car between one at y = 55x miles and one covering 170 miles in 3 hours.
  3. Worked exampleFind the faster savings plan between adding 12 dollars weekly and rising 50 dollars every 4 weeks.
  4. Going furtherWhat happens to the comparison when one function starts ahead but climbs more slowly?
Your turneasy1 of 3
Car B covers d miles in 3 hours at a steady speed. Write an expression, in terms of d, for the number of miles Car B covers in 10 hours.

Systems of Linear Equations on a Graph

  1. How it worksHow does graphing two lines turn a system into one point you can read off the grid?
  2. How it worksHow do parallel lines and identical lines picture no solution and endless solutions?
  3. Common mistakeWhy isn't a system with no solution the same as a system whose answer happens to be zero?
  4. Worked exampleFind the solution of y = x + 1 together with y = -2x + 7 by graphing both.
Your turneasy1 of 3
Graph y = x + 1 together with y = -7x + c. Write an expression for the x-coordinate of the point where the two lines cross, in terms of c.

Solving Systems by Substitution and Elimination

  1. How it worksHow does elimination line two equations up so one variable cancels the moment you add?
  2. How it worksHow do the coefficients tell you whether substitution or elimination will be faster?
  3. Worked exampleFind x and y for 3x + y = 5 and 2x - y = 10 by adding the two equations together.
  4. Worked exampleFind ticket prices when 2 adults and 3 children pay 58 dollars and 1 adult and 4 children pay 54.
  5. Going furtherWhat happens to elimination when neither variable starts with matching coefficients?
Your turneasy1 of 3
Solve the system 7x + y = 5 and 2x - y = c by adding the two equations. Write an expression for x, in terms of c.

Scatter Plots and Patterns of Association

  1. How it worksHow does a scatter plot turn two columns of numbers into one picture of association?
  2. How it worksHow do clusters, outliers and a curved shape each change what a scatter is telling you?
  3. Worked exampleFind the association between study hours and test scores when one point sits far off the trend.
  4. Common mistakeWhy doesn't a strong association between two quantities prove that one causes the other?
Your turneasy1 of 3
Two points on a scatter plot of study hours against test score are (2, 68) and (10, s). Write an expression for the rate of change between them, in score points per hour, in terms of s.

The Line of Best Fit and What Its Slope Means

  1. How it worksHow do you judge a fitted line by how closely the dots hug it above and below?
  2. Worked exampleFind what a slope of 1.5 centimeters per hour means for a plant given extra sunlight.
  3. Common mistakeWhy doesn't a best-fit line let you predict far outside the range of data that was collected?
  4. Worked exampleFind the score a best-fit line of y = 4x + 62 predicts for a student who studies 7 hours.
  5. Ties togetherHow does the slope of a best-fit line turn out to be the same rate of change any line carries?
Your turneasy1 of 3
A line of best fit is y = 4x + 64, where x is hours studied and y is the test score. What score does it predict for a student who studies 7 hours?