左側に日本語・右側に英語の『見開き 2 ページで日英両言語 ... · 1...
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左側に日本語・右側に英語の『見開き 2 ページで日英両言語参照型』を『SUHARA 式』と名付けてい
ます。中学 3年生で使用中の『SUHARA式 J - CLIL教育 テキスト』の抜粋です。
2015 年まで八尾市の公立中学校で採用されていました、啓林館の教科書と帰国子女用の教科書の合本
です。自塾でのこのような形式での使用許可を啓林館からいただいています。
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２０１７年度大阪府立高校入学試験 数学 C問題
左側は日本語版（新聞よりコピー）、右側はそれを英語に翻訳してくれたものです。
『見開き 2ページで日英両言語参照型』である『SUHARA式 J - CLIL教育 テキスト』の一例です。
解答編もあります。問題編の後にその解答部分を掲載しています。
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(1) Calculate (−1
3𝑎𝑏2)
2× (−2𝑎4𝑏) ÷
1
6(𝑎2𝑏)3
(2) Calculate (3√2+2)(3√2−2)
√6− (√
3
2+ √
2
3)
(3) Factor 𝑎𝑏2 − 2𝑎𝑏 − 2𝑏 + 4
(4) Solve the quadratic equation (𝑥 − 29)2 − 3(𝑥 − 30) − 31 = 0
(5) F’s high school had a two-day culture festival. F’s class sold candy and soda, and stocked
up with 140 pieces of candy and 240 cans of soda. On the first day, the candy was sold for
¥100 a piece, and the soda for ¥80 a piece, for a total of 𝑥 pieces of candy and 𝑦 cans of
soda. On the second day, a piece of candy and a can of soda were sold in combo sets of ¥160,
with nothing being sold separately. At the end of the second day, twelve pieces of candy
remained, but the soda was sold out. If the total sales of the two days were ¥30,560, what
are the values of 𝑥 and 𝑦? Please note that 𝑥, 𝑦 are natural numbers, and that sales tax
is ignored.
(6) The numbers 1 to 8 are each written on a single card, and contained in a box. You withdraw
two cards from the box simultaneously. Let 𝑎 be the product of the two numbers of the
cards you withdrew, and 𝑏 be the sum of the six remaining numbers on the cards in the
box. What is the probability that 𝑎 + 𝑏 is an even number greater than 40? Answer as if
each card has an equal likelihood of being withdrawn.
(7) In the graph on the right, 𝑛 represents 𝑦 = 𝑎𝑥2 (𝑎 > 0). A lies on the y-axis, and its y-
coordinate is 1. B is on 𝑛 and its x-intercept is positive. ℓ is a line the passes through A
and B, and has a positive slope. C is the point where ℓ intersects with the x-axis, and its
x-coordinate is 4 less than the x-coordinate of B. 𝑚 is the line that passes through B, and
has a slope of 1
2. D is the point where 𝑚 intersects the x-axis, and its x-coordinate is 3 less
than the x-coordinate of B. Find the value of 𝑎.
(8) Let 𝑛 be a natural number. Find all values of 𝑛 for which 𝑛+110
13 and
240−𝑛
7 are also
natural numbers. Please show your work.
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In figures 1 and 2, the quadrilateral ABCD has sides AB = 3cm and AD = 6cm. E is the
midpoint of side AD. E and B are connected. F is on AE without overlapping points A or E. G
is on the extended line formed by BF, on the opposite side of F from B. The triangle formed by
connecting the three points A, E, and G, is isosceles with AE = AG.
Answer the following questions. If your answer includes a radical, ensure that the number
inside the radical is as small as possible.
(1) In figure 1, prove that △ EFB~ △ GEB.
(2) Figure 2 is modified from figure 1, in that F is the midpoint of AE.
In figure 2, H is a point on the same side off AD as G., and the triangle formed by
connecting the three points D, E, and H, △DEH is congruent to △AEG. G and H, and G
and C are also connected. In this case, GH ∥ AD.
➀ Find the length of GE.
➁ Find the length of GH.
➂ Find the area of the quadrilateral GCDH.
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For figures I to III, the solid ABC-DEF is a triangular prism. △ABC and △DEF are
congruent, with AC = 4cm, BC = 8cm, and ∠ACB = 90°. The quadrilateral ACFD is a square,
and the quadrilaterals ABED and CBEF are rectangles. G is a point on BC, between B and C.
H is on EF, and HF = BG. G and H are connected.
Answer the following questions. If your answer contains a radical sign, ensure that the
number inside the radical is as small as possible.
(1) In figure I, I is a point on DE above H such that it runs parallel to DF. Let BG = HF = 𝑥cm,
where 0 < 𝑥 < 8.
➀ Express the length of IH in terms of 𝑥.
➁ Find the value of 𝑥, when the area of the quadrilateral CGHF is twice that of IHFD.
(2) In figure II, A and G, and A and H are connected. AG = AH. Find the area of △AGH.
(3) In figure III, BG = HF = 2cm. C and H are connected. J is on the extended line AC, on the
opposite side of A from C, and JA = 2cm. J and G, and J and H are connected. K is the point
of intersection of line JG and side AB. L is the point of intersection of line JH and rectangle
ABED. M is on AB such that LM is parallel to AD. In this case, the line LM is perpendicular
to the face ABC. A and L, and K and L are connected.
➀ Find the length of LM.
➁ Find the volume of the solid AKL-CGH.
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２０１７年度 センター試験問題 数Ⅰ・A 抜粋
『見開き 2 ページで日英両言語参照型』である『SUHARA 式 J - CLIL 教育 テキスト』の高校生用の一
例です。これも Seph さんが英語に翻訳してくれたものです。2017 年度分は、Ⅱ・B まで全問完成できて
います。現在、２０１８年度の分を作成中です。
２０１７年１２月、府立八尾高校ロングラン学習にて使用。熱心に取り組む高校生の姿に感激された校長先
生が、『校長メールマガジン』（府立高校の校長先生間のメール）にその様子を掲載してくださいました。『英
語でセンター試験』の学習は、府立高校では初めての画期的な試みでした。
３５ページ以降にその特集があります。
Question 1. (Compulsory. 30 points)
[1] Let 𝑥 be a positive real number that satisfies
{ 𝑥 squared, plus 4 over 𝑥 squared, equals 9}.
Thus,
{ 𝑥 plus 2 over 𝑥, all squared} = _____
and so,
{ 𝑥 plus 2 over 𝑥, equals root _____}.
Moreover,
{ 𝑥 cubed, plus 8 over 𝑥 cubed, equals 𝑥 plus 2 over 𝑥 ,
all times 𝑥 squared plus 4 over 𝑥 squared, minus _____,
which equals_____ times root_____ }.
Also,
{ 𝑥 to the fourth, plus 16 over 𝑥 to the fourth, equals _____}.
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〔2〕 Let p and q be conditions on the real number 𝑥 such that
p: { 𝑥 equals 1}
q: { 𝑥 squared equals 1}
Also, show the negative of p and q as p-bar, and q-bar, respectively.
(1) Pick from one of the four options from 0 to 3 that fill the blanks. However,
the same item may fill more than one blank.
q is _____ for p.
p-bar is _____ for q.
p or q-bar are _____ for q.
p-bar and q are _____ for q.
0. (a) necessary condition(s), but not (a) sufficient condition(s)
1. (a) sufficient condition(s), but not (a) necessary condition(s)
2. necessary and sufficient condition(s)
3. neither (a) necessary nor (a) sufficient condition(s)
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Question 1
[1]
𝑥2 +4
𝑥2= 9 (𝑥 > 0) …➀
(𝑥 +2
𝑥)
2
= 𝑥2 + 2 ∙ 𝑥 ∙2
𝑥+ (
2
𝑥)
2
= (x2 +4
x2) + 4
= 9 + 4 (from equation ➀)
= 13
And since 𝑥 > 0, 𝑥 +2
𝑥> 0 must also be true, so
𝑥 +2
𝑥= √13 …➁
Thus, we set the two equations, resulting in:
𝑥3 +8
𝑥3= (𝑥 +
2
𝑥) (𝑥2 +
4
𝑥2− 𝑎)
Expanding the right side, we get
𝑥3 +8
𝑥3= 𝑥3 + (2 − 𝑎)𝑥 + (4 − 2𝑎) ∙
1
𝑥+
8
𝑥3 …➂
By comparing like-terms on the two sides in ➂, we know that
2 − 𝑎 = 0 and 4 − 2𝑎 = 0
Therefore,
𝑎 = 2
Accordingly,
𝑥3 +8
𝑥3= (𝑥 +
2
𝑥) (𝑥2 +
4
𝑥2− 𝟐 )
= √13 ∙ (9 − 2) (from ➀ and ➁)
= 𝟕√𝟏𝟑
Moreover, since:
(𝑥2 +4
𝑥2)
2
= 𝑥4 + 2 ∙ 𝑥2 ∙4
𝑥2+
16
𝑥4
Then,
𝑥4 +16
𝑥4= (𝑥2 +
4
𝑥2)
2
− 8
= 92 − 8 (from ➀)
= 𝟕𝟑
[2] Conditions 𝑝, 𝑞 on the real number 𝑥:
𝑝: 𝑥 = 1
𝑞: 𝑥2 = 1, which means, 𝑥 = ±1
(1)
・Let’s determine what condition 𝑞 is on 𝑝.
Testing the veracity of 𝑞 ⇒ 𝑝, we find that it is false.
Testing the veracity of 𝑝 ⇒ 𝑞, we find that it is true. ➀
Therefore 𝑞 is a necessary but not sufficient condition on 𝑝. Therefore, (0.) is the best answer.
・Determine what condition 𝑝 is on 𝑞.
𝑝: 𝑥 ≠ 1, meaning, 𝑝: 𝑥 < 1, 1 < 𝑥. ➁
Testing the veracity of 𝑝 ⇒ 𝑞, we find that it is false.
Testing the veracity of 𝑞 ⇒ 𝑝, we find that it is false.
Therefore 𝑝 is neither a necessary nor a sufficient condition on 𝑞. Therefore, (3.) is the best answer.
・Determine what condition(s) 𝑝 or 𝑞 are on 𝑞.
𝑞: 𝑥2 ≠ 1, meaning, 𝑞: 𝑥 ≠ ±1
For 𝑝 and 𝑞, respectively:
𝑥 = 1 𝑜𝑟 𝑥 ≠ ±1, meaning, 𝑥 < −1, −1 < 𝑥
Testing the veracity of (𝑝 or 𝑞) ⇒ 𝑞, we find that it is false.
Testing the veracity of 𝑞 ⇒ (𝑝 or 𝑞), we find that it is false.
Therefore, (𝑝 or 𝑞) is neither a necessary nor sufficient condition on 𝑞. Therefore, (3.) is the best
answer.
・Determine what condition(s) 𝑝 and 𝑞 have on 𝑞.
From ➁, 𝑝 and 𝑞 yield:
(𝑥 < 1,1 < 𝑥) and 𝑥 = ±1, meaning, 𝑥 = −1
Testing the veracity of (𝑝 and 𝑞) ⇒ 𝑞, we find that it is true.
Testing the veracity of 𝑞 ⇒ (𝑝 and 𝑞), we find that it is false.
Accordingly, (𝑝 and 𝑞) is a sufficient, but not necessary condition on 𝑞. Therefore, (1.) is the best
answer.
(2) Condition 𝑟 on the real number 𝑥, is such that:
𝑟 ∶ 𝑥 > 0
Test the veracity of proposition A, (𝑝 and 𝑞) ⇒ 𝑟.
From 𝑝 and 𝑞, we know that:
𝑥 = 1 and 𝑥 = ±1, meaning, 𝑥 = 1
Thus,
A ∶ (𝑝 and 𝑞) ⇒ 𝑟 is true.
Testing the veracity of proposition B, 𝑞 ⇒ 𝑟, we find that it is false.
Test the veracity of proposition C: 𝑞 ⇒ 𝑝.
The contraposition of C is 𝑝 ⇒ 𝑞, which is true from ➀, so C ∶ 𝑞 ⇒ 𝑝 is also true.
Accordingly, (2.) is the best answer.
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『見開き 2ページで日英両言語参照型』以前の教材『関数総論』（抜粋）の一例です。
２０１５年１１月浪速高校勉強合宿と２０１６年１２月府立八尾高校ロングラン学習にて使っています。
関数（Functions）とグラフ（Graphs）
◎ 座標（Coordinates）
y 軸(y-axis)
第１象限
the second quadrant the first quadrant
( 4 , 3 ) coordinates four and three
x 軸(x-axis)
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原点(origin) 軸(axes ⇒単数形は axis)
the third quadrant the fourth quadrant
X-coordinate (abscissa, x-value) is four and y-coordinate (ordinate, y-value) is three.
◎ 関数（Functions）
関数とは何か。 ( What are functions? )
関数とは、ある値を一つ決めるとそれに対応して別の値が一つだけ決まる関係をいう。
通常、ある値を x 別の値を y で表すので、x の値を一つ決めると別の値 y が一つだけ決まる
とき、y は x の関数であるという。
Functions are the relations of one-to-one correspondence that when one value has been decided, another value is decided.
We usually use x as one value and y as the corresponding value. So, when the value of x is decided and there is only one
value of y, y is called a function of x.
◎ 定点を通る直線（a straight line through a fixed point ）
傾き m
y －ｂ
( x , y ) m ＝
x －a
定点 P ( a , b ) ・
∴ y －b ＝m ( x －a )
Therefore, y minus b equals m times open parenthesis x minus a close parenthesis.
このように、直線の傾き m と直線上の１点 P( a , b )がわかっているなら、その直線の式はその点と傾きを用い
て書ける。
In this way, if the slope m of a line, and the coordinates of a fixed point P ( a , b ) are known on the line, then an equation of
the line can be written by using the point-slope form.
２点 A ( x1 , y1 ) , B ( x2 , y2 )を通る直線の式は次のように表される。（但し、x1≠x2）
y－y1 y2－y1 the slope of the line
＝ increase the amount of y
x－x1 x2－x1 ＝
increase the amount of x
y2－y1
∴ y－y1 ＝ (x－x1 ) y minus y sub one over x minus x sub
x2－x1 one equals y sub two minus y sub one
over x sub two minus x sub one.
P( x ,y )
Therefore,
B( x2 , y2 ) y minus y sub one equals y sub two
minus y sub one over x sub two minus
A( x1 , y1 ) x sub one times (the quantity) x minus
x sub one (,close quantity).
An equation of a line may be written by using the two- point form: where A( x1 , y1 ) and B( x2 , y2 ) are two points on the
line.
Expressions：直線の方程式の標準形 (An equation of a line in standard form )
・傾きと切片 ( the gradient / the slope and the y-axis intercept )
y ＝m x＋b the slope-intercept form
・点と傾き ( the point and the gradient / the slope )
y－b＝m ( x－a ) the point-slope form
・２点形 ( two fixed points )
y2－y1
y－y1 ＝ (x－x1 ) the two-point form
x2－x1
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関数には、一次関数・二次関数・三次関数・分数関数・無理関数・指数関数・対数関数・三角関数など数多くのも
のがあり、また二次曲線には円や双曲線などの方程式のグラフもあります。微分や積分も含めますと、高校で学ぶ
数学の半分以上はグラフを使う学習内容です。基本のグラフは１２種類です。これらの式とそのグラフをセットで
覚えることが大切です。なぜならば、ほとんどのグラフはこれらの基本のグラフを以下の４つの操作で描けるから
です。
①平行移動する
②折り返す（対称移動）
③拡大・縮小する
④回転移動する
複雑な方程式のグラフも基本のグラフの式の変化の問題としてとらえることができるのです。グラフが描ければ
関数の理解が容易になり、高校数学が分かるようになります。さあ！グラフを描きましょう。
There are so many functions like the primary function (the linear function), the secondary function (the quadratic function),
the cubic function, the fractional function, the irrational function, the exponential function, the logarithmic function, and the
trigonometric function, and there are also so many quadratic curves such as an equation of a circle, an equation of an ellipse, an
equation of a hyperbola, and so on that we can say more than half of the mathematics learned at a high school is study using
graphs including differential and integral. There are 12 types of following basic graphs. It is important to memorize these
expressions and the graphs as a set. If you can understand these 12 basic graphs and their equations, and use the rules for curve
transformations, you can easily solve the problem of a graph of a complicated equation by using the expression of that basic
graph. The rules for curve transformations follow four path ways, they are;
① Parallel movement
② Symmetric movement
③ Enlargement and Reduction
④ Rotational movement
If you can draw a graph, you’ll be able to understand a function more easily, and be good at high school mathematics. Now,
let’s start drawing graphs!
The notation f (x) means ‘function of x’. A function of x is an expression which (usually) varies, depending on the value of x.
Any function f (x) can be transformed.
① 平行移動 (Translation / parallel movement)
y = f (x)上の点 A ( X , Y )を x 軸の方向に p、 y 軸の方向に q 平行移動した点を B ( x , y )とする。
点 A ( X , Y )は y = f (x)上の点だから、Y＝f (X)を満たす。
y y = f (x) を x 軸に p、y 軸に q 平行移動すると
x = X + p ⇒ X = x －p
y = Y + q ⇒ Y = y －q
Y＝f (X)に代入すると
∴ y －q ＝ f (x －p)
B ( x , y )
y = f (x)
A ( X , Y ) q
p
x
0
通常は元の関数 y = f (x)上の点 A ( x , y )を平行移動した点 B ( X , Y )を考えて説明するのが一般的である。これで
は最後に Y －q ＝ f (X －p)の式を( x , y )平面に戻し、X を x に Y を y に置き換えなければならない。したがって
上記のように、最初から y = f (x)上の点を A ( X , Y )として説明した方が分かりやすいと思う。
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使用テキスト（SUHARA 式）抜粋
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