9th Grade Building Functions Word Problems with Answers (PDF)

In Grade 9, students work with arithmetic sequences, lists of numbers where each term is found by adding the same fixed amount to the one before. They learn to describe these sequences in two ways: recursively, by saying how to get from one term to the next, and with an explicit formula that gives any term directly from its position. They also learn to move between the two forms and use them to model simple situations. The curriculum points for this skill do not give a specific number range for this grade, so this page describes only what the printed tasks ask for.

A common mistake is counting the steps wrong. When a sequence starts at the first term, the term in position n has been increased by the difference only n minus 1 times, so students often use n times the difference and land one step too far. Another frequent slip is mixing up the first term with the difference, or forgetting that the sum of the terms needs the first and last terms, not just one of them.

Practice in this area uses task templates such as "Arithmetic sequence: the value at a given step," "Arithmetic sequence: nth term," "Term of an arithmetic sequence step by step," and "Arithmetic sequence — sum of terms." Other templates ask students to find the missing difference or missing first term, to decide whether a statement about a sequence is true or false, or to find the mistake in a worked solution. Each template gives a short sequence or rule, and the child works through the numbers to reach an answer.

At this stage, the standards ask students to build a geometric sequence from a graph, a verbal description, a table, or two input-output pairs. They also ask students to write the sequence both recursively (each term from the one before, multiplied by a common ratio) and with an explicit formula, and to move between the two forms. For sums, students derive and use the formula for a finite geometric series when the common ratio is not 1. The standards give a mortgage payment as an example of this use. The curriculum points do not specify a number range, so ask your child's teacher which sizes of numbers are expected.

A common difficulty is mixing up the term number with the exponent. In the explicit formula, the exponent is one less than the term number, so the fifth term uses the ratio raised to the fourth power. Students also often confuse the common ratio with the common difference from arithmetic sequences, adding instead of multiplying. When adding up terms, the number of terms is easy to miscount, which throws off the whole answer.

In this grade, the printed worksheets include tasks such as Geometric sequence — nth term, Geometric sequence — sum of terms, and Sum of a geometric sequence: choose the answer. Other tasks ask students to find a missing ratio or first term, to work out how many terms a sum contains, or to compare two sequences. Some present a story, such as Geometric sequence: the total over several days, and others include extra information that must be ignored, such as Term of a geometric sequence: extra data.

By the end of grade 8, the standards ask students to build a linear function from two (x, y) values, reading them from a table or graph, and to find the rate of change and initial value. In grade 9, this skill is practiced with the tasks below, which give two points and ask for the slope, the intercept, or the full equation of the line through them.

A common mistake is mixing up the slope formula. Students often subtract the coordinates in a different order for the x-values than for the y-values, which flips the sign of the slope. Another frequent error with parallel lines is keeping the original slope but then reusing the given point's coordinates as the intercept, instead of solving for the intercept that makes the line pass through that point.

Typical grade 9 tasks in this area include Linear function through two points, where the child writes the full equation; Slope of a line through two points, offered both as a choose-the-answer question and as a true-or-false question; Intercept of a line through two points, where the child identifies where the line crosses the y-axis; and Parallel line through a point, where the child writes the equation of a line that runs parallel to a given line and passes through a new point.