At a glance. One lesson of about 60 minutes. It builds on inverse proportion and on the factorizing of lessons 1 to 3. Students sketch y = k/x, name its asymptotes, state the domain and range, and find k from a point.
Calculator: no calculator. No calculator: the values are chosen to divide exactly, and the shape comes from the rule.
What students will learn
Know
- y = k/x has two branches and asymptotes x = 0 and y = 0.
- The domain and the range both leave out 0.
- k = xy at every point, and its sign decides the quadrants.
Understand
- An asymptote is approached, not reached.
- The excluded value x = 0 is why there are two branches.
- Halving x doubles y: the product stays k.
Do
- Sketch y = k/x for positive and negative k.
- State asymptotes, domain and range.
- Find k from a point and use it to find a missing coordinate.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: values, k from a point (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe y as x gets close to 0 and as x gets large.
- Criterion C. Communicating: naming the asymptotes and stating the domain and range.
- Criterion D. Applying mathematics in real-life contexts: tank and journey times (hook, question 7).
The one thing to do with the figure
Ask for the prediction first. Set k = 3 and bring x down one step at a time. Ask students to say what y will be before the next step.
Ask: “What will y be if x is 0.01?” Then ask whether it can ever be exactly 0 or exactly infinite. The graph gets close but never meets the axis, and there is no value at x = 0.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tank; collect predictions for fast and slow pumps. |
| 12 min | Figure, key idea and warning card; sketch y = k/x on mini whiteboards. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what 5 ÷ 0.01 is; if D, ask what 5 ÷ 0 is. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 3; 30; x + 3. |
|---|---|
| Hook | Larger x makes T smaller and closer to 0; x close to 0 makes T very large. |
| Worked examples | 2, −4, 16 with x ≠ 0, y ≠ 0; k = 15 and x = −2.5; x = −4. |
| Faded example | y = −2 and x = 25. |
| 1. Values near the asymptotes | C. A multiplies by x; B loses the sign; D puts a point at x = 0. |
| 2. Evaluating y = k/x | −4.5. 18 ÷ (−4). |
| 3. Solving for x | 24. 18 ÷ 0.75. |
| 4. Finding k from a point | −24. 3 × (−8). |
| 5. Change in y | 6. 8 − 2. |
| 6. Two points on one graph | 3.2. 16 ÷ 5. |
| 7. A time model | 60. 4 − 3 hours. |
| 8. k and the quadrants | B. A loses the sign of k; C has the right k but the wrong quadrant; D adds the coordinates. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Multiplied by x, lost the sign, or put a point at x = 0. | Ask what 5 ÷ 0.01 and 5 ÷ 0 give. |
| 4.5 or −72 in question 2 | Lost the sign, or multiplied. | Divide, then check the sign rule. |
| 13.5 or 17.25 in question 3 | Multiplied or subtracted. | Rearrange 0.75 = 18/x for x. |
| 24 or −5 in question 4 | Lost the sign, or added coordinates. | Say: k is x times y. |
| 9 or 2 in question 5 | Gave the change in x, or only the end value. | Find y at both ends and subtract. |
| 1.25 or 80 in question 6 | Treated the graph as a line, or multiplied k by 5. | Say: k is the same at both points. |
| 1 or 20 in question 7 | Left the answer in hours, or gave the change in speed. | Underline the unit the question asks for. |
| A, C or D in question 8 | Lost the sign of k, used the wrong quadrant, or added the coordinates. | Ask for the sign of x times y at each point. |
Questions worth asking
- Why are there two branches? Listen for: there is no value at x = 0, and positive and negative x give different signs.
- What happens to y when x doubles? Listen for: y halves; the product stays k.
- Does the graph ever touch the x axis? Listen for: no, k/x is never 0.
- What does a negative k do? Listen for: the signs of x and y are opposite, so the second and fourth quadrants.
Differentiation
A way in for anyone stuck
- Give a table of x and y for x = −4, −2, −1, 1, 2, 4 with k given, and ask students to plot it.
- Provide axes with the asymptotes already drawn.
- Give the frame: “The asymptotes are …. The domain is …. The range is ….”
Stretch
- Find where y = 4/x meets y = x + 3.
- Find the equation of the line through (2, 8) and (8, 2) and where it meets y = 16/x.
Students learning in English as an additional language
“Asymptote” is a line a graph gets close to but never touches. “Branch” is one separate part of a graph. “Domain” is the set of x values allowed. “Range” is the set of y values the graph reaches.
| Word | Plain meaning | Example |
|---|---|---|
| asymptote | a line a graph approaches but never reaches | a line y = 2 that a graph gets closer and closer to |
| branch | one separate piece of a graph | the part with x > 0 |
| domain | all the values of x that can be used | every x ≥ 0 for y = √x |
| range | all the values of y the graph reaches | every y ≥ 0 for y = x2 |
| reciprocal | one divided by a number | the reciprocal of 4 is 1/4 |
Sentence starters for explaining aloud or in writing:
- As x gets closer to 0, y ….
- As x gets very large, y ….
- The graph lies in the … quadrants because k is ….
Two things not to say
“The graph hits the axis at infinity.” Say instead: it gets as close as you like but never reaches it.
“Join the two parts through the origin.” Say instead: there is no point at x = 0, so the branches are separate.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- EvaluateFor y = 12/x find y at x = 3 and at x = −6.
4 and −2. - StateWrite the domain and range of y = 7/x.
x ≠ 0 and y ≠ 0.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Find ky = k/x passes through (−4, 5). Find k and y when x = 10.
k = −20 and y = −2. - Earlier work (lesson 3)Write 1/(x + 1) + 1/(x − 1) as one fraction.
2x ÷ (x2 − 1).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why y = 6/x can never be 0.
A fraction with top 6 is 0 only if the top is 0. - ContextA grant of 9000 rupees is shared equally among x students. Write the share and say what happens as x grows.
9000/x; it gets smaller and closer to 0 but never reaches 0.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. “Show me the steps” gives the same states in words.
No calculator is needed.