Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 16 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Define numerator and denominator quantities for a rate
Define appropriate quantities for descriptive modeling.
Translate “per” into division before naming either part of the rate. The quantity before “per” belongs in the numerator and tells what is being counted, while the quantity after it …
Preview problemLabel graph axes from defined quantities
Define appropriate quantities for descriptive modeling.
First identify the direction of dependence: which quantity is the input, and which changes in response? Place the independent input on the horizontal axis and the dependent output on the …
Preview problemVerify that variables and units fit an equation
Define appropriate quantities for descriptive modeling.
Audit the equation term by term using both meaning and units. A rate multiplied by its input should produce the same unit as the fixed term, and their sum determines …
Preview problemRound a measurement result to the correct decimal place
Report quantities with accuracy appropriate to measurement limitations.
For addition of measurements, align the decimal points and carry out the arithmetic before rounding. The least precise decimal place supported by the inputs controls the final report, so preserve …
Preview problemChoose a precision rule for multiplying measurements
Report quantities with accuracy appropriate to measurement limitations.
Multiplication uses significant digits rather than matching decimal places. Count the meaningful digits in each measured factor, let the smallest count limit the product, and keep the calculator value unrounded …
Preview problemFind the possible true-value interval for a rounded measurement
Report quantities with accuracy appropriate to measurement limitations.
A rounded report represents a whole interval of possible true values. Move half of the rounding unit below and above the reported value to find the boundaries, then apply the …
Preview problemChoose practical precision for a context
Report quantities with accuracy appropriate to measurement limitations.
Practical precision has two filters: the reporting unit must measure the correct physical dimension, and its resolution must match what the task can actually use. A unit from the wrong …
Preview problemDecide whether rounding changes a conclusion
Report quantities with accuracy appropriate to measurement limitations.
Translate the decision rule into an inequality, paying attention to whether equality belongs at the cutoff. Test the exact value and the rounded display separately against that same boundary; rounding …
Preview problemIdentify overprecision in a measurement result
Report quantities with accuracy appropriate to measurement limitations.
A calculator reports arithmetic digits, not guaranteed measurement precision. Find the least significant-digit count supported by the measured factors, compare that limit with the raw product, and round once at …
Preview problemIdentify underprecision from excessive rounding
Report quantities with accuracy appropriate to measurement limitations.
Judge reporting precision against the task’s tolerance, not against a generic rounding habit. Compute the absolute difference between the report and the reference value, compare that error with the allowed …
Preview problemFind least and greatest possible true values from rounding
Report quantities with accuracy appropriate to measurement limitations.
Move half of the rounding unit to either side of the reported measurement to locate the two cutoffs. Then use the stated rounding convention to classify the endpoints: the lower …
Preview problemReport a calculated measurement using units and precision that fit the given measurements
Report quantities with accuracy appropriate to measurement limitations.
Set up the calculation so the quantity unit cancels against the denominator of the unit price, leaving the unit required for total cost. Compute with full precision, then apply the …
Preview problemCompare measurements using uncertainty intervals
Report quantities with accuracy appropriate to measurement limitations.
Treat each measurement as its full uncertainty interval rather than as a single central value. A definite ordering requires one interval to end before the other begins; compare the nearest …
Preview problemChoose graph axis precision that matches data precision
Report quantities with accuracy appropriate to measurement limitations.
Turn scale design into a calculation: divide the axis span by each proposed tick step to get its interval count. A workable step must align both endpoints, produce an allowed …
Preview problemCreate a dot plot from a small data set
Represent one-variable data with dot plots, histograms, and box plots.
Build a frequency tally across every equally spaced axis value, including positions with zero observations so gaps remain visible. Convert each tally into that many stacked dots, then add all …
Preview problemInterpret a dot plot from value counts
Represent one-variable data with dot plots, histograms, and box plots.
Expand the dot stacks into one ordered list before calculating center and spread. Apply the named quartile convention consistently, use the resulting IQR fences for formal outlier status, and only …
Preview problemCount how many data values fall in each histogram interval
Represent one-variable data with dot plots, histograms, and box plots.
Write the bin intervals with their endpoint rule visible, then assign each observation to exactly one bin. Boundary values deserve special attention because a single misplaced endpoint changes two bar …
Preview problemInterpret a histogram from bin counts
Represent one-variable data with dot plots, histograms, and box plots.
Read a histogram in layers: add bar heights for sample size, locate every tallest bar for the modal region, and then examine how occupied bins extend away from the peak. …
Preview problemCreate a box plot from a five-number summary
Represent one-variable data with dot plots, histograms, and box plots.
Verify that the five-number summary is in nondecreasing order before placing anything on the scale. The box runs from the first to the third quartile with the median inside, while …
Preview problemCreate a box plot summary from raw data
Represent one-variable data with dot plots, histograms, and box plots.
Order the raw data first, find the overall median, and follow the declared quartile convention when forming the lower and upper halves. Those five summary positions determine the entire display: …
Preview problemInterpret a box plot summary
Represent one-variable data with dot plots, histograms, and box plots.
Read the five marked positions first, then subtract adjacent positions to expose the two half-box lengths and two whisker lengths. The box width gives IQR and the full endpoint span …
Preview problemChoose the best display for one-variable data
Represent one-variable data with dot plots, histograms, and box plots.
Match the display to the information the task must preserve. Exact values and repeats require one mark per observation, whereas interval grouping sacrifices exact values and a five-number summary sacrifices …
Preview problemIdentify an error in a one-variable data display
Represent one-variable data with dot plots, histograms, and box plots.
Audit the source data and the display with the same invariant: one observation must produce one dot, so each stack height equals that value’s frequency. Compare counts value by value …
Preview problemFind the mean of two data sets and compare the means
Compare data sets using center and spread measures appropriate to distribution shape.
Compute each mean independently as that set’s total divided by its own observation count; comparing raw totals can be misleading, especially when sample sizes differ. Once both centers are in …
Preview problemFind the median of each data set and compare the medians
Compare data sets using center and spread measures appropriate to distribution shape.
Order each data set before looking for its center. With an odd number of observations, the median is the single value with equal counts on both sides; find that position …
Preview problemFind the range of each data set and compare the ranges
Compare data sets using center and spread measures appropriate to distribution shape.
For each set, ignore the interior values at first and identify only its minimum and maximum. Their difference is the range, so compare those two spans after computing them in …
Preview problemFind and compare interquartile ranges from Q1 and Q3
Compare data sets using center and spread measures appropriate to distribution shape.
Treat each middle-half interval as a width, not as a location on the number line. Subtract the first quartile from the third quartile for each data set, then compare the …
Preview problemChoose mean or median as the better measure of center
Compare data sets using center and spread measures appropriate to distribution shape.
Let distribution shape determine the center measure. The mean uses every value and works well when symmetry and the absence of outliers keep it representative, while the median’s resistance matters …
Preview problemChoose range or IQR as the better measure of spread
Compare data sets using center and spread measures appropriate to distribution shape.
Start with the reporting purpose: full endpoint-to-endpoint spread and typical middle spread answer different questions. Range directly uses both extremes, so an outlier can dominate it; IQR measures the middle …
Preview problemCompare two data sets from box plot summaries
Compare data sets using center and spread measures appropriate to distribution shape.
Compare like features rather than judging the plots by overall position or a single endpoint. Medians compare center, box widths compare middle-half spread, and maximum-minus-minimum spans compare full spread; calculate …
Preview problemCompare two data sets from visual summaries
Compare data sets using center and spread measures appropriate to distribution shape.
Treat center and spread as two separate questions about the same pair of distributions. First compare the medians to describe where typical values sit, then compare maximum-minus-minimum spans to describe …
Preview problemMeasure the effect of adding an outlier
Compare data sets using center and spread measures appropriate to distribution shape.
An extreme new value does not affect every summary in the same way. Recompute each statistic from its definition, then use the size of each change to distinguish summaries that …
Preview problemDecide which group has more variability
Compare data sets using center and spread measures appropriate to distribution shape.
Variability is about how widely values are spread, so larger spread measures point toward greater variability. Compare the groups on the same measure first, and use both the middle-half spread …
Preview problemInterpret center and spread comparisons in context
Compare data sets using center and spread measures appropriate to distribution shape.
A complete comparison needs two parallel stories: one about typical performance and one about consistency. Compute both differences in the same group order, then remember that a positive center difference …
Preview problemClassify the shape of a dot plot
Interpret differences in shape, center, spread, and outliers in context.
Read a dot plot as a frequency pattern, not just a cloud of marks. Locate any tallest stack, compare counts at values equally far from a possible center, and scan …
Preview problemClassify the shape of a histogram
Interpret differences in shape, center, spread, and outliers in context.
For equal-width histogram bins, the ordered bar heights carry the shape information. Find the tallest interval, compare bars that sit equally far from it, and check whether all heights are …
Preview problem