Test points: y < 2x + 1
(1, 2): 2 < 3, true: in
(0, 3): 3 < 1, false: out
(2, 5): 5 < 5, false: on the line, so not in
Inequalities on graphs Which side of the line is in the region, and is the line itself in?
Higher · AQA · 45 minutes
Students count the whole-number points under a moving line, then decide any region by testing one point, draw its line dashed or solid, and list the integer points where several inequalities overlap. Slides to present, a vote that shows who's stuck, practice and homework.
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The dots are the points with whole-number x and y from 1 to 6. Count the dots in the shaded region x + y ≤ c, the line included. Then count again without the dots on the line, which are marked: that is x + y < c. Move c. What patterns do the two counts make?
Whole-number points: (0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0)
The idea in one minute
An inequality in x and y is true at some points and false at others. Its boundary is the line where the two sides are equal, and every point on one side of it makes it true: test one point off the line to find which side. With < or > the points on the line are not in, so the line is dashed; with ≤ or ≥ they are, so it is solid. Several inequalities together give the region where all of them are true.
The lesson goes from points to regions. Students count the dots in a region as its edge moves, with the line in and then out, test points one at a time, then let one tested point decide a whole side. Above and below come last, and only when y is on its own: 2x + y < 6 and −y > x − 2 show why. Students list the integer points row by row, counting those on solid lines and leaving out those on dashed ones. It is Higher; the stretch counts the points in a growing triangle.
By the end, students can
- decide whether a point is in a region by putting its coordinates into each inequality
- draw a boundary dashed or solid, and choose its side by testing a point
- find the region where several inequalities are all true
- list the integer points in a region, counting those on solid lines only
- write the inequalities that describe a region
Key words
- inequality
- region
- boundary
- dashed line
- solid line
- integer point
Before this lesson
Students should draw straight lines such as y = 2x + 1, x = 3 and x + y = 5, and know what <, ≤, > and ≥ mean on a number line. Regions are taught here.
Taught in Simultaneous equations.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
How many dots are in?
Students count whole-number points in a region as its edge moves, with the line in and then out.
Say“The dots are the points with whole-number x and y from 1 to 6. Count the dots in x + y ≤ c, the line included, for c = 3, 4, 5 and 6. Then count them in x + y < c, with the line left out. Predict both counts for c = 7.”
Ask- Which dots are in one count but not the other? Why?
- What patterns do your two counts make?
- Why is the count for x + y < 5 the same as the count for x + y ≤ 4?
Watch forMixing the two counts: a dot on the line, such as (2, 3) when c = 5, is in x + y ≤ 5 but not in x + y < 5.
For the teacher
Give two minutes. Students count for c = 5 on paper, both ways, before you move c.
Don't explain yet. The questions are the point: how do you know a dot is in without looking, and when is a dot on the line in?
Write both lists of counts on the board. The dots that differ are the ones on the line: the next step draws the line solid for ≤ and dashed for <. For whole numbers, x + y < 5 is the same as x + y ≤ 4: each < count is the ≤ count of one less.
Shorter: Count for c = 5 only, both ways, then move c once (−2 min).
12 min 2 · Explain · you explain
Test a point, then a side
One tested point decides a whole side; the line's style says whether the line is in; then several at once.
Say“Put the point's x and y into the inequality. True: it is in. Every point on the same side of the line gives the same answer.”
Ask- (2, 5) is on the line y = 2x + 1. Is it in y < 2x + 1?
- −y > x − 2 has a >. Is its region above the line? Test (0, 0).
Watch forShading above for every >: x > 2 is to the right of its line, and 2x + y < 6 needs a point tested.
Several at once
y ≥ 1, y < x + 1 and x + y ≤ 5
Row y = 1: x from 1 to 4; (0, 1) is on the dashed line
Row y = 2: x = 2 and 3; (1, 2) is on the dashed line
Row y = 3: x over 2 and at most 2 at once: none
6 points: (1, 1), (2, 1), (3, 1), (4, 1), (2, 2), (3, 2)
For the teacher
First slide: test three points in y < 2x + 1. (1, 2) makes it true and (0, 3) false; (2, 5) gives 5 < 5, false, because it is on the line, and the line is not in. Every point below the line is in: one test decides a whole side.
Second slide: < and > give a dashed line, ≤ and ≥ a solid one. x ≥ 1 is to the right of the solid line x = 1; y < 3 is below the dashed line y = 3. A vertical line has no above or below. Each region is shaded; where the two shadings overlap, both are true.
Third slide: −y > x − 2. Its line is y = 2 − x, dashed. Test (0, 0), which is below the line: 0 > −2 is true, so the region is below, though the sign is >. Multiplying by −1 turns the sign round and gives the same: y < 2 − x. 2x + y < 6, through (0, 6) and (3, 0), is another to test. When the line goes through (0, 0), test another point, such as (1, 0).
Fourth slide: the region for several inequalities is where all of them are true. The slide shades each one; the region is where all three shadings overlap. Find its integer points row by row: those on a solid line count, those on a dashed line don't.
Vote before each test (Ask the class): in, out, or on the line?
Shorter: Skip the −y > x − 2 slide; say test a point, never above (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same.”
Ask- Is the point on a line? Then is that line dashed or solid?
Watch forWriting ≤ for a dashed line, or < for a solid one.
Write the inequality
The region below the dashed line y = x + 1
Below, and the line not in: y < x + 1
Then theirs
The region to the right of the solid line x = −2
Test a point
Is (3, 1) in 2x + y > 6?
2 × 3 + 1 = 7, and 7 > 6: yes
Then theirs
Is (1, 4) in y ≤ 3x?
List the integer points
x > 0, y > 0 and x + y < 4
All three lines are dashed: points on them don't count
(1, 1), (1, 2) and (2, 1)
Then theirs
x ≥ 0, y ≥ x and y < 3
For the teacher
Work the first example aloud, writing each line. Then students do its twin while you watch who writes < for a solid line.
Do the same for each pair. The third pair catches students who leave out the points on a solid line, such as those on y = x.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
25 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
For this length: The stretch, then: why do the hook's counts go up by 2, 3, 4, …?
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Which lines are dashed? Do the points on them count?
Watch forCounting points on a dashed line.
Show your working on paper: test a point for each side, and say whether each line is dashed or solid. The stretch is optional.
The question that shows who's stuck
- How many points with whole-number coordinates are in the region x > 1, y ≥ 1 and x + y < 5?
In your Classroom, as they answer (an example) - A10
- B9
- C3
- D3
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- Is the point (2, 3) in the region y < 2x + 1?
- The boundary line of the region x + 2y ≥ 4 is drawn…
- Write the inequality for the region to the left of the dashed line x = 3.
- Which side of the line 3x + 2y = 12 is in the region 3x + 2y < 12?
- The region −y < 2x − 3 is on which side of the line y = 3 − 2x?
- To shade the region y > 3x, can you test the point (0, 0)?
- How many points with whole-number coordinates are in the region y ≥ 2x, x > 0 and y < 6?
- A triangle has the solid line y = 1 along its bottom, the dashed line x = 4 on its right and the solid line y = x along its top. Which inequalities describe the region inside it?
- Stretch: the region x ≥ 0, y ≥ 0 and x + y ≤ k has exactly 10 points with whole-number coordinates. k is a whole number. What is k?
Support · For question 4: put x = 0 and y = 0 into 3x + 2y < 12. True or false? That decides which side of the line is in.
For the teacher
Everyone chooses before anyone explains. Then ask every student, whatever they chose: is a point on the line x + y = 5 in? One on the line x = 1? One on the line y = 1? Which of the three lines are solid?
Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7–8 are exam-style. The stretch counts the points in a growing triangle.
Students show their working on paper or the board; the quiz keeps their final answers.
If the votes are split, ask each line in turn: dashed or solid, and do the points on it count? Then let them vote again before they practise.
Shorter: After the vote, questions 1–4 only (−6 min).
Longer: The stretch, then: why do the hook's counts go up by 2, 3, 4, …? (+10 min).
10 min 5 · Exit · students show
Find the region, list its points
One region drawn and its points listed show who handles dashed and solid lines.
For this length: Find the region's corners, and say which of them are in
Say“Draw the three lines, dashed or solid, mark the region and list its integer points.”
Watch forCounting points on the dashed line x = 0, or leaving out those on the solid line y = x − 1.
Draw it on paper or the board; list the points and write one sentence.
- Draw the region x > 0, y ≥ x − 1 and y < 3, each line dashed or solid. List every point in it with whole-number coordinates. Which of your points lie on a line, and why do they count?
For the teacher
The region is x > 0, y ≥ x − 1 and y < 3. Draw it, then list every point in it with whole-number coordinates.
Then say in one sentence which of your points lie on a line, and why they count.
The lists show who still counts points on dashed lines: start the next lesson with the hook's dots if many do. Set the homework.
Longer: Find the region's corners, and say which of them are in (+5 min).
One to one: let the student test points aloud before drawing anything, and ask each time whether the point is on the line, and so whether it counts.
Homework: regions and points
Show your working: say whether each line is dashed or solid, and test a point to choose each side.
- Which of the points (1, 2), (2, 4) and (0, −3) are in the region y > 3x − 2? Test each one.
- Write the inequality for each region: (a) to the left of the solid line x = 4; (b) below the dashed line y = 2x.
- Draw the region x + 2y ≤ 6. Is its line dashed or solid? Test (0, 0) to choose the side.
- List every point with whole-number coordinates in the region x ≥ 1, y > 1 and x + y ≤ 5.
- A region is inside the triangle made by the solid x-axis, the solid y-axis and the dashed line x + y = 4. Write the three inequalities that describe it.
- The dashed lines y = 2x + 1 and y = 4 − x cross. Find where. Is that point in the region y < 2x + 1 and y > 4 − x? Say why.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching inequalities on graphs: common questions
How do you teach which side of the line to shade?
Test a point. Put a point that is not on the line, often (0, 0), into the inequality: if it is true, that point's side is in; if not, the other side is. Above and below work only when y is on its own: 2x + y < 6 and −y > x − 2 catch students who rely on them.
Why is the line dashed for < and solid for ≤?
Points on the line make the two sides equal. With ≤ or ≥ equal counts, so those points are in and the line is solid. With < or > it doesn't, so they are out and the line is dashed. It matters most when listing integer points: those on a dashed line never count.
Should students shade the region they want, or the parts they don't want?
Either works if it is clear which: label the region R. Shading the parts not wanted keeps the region clean when three or more lines meet. Follow what the question asks, and say which way the class does it.
Is inequalities on graphs on the Foundation or Higher tier?
Higher. Foundation students solve inequalities in one variable and show them on a number line; inequalities in x and y shown as a region on a graph are Higher. Students should draw straight lines such as x + y = 5 first.
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Curriculum reference: AQA GCSE Mathematics 8300: A22 (linear inequalities in two variables, their solution sets shown on a graph, Higher).
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