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point"],"",1],["s","degree-sign",["度（°）"],"30 度",["thirty degrees","a thirty-degree angle","degree sign"],"",1],["s","delta-lowercase",["ギリシャ文字 δ"],"デルタ",["delta","delta (lowercase)"],"",1],["s","delta-x",["Δx（x の増分）"],"x の増分",["delta x","change in x","delta x (change in x)"],"",1],["s","derivative-evaluated-at",["x = a での微分係数"],"x = a における dy/dx の値",["d y d x evaluated at x equals a","d y d x at x equals a","derivative evaluated at a point"],"",1],["s","derivative-leibniz",["導関数（ライプニッツの記法）"],"ディーワイ ディーエックス",["d y d x","d y over d x","derivative (Leibniz notation)"],"",1],["s","derivative-prime",["導関数（プライム記法）"],"エフ ダッシュ エックス",["f prime of x","f prime","derivative (prime notation)"],"",1],["s","determinant-notation",["行列式（det A、|A|）"],"行列式 A ／ デターミナント A",["the determinant of A","determinant"],"",1],["s","df-degrees-of-freedom",["自由度（df）"],"自由度",["degrees of freedom","degrees of freedom (df)"],"",1],["s","difference-quotient-limit",["導関数の定義の極限"],"リミット h → 0 h 分の f(x + h) − f(x)",["the limit as h approaches zero of f of x plus h minus f of x, all over h","the limit as h goes to zero of the quantity f of x plus h minus f of x over h","limit of the difference quotient"],"",1],["s","directional-derivative-notation",["方向微分"],"u 方向の方向微分",["the directional derivative of f in the direction of u","directional derivative"],"",1],["s","distributed-as-tilde",["確率変数が分布に従う（X ~ N(μ, σ)）"],"X は N(μ, σ) に従う",["X is normally distributed with mean mu and standard deviation sigma","X follows a normal distribution","distributed as (tilde)"],"",1],["s","divergence-del-dot",["発散（div）"],"ダイバージェンス F ／ ナブラ ドット F",["the divergence of F","div F","del dot F","divergence"],"",1],["s","divides-bar",["割り切る（|）"],"a は b を割り切る ／ a は b の約数",["a divides b","divides"],"",1],["s","division-sign",["わる（÷）"],"a わる b",["a divided by b","division sign"],"",1],["s","dne-abbreviation",["存在しない（DNE）"],"（極限が）存在しない",["does not exist","D N E","does not exist (DNE)"],"",1],["s","dot-product-dot",["内積（a⃗ · b⃗）"],"a ドット b ／ a と b の内積",["a dot b","a dotted with b","the dot product of a and b","dot product"],"",1],["s","double-integral-sign",["二重積分"],"二重積分 ∬ f(x, y) dA",["the double integral over R of f of x y d A","double integral"],"",1],["s","e-constant",["自然対数の底 e"],"e ／ ネイピア数",["the number e","the number e"],"",1],["s","e-to-the-x",["e の x 乗"],"e の x 乗",["e to the x","e to the power of x","e to the x power","e to the x"],"",1],["s","element-of-sign",["属する（∈）"],"a は A に属する ／ a は A の要素",["a is a member of A","a is an element of A","a belongs to A","element of"],"",1],["s","ellipsis",["…（省略記号）"],"…（てんてんてん）",["and so on","all the way up to","dot dot dot","ellipsis"],"",1],["s","empty-set-symbol",["空集合（∅）"],"空集合",["the empty set","empty set"],"",1],["s","epsilon",["ギリシャ文字 ε"],"イプシロン",["epsilon","epsilon"],"",1],["s","equals-question-mark",["検算の等号（≟）"],"（検算で使う等号）",["does three times two plus one equal seven","equals sign with a question mark"],"",1],["s","equals-sign",["等号（＝）"],"a イコール b ／ a は b に等しい",["a is equal to b","a equals b","equal sign"],"",1],["s","evaluation-bar",["定積分の計算の縦線"],"F(x) の a から b ／ [F(x)] a から b",["F of x evaluated from a to b","F of x evaluated at b minus F of x evaluated at a","evaluation bar"],"",1],["s","exp-function",["exp"],"エクスポネンシャル x",["exp of x","the exponential of x","exponential function (exp)"],"",1],["s","expected-value-e-of-x",["期待値（E(X)、μ_X）"],"X の期待値 ／ E(X)",["the expected value of X","the mean of X","expected value"],"",1],["s","expression-exponent",["指数が式のとき"],"2 の x+1 乗",["two to the x plus one","two to the x plus one power","two raised to the quantity x plus one","exponent that is an expression"],"",1],["s","factorial-n",["階乗（!）"],"n の階乗",["n factorial","factorial"],"",1],["s","floor-brackets",["床関数（⌊x⌋）"],"x の床関数",["the floor of x","floor function"],"",1],["s","for-all-quantifier",["すべての（∀）"],"すべての x について",["for all x","for every x","for each x","for all"],"",1],["s","fraction-a-over-b",["分数"],"b 分の a",["a over b","a divided by b","fraction"],"",1],["s","fractional-exponent",["分数の指数"],"a の n 分の 1 乗 ／ a の n 分の m 乗",["a to the one half","a to the m over n","rational exponent"],"",1],["s","function-f-of-x",["関数記号 f(x)"],"エフ エックス",["f of x","function notation"],"",1],["s","function-maps-arrow",["写像の矢印"],"f は A から B への写像 ／ x を f(x) にうつす",["f maps A to B","function arrow"],"",1],["s","gamma",["ギリシャ文字 γ"],"ガンマ",["gamma","gamma"],"",1],["s","gauss-bracket-jp",["ガウス記号"],"ガウス x ／ x を超えない最大の整数",["the greatest integer less than or equal to x","the floor of x","Gauss bracket (Japanese notation)"],"",1],["s","gcd-notation",["最大公約数"],"a と b の最大公約数",["the greatest common divisor of a and b","the greatest common factor of a and b","the GCF of a and b","the gcd of a and b","greatest common divisor"],"",1],["s","geq-sign",["以上（≧）"],"a 大なりイコール b ／ a は b 以上",["a is greater than or equal to b","a is at least b","greater-than-or-equal-to sign"],"",1],["s","greater-than-sign",["大なり（>）"],"a 大なり b ／ a は b より大きい",["a is greater than b","greater-than sign"],"",1],["s","h-naught-null",["帰無仮説（H₀）"],"帰無仮説 H₀ ／ エイチ ゼロ",["the null hypothesis","H sub zero","null hypothesis (H-naught)"],"",1],["s","h-sub-a-alternative",["対立仮説（Hₐ、H₁）"],"対立仮説 H₁ ／ エイチ ワン",["the alternative hypothesis","H sub one","H sub a","alternative hypothesis"],"",1],["s","half-open-interval",["半開区間"],"a 以上 b 未満の区間",["the half-open interval from a to b","from a to b, including a but not b","half-open interval"],"",1],["s","hyperbolic-functions-notation",["双曲線関数"],"ハイパボリックサイン x ／ ハイパボリックコサイン x",["cosh x","hyperbolic sine of x","cinch x","hyperbolic functions"],"",1],["s","identity-matrix-i",["単位行列（I、Iₙ、E）"],"単位行列 E",["the identity","the identity matrix","identity matrix"],"",1],["s","iff-arrow",["同値（⇔）"],"p と q は同値 ／ p ⇔ q",["p if and only if q","p is equivalent to q","if and only if"],"",1],["s","imaginary-unit-i",["虚数単位（i）"],"アイ ／ 虚数単位 i",["i","the imaginary unit","imaginary unit"],"",1],["s","implies-arrow",["ならば（⇒）"],"p ならば q",["p implies q","implies"],"",1],["s","indeterminate-forms",["不定形"],"0 分の 0 ／ ∞ 分の ∞",["zero over zero","infinity over infinity","indeterminate forms"],"",1],["s","infinity-symbol",["無限大（∞）"],"無限大 ／ マイナス無限大",["infinity","negative infinity","minus infinity","infinity"],"",1],["s","integers-symbol",["整数全体の集合（ℤ）"],"整数全体の集合 Z",["the integers","integers"],"",1],["s","integral-definite",["定積分"],"インテグラル a から b、f(x) dx ／ f(x) の a から b までの定積分",["the integral from a to b of f of x d x","the integral of f of x from a to b","definite integral"],"",1],["s","integral-indefinite",["不定積分"],"インテグラル f(x) dx ／ f(x) の不定積分",["the integral of f of x d x","the indefinite integral of f of x","the integral of f of x with respect to x","indefinite integral"],"",1],["s","intersection-sign",["共通部分（∩）"],"A キャップ B ／ A と B の共通部分",["A intersect B","the intersection of A and B","intersection"],"",1],["s","inverse-cosine-notation",["逆余弦（アークコサイン）"],"アークコサイン x",["arccosine of x","inverse cosine of x","cosine inverse of x","inverse cosine"],"",1],["s","inverse-function-notation",["逆関数の記号"],"f インバース x",["f inverse","f inverse of x","the inverse of f","inverse function notation"],"",1],["s","inverse-sine-notation",["逆正弦（アークサイン）"],"アークサイン x",["arcsine of x","sine inverse of x","inverse sine of x","inverse sine"],"",1],["s","inverse-tangent-notation",["逆正接（アークタンジェント）"],"アークタンジェント x",["arctangent of x","inverse tangent of x","tangent inverse of x","inverse tangent"],"",1],["s","iqr-abbreviation",["四分位範囲（IQR）"],"四分位範囲",["the interquartile range","I Q R","interquartile range (IQR)"],"",1],["s","jacobian-notation",["ヤコビアン"],"ヤコビアン",["the Jacobian of x and y with respect to u and v","Jacobian"],"",1],["s","lambda",["ギリシャ文字 λ"],"ラムダ",["lambda","lambda"],"",1],["s","lcm-notation",["最小公倍数"],"a と b の最小公倍数",["the least common multiple of a and b","the lcm of a and b","least common multiple"],"",1],["s","length-ab",["線分の長さ（AB）"],"線分 AB の長さ ／ AB",["the length of segment AB","the length of AB","length of a segment"],"",1],["s","leq-sign",["以下（≦）"],"a 小なりイコール b ／ a は b 以下",["a is less than or equal to b","a is at most b","less-than-or-equal-to sign"],"",1],["s","less-than-sign",["小なり（<）"],"a 小なり b ／ a は b より小さい ／ a は b 未満",["a is less than b","less-than sign"],"",1],["s","limit-from-left",["左側極限"],"x が a に左から近づくときの極限 ／ リミット x → a マイナス 0",["the limit as x approaches a from the left of f of x","the left-hand limit of f at a","limit from the left"],"",1],["s","limit-from-right",["右側極限"],"x が a に右から近づくときの極限 ／ リミット x → a プラス 0",["the limit as x approaches a from the right of f of x","the right-hand limit of f at a","limit from the right"],"",1],["s","limit-x-to-a",["極限（lim）"],"リミット x → a f(x) ／ x が a に限りなく近づくときの f(x) の極限値",["the limit as x approaches a of f of x","the limit of f of x as x approaches a","the limit as x goes to a of f of x","limit"],"",1],["s","line-ab-arrow",["直線の記号（両矢印の AB）"],"直線 AB",["line AB","line notation"],"",1],["s","line-integral-sign",["線積分"],"線積分",["the integral over C of F dot d r","the integral around C","the line integral over C of F dot d r","the line integral of F along C","line integral"],"",1],["s","log-base-a",["a を底とする対数"],"a を底とする x の対数",["log base a of x","the logarithm of x to the base a","logarithm with base a"],"",1],["s","log-e-jp",["数III の log x（底 e）"],"ログ x（自然対数）",["the natural log of x","log base e of x","natural log written log x (Japanese 数III)"],"",1],["s","logical-and-wedge",["かつ（∧）"],"p かつ q",["p and q","logical and"],"",1],["s","logical-or-vee",["または（∨）"],"p または q",["p or q","logical or"],"",1],["s","long-division-bracket",["割り算の筆算"],"20 ÷ 5 の筆算",["twenty divided by five","five goes into twenty","long division bracket"],"",1],["s","matrix-brackets",["行列（[ ] と ( )）"],"行列 a b c d",["the matrix a, b, c, d","the two-by-two matrix with entries a, b, c, d","matrix (brackets)"],"",1],["s","matrix-dimensions-by",["行列の型（m × n）"],"2 行 3 列の行列 ／ m 行 n 列",["a two-by-three matrix","a matrix with two rows and three columns","matrix dimensions"],"",1],["s","matrix-entry-aij",["行列の (i, j) 成分（aᵢⱼ）"],"a i j ／ (i, j) 成分",["the entry in row i, column j","the i j entry of A","matrix entry"],"",1],["s","matrix-inverse-notation",["逆行列（A⁻¹）"],"A インバース ／ A の逆行列",["A inverse","the inverse of A","inverse matrix"],"",1],["s","max-min-notation",["max・min の記号"],"a と b の大きい方",["the maximum of a and b","max and min notation"],"",1],["s","measure-of-angle",["角の大きさ（m∠ABC）"],"角 ABC の大きさ",["the measure of angle ABC","measure of an angle"],"",1],["s","minus-plus-sign",["複号（∓）"],"マイナスプラス（複号同順）",["minus or plus","minus-plus sign"],"",1],["s","minus-sign",["ひく（−）"],"a ひく b ／ a マイナス b",["a minus b","minus sign"],"",1],["s","mixed-number",["帯分数"],"2 と 4 分の 3",["two and three fourths","two and three quarters","mixed number"],"",1],["s","mu-population-mean",["母平均（μ、m）"],"ミュー ／ 母平均 m",["mu","the population mean","population mean (mu)"],"",1],["s","nabla",["ナブラ"],"ナブラ エフ",["del f","the gradient of f","del (nabla)"],"",1],["s","natural-log-ln",["自然対数（ln）"],"x の自然対数",["the natural log of x","l n x","natural logarithm"],"",1],["s","natural-numbers-symbol",["自然数全体の集合（ℕ）"],"自然数全体の集合 N",["the natural numbers","natural numbers"],"",1],["s","negation-sign",["否定"],"p でない ／ p バー ／ p の否定",["not p","the negation of p","negation"],"",1],["s","negative-exponent-power",["負の指数"],"x のマイナス 1 乗 ／ x のマイナス n 乗",["x to the negative one","x to the negative first power","negative exponent"],"",1],["s","negative-sign",["負の符号（−）"],"マイナス 3 ／ 負の 3 ／ マイナス x",["negative three","minus three","the opposite of x","negative sign"],"",1],["s","normal-distribution-n",["正規分布の記号（N(μ, σ)）"],"正規分布 N(m, σ²)",["normal with mean mu and standard deviation sigma","normal distribution notation"],"",1],["s","not-element-of-sign",["属さない（∉）"],"a は A に属さない",["a is not an element of A","a is not a member of A","not an element of"],"",1],["s","not-equal-sign",["等しくない（≠）"],"a は b に等しくない",["a does not equal b","a is not equal to b","not-equal sign"],"",1],["s","nth-derivative-notation",["第 n 次導関数"],"f の n 次導関数",["the nth derivative of f","f triple prime of x","nth derivative (higher-order derivatives)"],"",1],["s","nth-root-radical",["n 乗根"],"n 乗根 x ／ x の n 乗根",["the nth root of x","nth root of x","nth root"],"",1],["s","numeric-fraction",["数の分数"],"4 分の 3 ／ 2 分の 1",["one half","three over four","three fourths","three quarters","numerical fraction"],"",1],["s","omega",["ギリシャ文字 ω"],"オメガ",["omega","omega"],"",1],["s","open-interval-parentheses",["開区間"],"開区間 a b ／ a より大きく b より小さい",["the open interval from a to b","open interval"],"",1],["s","ordered-pair",["座標（(x, y)）"],"点 (x, y) ／ 点 P(3, 4)",["the point three comma four","the ordered pair x, y","ordered pair"],"",1],["s","p-hat-sample-proportion",["標本比率（p̂）"],"p ハット ／ 標本比率",["p hat","the sample proportion","sample proportion (p-hat)"],"",1],["s","parallel-sign",["平行の記号（∥ ／ //）"],"AB 平行 CD ／ AB と CD は平行",["AB is parallel to CD","parallel sign"],"",1],["s","parallelogram-abcd",["平行四辺形の記号（▱ABCD）"],"平行四辺形 ABCD",["parallelogram ABCD","parallelogram notation"],"",1],["s","parenthesized-quantity",["かっこでくくった式"],"かっこ a たす b の 2 乗",["the quantity a plus b, squared","a plus b, all squared","the quantity (parentheses read aloud)"],"",1],["s","partial-derivative-leibniz",["偏微分（∂）"],"ラウンド f ラウンド x ／ デル f デル x",["the partial derivative of f with respect to x","the partial of f with respect to x","partial f partial x","del f del x","partial derivative"],"",1],["s","partial-derivative-subscript",["偏導関数の添字記法"],"f x（偏導関数）",["f sub x","partial derivative (subscript notation)"],"",1],["s","percent-sign",["パーセント（%）"],"25 パーセント",["twenty-five percent","percent sign"],"",1],["s","permutation-npr",["順列の数（ₙPᵣ）"],"エヌ ピー アール",["the number of permutations of n things taken r at a time","n P r","number of permutations (nPr)"],"",1],["s","perpendicular-sign",["垂直の記号（⊥）"],"AB 垂直 CD ／ AB と CD は垂直",["AB is perpendicular to CD","perpendicular sign"],"",1],["s","phi",["ギリシャ文字 φ"],"ファイ",["phi","phi"],"",1],["s","pi-symbol",["円周率（π）"],"パイ ／ 円周率",["pi","pi"],"",1],["s","piecewise-brace",["場合分けで定義した関数"],"f(x) は x ≥ 0 のとき x²、x < 0 のとき −x",["f of x equals x squared when x is greater than or equal to zero, and negative x when x is less than zero","x squared if x is greater than or equal to zero","x squared for x greater than or equal to zero","piecewise-defined function"],"",1],["s","plus-c-constant",["積分定数（+ C）"],"プラス C（C は積分定数）",["plus C","plus a constant","constant of integration"],"",1],["s","plus-minus-sign",["複号（±）"],"プラスマイナス",["plus or minus","plus-minus sign"],"",1],["s","plus-sign",["たす（＋）"],"a たす b ／ a プラス b",["a plus b","plus sign"],"",1],["s","polar-form-cis",["複素数の極形式（r(cos θ + i sin θ)、r cis θ）"],"極形式 r(cos θ + i sin θ)",["r cis theta","r times cosine theta plus i sine theta","polar form of a complex number"],"",1],["s","power-cubed",["3 乗"],"x の 3 乗",["x cubed","x to the third","x to the third power","cube (third power)"],"",1],["s","power-fourth",["4 乗"],"x の 4 乗",["x to the fourth","x to the fourth power","x to the power of four","fourth power"],"",1],["s","power-n",["n 乗"],"x の n 乗",["x to the n","x to the nth power","x to the nth","x to the power of n","nth power"],"",1],["s","power-set-notation",["べき集合"],"A のべき集合",["the power set of A","power set"],"",1],["s","power-squared",["2 乗"],"x の 2 乗",["x squared","x to the second power","square (second power)"],"",1],["s","prime-label-image",["移動後の点（A′、A″）"],"A ダッシュ ／ A ツーダッシュ",["A prime","prime (image of a point)"],"",1],["s","probability-of-a",["事象 A の確率（P(A)）"],"A の確率 ／ P(A)",["the probability of A","P of A","probability of an event"],"",1],["s","probability-of-intersection",["積事象の確率（P(A ∩ B)）"],"A かつ B の確率 ／ P(A キャップ B)",["the probability of A and B","the probability of A intersect B","probability of A and B"],"",1],["s","probability-of-union",["和事象の確率（P(A ∪ B)）"],"A または B の確率 ／ P(A カップ B)",["the probability of A or B","the probability of A union B","probability of A or B"],"",1],["s","probability-x-equals-k",["確率変数が値をとる確率（P(X = k)）"],"X = k となる確率",["the probability that X equals k","P of X equals k","probability that X equals k"],"",1],["s","product-pi-notation",["総乗（∏）"],"パイ（総乗）、i = 1 から n まで a i",["the product from i equals one to n of a sub i","product notation"],"",1],["s","projection-notation",["正射影ベクトル（proj_v u）"],"u の v への正射影ベクトル",["the projection of u onto v","vector projection"],"",1],["s","proper-subset-sign",["真部分集合（⊊）"],"A は B の真部分集合",["A is a proper subset of B","proper subset"],"",1],["s","proportion-colon",["比例式"],"a 対 b イコール c 対 d",["a is to b as c is to d","proportion"],"",1],["s","qed-end-of-proof",["証明終わりの印"],"証明終わり",["end of proof","Q.E.D.","end-of-proof mark"],"",1],["s","quartile-q1-q3",["四分位数（Q₁、Q₃）"],"第 1 四分位数 Q₁",["Q one","the first quartile","quartiles (Q1, Q3)"],"",1],["s","r-squared",["決定係数（r²）"],"r の 2 乗 ／ 決定係数",["r squared","the coefficient of determination","coefficient of determination (r-squared)"],"",1],["s","radian-unit",["ラジアン"],"2 分の π ラジアン",["pi over two radians","radian"],"",1],["s","ratio-colon",["比（:）"],"a 対 b",["the ratio of a to b","ratio"],"",1],["s","rational-expression-fraction",["式の分数"],"x − 1 分の x + 1",["x plus one, all over x minus one","the quantity x plus one over the quantity x minus one","fraction with expressions"],"",1],["s","rationals-symbol",["有理数全体の集合（ℚ）"],"有理数全体の集合 Q",["the rational numbers","rational numbers"],"",1],["s","ray-ab-arrow",["半直線の記号（矢印の AB）"],"半直線 AB",["ray AB","ray notation"],"",1],["s","reals-symbol",["実数全体の集合（ℝ）"],"実数全体の集合 R ／ R 2",["the real numbers","the reals","real numbers"],"",1],["s","repeated-combination-h-jp",["重複組合せの記号（ₙHᵣ）"],"エヌ エイチ アール ／ 重複組合せの数",["the number of combinations with repetition of n things taken r at a time","combinations with repetition (Japanese notation)"],"",1],["s","repeating-decimal-bar",["循環小数の上線"],"れい てん さん さん …",["zero point three repeating","zero point three with a bar over the three","repeating decimal bar"],"",1],["s","rho",["ギリシャ文字 ρ"],"ロー",["rho","rho"],"",1],["s","roster-braces",["要素を列挙して表す集合"],"1, 2, 3 を要素とする集合",["the set containing one, two, and three","the set one, two, three","roster notation"],"",1],["s","s-sample-sd",["標本標準偏差（s、sₓ）"],"s ／ 標本標準偏差",["the sample standard deviation","sample standard deviation"],"",1],["s","scientific-notation-form",["a × 10ⁿ の形"],"3.2 かける 10 の 5 乗",["three point two times ten to the fifth","scientific notation"],"",1],["s","se-standard-error",["標準誤差（SE）"],"標準誤差",["the standard error","standard error (SE)"],"",1],["s","secant-of-theta",["セカント"],"セカント シータ",["secant theta","secant of theta","sec theta","secant"],"",1],["s","second-derivative-leibniz",["第 2 次導関数（ライプニッツの記法）"],"ディー 2 ワイ ディーエックス 2 乗",["the second derivative of y with respect to x","d squared y d x squared","second derivative (Leibniz notation)"],"",1],["s","segment-ab-overline",["線分の記号（上線の AB）"],"線分 AB",["segment AB","line segment AB","line segment notation"],"",1],["s","sequence-braces",["数列の記号"],"数列 {aₙ} ／ 数列 a n",["the sequence a sub n","the sequence a n","sequence notation"],"",1],["s","set-builder-braces",["条件で表す集合（内包的記法）"],"x > 0 を満たす x 全体の集合",["the set of all x such that x is greater than zero","set-builder notation"],"",1],["s","set-difference",["差集合"],"A から B を除いた集合",["the set difference of A and B","set difference"],"",1],["s","similar-sign",["相似の記号（日本は ∽、米国は ∼）"],"三角形 ABC 相似 三角形 DEF",["triangle ABC is similar to triangle DEF","similarity sign"],"",1],["s","sine-of-theta",["サイン"],"サイン シータ",["sine of theta","sine theta","the sine of theta","sine"],"",1],["s","sine-squared",["sin² θ"],"サイン 2 乗 シータ",["sine squared theta","sine squared of theta","sine of theta, squared","sine squared"],"",1],["s","square-brackets",["角かっこ（[ ]）"],"大かっこ ／ ブラケット",["brackets","square brackets","brackets"],"",1],["s","square-root",["平方根（ルート）"],"ルート x の 2 乗 プラス 1",["the square root of x squared plus one","root x squared plus one","square root"],"",1],["s","standard-deviation-sigma",["標準偏差（σ、σ(X)、σ_X）"],"シグマ ／ X の標準偏差",["the standard deviation of X","sigma","standard deviation (sigma)"],"",1],["s","subset-sign",["部分集合（⊆）"],"A は B の部分集合 ／ A は B に含まれる",["A is a subset of B","A is contained in B","subset"],"",1],["s","summation-sigma",["総和（シグマ）"],"シグマ、k イコール 1 から n まで、a k",["the sum from k equals one to n of a sub k","the sum of a k, k from one to n","summation"],"",1],["s","tangent-of-theta",["タンジェント"],"タンジェント シータ",["tangent theta","tangent of theta","tan theta","the tangent of theta","tangent"],"",1],["s","there-exists-quantifier",["存在する（∃）"],"ある x が存在して",["there exists an x","there is an x such that","there exists"],"",1],["s","therefore-sign",["ゆえに（∴）"],"ゆえに",["therefore","therefore sign"],"",1],["s","theta",["θ（角を表す文字）"],"シータ",["theta","theta (a letter for an angle)"],"",1],["s","times-sign",["かける（× ・）"],"a かける b",["a times b","a multiplied by b","multiplication sign"],"",1],["s","transpose-notation",["転置行列（Aᵀ、ᵗA）"],"A の転置 ／ A トランスポーズ ／ t A",["A transpose","the transpose of A","transpose"],"",1],["s","triangle-abc",["三角形の記号（△ABC）"],"三角形 ABC",["triangle ABC","triangle sign"],"",1],["s","triple-integral-sign",["三重積分"],"三重積分",["the triple integral over E of f d V","the triple integral of f over E","triple integral"],"",1],["s","union-sign",["和集合（∪）"],"A カップ B ／ A と B の和集合",["A union B","the union of A and B","union"],"",1],["s","unit-vector-hat",["単位ベクトルのハット"],"i ハット ／ 単位ベクトル",["i hat","unit vector hat"],"",1],["s","variance-of-x",["分散（V(X)、Var(X)、σ²）"],"X の分散 ／ V(X)",["the variance of X","sigma squared","variance"],"",1],["s","vector-ab-arrow",["ベクトル AB"],"ベクトル AB",["vector AB","the vector from A to B","vector from A to B"],"",1],["s","vector-arrow-notation",["ベクトルの記号"],"ベクトル a",["vector v","the vector v","v vector","vector notation"],"",1],["s","vector-magnitude-bars",["ベクトルの大きさ（‖v‖、|a⃗|）"],"ベクトル a の大きさ",["the magnitude of v","the length of v","the norm of v","magnitude of a vector"],"",1],["s","wave-dash-range-jp",["範囲の〜"],"3 から 5",["from three to five","between three and five","range written with a wave dash (Japanese)"],"",1],["s","x-bar-sample-mean",["標本平均（x̄、X̄）"],"x バー ／ 標本平均 X̄",["the sample mean","x bar","sample mean (x-bar)"],"",1],["s","x-naught",["添字 0"],"x ゼロ ／ x 1",["x sub zero","x naught","x zero","x naught (subscript zero)"],"",1],["s","y-hat-predicted-value",["回帰直線による予測値（ŷ）"],"y ハット ／ 予測値",["y hat","predicted value (y-hat)"],"",1],["s","z-star-critical-value",["臨界値（z*、t*）"],"臨界値 z",["z star","the critical value","critical value (z-star)"],"",1],["p","add-up-thin-disks",["回転体の体積を、薄く切った円板の体積を足し合わせるものとして説明する","立体を薄く切って、1 枚 1 枚の体積を足し合わせるイメージです。"],"",["Think of it as cutting the solid into a bunch of thin slices and adding up their volumes.","We're adding up a bunch of thin disks."],"class-listening",1],["p","apply-the-theorem",["定理を使う場面だと伝える","ここで平均値の定理が使えます。"],"",["Now we can use the mean value theorem.","Now we can apply the intermediate value theorem."],"class-listening",1],["p","are-equal-respectively",["対応する辺がそれぞれ等しいと書く","3 組の辺がそれぞれ等しい。"],"",["AB, BC, and CA are equal to DE, EF, and FD, respectively.","AB = DE, BC = EF, and CA = FD."],"written-solution",1],["p","area-is-never-negative",["面積は負にならないこと、負になったら計算を見直すことを伝える","面積は負になりません。"],"",["Area is never negative."],"class-listening",1],["p","assume-it-holds-for-n-k",["数学的帰納法で、n = k のとき成り立つと仮定する","n = k のとき成り立つと仮定する。"],"",["Inductive hypothesis: assume the statement is true for n = k.","Assume the statement is true for some k ≥ 1."],"written-solution",1],["p","by-definition",["定義から直接言えることを書く","定義より、"],"",["By definition,","By the definition of the derivative,"],"written-solution",1],["p","by-hypothesis",["仮定（与えられた条件）を理由として書く","仮定より AB = CD である。"],"",["By assumption, AB = CD.","By hypothesis, AB = CD.","Since AB = CD is given,"],"written-solution",1],["p","check-by-differentiating",["不定積分の答えは微分して確かめられると伝える","答えは微分して確かめられます。"],"",["You can always check your answer by taking the derivative.","Take the derivative, and you should get back the integrand."],"class-listening",1],["p","check-the-sign",["各区間で f′ の符号を調べる","各区間で f′ の符号を調べましょう。"],"",["Let's check the sign of f prime on each interval.","Check the sign."],"class-listening",1],["p","class-asking-another-example",["例題をもう 1 つやってほしいと頼む","例をもう 1 つやってもらえますか。"],"",["Could you do another example?","Can you show us one more?","Could we do one more example?"],"class-asking",1],["p","class-asking-calculator-on-the-test",["テストで電卓を使えるかを聞く","次のテストで電卓は使えますか。"],"",["Can we use a calculator on the test?","Is the test calculator or non-calculator?"],"class-asking",1],["p","class-asking-can-i-write-it-this-way",["自分の書き方でもよいか確かめる","こう書いても大丈夫ですか。"],"",["Is it okay if I write it like this?","I wrote it as 2(x + 1) — is that okay?","Can I just write it as 2x + 2?"],"class-asking",1],["p","class-asking-could-we-use-another-method",["先生と違う方法（別の公式や定理）を使ってもよいかを聞く","ここで解の公式を使ってはだめですか。"],"",["Couldn't we just use the quadratic formula?","Can I use L'Hôpital's rule here instead?"],"class-asking",1],["p","class-asking-difference-between",["2 つの用語・概念の違いを聞く","極大値と最大値の違いは何ですか。"],"",["What's the difference between a local max and an absolute max?","How is this different from a local max?"],"class-asking",1],["p","class-asking-do-we-need-to-memorize",["公式を覚える必要があるか（公式集が配られるか）を聞く","この公式は覚える必要がありますか。"],"",["Do we need to memorize this formula?","Will we get a formula sheet?"],"class-asking",1],["p","class-asking-does-it-still-work-if",["条件を変えても成り立つかを聞く","x が負だったらどうなりますか。"],"",["What if x is negative?","Does this still work if x is negative?"],"class-asking",1],["p","class-asking-go-back",["説明が先に進んでしまったので、前の画面に戻ってほしい","前のスライドに戻ってもらえますか。"],"",["Could you go back to the previous slide?"],"class-asking",1],["p","class-asking-how-do-you-read-this",["記号の読み方を聞く","この記号は何と読むんですか。"],"",["How do you say this symbol?"],"class-asking",1],["p","class-asking-i-got-a-different-answer",["自分の答えが先生の答えと違ったことを伝え、どこが違うか聞く","違う答えになったんですが。"],"",["I got a different answer.","What am I doing wrong?"],"class-asking",1],["p","class-asking-is-there-a-name-for-this",["それの名前（英語で何と呼ぶか）を聞く","これって何て言うんですか。"],"",["What's this called?","What do you call this?"],"class-asking",1],["p","class-asking-is-there-an-easier-way",["もっと楽な解き方がないかを聞く","もっと楽な解き方はありますか。"],"",["Is there an easier way to do this?","Is there a shortcut?"],"class-asking",1],["p","class-asking-repeat",["聞き取れなかったので、もう一度言ってほしい","すみません、もう一度言っていただけますか。"],"",["Sorry, could you say that again?","Could you repeat that last part?","Sorry, what did you say?"],"class-asking",1],["p","class-asking-simplify-further",["答えをさらに簡単にする必要があるか、これで最終的な答えかを聞く","これ以上簡単にする必要はありますか。"],"",["Do we need to simplify this further?","Is that as simple as it gets?"],"class-asking",1],["p","class-asking-slow-down",["説明が速すぎるので、ゆっくり話してほしい","もう少しゆっくりお願いできますか。"],"",["Could you slow down a little?"],"class-asking",1],["p","class-asking-typo-on-the-board",["板書の誤りかもしれない箇所を確かめる","そこ、マイナスじゃないですか。"],"",["Should that be a minus?","Is that a typo?"],"class-asking",1],["p","class-asking-when-is-it-due",["課題の締め切りを確かめる","提出はいつまでですか。"],"",["When is this due?","Is that due Friday or Monday?"],"class-asking",1],["p","class-asking-where-did-that-come-from",["板書の式がどこから来たのか・どう変形したのかを聞く","この行から次の行へは、どうやって変形するんですか。"],"",["How do you get from this line to that one?","Where did the 2 come from?"],"class-asking",1],["p","class-asking-which-problems",["宿題の範囲（どの問題か）を確かめる","宿題はどの問題ですか。"],"",["Which problems are we supposed to do?","Was that odds only?"],"class-asking",1],["p","class-asking-why-can-we",["理由を聞く（その操作をしてよい理由など）","それはどうしてですか。"],"",["Why is that?","How come we can divide by x here?"],"class-asking",1],["p","class-asking-will-this-be-on-the-test",["今の内容がテストの範囲かを聞く","これ、試験で必要ですか。"],"",["Do we need to know this for the exam?","Will this be on the test?"],"class-asking",1],["p","class-listening-answers-in-the-back",["奇数番の答えは教科書の巻末にあると伝える","奇数番の答えは教科書の後ろに載っています。"],"",["The answers to the odd-numbered problems are in the back of the book."],"class-listening",1],["p","class-listening-common-mistake",["よくある間違いに注意させる","これは本当によくある間違いです。"],"",["This is a really common mistake.","Watch out — this is where people lose points."],"class-listening",1],["p","class-listening-does-that-make-sense",["説明が伝わったかをクラスに確かめる","次に進む前に、何か質問はありますか？"],"",["Any questions before we move on?","Does that make sense?"],"class-listening",1],["p","class-listening-extra-credit",["加点の問題だと伝える","これは加点問題です。"],"",["This one's for extra credit.","This is a bonus question."],"class-listening",1],["p","class-listening-final-is-cumulative",["期末試験が全範囲から出ると伝える","期末試験は全範囲です。"],"",["The final is cumulative.","The final covers everything."],"class-listening",1],["p","class-listening-homework-is",["宿題の範囲を伝える","宿題は 3.2 節の 1〜25 の奇数番です。"],"",["For homework, do section 3.2, problems 1 through 25, odds.","Tonight's homework is 3.2, odd problems 1 to 25."],"class-listening",1],["p","class-listening-in-your-own-words",["なぜそうなるかを自分の言葉で説明させる","なぜそうなるのか説明できますか？"],"",["Can you explain why that works?","Explain it in your own words."],"class-listening",1],["p","class-listening-its-due",["提出の締め切りを伝える","提出は金曜日です。"],"",["It's due Friday.","Turn it in at the start of class on Friday."],"class-listening",1],["p","class-listening-lets-go-over-the-homework",["宿題の答え合わせをする","宿題の答え合わせをしましょう。"],"",["Let's go over the homework."],"class-listening",1],["p","class-listening-lowest-quiz-dropped",["小テストの最低点を成績から除くと伝える","小テストの一番低い点は成績に入れません。"],"",["Your lowest quiz score will be dropped."],"class-listening",1],["p","class-listening-notice-that",["注目すべき点を示す","2 つの三角形が 1 辺を共有していることに注目してください。"],"",["Notice that the two triangles share a side.","The key thing to see here is that they share a side."],"class-listening",1],["p","class-listening-office-hours-are",["オフィスアワーの時間を伝える","オフィスアワーは火曜の 2 時から 4 時です。"],"",["My office hours are Tuesdays from 2 to 4.","I have office hours on Tuesdays from 2 to 4."],"class-listening",1],["p","class-listening-oops-good-catch",["板書の誤りを認めて直す","おっと、間違えました。ここはマイナスですね。"],"",["Oops, my mistake — that should be a minus.","Sorry, my bad — that should be a minus.","Good catch, thank you."],"class-listening",1],["p","class-listening-pass-your-papers-forward",["答案・プリントを前に回させる","答案（プリント）を前に回してください。"],"",["Pass your papers forward."],"class-listening",1],["p","class-listening-raise-your-hand-if",["手を挙げさせて答えを確かめる","5 になった人は手を挙げてください。"],"",["Raise your hand if you got 5.","Thumbs up if you got 5."],"class-listening",1],["p","class-listening-recall-that",["前に習ったことを思い出させる","sin x の導関数は cos x でしたね。"],"",["Remember that the derivative of sin x is cos x.","Recall that the derivative of sin x is cos x."],"class-listening",1],["p","class-listening-same-idea-as-before",["前の問題と同じ考え方だと伝える","さっきと同じ考え方です。"],"",["This is the same idea as before.","It's just like the last problem."],"class-listening",1],["p","class-listening-sanity-check",["答えが妥当かを確かめさせる","答えが妥当か、さっと確かめてみましょう。"],"",["Let's do a quick sanity check.","Does this answer make sense?"],"class-listening",1],["p","class-listening-take-out-a-sheet-of-paper",["小テストのために紙を出させる","紙を 1 枚出してください。"],"",["Take out a sheet of paper."],"class-listening",1],["p","class-listening-this-will-be-on-the-test",["試験に出ると予告する","ここはテストに出ます。"],"",["This will be on the exam.","This will be on the test."],"class-listening",1],["p","class-listening-try-this-one",["例題を自分で解かせる","この問題をやってみてください。少し時間をとります。"],"",["Go ahead and try this one — I'll give you a couple of minutes."],"class-listening",1],["p","class-listening-turn-to-page",["教科書の指定のページを開かせる","45 ページを開いてください。"],"",["Turn to page 45.","Open your books to page 45."],"class-listening",1],["p","class-listening-warm-up",["授業の始めに短い問題をさせる","まずは軽いウォームアップから始めましょう。"],"",["Let's start with a quick warm-up."],"class-listening",1],["p","class-listening-well-come-back-to-this",["今は深入りせず、後で扱うと伝える","これは後でまた扱います。"],"",["We'll come back to this later."],"class-listening",1],["p","class-listening-what-do-you-notice",["気づいたことを言わせる","何か気づいたことはありますか？"],"",["What do you notice?"],"class-listening",1],["p","class-listening-whats-the-next-step",["次の手順をクラスに考えさせる","次はどうしますか？"],"",["What's the next step?","What do we do now? Anybody?"],"class-listening",1],["p","class-listening-who-wants-to-come-up",["黒板で解く人を募る","誰か前に出て解いてくれる人はいますか？"],"",["Can I get a volunteer?","Who wants to come up and do this one?"],"class-listening",1],["p","class-listening-work-with-a-partner",["隣の人と答えを比べさせる・ペアで取り組ませる","ペアで取り組んでください。"],"",["Work with a partner.","Turn to your neighbor and compare."],"class-listening",1],["p","class-listening-write-this-down",["ノートに書いておくよう指示する","これはノートに書いておいてください。"],"",["Write this down.","You'll want this in your notes."],"class-listening",1],["p","class-listening-you-dont-need-to-memorize",["公式集が配られるので暗記しなくてよいと伝える","公式集が配られるので、これは覚えなくていいです。"],"",["You'll get a formula sheet, so you don't need to memorize this.","You don't have to memorize this one."],"class-listening",1],["p","clearly",["自明なことを「明らかに」と書く","明らかに f(0) = 1 である。"],"",["Clearly, f(0) = 1.","It is clear that f(0) = 1."],"written-solution",1],["p","continuous-but-not-differentiable",["連続だが微分できない点があることを示す","x = 0 で連続ですが、微分はできません。"],"",["It's continuous at 0, but it's not differentiable there.","It's continuous, but there's a sharp corner."],"class-listening",1],["p","conversely",["逆向きの主張を示すときに「逆に」と書く","逆に、a² + b² = c² ならば、その三角形は直角三角形である。"],"",["Conversely, if a² + b² = c², then the triangle is a right triangle.","Now for the other direction,"],"written-solution",1],["p","differentiate-the-outside-first",["合成関数の微分で、まず外側の関数を微分する（中身はそのまま）","まず外側を微分します。"],"",["Take the derivative of the outside first.","Differentiate the outside and leave the inside alone."],"class-listening",1],["p","discord-anyone-get",["クラスのチャットで、ある問題が解けた人がいるか聞く","3 番解けた人いる？"],"",["Did anyone get #3?","anyone figure out 3?"],"discord",1],["p","discord-can-someone-explain",["クラスのチャットで説明を頼む","ここで極限が 0 になる理由、誰か説明してくれない？"],"",["Can someone explain why the limit is 0 here?"],"discord",1],["p","discord-due-tonight",["締め切りを確認する","課題って今夜締め切り？"],"",["Is the homework due tonight?"],"discord",1],["p","discord-here-is-my-work",["自分の途中式を貼って、どこで間違えたか聞く","途中までの計算貼るね。どこで間違えた？"],"",["Here's my work so far — where did I go wrong?","Where did I go wrong?"],"discord",1],["p","discord-hint-no-spoilers",["答えではなくヒントだけを頼む","ヒントだけちょうだい。答えは書かないで。"],"",["Can someone give me a hint? No full solutions pls.","Just a nudge please, no spoilers."],"discord",1],["p","discord-notes-from-today",["今日の講義ノートを誰かに共有してほしいと頼む","今日のノート、誰か共有してくれない？"],"",["Can someone share their notes from today?"],"discord",1],["p","discord-nvm-figured-it-out",["質問を取り下げる（自己解決した）","ごめん、自己解決した。"],"",["never mind, got it","nvm, figured it out"],"discord",1],["p","discord-office-hours-today",["今日のオフィスアワーがあるか確認する","今日のオフィスアワー、予定どおりやる？"],"",["Are you still having office hours today?","Are office hours still happening today?"],"discord",1],["p","discord-quiz-covers",["小テストの範囲を確認する","小テストって 4.3 も入る？"],"",["Is 4.3 on the quiz too?","Does the quiz cover 4.3?"],"discord",1],["p","discord-right-channel",["質問する場所がここでよいか確認する","微積の質問ってここでいい？"],"",["Is this the right channel for calc questions?","Is this the right place to ask?"],"discord",1],["p","discord-same-answer",["自分も同じ答えになったと言う","自分も 12 になった。"],"",["Same, I got 12 too.","I got the same answer — 12.","+1"],"discord",1],["p","discord-study-on-a-call",["通話しながら一緒に勉強しようと誘う","通話で勉強しない？"],"",["Anyone want to hop on a call and study?"],"discord",1],["p","discord-thanks",["チャットで手短にお礼を言う","ありがとう、助かった！"],"",["thanks, that helped","ty!"],"discord",1],["p","discord-typo-in-the-pset",["課題の誤植を指摘する","課題の 5 番、誤植っぽくない？"],"",["I think there's a typo in #5 on the pset."],"discord",1],["p","discord-when-is-the-midterm",["中間試験の日を聞く","中間試験っていつ？"],"",["When is the midterm?"],"discord",1],["p","divide-numerator-and-denominator-by-n",["極限の計算で、分子と分母を最高次の n で割る","分子と分母を n で割ります。"],"",["Divide the numerator and denominator by n.","Divide top and bottom by n."],"class-listening",1],["p","divide-the-figure",["面積を求めるために図形を分割する","図形を長方形と三角形に分けましょう。"],"",["Break the figure up into rectangles and triangles.","Break it up into smaller shapes."],"class-listening",1],["p","dont-forget-the-plus-c",["不定積分の答えに積分定数を付け忘れないよう注意する","積分定数の C を忘れないでね。"],"",["Don't forget the plus C.","Don't forget your constant of integration."],"class-listening",1],["p","email-attached",["添付ファイルがあることを書く","解答を PDF で添付しました。"],"",["I've attached my work as a PDF.","Please see the attached file."],"email",1],["p","email-cannot-make-office-hours",["オフィスアワー以外の時間に会えるか頼む","オフィスアワーの時間は授業があるため、別の時間にお会いできないでしょうか。"],"",["I have a class during your office hours. Would it be possible to meet at another time?"],"email",1],["p","email-confirm",["範囲などを確認させてもらう","金曜日の小テストに 4.3 節も含まれるか確認させてください。"],"",["Could you confirm whether the quiz on Friday covers Section 4.3?"],"email",1],["p","email-describe-where-stuck",["メールでどこまでできて、どこで詰まったかを書く","3 番は積分の式を立てるところまではできたのですが、絶対値の扱いが分かりません。"],"",["For problem 3, I got as far as setting up the integral, but I'm not sure how to handle the absolute value.","I set up the integral, but I got stuck on the absolute value."],"email",1],["p","email-dictionary-during-exam",["英語が母語でないことを伝え、試験での辞書使用を相談する","英語が母語ではないため、試験中に紙の英和辞書を使ってもよいか伺いたいです。"],"",["English is not my first language. Would it be possible for me to use a paper bilingual dictionary during the exam?"],"email",1],["p","email-exam-conflict",["試験の時間が重なっていることを伝え、別の時間を頼む","期末試験の時間が別の予定と重なっています。別の時間に受けることはできますか。"],"",["I have a conflict with the final exam time. Would it be possible to take it at a different time?","Could I take the exam at another time?","I have two finals scheduled at the same time."],"email",1],["p","email-extension",["提出期限の延長を頼む","課題の提出期限を延ばしていただくことは可能でしょうか。"],"",["Would it be possible to get an extension on the homework?"],"email",1],["p","email-follow-up",["返事のないメールについて改めて連絡する","先週お送りしたメールの件で、改めてご連絡しました。"],"",["I just wanted to follow up on my email from last week."],"email",1],["p","email-greeting",["教員へのメールの書き出し","〜先生（メールの書き出し）"],"",["Dear Professor Smith,","Hi Professor Smith,"],"email",1],["p","email-introduce-yourself",["メールで名前と受講しているセクションを名乗る","月水金 10 時の MATH 221 を受講している山田太郎です。"],"",["My name is Taro Yamada, and I'm in your MATH 221 section that meets MWF at 10."],"email",1],["p","email-missed-class",["欠席を伝え、補うべきことを聞く","水曜日の授業を欠席してしまい、申し訳ありません。何か補っておくべきことはありますか。"],"",["I'm sorry I had to miss class on Wednesday. Is there anything I should do to catch up?","Is there anything I need to catch up on?"],"email",1],["p","email-prerequisite",["日本で履修した科目が前提科目を満たすか聞く","日本で数学 III を履修しましたが、MATH 221 の前提科目を満たしますか。"],"",["I took calculus in Japan (Math III). Would that satisfy the prerequisite for MATH 221?"],"email",1],["p","email-regrade-request",["メールで採点の見直しを頼む","中間試験の 2 番の採点について、見直しをお願いできないでしょうか。答案のスキャンを添付します。"],"",["I'd like to ask about the grading on problem 2 of the midterm. I've attached a scan of my work.","Could you take another look at problem 2?"],"email",1],["p","email-sign-off",["メールの結びと署名","（結び・署名）"],"",["Best, Taro Yamada","Best regards, Taro Yamada","Sincerely, Taro Yamada"],"email",1],["p","email-subject-line",["メールの件名に授業・セクション・用件を書く","件名：MATH 221 セクション 3 ― 課題 5 についての質問"],"",["Subject: MATH 221 Sec. 3 – Question about HW 5"],"email",1],["p","email-thank-you-for-your-time",["メールの最後にお礼を書く","お忙しいところありがとうございます。"],"",["Thank you for your time.","Thanks in advance."],"email",1],["p","email-writing-to-ask",["メールの用件を切り出す","課題の 4 番についてお伺いしたく、ご連絡しました。"],"",["I'm writing to ask about problem 4 on the homework.","I have a question about problem 4 on the homework."],"email",1],["p","evaluate-critically",["推論や結論が正しいかを批判的に検討させる","この考え方は正しいと思いますか。理由も説明してください。"],"",["Do you agree with this reasoning? Why or why not?","Critique the reasoning.","Does this conclusion hold up?"],"class-listening",1],["p","exam-ask-can-i-write-on-the-back",["答案用紙の裏に書いてよいか聞く","裏に書いてもいいですか。"],"",["Can I write on the back?"],"exam",1],["p","exam-ask-how-much-time",["試験の残り時間を聞く","あと何分ありますか。"],"",["How much time do we have left?"],"exam",1],["p","exam-ask-scratch-paper",["試験中に計算用紙をもらう","計算用紙をもう 1 枚いただけますか。"],"",["Could I have another sheet of scratch paper?","Can I get some more paper?","Can I get another piece of scratch paper?"],"exam",1],["p","exam-ask-typo",["試験中に問題文の誤植を確認する","ここは f(3) ではなく f(2) ではありませんか。"],"",["Should this be f(2) instead of f(3)?","Is there a typo in number 4?"],"exam",1],["p","exam-box-your-answer",["問題文の「最終的な答えを丸（四角）で囲め」","最終的な答えを丸（四角）で囲め。"],"",["Circle your final answer.","Box your answer."],"exam",1],["p","exam-calculator-allowed",["試験で電卓を使えるかどうかを伝える","この部分は電卓使用可（不可）です。"],"",["You may use a calculator on this part.","Calculator not permitted"],"exam",1],["p","exam-clarify-instruction",["試験中に、問題文の指示の意味を確認する","ここでの simplify はどういう意味ですか。"],"",["What do you mean by \"simplify\" here?","When it says \"simplify,\" do you want us to rationalize the denominator?","Are we supposed to rationalize the denominator?"],"exam",1],["p","exam-describe-the-distribution",["問題文の「分布を述べよ」（形・中心・散らばり）","分布の特徴を述べよ。"],"",["Describe the distribution.","Describe the shape, center, and variability of the distribution."],"exam",1],["p","exam-determine-whether",["問題文の「〜かどうか判定せよ」","この級数が収束するか発散するか判定せよ。"],"",["Determine whether the series converges or diverges.","Determine if the series converges or diverges.","Decide whether the series converges or diverges.","Tell whether the series converges or diverges."],"exam",1],["p","exam-exact-form",["問題文の「正確な値で答えよ」","近似値ではなく、正確な値で答えよ。"],"",["Give the exact value, not a decimal approximation.","Leave your answer in exact form."],"exam",1],["p","exam-explain-your-reasoning",["問題文の「考え方を説明せよ」","考え方を説明せよ。"],"",["Explain your reasoning.","Explain how you know."],"exam",1],["p","exam-express-in-terms-of",["問題文の「〜を…の式で表せ」","y を x の式で表せ。"],"",["Express y in terms of x.","Write your answer in terms of x."],"exam",1],["p","exam-indicate-units",["問題文の「単位をつけて答えよ」","単位をつけて答えよ。"],"",["Include units in your answer.","Indicate units of measure.","Use appropriate units."],"exam",1],["p","exam-interpret-in-context",["問題文の「状況に即して意味を説明せよ」","f′(5) の意味を、問題の状況に即して説明せよ。"],"",["Interpret the meaning of f′(5) in context.","Interpret the meaning of f′(5) in the context of the problem."],"exam",1],["p","exam-is-it-curved",["試験の得点調整（カーブ）があるか聞く","この試験は得点調整（カーブ）がありますか。"],"",["Is the exam curved?"],"exam",1],["p","exam-justify-your-answer",["問題文の「理由を述べよ」","理由を述べよ。"],"",["Justify your answer.","Give a reason for your answer."],"exam",1],["p","exam-label-your-axes",["問題文の「軸に名前と目盛りを書け」","軸に名前と目盛りを書け。"],"",["Label and scale the axes.","Label the axes."],"exam",1],["p","exam-multiple-choice-and-free-response",["試験の構成（選択問題と記述問題）を説明する","試験は選択問題と記述問題に分かれています。"],"",["The exam has a multiple-choice section and a free-response section."],"exam",1],["p","exam-notes-allowed",["試験に持ち込めるものを伝える","ノート 1 枚（両面）まで持ち込みできます。"],"",["You can bring one sheet of notes, front and back.","You're allowed a cheat sheet."],"exam",1],["p","exam-partial-credit",["部分点があるので書けるところは書くよう促す","途中の式も書きなさい。部分点があります。"],"",["Show your work — you can get partial credit."],"exam",1],["p","exam-pencils-down",["試験の終了を告げる","時間です。筆記用具を置いてください。"],"",["Time's up. Pencils down."],"exam",1],["p","exam-set-up-but-do-not-evaluate",["問題文の「式を立てるだけでよい（計算しなくてよい）」","体積を表す積分の式を立てよ。ただし、計算はしなくてよい。"],"",["Set up, but do not evaluate, an integral for the volume.","Write, but do not evaluate, an integral expression that gives the area of R."],"exam",1],["p","exam-show-that",["問題文の「〜であることを示せ」","f が [0, 1] に零点をもつことを示せ。"],"",["Show that f has a zero on [0, 1]."],"exam",1],["p","exam-show-your-work",["問題文の「途中の過程も書け」","途中の過程も書け。"],"",["Show your work.","Show the work that leads to your answer.","Answers without supporting work will receive no credit."],"exam",1],["p","exam-simplest-form",["問題文の「最も簡単な形で答えよ」","最も簡単な形で答えよ。"],"",["Write your answer in simplest form.","Express your answer as a fraction in lowest terms."],"exam",1],["p","exam-three-decimal-places",["問題文の「小数第 3 位まで答えよ」","（電卓問題）答えは小数第 4 位を四捨五入して、小数第 3 位まで求めよ。"],"",["Round your answer to three decimal places.","Give your answer correct to three decimal places.","Your answer should be accurate to three places after the decimal point."],"exam",1],["p","exam-time-remaining",["試験の残り時間を知らせる","残り 10 分です。"],"",["You have 10 minutes left.","Ten minutes remaining."],"exam",1],["p","exam-use-the-table-to-approximate",["問題文の「表のデータを用いて近似値を求めよ」","表のデータを用いて W′(12) の近似値を求めよ。"],"",["Use the table to approximate W′(12).","Use the data in the table to approximate W′(12)."],"exam",1],["p","exam-when-do-we-get-it-back",["テストの返却時期を聞く","テストはいつ返却されますか。"],"",["When will we get the exams back?","When are we getting our midterms back?"],"exam",1],["p","exam-which-of-the-following",["選択問題の「次のうち〜を選べ」","次のうち正しいものを選べ。"],"",["Which of the following is true?","Which of the following could be the graph of f?"],"exam",1],["p","exam-write-an-equation-for-the-tangent-line",["問題文の「接線の方程式を求めよ」","x = 2 における y = f(x) の接線の方程式を求めよ。"],"",["Find the equation of the tangent line at x = 2.","Write an equation for the line tangent to the graph of f at x = 2."],"exam",1],["p","explaining-solution-another-way",["別の解き方もあると示す","平方完成して解くこともできます。"],"",["You could also complete the square.","Another way to do it is to complete the square."],"explaining-solution",1],["p","explaining-solution-by-symmetry",["対称性を使ったと説明する","対称性から残りの半分も同じなので、2 倍しました。"],"",["By symmetry, the other half is the same, so I just doubled it.","Since it's symmetric, I found one half and doubled it."],"explaining-solution",1],["p","explaining-solution-checked-the-endpoints",["区間の端点の値も調べたと言う","端点での値も調べました。"],"",["I also checked the endpoints.","I plugged in the endpoints too."],"explaining-solution",1],["p","explaining-solution-derivative-equal-to-zero",["導関数を 0 とおいて臨界点を求めたと説明する","微分して 0 とおき、臨界点を求めました。"],"",["I found the critical points by setting the derivative equal to zero.","I took the derivative and set it equal to zero."],"explaining-solution",1],["p","explaining-solution-drew-a-picture",["まず図やグラフを描いたと説明する","まず図を描きました。"],"",["I started by drawing a picture.","I sketched the graph to see what was going on."],"explaining-solution",1],["p","explaining-solution-factored-and-set-to-zero",["因数分解して、各因数を 0 とおいたと説明する","因数分解して、それぞれの因数を 0 とおきました。"],"",["I factored it and set each factor equal to zero.","Then I used the zero product property."],"explaining-solution",1],["p","explaining-solution-first-step",["自分の解き方を、順を追って説明し始める","まず 2 つの式を等号で結んで、x について解いて、最後に答えを確かめました。"],"",["First I set the two expressions equal, then I solved for x and checked the answer.","What I did was set them equal and solve for x."],"explaining-solution",1],["p","explaining-solution-flipped-the-inequality",["負の数で割ったので不等号の向きを変えたと説明する","負の数で割ったので、不等号の向きを逆にしました。"],"",["I divided by a negative, so I flipped the inequality sign.","Dividing by a negative switches the inequality."],"explaining-solution",1],["p","explaining-solution-im-assuming",["解法の前提（仮定）を述べる","速さが一定だと仮定しています。"],"",["I'm assuming the speed is constant.","This only works if the speed is constant."],"explaining-solution",1],["p","explaining-solution-in-context",["求めた値が問題の状況で何を意味するかを述べる","つまり、t = 5 のとき水槽の水は毎分 3 リットルの割合で減っているということです。"],"",["So this means the tank is draining at 3 liters per minute at t = 5.","In context, that means the tank is losing 3 liters per minute."],"explaining-solution",1],["p","explaining-solution-isolated-x",["方程式を x について解いたと説明する","それから x について解きました。"],"",["Then I solved for x.","I got x by itself.","Then I isolated x."],"explaining-solution",1],["p","explaining-solution-let-me-back-up",["説明の途中で誤りに気づき、少し戻る","ちょっと戻ります。ここで間違えました。"],"",["Wait, let me back up — I made a mistake here.","Wait, I take that back."],"explaining-solution",1],["p","explaining-solution-let-x-be",["何を文字 x でおいたかを説明する","幅は x とおきました。"],"",["For the width, I just called it x.","I let x be the width of the rectangle."],"explaining-solution",1],["p","explaining-solution-matches-the-other-way",["別の方法で出した答えと一致したと言う","前に（別の方法で）出した答えと同じになったので、たぶん合っています。"],"",["I got the same answer as before, so it's probably right.","This matches what we got the other way."],"explaining-solution",1],["p","explaining-solution-moved-everything-to-one-side",["すべての項を片方の辺に移したと説明する","全部片方の辺に移項しました。"],"",["I moved everything to one side.","I got everything on the left side."],"explaining-solution",1],["p","explaining-solution-multiplied-both-sides",["両辺に同じ操作をしたことと、それが許される理由を説明する","両辺を x で割りました。x は 0 ではないので大丈夫です。"],"",["I divided both sides by x, which is okay since x isn't zero.","I multiplied both sides by 2."],"explaining-solution",1],["p","explaining-solution-not-sure-about-this-step",["自信のないステップがあると伝える","このステップにはあまり自信がありません。"],"",["I'm not totally sure about this step.","This part might be wrong."],"explaining-solution",1],["p","explaining-solution-otherwise-contradiction",["成り立たないと矛盾が出るので成り立つ、と説明する","もし成り立たないとすると矛盾が出るので、成り立ちます。"],"",["If it weren't true, we'd get a contradiction, so it has to be true.","Assuming the opposite leads to a contradiction."],"explaining-solution",1],["p","explaining-solution-overall-plan",["解き方の方針を最初に述べる","方針としては、まず交点を求めて、それから積分します。"],"",["The idea is to find the intersection points first and then integrate.","My plan was to find where the curves meet and then integrate."],"explaining-solution",1],["p","explaining-solution-plugging-back-in",["求めた値を元の式に代入して確かめたと説明する","元の式に代入すると 17 になるので、合っています。"],"",["When I plugged it back in, I got 17, so it checks out.","I substituted it back into the original equation."],"explaining-solution",1],["p","explaining-solution-rewrote-it-as",["式を別の形に書き直したと説明する","(x − 3)² + 1 の形に書き直しました。"],"",["I wrote it as (x − 3)² + 1.","I rewrote it as (x − 3)² + 1."],"explaining-solution",1],["p","explaining-solution-rounded-at-the-end",["途中で丸めず最後に丸めたと言う","途中では丸めず、最後に丸めました。"],"",["I didn't round until the very end."],"explaining-solution",1],["p","explaining-solution-set-up-an-equation",["問題の条件から方程式を立てたと説明する","まず、問題の条件から方程式を立てました。"],"",["First I wrote an equation from the given information.","I set up an equation from what the problem says."],"explaining-solution",1],["p","explaining-solution-so-the-answer-is",["最終的な答えを述べる","なので、答えは 12 です。"],"",["So the answer is 12.","So I got 12.","And that gives us 12."],"explaining-solution",1],["p","explaining-solution-that-cant-be-right",["自分の答えがおかしいと気づいて言う","あれ、おかしい。確率が 1 より大きくなるはずがありません。"],"",["Wait, that can't be right — a probability can't be bigger than 1.","Hmm, that doesn't seem right."],"explaining-solution",1],["p","explaining-solution-that-makes-sense",["答えが妥当だと理由をつけて確かめる","10 より少し小さいはずなので、妥当な答えです。"],"",["That makes sense, because the answer should be a little less than 10.","That seems reasonable, since it should be a little less than 10."],"explaining-solution",1],["p","explaining-solution-thats-where-it-comes-from",["ある数や項がどこから出てきたかを示す","その 2 はここから来ています。"],"",["That's where the 2 comes from."],"explaining-solution",1],["p","explaining-solution-the-key-was",["解き方の要点（ひらめき）を示す","2 つの三角形が相似だと気づくのがポイントでした。"],"",["The key was noticing that the two triangles are similar.","The trick is to see that the triangles are similar."],"explaining-solution",1],["p","explaining-solution-these-cancel",["項が打ち消し合うことを示す","この項とこの項が打ち消し合います。"],"",["These two terms cancel out.","Everything in the middle cancels."],"explaining-solution",1],["p","explaining-solution-threw-out-a-solution",["条件に合わない解を除いたと説明する","長さは負にならないので、負の解は捨てました。"],"",["I threw out the negative solution because a length can't be negative.","I rejected the negative solution because a length can't be negative."],"explaining-solution",1],["p","explaining-solution-tried-small-cases",["まず具体的な数や簡単な場合を試して様子を見たと説明する","まずいくつか数を代入してみて、感じをつかみました。"],"",["I plugged in some numbers first to get a feel for it.","I tried a few simple cases first to see what happens."],"explaining-solution",1],["p","explaining-solution-used-a-calculator",["電卓を使った箇所を説明する","最後は電卓で近似値を出しました。"],"",["I used my calculator to get a decimal approximation at the end.","At the end, I put it into my calculator."],"explaining-solution",1],["p","explaining-solution-used-the-rule",["どの公式・法則を使ったかを示す","ここで合成関数の微分（連鎖律）を使いました。"],"",["I used the chain rule here.","Here I applied the product rule."],"explaining-solution",1],["p","explaining-solution-with-units",["答えを単位つきで述べる","速度は毎秒 4 メートルです。"],"",["So the velocity is 4 meters per second.","The units are meters per second."],"explaining-solution",1],["p","explaining-solution-worked-backwards",["示したい式から逆にたどったと説明する","示したい式から逆にたどりました。"],"",["I worked backwards from what we want to show."],"explaining-solution",1],["p","f-double-prime-is-positive",["f″(x) > 0 からグラフが下に凸（concave up）だと書く","f″(x) > 0 より、f のグラフは下に凸である。"],"",["Since f″(x) > 0, the graph of f is concave up."],"written-solution",1],["p","find-the-area-by-integration",["面積を積分で求めると伝える","面積は積分で求めます。"],"",["Let's set up an integral for the area.","We'll find the area by integrating."],"class-listening",1],["p","find-the-intersections-to-get-the-limits",["2 曲線の交点を求めて、それを積分区間にすると伝える","まず交点を求めて、それを積分区間にします。"],"",["First find where the curves intersect — those are your bounds of integration.","Set them equal to get the bounds."],"class-listening",1],["p","find-the-pattern",["規則性に気づかせる","何か規則が見えますか？"],"",["Do you see a pattern?","Does anyone notice a pattern?"],"class-listening",1],["p","group-study-answer-key-wrong",["解答が間違っているのではと言う","解答が間違ってる気がする。"],"",["I think the answer key might be wrong.","The back of the book says 12, but I keep getting 13."],"group-study",1],["p","group-study-ask-the-ta",["TA やオフィスアワーで聞こうと提案する","TA に聞いてみようか。"],"",["Should we ask the TA?","Let's ask at office hours."],"group-study",1],["p","group-study-can-i-see-your-notes",["友達のノートを見せてもらう","ノート見せてもらってもいい？"],"",["Can I see your notes?"],"group-study",1],["p","group-study-do-you-see-what-i-mean",["自分の説明が伝わっているか確かめる","言いたいこと、伝わってる？"],"",["Do you see what I mean?","Do you know what I'm saying?","Does that make sense?"],"group-study",1],["p","group-study-does-that-mean",["聞いたことから分かることを確かめる","ということは、x = 0 では微分できないってこと？"],"",["Does that mean it's not differentiable at x = 0?","So that means the limit doesn't exist?"],"group-study",1],["p","group-study-how-did-you-get-that",["違う答えになったので、相手の解き方を聞く","私は違う答えになった。どうやって出したの？"],"",["I got something different — how did you get that?","Can you show me how you got that?"],"group-study",1],["p","group-study-i-thought-it-was",["自分の思っていたことと違うと言う","あれ、答えは負の数だと思ってたんだけど。"],"",["Wait, I thought it was negative.","I thought you said the answer was 5."],"group-study",1],["p","group-study-is-that-right",["自分の答えが合っているか確かめる","答えは 3 で合ってる？"],"",["The answer is 3, is that right?","Is it just 2x?"],"group-study",1],["p","group-study-not-sure-but",["自信がないが自分の考えを言う","自信ないけど、4 だと思う。"],"",["I'm not sure, but I think it's 4.","Don't quote me on this, but I think it's 4."],"group-study",1],["p","group-study-oh-that-makes-sense",["説明を聞いて納得したと伝える","あ、なるほど！"],"",["Oh, I see.","Okay, that makes sense."],"group-study",1],["p","group-study-practice-exam",["時間を計って模擬試験を解こうと誘う","時間を計って模擬試験を解こう。"],"",["Let's do the practice exam under timed conditions.","Let's time ourselves."],"group-study",1],["p","group-study-quiz-each-other",["問題を出し合おうと誘う","問題出し合おう。"],"",["Let's quiz each other."],"group-study",1],["p","group-study-so-youre-saying",["相手の言ったことを言い直して確かめる","つまり、先に通分するってこと？"],"",["Oh, so we find a common denominator first?","So you're saying we should find a common denominator first?"],"group-study",1],["p","group-study-split-them-up",["問題を分担しようと提案する","何問かずつ分担して、あとで説明し合おう。"],"",["Let's each do a few and then explain them to each other.","Should we split them up?","Should we split up the problems?"],"group-study",1],["p","group-study-what-did-you-get",["友達に答えを聞く","5 番、答え何になった？"],"",["What did you get for number 5?"],"group-study",1],["p","group-study-what-do-you-mean",["相手の言ったことの意味を聞き返す","「ここで打ち消し合う」ってどういう意味？"],"",["What do you mean by \"they cancel here\"?","What does that mean?"],"group-study",1],["p","group-study-what-is-it-asking",["問題が何を聞いているのか分からないと言う","この問題、何を聞いてるの？"],"",["What is this question even asking?","What are they asking for here?"],"group-study",1],["p","group-study-where-do-we-start",["どこから手をつけるか相談する","これ、どこから手をつける？"],"",["Where do we even start with this one?","Any ideas for number 7?"],"group-study",1],["p","group-study-where-to-meet",["勉強会の時間と場所を決める","7 時に図書館で会わない？"],"",["Do you want to meet at the library at 7?","Does 7 at the library work?","Let's meet at the library at 7."],"group-study",1],["p","group-study-which-one-do-you-mean",["相手がどれのことを言っているか確かめる","どれのこと？ 2 本目の式？"],"",["Which one? The second equation?","Are you talking about the second equation?","Do you mean the second equation?"],"group-study",1],["p","group-study-work-together",["一緒に勉強しよう（課題をやろう）と誘う","一緒に課題やらない？"],"",["Do you want to work on the homework together?","Do you want to work on the problem set together?","Want to study together for the midterm?"],"group-study",1],["p","group-study-write-up-our-own",["答案は各自で書く決まりを確認する","相談はいいけど、答案はそれぞれ自分で書かないといけないよ。"],"",["We can talk about the problems, but we have to write up our own solutions."],"group-study",1],["p","group-study-you-dropped-a-sign",["相手の計算ミスを指摘する","ここ、マイナスが抜けてると思う。"],"",["I think you dropped a negative sign.","I think you forgot the 2 here."],"group-study",1],["p","in-general",["一般には成り立たないこと・一般の場合を述べる","一般には、これは成り立たない。"],"",["In general, this is not true."],"written-solution",1],["p","in-order",["計算を左から順に進めると伝える","左から順に計算します。"],"",["Work from left to right."],"class-listening",1],["p","integrate-by-parts-repeatedly",["部分積分を 2 回使う必要があると伝える","部分積分を 2 回使います。"],"",["You'll have to integrate by parts twice.","Just integrate by parts again."],"class-listening",1],["p","integrate-the-inequality",["不等式の各辺を積分して、新しい不等式を得る","不等式の両辺を 0 から 1 まで積分すると、"],"",["Integrating both sides of the inequality from 0 to 1, we get"],"written-solution",1],["p","involves-imaginary-numbers",["2 次方程式の解が実数でなく虚数（複素数）になると伝える","解は虚数になります（実数解はありません）。"],"",["You get complex roots.","You get imaginary solutions.","There are no real solutions."],"class-listening",1],["p","it-also-holds-for-n-k-1",["数学的帰納法で、n = k + 1 のときも成り立つことを示す段階を書く","n = k + 1 のときも成り立つことを示す。"],"",["Inductive step: we show that the statement also holds for n = k + 1.","Now we show that it is also true for n = k + 1."],"written-solution",1],["p","left-side-minus-right-side",["不等式の証明で、左辺から右辺を引いた差を調べる","左辺から右辺を引く。"],"",["Subtract the right side from the left side."],"written-solution",1],["p","let-h-go-to-zero",["差分商で h を 0 に近づける","では h を 0 に近づけます。"],"",["Now take the limit as h approaches zero.","Now let h go to zero."],"class-listening",1],["p","let-p-be-the-position-vector-of-p",["点の位置ベクトルを文字でおく","点 P の位置ベクトルを p とする。"],"",["Let p be the position vector of P.","Let p = OP⃗."],"written-solution",1],["p","multiply-by-r-and-subtract",["等比数列の和を、両辺に r を掛けて引き算して求める","両辺に r を掛けて、2 つの式を引き算します。"],"",["Multiply both sides by r, then subtract the two equations.","Multiply every term by r.","Subtract one equation from the other."],"class-listening",1],["p","multiply-by-the-derivative-of-the-inside",["合成関数の微分で、最後に中身の微分を掛ける","そして中身の微分を掛けます。"],"",["Then multiply by the derivative of the inside."],"class-listening",1],["p","office-hours-am-i-on-the-right-track",["今の方針で合っているかを確かめる","この方針で合っていますか。"],"",["Am I on the right track?","Is this approach going to work?"],"office-hours",1],["p","office-hours-can-i-come-back",["また行き詰まったら、もう一度来てよいかを聞く","また行き詰まったら、もう一度来てもいいですか。"],"",["Can I come back if I get stuck again?"],"office-hours",1],["p","office-hours-can-i-show-you-what-i-tried",["自分で途中までやったものを見てもらう","自分でやったところを見ていただけますか。"],"",["Can I show you what I tried?","Here's what I have so far."],"office-hours",1],["p","office-hours-do-you-have-a-minute",["先生の部屋を訪ねて、今話してよいかを聞く","今少しお時間よろしいですか。"],"",["Do you have a minute?","Is now a good time?","Do you have a few minutes?"],"office-hours",1],["p","office-hours-english-terms-are-new",["日本語で習ったので、英語の数学用語に慣れていないと伝える","日本語で習ったので、英語の数学用語にまだ慣れていません。"],"",["I learned this in Japanese, so I'm still getting used to the English terms.","I understand the math, but I don't know the English terms yet."],"office-hours",1],["p","office-hours-extra-practice",["練習用の問題や過去の試験があるかを聞く","練習に使える過去の試験はありますか。"],"",["Do you have any old exams I could practice with?","Are there any extra practice problems I could do?"],"office-hours",1],["p","office-hours-hint-not-the-answer",["答えそのものではなく、ヒントだけほしいと頼む","答えではなく、ヒントだけいただけますか。"],"",["Could you give me a hint instead of the answer?","Could you point me in the right direction?"],"office-hours",1],["p","office-hours-how-to-study",["試験に向けた勉強の仕方を相談する","中間試験に向けて、どう勉強すればいいでしょうか。"],"",["How would you recommend studying for the midterm?","What's the best way to study for the exam?"],"office-hours",1],["p","office-hours-i-was-wondering",["丁寧に頼みごとを切り出す","2 番の答えを見ていただけないかと思いまして。"],"",["I was wondering if you could look over my answer to number 2.","I was wondering whether you could take a look at my proof."],"office-hours",1],["p","office-hours-intuition",["計算は追えるが、直感的な意味が分からないと伝える","計算は追えるのですが、直感的な意味が分かりません。"],"",["I can follow the algebra, but I don't get the intuition behind it.","What's the big picture here?"],"office-hours",1],["p","office-hours-is-this-rigorous-enough",["自分の証明が十分に厳密かを見てもらう","この証明、これで十分厳密か見ていただけますか。"],"",["Could you look over my proof and tell me if it's rigorous enough?","Is this rigorous enough?"],"office-hours",1],["p","office-hours-question-about-homework",["宿題の特定の問題について質問しに来たと切り出す","宿題の 3 番について質問があって来ました。"],"",["I had a question about number 3 on the homework.","I wanted to ask about problem 3."],"office-hours",1],["p","office-hours-regrade",["採点を見直してほしいと丁寧に頼む","採点が間違っているかもしれないのですが、もう一度見ていただけませんか。"],"",["I think this might have been graded incorrectly. Could you take another look?"],"office-hours",1],["p","office-hours-stuck-at-step",["どこで詰まっているかを伝える","この行から次の行にどうやって変形したのかが分かりません。"],"",["I'm confused about how you got from this line to the next one.","I don't see how you got this line.","I don't understand how you got this line."],"office-hours",1],["p","office-hours-thanks-that-helps",["説明で分かったとお礼を言う","ありがとうございます、おかげでよく分かりました。"],"",["Thanks, that really helps.","Thanks, that makes a lot more sense now."],"office-hours",1],["p","office-hours-understand-in-class-not-alone",["授業では分かるのに、一人で解くと手が止まることを相談する","授業では分かるのに、自分でやると手が止まります。"],"",["I understand it in class, but I get stuck when I try it on my own.","I can follow it in class, but I can't do it on my own.","It makes sense in lecture, but I keep getting stuck on the homework."],"office-hours",1],["p","office-hours-when-to-use-which",["2 つの方法をどう使い分けるかを聞く","どの方法を使えばいいか、どうやって見分ければいいですか。"],"",["How do I know which method to use?","How do I know when to use substitution and when to use integration by parts?"],"office-hours",1],["p","office-hours-which-course-next",["次の学期にどの科目を取るべきか相談する","次の学期は線形代数と Calc II のどちらを取るべきでしょうか。"],"",["Should I take Linear Algebra or Calc II next semester?","Which one would you recommend taking next semester?"],"office-hours",1],["p","office-hours-why-did-i-lose-points",["返ってきた答案で減点された理由を聞く","ここでなぜ減点されたのか分からなくて。"],"",["I'm not sure why I lost points here.","Could you explain what was wrong with my answer?"],"office-hours",1],["p","omit",["式を書くとき掛け算の記号を省略すると伝える","掛け算の記号はふつう省略します。"],"",["We usually leave out the multiplication sign."],"class-listening",1],["p","on-the-interval-from-0-to-2",["方程式を解く範囲（区間）を指定する（問題文）","0 ≦ θ < 2π の範囲で解け。"],"",["Find all solutions on the interval [0, 2π).","Find all solutions in the interval [0, 2π)."],"exam",1],["p","organize-in-a-table",["表に整理する","表にまとめましょう。"],"",["Let's make a table.","Let's set up a table."],"class-listening",1],["p","organize-the-data",["データを整理する（小さい順に並べる）","データを小さい順に並べましょう。"],"",["Put the data in order from least to greatest.","Put the numbers in order from smallest to largest."],"class-listening",1],["p","probabilities-sum-to-1",["確率の合計が 1 になることを確かめさせる","確率を全部足すと 1 になります。"],"",["The probabilities have to add up to one.","The probabilities sum to one."],"class-listening",1],["p","rationalize-and-take-the-limit",["根号を含む式を有理化してから極限をとる","有理化してから極限をとります。"],"",["Rationalize first, then take the limit.","Multiply by the conjugate, and then take the limit."],"class-listening",1],["p","represent-with-a-graph",["グラフに描いて様子を見る","グラフに描いて様子を見てみましょう。"],"",["Let's graph it and see what it looks like.","Let's draw a graph."],"class-listening",1],["p","rewrite-the-expression",["式を扱いやすい形に書き直す","これを 1 つの分数に書き直しましょう。"],"",["Let's rewrite this as a single fraction.","Let's rewrite the expression."],"class-listening",1],["p","since",["答案で理由を「〜なので」と書く","f′(x) > 0 なので、f は増加する。"],"",["Since f′(x) > 0, f is increasing.","Because f′(x) > 0, f is increasing."],"written-solution",1],["p","split-at-the-median",["中央値でデータを下半分と上半分に分ける（四分位数を求める）","中央値でデータを下半分と上半分に分けます。"],"",["The median splits the data into a lower half and an upper half."],"class-listening",1],["p","split-the-integral",["途中で式が変わるので、積分区間を分けて計算すると伝える","ここで積分区間を分けます。"],"",["We need to split the integral here.","Let's break it into two intervals at x = 0."],"class-listening",1],["p","square-and-add",["2 つの式を 2 乗して足し、θ を消去する","両方の式を 2 乗して足します。"],"",["Square both equations and add them.","Square and add."],"class-listening",1],["p","squares-are-nonnegative",["実数の 2 乗は 0 以上であることを理由に使う","実数の 2 乗は 0 以上なので、(x − 1)² ≧ 0 である。"],"",["Since a square is never negative, (x − 1)² ≥ 0.","Squares of real numbers are always nonnegative."],"written-solution",1],["p","take-positive-values",["関数が常に正の値をとると書く","f(x) は常に正の値をとる。"],"",["f(x) is always positive.","f takes only positive values."],"written-solution",1],["p","the-area-of-the-region-bounded-by",["曲線で囲まれた部分の面積を求めさせる（問題文）","曲線 y = x² と直線 y = x + 2 で囲まれた部分の面積を求めよ。"],"",["Find the area of the region bounded by y = x² and y = x + 2.","Find the area of the region enclosed by the two curves."],"exam",1],["p","the-base-is-greater-than-1",["底が 1 より大きいので、不等号の向きは変わらないと書く","底が 1 より大きいので、不等号の向きは変わらない。"],"",["Since the base is greater than 1, the inequality sign stays the same.","Since the base is greater than 1, the direction of the inequality does not change."],"written-solution",1],["p","the-common-ratio-is-less-than-1",["公比の絶対値が 1 より小さいので等比級数が収束すると書く","|r| < 1 なので、この級数は収束する。"],"",["Since |r| < 1, the series converges."],"written-solution",1],["p","the-derivative-is-zero",["その点で微分係数が 0 になる（接線が水平になる）と書く","f のグラフは x = a で水平な接線をもつ。"],"",["The graph of f has a horizontal tangent at x = a.","The derivative is zero at x = a."],"written-solution",1],["p","the-equation-holds",["等式が成り立つことを示し終えたと書く","よって、等式は成り立つ。"],"",["Thus, the identity is verified.","Therefore, the equation holds."],"written-solution",1],["p","the-one-sided-limits-agree",["左側極限と右側極限が一致するので極限が存在すると書く","左側極限と右側極限が等しいので、極限は存在する。"],"",["Since the left-hand and right-hand limits are equal, the limit exists.","Since the one-sided limits are equal, the limit exists."],"written-solution",1],["p","the-tangent-line-passes-through",["接線がある点を通ることを条件として使う","接線が点 (0, −1) を通るので、"],"",["Since the tangent line passes through (0, −1),","The tangent line goes through (0, −1)."],"written-solution",1],["p","top-minus-bottom",["2 つの曲線の間の面積は、上の関数から下の関数を引いて積分すると伝える","上の曲線から下の曲線を引きます。"],"",["It's top minus bottom.","Subtract the bottom curve from the top curve."],"class-listening",1],["p","what-we-want-to-show",["何を示せばよいかを書く","示すべきことは、すべての x について f(x) > 0 となることである。"],"",["We need to show that f(x) > 0 for all x.","We want to show that f(x) > 0 for all x."],"written-solution",1],["p","where",["式の後に文字の条件を書き添える（ただし〜）","x = π/2 + kπ（ただし k は整数）"],"",["x = π/2 + kπ, where k is an integer."],"written-solution",1],["p","write-as-a-limit-of-a-sum",["定積分を和の極限の形で書き直す","和の極限の形に書き直しましょう。"],"",["Write it as the limit of a Riemann sum.","Let's write the integral as the limit as n approaches infinity of a sum."],"class-listening",1],["p","written-solution-as-desired",["証明の最後に「以上で示された」「（証明終わり）」と書く","よって、a + b は偶数である。以上で示された。"],"",["Hence a + b is even, as desired.","This completes the proof.","Hence a + b is even, as required.","Hence a + b is even, which is what we wanted to show.","∎"],"written-solution",1],["p","written-solution-by-induction",["帰納法の証明の結びを書く","以上より、数学的帰納法によってすべての自然数 n について成り立つ。"],"",["By induction, the statement holds for all n ≥ 1.","By the principle of mathematical induction, the statement holds for all n ≥ 1."],"written-solution",1],["p","written-solution-by-the-theorem",["答案で定理を根拠に挙げる（〜の定理より）","三平方の定理より AC = 5。"],"",["By the Pythagorean Theorem, AC = 5.","Using the Pythagorean Theorem, AC = 5.","It follows from the Pythagorean Theorem that AC = 5."],"written-solution",1],["p","written-solution-case",["答案で場合分けを書く","x ≥ 0 のときと x < 0 のときの 2 つの場合に分ける。"],"",["There are two cases: x ≥ 0 and x < 0.","Case 1: x ≥ 0."],"written-solution",1],["p","written-solution-combining",["答案で「(1)、(2) より」と結果を合わせる","以上より（(1)、(2) より）、x = 2 である。"],"",["Putting this together, x = 2.","Combining these two results, we get x = 2.","From (1) and (2), x = 2."],"written-solution",1],["p","written-solution-conclusion-because-reason",["答案で結論と理由を 1 文で書く（極値の判定）","f は x = 2 で増加から減少に変わるので、x = 2 で極大となる。"],"",["f changes from increasing to decreasing at x = 2, so f has a relative maximum there.","f has a relative maximum at x = 2 because f′ changes from positive to negative there."],"written-solution",1],["p","written-solution-final-answer",["答案の最後に答えを書く","答　x = 3"],"",["The solution is x = 3.","Answer: x = 3"],"written-solution",1],["p","written-solution-for-some-integer",["答案で「k は整数」と条件を添える","n = 2k + 1（k は整数）と表せる。"],"",["n = 2k + 1, where k is an integer.","n = 2k + 1 for some integer k."],"written-solution",1],["p","written-solution-from-the-graph",["答案で「グラフより」と根拠を示す","グラフより、f(2) = 3 である。"],"",["From the graph, f(2) = 3.","The graph shows that f(2) = 3."],"written-solution",1],["p","written-solution-given-prove",["幾何の証明で仮定と結論を書く","仮定：AB = CD, AB // CD　結論：△ABC ≡ △CDA"],"",["Given: AB = CD, AB ∥ CD. Prove: △ABC ≅ △CDA."],"written-solution",1],["p","written-solution-hypotheses-are-met",["定理の仮定が満たされることを書く","f は [1, 3] で連続、(1, 3) で微分可能なので、平均値の定理が使える。"],"",["Since f is continuous on [1, 3] and differentiable on (1, 3), the Mean Value Theorem applies.","The hypotheses of the Mean Value Theorem are satisfied."],"written-solution",1],["p","written-solution-i-e",["答案で「すなわち」と言い換える","x = 1 における接線は水平、すなわち f′(1) = 0 である。"],"",["The tangent line at x = 1 is horizontal; that is, f′(1) = 0.","The tangent line at x = 1 is horizontal, i.e., f′(1) = 0."],"written-solution",1],["p","written-solution-it-suffices-to-show",["答案で「〜を示せば十分である」と書く","f′(x) > 0 を示せば十分である。"],"",["It suffices to show that f′(x) > 0.","It is enough to show that f′(x) > 0."],"written-solution",1],["p","written-solution-let-epsilon-be-given",["ε-δ の証明を書き始める","任意の ε > 0 をとる。"],"",["Let ε > 0 be given."],"written-solution",1],["p","written-solution-let",["答案で文字を導入する（〜を x とする）","売れたチケットの枚数を x とする。"],"",["Let x be the number of tickets sold.","Let x represent the number of tickets sold.","Let x denote the number of tickets sold."],"written-solution",1],["p","written-solution-no-solution",["答案で「解なし」と書く","よって解なし。"],"",["Therefore, there is no solution.","The equation has no real solutions."],"written-solution",1],["p","written-solution-note-that",["答案で「〜に注意する」と書く","ここで、すべての x について x² + 1 > 0 であることに注意する。"],"",["Notice that x² + 1 > 0 for all x.","Note that x² + 1 > 0 for all x."],"written-solution",1],["p","written-solution-on-the-other-hand",["答案で「一方、」と対比する","一方、f(3) < 0 である。"],"",["On the other hand, f(3) < 0."],"written-solution",1],["p","written-solution-reject-extraneous",["答案で「x = −2 は不適」と書く","x = −2 は不適。"],"",["x = −2 is an extraneous solution, so we reject it.","x = −2 does not satisfy the original equation."],"written-solution",1],["p","written-solution-similarly",["答案で「同様にして」と書く","同様にして BD = CE である。"],"",["Similarly, BD = CE.","By the same argument, BD = CE."],"written-solution",1],["p","written-solution-suppose-for-contradiction",["背理法の仮定を書く","√2 が有理数であると仮定する（背理法）。"],"",["Suppose, for the sake of contradiction, that √2 is rational.","Assume to the contrary that √2 is rational.","Suppose not. Then √2 is rational."],"written-solution",1],["p","written-solution-suppose",["答案で仮定を置く","f(a) = f(b) とする（と仮定する）。"],"",["Suppose that f(a) = f(b).","Assume that f(a) = f(b)."],"written-solution",1],["p","written-solution-taking-the-limit",["答案で「n → ∞ とすると」と極限をとる","n → ∞ とすると、L = 1/2 を得る。"],"",["Taking the limit as n → ∞, we get L = 1/2.","Letting n approach infinity, we get L = 1/2."],"written-solution",1],["p","written-solution-then",["答案で「すると」と式をつなぐ","すると f′(x) = 2x − 4 である。"],"",["This gives f′(x) = 2x − 4.","Then we have f′(x) = 2x − 4."],"written-solution",1],["p","written-solution-therefore",["答案で「よって」「したがって」と書く","よって x = 3 が唯一の解である。"],"",["Therefore x = 3 is the only solution.","Thus x = 3 is the only solution.","Hence x = 3 is the only solution.","So x = 3 is the only solution."],"written-solution",1],["p","written-solution-we-have",["答案で「〜であるから」と成り立つ事実を示す","f(2) = 5 であるから、点 (2, 5) は f のグラフ上にある。"],"",["We have that f(2) = 5, so the point (2, 5) is on the graph of f.","We get that f(2) = 5."],"written-solution",1],["p","written-solution-which-implies",["答案で「〜となり、…である」と結論を導く","2x = 6 となり、x = 3 である。"],"",["2x = 6. It follows that x = 3.","2x = 6, which implies x = 3.","2x = 6, so x = 3.","2x = 6 ⟹ x = 3"],"written-solution",1],["c","accumulation-function-scope",["定積分で表された関数の呼び方"],"",["Accumulation functions"],"",1],["c","antiderivative-vs-indefinite-integral",["原始関数と不定積分"],"",["Antiderivative and indefinite integral"],"",1],["c","approximately-equal-notation",["近似の記号"],"",["Approximately equal sign"],"",1],["c","arc-measure-notation",["弧の大きさ"],"",["Arc measure"],"",1],["c","area-and-base-letters",["底面積の文字"],"",["Letter for base area"],"",1],["c","calculator-instead-of-tables",["数表と電卓"],"",["Tables and calculators"],"",1],["c","circle-and-sphere-boundary",["球と sphere"],"",["Sphere: surface or solid"],"",1],["c","combination-permutation-notation",["ₙCᵣ と C(n, r)"],"",["Combinations and permutations notation"],"",1],["c","complement-bar-vs-c-prime",["補集合・余事象の記号"],"",["Complement notation"],"",1],["c","conditional-probability-notation",["条件付き確率の書き方"],"",["Conditional probability notation"],"",1],["c","congruence-abbreviations-in-proofs",["証明の理由の書き方"],"",["Reasons in congruence proofs"],"",1],["c","congruence-symbol",["合同の ≡ と ≅"],"",["Congruence sign"],"",1],["c","constant-of-integration-remark",["「C は積分定数」の書き添え"],"",["Saying what C is"],"",1],["c","descriptive-statistics-coverage",["平均絶対偏差"],"",["Mean absolute deviation"],"",1],["c","discriminant-letter-d",["判別式の D"],"",["The letter D for the discriminant"],"",1],["c","dne-in-answers",["「存在しない」と DNE"],"",["DNE"],"",1],["c","euler-trail-scope",["一筆書きと Euler trail"],"",["Euler trails and circuits"],"",1],["c","factoring-trinomials-method",["二次式の因数分解の方法"],"",["Factoring trinomials"],"",1],["c","figure-and-measure-notation",["図形と大きさの書き分け"],"",["Figures and their measures"],"",1],["c","floor-function-notation",["ガウス記号と床関数"],"",["Gauss bracket and floor function"],"",1],["c","graph-transformations",["グラフの移動の言い方"],"",["Graph transformations"],"",1],["c","hensachi",["偏差値"],"",["Hensachi (T-score)"],"",1],["c","imaginary-number-scope",["虚数の範囲"],"",["Imaginary numbers"],"",1],["c","inequality-graph-boundary",["不等式の領域の境界"],"",["Boundary of an inequality's graph"],"",1],["c","inequality-symbols",["≦ と ≤"],"",["Inequality signs"],"",1],["c","inequality-terms-scope",["一次不等式の範囲"],"",["Linear inequalities"],"",1],["c","interval-notation-vs-inequalities",["区間の書き方"],"",["Interval notation"],"",1],["c","inverse-trig-functions",["逆三角関数"],"",["Inverse trig functions"],"",1],["c","jp-named-techniques-unnamed-in-us",["日本だけが名前を付ける手法"],"",["Techniques Japan names"],"",1],["c","jp-only-algebra-and-discrete-topics",["日本だけの内容（二重根号・階差数列）"],"",["Nested radicals and sequences of differences"],"",1],["c","jp-only-number-theory",["日本の数学Aだけの整数の内容"],"",["Number theory in Japan's Math A"],"",1],["c","jp-only-synthetic-geometry",["チェバ・メネラウスの定理"],"",["Ceva's and Menelaus's theorems"],"",1],["c","lhopitals-rule-use",["ロピタルの定理"],"",["L'Hôpital's Rule"],"",1],["c","line-standard-form",["直線の方程式の一般形"],"",["Standard form of a line"],"",1],["c","log-and-ln",["log と ln"],"",["log and ln"],"",1],["c","logic-in-geometry-course",["論理を教える科目"],"",["Where logic is taught"],"",1],["c","logic-notation",["否定と「かつ」「または」の書き方"],"",["Negation and connectives"],"",1],["c","mapping-rule-notation",["平行移動の書き方"],"",["Mapping rule"],"",1],["c","negative-vs-minus",["マイナスと negative"],"",["Minus and negative"],"",1],["c","normal-distribution-parameter",["正規分布の 2 つ目の値"],"",["Second parameter of the normal distribution"],"",1],["c","normal-table-format",["正規分布表の形"],"",["Standard normal table format"],"",1],["c","one-sided-limit-notation",["片側極限の書き方"],"",["One-sided limit notation"],"",1],["c","probability-and-or-notation",["積事象・和事象の書き方"],"",["And and or in probability"],"",1],["c","proposition-vs-condition",["「命題」と「条件」"],"",["Statements and predicates"],"",1],["c","quadrant-roman-numerals",["象限の番号"],"",["Quadrant numbers"],"",1],["c","quadratic-forms-naming",["二次関数の式の形の名前"],"",["Names of quadratic forms"],"",1],["c","reciprocal-trig-functions",["sec・csc・cot"],"",["Reciprocal trig functions"],"",1],["c","regression-line-letters",["直線の式の a と b"],"",["a and b in a line's equation"],"",1],["c","repeated-combination-notation",["重複組合せの記号"],"",["Combinations with repetition"],"",1],["c","rounding-place",["四捨五入の位の言い方"],"",["Rounding place"],"",1],["c","sample-variance-n-minus-1",["分散を n で割るか n − 1 で割るか"],"",["Dividing by n or n − 1"],"",1],["c","scale-factor-direction",["相似比と scale factor"],"",["Similarity ratio and scale factor"],"",1],["c","sequence-starting-index",["数列の最初の番号"],"",["Starting index of a sequence"],"",1],["c","series-includes-finite-sums",["級数の範囲"],"",["Series"],"",1],["c","similarity-symbol",["相似の ∽ と ∼"],"",["Similarity sign"],"",1],["c","slant-height-vs-generatrix",["母線と slant height"],"",["Generatrix and slant height"],"",1],["c","slope-intercept-form",["y = ax + b と y = mx + b"],"",["Slope-intercept form"],"",1],["c","square-and-cubic-units",["平方単位・立方単位"],"",["Square units and cubic units"],"",1],["c","subset-symbol",["部分集合の記号"],"",["Subset symbol"],"",1],["c","substitution-variable-u",["置換積分の文字"],"",["Letter in substitution"],"",1],["c","therefore-because-symbols",["∴ と ∵"],"",["Therefore and because signs"],"",1],["c","transformations-define-congruence-and-similarity",["移動と合同・相似"],"",["Transformations, congruence and similarity"],"",1],["c","trapezoid-definition",["台形の定義"],"",["Definition of a trapezoid"],"",1],["c","triangle-congruence-criteria",["三角形の合同条件"],"",["Triangle congruence criteria"],"",1],["c","two-column-proof-format",["証明の書き方"],"",["Two-column proof"],"",1],["c","universal-set-and-sample-space",["全体集合と標本空間の文字"],"",["Universal set and sample space"],"",1],["c","us-named-rules-and-tests",["米国だけが名前を付ける規則・判定法"],"",["Rules and tests the US names"],"",1],["c","us-named-theorems-and-reasons-in-geometry",["幾何の定理・公準の名前"],"",["Named postulates and theorems in geometry"],"",1],["c","us-only-calculus-topics",["AP Calculus だけの内容"],"",["Topics in AP Calculus only"],"",1],["c","us-only-precalculus-topics",["米国の Algebra・Precalculus だけの内容"],"",["Topics in US algebra and precalculus only"],"",1],["c","vertex-and-center-letters",["頂点の文字"],"",["Letters for the vertex"],"",1],["c","zeros-roots-intercepts",["解・共有点と zero・x-intercept"],"",["Zeros, roots and intercepts"],"",1]]}