Dashboard

Problem-Solving & Data Analysis

Ratios · Rates · Units
CORD — C Concept O Operation · Desmos R Recognition D Drill
P1 Ratios & proportions — cross-multiply Drill 30
Desmos
  1. Type the equation with the actual numbers: e.g. 3/4 = 9/x.
  2. Desmos draws a vertical line at the solution.
  3. Compare the line's x-position to the answer choices — pick the one that matches.
  4. For an exact value: click the line, Desmos labels the point.
Concept
  • Cross-multiply: $\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc$.
  • Scale to total: ratio $a:b$ with total $T$ → first part = $\dfrac{a}{a+b} \cdot T$.
  • "Ratio of A to B is k": $A = kB$, OR set $A = kn$ and $B = n$ for whole-number cases.
  • Equivalent ratios: $a:b = ka:kb$ for any $k$.
Match the order. "Ratio of red to blue" means $\dfrac{\text{red}}{\text{blue}}$, not the reverse. Read the question twice.
RecognitionFlipping the ratio order — 'red to blue' is red/blue, not blue/red.
P2 Rates & unit conversion Drill 30
Desmos
  1. Desmos has no unit awareness — words like "miles" are just unknown variables.
  2. Strip units, type the arithmetic only: 60 * 5 → 300.
  3. Track units in your head (or on scratch) — Desmos won't help cancel them.
Concept
  • Rate × time = distance: $r \cdot t = d$. Solve for whichever is missing.
  • Unit factor chain: multiply by ratios where the unit you want to cancel is on the bottom. e.g. miles → feet: $\times \dfrac{5280 \text{ ft}}{1 \text{ mile}}$.
  • Speed = distance / time. Match units (mph wants miles + hours).
  • Density / concentration: $\dfrac{\text{quantity}}{\text{volume}}$ stays constant — same ratio for any sample.
Always cancel units to check. If your final units don't match the question's units, redo.
RecognitionForgetting to convert units — final answer in wrong unit (feet vs. miles).
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Problem-Solving & Data Analysis

Percent · Reverse Percent
P3 Percent of · percent change · increase/decrease Drill 45
Desmos
  1. Convert percent to decimal first: 15% → 0.15. Type 0.15 * 80.
  2. The row shows the expression — expand it (down-arrow next to row) to see the numerical value.
  3. For SAT-typical percent, mental math + multiplier method is usually faster than Desmos.
Concept
  • Percent of: $p\% \text{ of } x = \dfrac{p}{100} \cdot x$.
  • Percent change: $\dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100$.
  • Increase by $r\%$: multiply by $(1 + r/100)$. e.g. +20% → $\times 1.20$.
  • Decrease by $r\%$: multiply by $(1 - r/100)$. e.g. -25% → $\times 0.75$.
  • Compound: +20% then -10% → $\times 1.20 \times 0.90 = \times 1.08$ (net +8%).
"4% increase" means multiply by 1.04, not 0.04. The "$1 +$" is the #1 percent trap.
RecognitionUsing 0.04 instead of 1.04 for a 4% increase multiplier.
P4 Reverse percent — find original value Drill 7
Desmos
  1. Type the equation: for "after 20% increase price is 60", type 1.20 * x = 60.
  2. Desmos draws a vertical line at the solution — compare to answer choices.
  3. Or just compute: 60 / 1.20 and expand the row to read the value (50).
Concept
  • Reverse percent rule: divide the new value by $(1 \pm r/100)$ to get the original.
  • e.g. "60 after 20% increase": original = 60 / 1.20 = 50.
  • e.g. "60 after 20% discount": original = 60 / 0.80 = 75.
  • Tax / tip: "54 total includes 8% tax" → pre-tax = 54 / 1.08 = 50.
Don't subtract the percent. 20% off 60 is NOT 60 − 12 = 48 to get the original. Original was 75, marked down to 60.
RecognitionSubtracting the percent off the new price instead of dividing by the multiplier.
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Problem-Solving & Data Analysis

Stats · Distribution
P5 Mean · median · stdev — Desmos list functions Drill 29
Desmos
  1. Type the data as a list with square brackets: L = [141, 143, 145, 151, 154, 156, 161, 162].
  2. New line: mean(L) → Desmos prints the mean.
  3. New line: median(L) → Desmos prints the median.
  4. Also available: stdev(L), total(L), count(L), quartile(L, 0.25).
  5. Visualize: boxplot(L) draws Min · Q1 · Median · Q3 · Max.
Lesson · Mean & Median →
Concept
  • Mean: sum of values / count. Sensitive to outliers.
  • Median: middle value when sorted. Robust to outliers.
  • Mode: most frequent value.
  • Range: max − min.
  • For tables / frequency: expand the list (a value with frequency 3 appears 3 times).
RecognitionForgetting frequency tables expand — treating each row as one data point.
P6 Distribution shape — skew, mean vs median, spread Drill 20
Desmos
  1. These are reading + reasoning questions. No Desmos shortcut.
  2. If verifying: enter the lists and compute mean(L1), mean(L2), stdev(L1), stdev(L2) — but the conceptual rule is faster.
Concept
  • Skewed left (long tail left) → Mean < Median.
  • Skewed right (long tail right) → Mean > Median.
  • Symmetric → Mean = Median.
  • Stdev = spread. More spread out → bigger stdev (like range).
  • Transforming data: add/subtract constant → mean shifts, stdev unchanged. Multiply by $k$ → both scale by $k$.
  • Outliers pull the mean much more than the median.
Same mean, more spread → larger stdev. The "wider" distribution always has higher stdev — even if mean is identical.
RecognitionPicking the taller distribution as larger stdev instead of the wider one.
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Problem-Solving & Data Analysis

Scatter regression · Probability
P7 Scatter plot best-fit — Desmos regression Drill 34
Desmos
  1. Click + → table. Enter all data points: $x_1$ column = x-values, $y_1$ column = y-values.
  2. New line: type y_1 ~ a x_1 + b (the squiggle ~ runs linear regression).
  3. Desmos prints a = slope and b = y-intercept. Reads off the trend line.
  4. For predictions: type a * (value) + b or click along the red regression line.
Lesson · Scatter Best-Fit →
Concept
  • Best-fit line minimizes total squared distance from points.
  • Slope = rate of change of the y-variable per unit x.
  • y-intercept = predicted value when x = 0 (may be physically meaningful or not).
  • Positive correlation: upward trend. Negative: downward.
  • No correlation: points scattered with no pattern.
  • Predictions outside the data range are unreliable (extrapolation).
RecognitionTrusting predictions far outside the data range (extrapolation).
P8 Probability — two-way table · Venn · conditional Drill 35
Desmos
  1. Don't open Desmos. These are reading + logic problems — Desmos can't read a table.
  2. Only use Desmos to convert the final fraction to a decimal, e.g. 34/52.
Concept
  • Basic: P(event) = favorable / total.
  • Two-way table: circle the row/column matching the condition. New denominator = that row/column total.
  • Union: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
  • "Neither": $P(\text{neither}) = 1 - P(A \cup B)$.
  • Conditional: $P(A \mid B) = \dfrac{n(A \cap B)}{n(B)}$. When question says "given that" or "if it is", use this.
"Given that" changes the denominator to the condition's total — not the grand total.
RecognitionUsing the grand total as denominator on a 'given that' conditional question.
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Problem-Solving & Data Analysis

Sampling · Margin of error · Confidence interval
P9 Sample stats · sampling generalization Drill 19
Desmos
  1. Type the proportion as arithmetic: e.g. for $6/20$ sample success in population of $50$, type (6/20) * 50.
  2. Expand the row to read the value (= 15).
Concept
  • Estimate total: sample rate $\times$ population, where sample rate = successes / sample size.
  • Generalization rule: conclusions are limited to the population that was randomly sampled. If you sampled from "all students at school X", you can ONLY generalize to school X — not other schools.
  • Random sampling is required to generalize beyond the sample.
  • Non-random / voluntary samples introduce bias — conclusions don't generalize.
  • Random assignment (different from random sampling) is needed to make causal claims.
Read the sampling frame carefully. "Random sample of 3 schools, 80 students each" generalizes to those 3 schools — NOT the whole district.
RecognitionGeneralizing beyond the sampled population (3 schools to whole district).
P10 Margin of error · confidence interval Drill 10
Desmos
  1. Mostly conceptual — Desmos has no role here.
  2. If you need the endpoint range, type 0.57 - 0.06 and 0.57 + 0.06 on separate lines, expand each row to read the values.
Concept
  • Confidence interval = estimate ± MOE. e.g. $0.57 \pm 0.06$ → range $[0.51, 0.63]$.
  • "95% confident" means: if we repeated the sampling many times, 95% of those intervals would contain the true value.
  • Larger sample → smaller MOE. More data tightens the estimate.
  • Higher confidence level → larger MOE. Being more confident requires a wider net.
  • Valid claims: "The true value is plausibly between $a$ and $b$" — NOT "$X\%$ of values fall in this range".
CI is about the parameter, not individual data points. "95% CI [0.51, 0.63]" means we're 95% confident the true proportion is in that range — not that 95% of voters are in it.
RecognitionReads a confidence interval as where 95% of individual values fall.
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