Unit 11 · Geometry and the Coordinate Plane

Touch the tip of your pencil to paper and lift it. That dot is a point. A point has a location but no size. We name points with capital letters, like point A. Now draw a straight path through A that goes on forever in both directions. That is a line. A line has no ends. We draw arrows on both sides to show it keeps going. We name a line by two points on it, like line AB.

Most straight things we see do not go on forever. A crosswalk stripe starts and stops. That is a line segment, a piece of a line with two endpoints. Segment AB has endpoint A and endpoint B, and it has a length you can measure. A ray is different. It starts at one endpoint and goes on forever in one direction, like the beam from a flashlight. Ray AB starts at A and passes through B.

Here is a quick way to tell them apart. Count the endpoints. Zero endpoints: a line. One endpoint: a ray. Two endpoints: a segment. A common mistake is to name a ray backward. Ray AB and ray BA are not the same, because the first letter must be the endpoint. Ray AB starts at A. Ray BA starts at B and heads the other way.

Open a book partway. The two covers meet at the spine. That opening is an angle. In geometry, an angle is made of two rays that share the same endpoint. The shared endpoint is the vertex. The two rays are the sides. We name an angle with three points and put the vertex in the middle. Angle ABC has its vertex at B.

The size of an angle is the amount of turn between its two sides. Think of a clock. At 3:00 the hands make a square corner. At 1:00 the hands are close together, a small opening. The unit for angle size is the degree. A full turn all the way around is 360 degrees, written 360°. A square corner is one quarter of a full turn: 360 ÷ 4 = 90°. Half a turn, a straight line, is 180°.

The most common mistake is to think a longer side means a bigger angle. It does not. Draw a small V and a giant V with the same opening. Both angles are the same size, because the turn between the sides is the same. Only the opening counts. If you stretch the sides of a 40° angle to twice their length, it is still a 40° angle.

A protractor is a half-circle tool marked from 0 to 180 degrees. Most protractors have two rows of numbers. Each row is a scale. One runs left to right and one runs right to left. To measure an angle, follow three steps. Step 1: put the small hole or dot at the center of the protractor exactly on the vertex. Step 2: turn the protractor so one side of the angle lies along the 0 line. Step 3: find where the other side crosses the curved edge and read the number.

The two scales cause most mistakes. Suppose one side sits on the 0 mark of the inside scale. The other side crosses the edge where the inside scale says 50 and the outside scale says 130. Which is right? Use the scale that started at 0 on your first side. That gives 50°. Now check with your eyes. Does the angle look smaller than a square corner? If yes, the answer must be less than 90, so 50° makes sense and 130° does not.

Sometimes the sides of a drawn angle are too short to reach the edge of the protractor. Extend them with a straightedge. Making a side longer does not change the angle. You can also draw an angle. To draw a 65° angle, draw a ray first. Put the center on its endpoint, mark a dot at 65 on the correct scale, and connect the endpoint to the dot.

A square corner, like the corner of a sheet of paper, is a right angle. It measures exactly 90°. We mark a right angle with a small square drawn in the corner. Any angle smaller than a right angle is an acute angle. Acute angles are sharp and narrow, between 0° and 90°. The tip of a pizza slice is acute. Any angle bigger than a right angle but smaller than a straight line is an obtuse angle, between 90° and 180°.

Sort these angles: 35°, 90°, 125°, 89°, 91°. The 35° and 89° angles are acute, because they are less than 90. The 90° angle is right. The 125° and 91° angles are obtuse, because they are more than 90 but less than 180. Notice that 89° and 91° look almost the same. That is why a protractor matters. A straight angle is exactly 180°. Its two sides form a straight line.

A common mistake is to guess from how the drawing is turned. An obtuse angle that opens downward is still obtuse. Another mistake is calling any big-looking angle a right angle. Use the corner of an index card as a test. If the card fits exactly, the angle is right. If the card covers more than the angle, the angle is acute. If the angle is wider than the card, the angle is obtuse.

Angles add just like lengths. Picture a 90° corner. Draw a ray from the vertex through the middle of it. Now there are two smaller angles. If one is 30°, the other must be 90 − 30 = 60°. Together, 30 + 60 = 90, the whole corner. Angle measure is additive: the parts add up to the whole angle. This lets you find a missing angle without measuring it.

Try this one. Angle ABC is 140°. Ray BD splits it into angle ABD and angle DBC. Angle ABD measures 55°. What is angle DBC? Subtract: 140 − 55 = 85°. Check by adding: 55 + 85 = 140. It works. The same idea works on a straight line. The angles along a straight line add to 180°. If one angle on the line is 110°, the other is 180 − 110 = 70°.

Two mistakes are common. One is subtracting from 90 when the whole angle is 180, or from 180 when the whole is 90. Always ask first: what is the whole? The other is subtracting the wrong way, like 55 − 140. The part is always smaller than the whole, so subtract the part from the whole. When two parts are equal, divide the whole by 2. A 90° corner split into two equal angles gives 45° each.

Look at railroad tracks. The two rails run side by side and never touch, no matter how far they go. Lines like this are parallel. Parallel lines stay the same distance apart everywhere. On a map of Chicago, most north-south streets are parallel to each other, and most east-west streets are parallel to each other. We write that line AB is parallel to line CD with the symbol ∥.

Now look at where a north-south street crosses an east-west street. They meet at a square corner, a right angle. Lines that cross at 90° are perpendicular. The symbol is ⊥. The edges of a door, a window or a page are perpendicular where they meet. Two lines can also cross without being perpendicular. A slanted street crossing a grid street makes one acute and one obtuse angle, so those lines intersect but are not perpendicular.

A common mistake is to think lines that do not cross on the page are parallel. Two lines can be slightly tilted and cross far off the page. Real parallel lines never cross, so check that the distance between them is the same at both ends. Another mistake is calling any crossing lines perpendicular. Check for the square corner. Perpendicular lines always intersect, but intersecting lines are not always perpendicular.

A triangle has three straight sides and three angles. Look at its sides first. If all three sides are the same length, it is equilateral. A yield sign is one. If exactly two sides are equal, it is isosceles. If no sides are equal, it is scalene. Tick marks on a drawing show which sides are equal. Sides with the same number of ticks are the same length.

Now look at the angles. A triangle can have at most one right angle or one obtuse angle. If one angle is exactly 90°, it is a right triangle. If one angle is more than 90°, it is an obtuse triangle. If all three angles are less than 90°, it is an acute triangle. A triangle gets both names at once. A triangle with sides 3 cm, 4 cm and 5 cm and one 90° corner is a scalene right triangle.

A helpful fact: the three angles of any triangle add up to 180°. So if two angles are 50° and 60°, the third is 180 − 50 − 60 = 70°, and the triangle is acute. Common mistake: calling a tipped triangle a different kind. Turn the page; the sides and angles do not change. Another: thinking an equilateral triangle can have a right angle. It cannot, because 180 ÷ 3 = 60, so all three of its angles are 60°.

A quadrilateral is any closed shape with four straight sides. Its four angles always add up to 360°. Sort quadrilaterals by asking three questions. Are any sides parallel? Are any sides equal? Are any angles right angles? A trapezoid has at least one pair of parallel sides. A parallelogram has two pairs of parallel sides, and its opposite sides are equal.

Inside the parallelogram family are three famous shapes. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four equal sides, like a tilted diamond. A square is a parallelogram with four right angles and four equal sides. A kite is different. It has two pairs of equal sides that sit next to each other, not across from each other, and it is not a parallelogram.

Try sorting. Shape 1 has sides 4, 6, 4, 6 and four right angles: a rectangle. Shape 2 has four sides of 5 and no right angles: a rhombus. Shape 3 has only one pair of parallel sides: a trapezoid. Common mistake: thinking a tilted square is a rhombus but not a square. Turning a shape does not change it. If it has four equal sides and four right angles, it is a square, however it sits.

Here is a puzzle. Is a square a rectangle? A rectangle is a quadrilateral with four right angles. A square has four right angles. So yes, a square is a rectangle. It is a special rectangle whose sides all happen to be equal. But a rectangle is not always a square, because a rectangle’s sides can be different lengths. This is a hierarchy: groups inside groups, like dogs inside mammals inside animals.

Read the family tree from the top. All quadrilaterals. Inside them, parallelograms, with two pairs of parallel sides. Inside parallelograms, rectangles (four right angles) and rhombuses (four equal sides). Where rectangles and rhombuses overlap sits the square. So a square is a rhombus, a rectangle, a parallelogram and a quadrilateral, all at once. Any property of every parallelogram, like opposite sides being equal, is also true of every square.

Test yourself with the words always, sometimes and never. A square is always a rectangle. A rectangle is sometimes a square. A rhombus is sometimes a rectangle, when it is a square. A trapezoid with only one pair of parallel sides is never a parallelogram. The most common mistake is reading the hierarchy backward and saying every rectangle is a square. Check the rule. Does every rectangle have four equal sides? No. So the claim is false.

Fold a paper heart down the middle. The two halves land exactly on top of each other. That fold is a line of symmetry. A shape is symmetric if you can fold it along a line and the halves match perfectly. The letter A has one line of symmetry, straight down the middle. The letter H has two, one up and down and one across. The letter F has none.

Count the lines of symmetry in common shapes. A rectangle that is not a square has 2: one through the middle of the long sides and one through the middle of the short sides. A diagonal of the rectangle is not a line of symmetry. Fold along one and the corners stick out. A square has 4: two through the middles of the sides and two diagonals. An equilateral triangle has 3. An isosceles triangle has 1. A scalene triangle has 0.

Two mistakes are common. First, students draw a line that cuts a shape into two equal pieces and call it symmetry. Cutting a rectangle corner to corner makes two equal triangles, but the halves do not fold onto each other, so it is not a line of symmetry. Second, students forget to test diagonal lines. Test every line by folding, on real paper or in your mind. Symmetry shows up in butterflies, snowflakes, faces and buildings.

Half of a shape is drawn on grid paper, with a line of symmetry down the middle. To finish it, take each corner of the half and count its distance from the line. Then mark a matching point, its mirror point, the same distance away on the other side. If a corner is 3 squares left of the line and 2 squares up, its mirror point is 3 squares right of the line and 2 squares up. Connect the mirror points in the same order.

Example. The line of symmetry runs up and down. The half shape has corners at 1 square left and 0 up, 4 squares left and 0 up, and 1 square left and 5 up. The mirror points are 1 right and 0 up, 4 right and 0 up, and 1 right and 5 up. Draw the whole shape and you get a symmetric figure like a wide arrowhead. Fold along the line in your mind. Every corner meets its twin.

Common mistake: copying the half by sliding it over instead of flipping it. A slide keeps a shape facing the same way. A flip turns it around, like a reflection in a mirror. Points close to the line stay close, and points far from the line stay far. Another mistake: measuring from the edge of the paper instead of from the line of symmetry. Always measure from the line.

A coordinate plane is a grid made by two number lines that cross at a right angle. The line going across is the x-axis. The line going up is the y-axis. They cross at a point called the origin, labeled (0, 0). In this chapter we use only the part of the grid where both numbers are 0 or bigger. That part is called the first quadrant.

Every point on the grid has an address made of two numbers, called an ordered pair. The first number is the x-coordinate: how many units to move right from the origin. The second number is the y-coordinate: how many units to move up. To find (3, 5), start at the origin, move 3 right, then 5 up. Say it as ’over 3, up 5’. The x comes first, just as x comes before y in the alphabet.

The order is the whole point. (3, 5) and (5, 3) are different spots. (3, 5) is over 3 and up 5. (5, 3) is over 5 and up 3. Common mistake: reading the y first because the y-axis stands up tall. Another: mixing up points on the axes. (4, 0) sits on the x-axis, 4 units right and 0 up. (0, 4) sits on the y-axis, 0 right and 4 up. The origin, (0, 0), is where you always begin.

To plot is to go from numbers to a dot. To name is to go from a dot to numbers. To name a point, put your finger on it. Slide straight down to the x-axis and read that number: the x-coordinate. Go back to the point and slide straight left to the y-axis: the y-coordinate. Write them as (x, y). A dot that is 6 over and 2 up is named (6, 2).

Plot these four points and connect them in order: A (1, 1), B (5, 1), C (5, 4), D (1, 4), then back to A. A and B are both 1 up, so segment AB is flat, or horizontal. B and C are both 5 over, so BC goes straight up; it is vertical. The shape is a rectangle, 4 units wide and 3 units tall. Points that share a y-coordinate make a horizontal segment. Points that share an x-coordinate make a vertical segment.

Common mistakes when naming: counting the grid lines instead of the spaces, or starting the count at 1 instead of 0. The origin is 0, and the first line to the right of it is 1. Another mistake: reading a label above the point instead of below it. Always trace down to the x-axis first. Then trace left to the y-axis.

How far is it from (2, 3) to (9, 3)? Both points are 3 up, so they sit on the same horizontal line. Count the squares between them, or just subtract the x-coordinates: 9 − 2 = 7 units. Now try (4, 1) to (4, 10). Same x, so the points are on a vertical line. Subtract the y-coordinates: 10 − 1 = 9 units. The distance is the difference of the coordinates that are not the same.

For a trip that turns a corner, add the legs. From (1, 2) go to (6, 2), then up to (6, 8). First leg: 6 − 1 = 5. Second leg: 8 − 2 = 6. Total: 5 + 6 = 11 units. On a city grid, this is how far you walk when you cannot cut through buildings. If each unit is one block, that trip is 11 blocks.

Common mistake: subtracting the wrong pair. From (2, 3) to (9, 3), a student computes 9 − 3 = 6, mixing an x with a y. Always subtract x from x or y from y. Another mistake: counting grid lines instead of spaces, which gives one too many. From 2 to 9 there are 7 spaces, not 8. Check with a tiny case: from 2 to 3 is 1 space, and 3 − 2 = 1.

Marta saves $3 every week. After 1 week she has $3, after 2 weeks $6, after 3 weeks $9, after 4 weeks $12. Make a table with two columns: week and dollars. Now turn each row into an ordered pair: (1, 3), (2, 6), (3, 9), (4, 12). Plot them, with weeks on the x-axis and dollars on the y-axis. The points march up in a straight line.

Why a straight line? Each step right (one more week) goes up the same amount (3 dollars). A pattern that adds the same number each time always plots as points in a straight line. The steeper the line, the bigger the number being added. Saving $5 a week would make a steeper line: (1, 5), (2, 10), (3, 15). Saving $1 a week would make a line that is nearly flat.

Read the graph backward too. A point at (6, 18) says: after 6 weeks, $18. To find the amount after 10 weeks, extend the pattern: 10 × 3 = 30, the point (10, 30). Common mistake: swapping the axes and plotting (3, 1) for week 1. Put the week first, because it is what you choose, and the dollars depend on it. Another mistake: drawing a solid line when only whole weeks make sense. Points are enough.

Start two patterns at 0. Pattern A adds 3 each step: 0, 3, 6, 9, 12. Pattern B adds 6 each step: 0, 6, 12, 18, 24. Line them up. Step 1: 3 and 6. Step 2: 6 and 12. Step 3: 9 and 18. Each B value is exactly double the corresponding A value. That makes sense. B adds twice as much every time, so after any number of steps B has twice as much.

Now graph the pairs (A, B): (0, 0), (3, 6), (6, 12), (9, 18). These points also fall in a straight line, and every point has a y that is twice its x. Try another pair. Pattern C adds 2: 0, 2, 4, 6. Pattern D adds 10: 0, 10, 20, 30. Each D value is 5 times the matching C value, because 10 is 5 times 2.

Common mistake: saying B is always 3 more than A. That is only true at step 1. At step 4, A is 12 and B is 24, a difference of 12, not 3. The relationship that stays true at every step is that B is 2 times A. Ask: what do I multiply A by to get B? If the patterns do not both start at 0, the rule is different, so check the starting numbers first.

A rule is an instruction that makes a pattern. ’Start at 5, add 4’ gives 5, 9, 13, 17, 21. Each number is a term. The first term is 5. To get the next term, add 4. To find the 6th term, keep going: 21 + 4 = 25. Or count the jumps: five jumps of 4 after the start. 5 + 5 × 4 = 5 + 20 = 25. Either way, the 6th term is 25.

Rules can subtract too. ’Start at 50, subtract 7’ gives 50, 43, 36, 29, 22. Check: 43 − 7 = 36, and 36 − 7 = 29. The terms go down by the same amount each time. To find a rule from a pattern, subtract neighbors. In 8, 14, 20, 26, each gap is the same: 14 − 8 = 6, 20 − 14 = 6, 26 − 20 = 6. Rule: start at 8, add 6.

Common mistakes: forgetting the start, so ’start at 5, add 4’ turns into 4, 8, 12. Or adding one extra jump: the 6th term has 5 jumps after the first term, not 6. Count the jumps on your fingers. Term 1 to term 2 is one jump. Another mistake when finding a rule: subtracting in the wrong order. Later term minus earlier term gives the gap.

’Start at 1, multiply by 2’ gives 1, 2, 4, 8, 16, 32. Each term is double the one before. Compare that with ’start at 1, add 2’, which gives 1, 3, 5, 7, 9, 11. The add pattern climbs steadily. The multiply pattern climbs slowly at first and then races ahead. By the 6th term, the doubling pattern is at 32 while the add pattern is at 11. Folding paper gets thick so fast for the same reason. Each fold doubles the layers.

To find a multiply rule from a pattern, divide neighbors. In 3, 12, 48, 192: 12 ÷ 3 = 4, 48 ÷ 12 = 4, 192 ÷ 48 = 4. Rule: start at 3, multiply by 4. Every term is a multiple of the one before it. Check the next term: 192 × 4 = 768. Rules can divide too. ’Start at 80, divide by 2’ gives 80, 40, 20, 10, 5.

On a graph, an add pattern makes points in a straight line. A multiply pattern makes points that curve upward, getting steeper. Common mistake: testing only the first gap. In 2, 4, 8 a student sees 4 − 2 = 2 and says ’add 2’. But 8 − 4 = 4, not 2. Check at least two gaps. Then try division: 4 ÷ 2 = 2 and 8 ÷ 4 = 2. The rule is multiply by 2.

Once you have a pattern, describe what you notice. Rule: start at 3, add 3. Terms: 3, 6, 9, 12, 15, 18. Notice that the terms switch between odd and even. Every term is a multiple of 3. Every other term is a multiple of 6. Each of these is a feature of the pattern. The rule does not say it out loud, but you can explain it. Adding 3 to an odd number gives an even number, and adding 3 to an even number gives an odd number. So the terms must alternate.

Rule: start at 4, add 4. Terms: 4, 8, 12, 16, 20. Every term is even. Why? 4 is even, and even plus even is always even, so the pattern can never reach an odd number. Rule: start at 5, add 10. Terms: 5, 15, 25, 35. Every term ends in 5. Why? Adding 10 changes the tens digit but never the ones digit.

Common mistake: describing only the first few terms. ’The terms are less than 20’ is true for 4, 8, 12, 16 but false for 20. A good feature holds for every term, and you can say why. Another mistake: saying two patterns are the same because they share a few numbers. ’Add 3’ and ’add 6’ both contain 6 and 12, but they are different patterns with different rules.

Chicago’s street addresses work like a coordinate plane. The origin is the corner of State Street and Madison Street downtown. State Street splits the city into east and west. Madison Street splits it into north and south. An address number grows as you move away from that corner. In most of the city, 800 address numbers equal about 1 mile. So an address of 1600 North is about 2 miles north of Madison Street.

Put a simple town on a grid. Each unit is 1 block. The school is at (2, 3), the library at (2, 9) and the pool at (8, 9). School to library: same x, so 9 − 3 = 6 blocks north. Library to pool: same y, so 8 − 2 = 6 blocks east. A walk from the school to the pool by way of the library is 6 + 6 = 12 blocks. Coordinates turn a map question into subtraction.

Grids show up in other places too. A spreadsheet names each cell by a column letter and a row number, like B4. A chessboard names each square the same way, from a1 to h8. In the game Battleship, you call out a letter and a number. Common mistake with real grids: forgetting which direction comes first. A chessboard reads column then row, just like (x, y). Check the labels before you count.

Plot A (2, 1), B (7, 1) and C (7, 4). Connect them and you have a right triangle. Side AB is horizontal: 7 − 2 = 5 units. Side BC is vertical: 4 − 1 = 3 units. The right angle is at B, the vertex where the horizontal and vertical sides meet. Coordinates let you find lengths without a ruler, as long as the sides run along grid lines.

Find a missing corner. Three corners of a rectangle are (1, 2), (6, 2) and (6, 5). The fourth corner must be straight above (1, 2) and straight left of (6, 5). So it shares x = 1 with the first point and y = 5 with the third: (1, 5). The rectangle is 6 − 1 = 5 units wide and 5 − 2 = 3 units tall. Its perimeter, the distance all the way around, is 5 + 3 + 5 + 3 = 16 units.

Common mistake: picking a missing corner because it looks about right. Use the shared coordinates instead. Another mistake: finding the width as 6 + 1 instead of 6 − 1. The width is a difference. Also remember that a slanted side, like the side from A to C in the triangle, cannot be found by subtracting coordinates. Only horizontal and vertical sides work that way.