Maths in Germany, Klasse 4 (Jahrgangsstufe 4) to Klasse 10, and where our lectures fit

Hook

Look at this pen for a pony. Its sides are written with x. To find the length of fence, we must build an expression and make it simpler.

But first, the words of algebra. Every other lecture uses them.

Explanation

1. Algebra has its own words. A term is a piece that is added or taken away. In three x plus six, the terms are three x and six.
2. A factor is a piece that is multiplied. In three x, the factors are three and x. In three times x plus two, the factors are three and the bracket.
3. An expression is terms put together, like three x plus six. It has no equals sign. You can simplify an expression, but you cannot solve it.
4. An equation has an equals sign, like three x plus six equals twelve. It is true only for some values of x, and solving finds them.
5. An identity is true for every value of x, like two times x plus three, equals two x plus six. A formula links quantities, like area equals length times width.
6. And the golden rule for fractions: you may cancel factors, never terms. Two x plus six, over two, is not x plus six. The two divides every term, so it is x plus three.

Demonstration

1. Let us simplify an expression: three times x plus two, plus two x. On the left is the question, and it stays the same. I make the copy after the equals sign simpler.
2. The three is a factor of the whole bracket, so it multiplies every term inside: three x, and three times two.
3. Three times two is six.
4. Three x and two x are like terms: both are a number times x. Together they make five x. Six has no x, so it stays on its own.
5. Five x plus six. The like terms are joined and the bracket is gone, so it is as simple as it gets. It is an identity: true for every x.

Exercise

Your turn. Tap the three to multiply out the bracket. Then join the like terms.

Well done. Three times x plus two, plus two x, is five x plus six.

Hook

Every day we meet problems like this one. A taxi, a phone bill, a plant that grows. To solve them, we use equations.

But first, you need to know how to solve the equation. Let me show you how.

Explanation

1. Today I am going to show you how to solve an equation with fractions. First I will explain. Then I will do it in front of you. Then you will do it on your own.
2. Here is our equation: x divided by three, plus two, equals four.
3. The equals sign means both sides are the same. If we change one side, we must change the other side in the same way.
4. x over three means x is cut into three equal parts, and we have one part. Then two is added to it.
5. First we get rid of the fraction. Multiply everything on both sides by three. Then x over three becomes just x.
6. Next we get rid of the number next to x. Take it away from both sides. Now x is on its own.
7. So the plan is: clear the fraction, then clear the extra number. Watch me do it once. Then you try.

Demonstration

1. Let us start. x over three, plus two, equals four.
2. I pick up a three. Every part of both sides gets multiplied by it.
3. Three on the left side.
4. And three on the right side. Both sides are three times bigger, so they are still equal.
5. The three multiplies each part inside the bracket.
6. Three divided by three is one. So x over three, times three, is just x.
7. Three times two is six.
8. Three times four is twelve. No more fraction. Now it says x plus six equals twelve.
9. Now I take six away from both sides.
10. Six take away six is zero. It is gone.
11. Twelve take away six is six. So x is six.
12. Let us check. Six divided by three is two. Two plus two is four. It works. Try moving the numbers yourself. If you get stuck, ask me a question.

Exercise

Your turn. Take the three from the tray and multiply both sides. Then take away the six. I will watch.

Well done. You found that x is six. Great work.

Hook

Look at this problem. A rectangle's area and width are written with x. To find its length, we must divide one expression by another, and then make the answer simpler.

But first, let me show you how to simplify a fraction with letters in it.

Explanation

1. This fraction has letters in it. On top: x squared plus five x plus six. On the bottom: x plus two. We will make it simpler.
2. A fraction gets simpler when the same thing is on the top and on the bottom. But we cannot cross out a piece of a sum. First we turn the top into brackets.
3. To do that, find two numbers that multiply to make six and add to make five. They are two and three. So the top is x plus two, times x plus three.
4. Now x plus two is on the top and on the bottom. Anything divided by itself is one. So they cancel, and x plus three is left.
5. One warning. If x was minus two, the bottom would be zero. You cannot divide by zero. So we say x is not allowed to be minus two.
6. So: make brackets, cancel the match, and note the value that is not allowed. Watch me, then you try.

Demonstration

1. Here is the fraction, written twice. On the left is the question, and it never changes. After the equals sign is a copy. I make the copy simpler, one step at a time.
2. Now the brackets go back into the fraction, in place of the top.
3. x plus two is on the top and the bottom. They cancel. And I note that x cannot be minus two.
4. What is left is x plus three. So the fraction is equal to x plus three, for every x except minus two. If you get stuck, ask me.

Exercise

Your turn. Tap the top of the fraction to make brackets. Then drag the matching bracket onto the one below the line.

Well done. You made brackets and cancelled the match. The fraction is x plus three, and x may not be minus two.

Hook

Look at this garden. Its length and its width are fractions with x in them. To find its area, we must multiply two fractions.

But first, let me show you how to multiply fractions with letters.

Explanation

1. Today we multiply two fractions with letters. x plus one over two x, times four x over x plus one.
2. Multiplying fractions is easy. Top times top, and bottom times bottom. You get one fraction.
3. Then we make it simpler. Anything that is on both the top and the bottom cancels. Each cancelled bracket gives a value that is not allowed.
4. Dividing has one extra step. To divide by a fraction, turn it upside down and multiply. Dividing by a half is the same as multiplying by two.
5. So: multiply tops and bottoms, cancel the matches, note the values that are not allowed. Watch.

Demonstration

1. Here are the two fractions, multiplied together. On the left is the question, and it stays as it is. I make the copy after the equals sign simpler.
2. Top times top, bottom times bottom. Now it is one fraction.
3. There is an x on the top and an x on the bottom. They cancel. So x cannot be zero.
4. And x plus one is on the top and the bottom. They cancel too. So x cannot be minus one.
5. Four over two is two. So the whole thing is just two.
6. So the whole product is just two, for every x except zero and minus one. If you get stuck, ask me.

Exercise

Your turn. Tap the fractions to multiply them. Then drag each matching piece from the top onto the same piece below the line.

Well done. You multiplied and cancelled. It is just two, and x may not be zero or minus one.

Hook

Here is a journey in two parts. Each part takes a fraction of an hour with x in it. To find the total time, we must add two fractions.

But first, let me show you how to add fractions with letters.

Explanation

1. Today we add two fractions with letters. Two over x plus one, plus three over x minus two.
2. You can only add fractions when the bottoms are the same. Halves add to halves. Thirds add to thirds. Here the bottoms are different, so first we make them the same.
3. Each fraction is missing the other one's bottom. So we multiply the top and the bottom of each fraction by the bracket it is missing. The value stays the same.
4. Now both bottoms are x plus one, times x minus two. So we add the tops. Then we open the brackets and collect the like terms.
5. Taking away works the same way. Just keep the minus sign on the top. Watch me add these two.

Demonstration

1. Two over x plus one, plus three over x minus two. The question stays on the left. I work on the copy after the equals sign.
2. I give each fraction the other one's bottom, on the top and the bottom. Now the bottoms match.
3. The bottoms match, so I add the tops over the same bottom.
4. Now I open each bracket on the top.
5. Two x and three x make five x. Minus four and three make minus one.
6. Five x minus one, over x plus one times x minus two. One fraction, and it cannot be made simpler. If you get stuck, ask me.

Exercise

Your turn. Drag one fraction onto the other to make the bottoms match. Do it again to add them. Tap the brackets on top to open them. Then join the like terms.

Well done. Match the bottoms, add, open the brackets, collect. You added fractions with letters.

Hook

Here is a rectangle. We know its area, and its sides are written with x. That gives an equation with x squared in it.

But first, let me show you how to solve an equation with x squared in it.

Explanation

1. An equation with x squared in it is called a quadratic equation. It usually has two answers.
2. Here is the key idea. If two numbers multiply to make zero, one of them must be zero. Three times zero is zero. Zero times five is zero.
3. So first, get zero on one side. Then factorise the other side into two brackets, just like in the simplifying lecture.
4. Now either bracket could be the zero. Set each bracket equal to zero and solve it. Each one gives an answer.
5. Always check each answer in the equation. And in a real problem, ask if the answer makes sense. A length cannot be negative.

Demonstration

1. Here is x squared plus five x plus six, equals zero. Zero is already on one side.
2. Two and three multiply to six and add to five. So it is x plus two, times x plus three, equals zero.
3. The two brackets multiply to make zero, so one of them is zero. First, x plus two equals zero.
4. Take two from both sides, and x is on its own.
5. So x is minus two. Keep that answer. Now the other bracket: x plus three equals zero.
6. Take three from both sides.
7. So x is minus two, or minus three. Check minus two: four, minus ten, plus six, is zero.

Exercise

Your turn. Factorise, set each bracket equal to zero, and solve each one.

Well done. x is minus two or minus three. Two answers, one from each bracket.

Hook

A boat goes upstream and downstream in the same time. That gives us an equation with fractions in it, and we must find x.

But first, let me show you how to solve an equation with fractions in it.

Explanation

1. Our last lesson. Three over x plus one, equals one over x minus one. This time we find x.
2. The bottoms are brackets with x in them. To clear a bracket from the bottom, multiply both sides by it. It cancels on one side and appears on the other.
3. Every time a bracket cancels, note the value that is not allowed. x plus one is zero when x is minus one. x minus one is zero when x is one.
4. When both bottoms are gone, a normal equation is left. Open the brackets, collect the x terms, and divide.
5. At the end, compare your answer with the values that are not allowed. If they are the same, the answer is wrong. Watch.

Demonstration

1. Three over x plus one, equals one over x minus one.
2. I multiply both sides by x plus one.
3. On the left, the bracket cancels with the bottom. So x cannot be minus one. Three is left.
4. Now I multiply both sides by x minus one.
5. It cancels on the right. So x cannot be one. No more fractions.
6. Open the bracket. Three x minus three equals x plus one.
7. Take x away from both sides, and collect.
8. Add three to both sides, and collect again.
9. Two x equals four. Divide both sides by two.
10. x is two. Now check. Two is not minus one, and two is not one. So the answer is good. Try moving the numbers yourself. If you get stuck, ask me a question.

Exercise

Your turn. Drag a bottom bracket to the Multiply spot. Cancel it with its match. Do the same for the other one. Then finish the equation and check the values that are not allowed.

Well done. You solved it, and x is allowed: it does not make any bottom zero. Now try one on your own.

Hook

Look up at the end of the roof. This triangle is called the pediment. Its top angle is one hundred and fifty-four degrees. How big are the two angles at the bottom?

But first, the angle facts. Every geometry question uses them.

Explanation

1. Angles on a straight line add up to one hundred and eighty degrees. Angles all the way round a point add up to three hundred and sixty.
2. The three angles inside a triangle add up to one hundred and eighty degrees. In an isosceles triangle, two sides are equal, and so are the two angles at their feet.
3. When a line crosses two parallel lines, alternate angles are equal: they make a Z shape. Corresponding angles are equal too: they make an F shape.
4. Co-interior angles sit between the parallel lines, on the same side. They make a C shape, and they add up to one hundred and eighty degrees.
5. In the exam, every angle you find needs a reason. Say which fact you used, not just the number.

Demonstration

1. Here is the pediment. The two bottom angles are equal, so call each one x. The angles in a triangle add up to one hundred and eighty, so two x plus one hundred and fifty-four equals one hundred and eighty.
2. So each bottom angle is thirteen degrees. The reason: angles in a triangle add up to one hundred and eighty, and the base angles of an isosceles triangle are equal.

Exercise

Your turn. Take one hundred and fifty-four away from both sides, then share both sides by two.

Well done. x is thirteen degrees. Check: thirteen, plus thirteen, plus one hundred and fifty-four, is one hundred and eighty.

Hook

Look at this column. Its surface is cut into twenty grooves called flutes, so its cross-section is a regular polygon with twenty sides. How big is each corner?

But first, let me show you the angles of any polygon.

Explanation

1. A polygon is a flat shape with straight sides. A pentagon has five sides, a hexagon six, and an octagon eight.
2. Split a polygon into triangles from one corner. There are always two fewer triangles than sides. So the angles inside add up to n minus two, times one hundred and eighty.
3. Walk round the outside of any polygon, and you turn all the way round once. So the outside angles always add up to three hundred and sixty.
4. In a regular polygon, all the sides and all the angles are equal. Share the total between the angles.

Demonstration

1. Here is a regular octagon, with eight equal angles. Eight sides make six triangles, and six times one hundred and eighty is one thousand and eighty. So eight x equals one thousand and eighty.
2. Each angle is one hundred and thirty-five degrees.

Exercise

Your turn. Share both sides by eight.

Well done. Each angle of a regular octagon is one hundred and thirty-five degrees.

Hook

The builders of this temple needed perfect right angles at its corners. They used a rope with knots in it, making a triangle with sides three, four and five. Why does that work?

The answer carries my name. Let me show you my theorem.

Explanation

1. A right-angled triangle has one square corner. The longest side is opposite the right angle. It is called the hypotenuse.
2. Draw a square on each side. The square on the hypotenuse is exactly as big as the other two squares together. We write a squared plus b squared equals c squared.
3. Three, four, five: nine plus sixteen is twenty-five. That is why the builders' rope always makes a right angle.
4. To find the hypotenuse, square the two short sides, add them, and take the square root. To find a short side, take away instead of adding.

Demonstration

1. Here is a right-angled triangle. The short sides are six and eight, and the hypotenuse is x. So x squared equals six squared plus eight squared.
2. The hypotenuse is ten. Check: thirty-six plus sixty-four is one hundred, and ten squared is one hundred.

Exercise

Your turn. Work out the squares, add them, then take the square root of both sides.

Well done. The hypotenuse is ten.

Hook

How tall is a column? You cannot climb it with a tape measure. But you can stand back, measure the angle up to its top, and let a triangle do the rest.

First, the three ratios of a right-angled triangle.

Explanation

1. Pick an angle in a right-angled triangle. The side across from it is the opposite. The side next to it is the adjacent. The longest side is the hypotenuse.
2. Sine is opposite over hypotenuse. Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. Remember it as SOH, CAH, TOA.
3. Some values are exact, and you need them without a calculator. Sine of thirty degrees is one half. Cosine of sixty degrees is one half. Tangent of forty-five degrees is one.
4. Write the ratio as an equation, put in the exact value, and solve it, just like an equation with a fraction.

Demonstration

1. The hypotenuse is twelve, the angle is thirty degrees, and we want the opposite side, x. Opposite and hypotenuse: that is sine. Sine of thirty is one half, so x over twelve equals one half.
2. So x is six. Half the hypotenuse, because sine of thirty is one half.

Exercise

Your turn. Multiply both sides by twelve to clear the fraction.

Well done. The opposite side is six.

Hook

Long ago, Thales measured a pyramid without climbing it. He used its shadow. We can do the same with a column of this temple.

First, what it means for two shapes to be similar.

Explanation

1. Two shapes are similar when one is an enlargement of the other. The angles stay the same, and every side is multiplied by the same scale factor.
2. Match the sides that sit in the same place. The ratio of matching sides is the same all the way round.
3. Write one ratio equal to the other, as an equation with fractions. Then solve it.
4. Careful with areas and volumes. If the lengths are multiplied by k, areas are multiplied by k squared, and volumes by k cubed.

Demonstration

1. Here are two similar triangles. The small one has sides four and ten. The big one has matching sides six and x. So x over ten equals six over four.
2. So x is fifteen. The scale factor is one and a half, and ten times one and a half is fifteen.

Exercise

Your turn. Clear the fractions, then find x.

Well done. x is fifteen.