Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Compare output changes over equal input intervals.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For x=0,1,2,3, the outputs are 2,5,10,17. Is this table compatible with one linear rule?
Why this math matters
Use a table's rate structure to choose a candidate model and state the modeling assumption. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Test whether a table has constant rate
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For x=0,1,2,3, the outputs are 2,5,10,17. Is this table compatible with one linear rule?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
First differences: 3, 5, 7
The input steps are all one, so output increments can be compared directly.
Develop the calculation
The first differences are not constant
A linear rule would add the same amount at every unit input step.
Check and interpret
The data are nonlinear; y=x²+2x+2 fits all four points
Second differences are two, consistent with a quadratic pattern, but finitely many points do not uniquely determine a global rule.
The result
The data are nonlinear; y=x²+2x+2 fits all four points
Second differences are two, consistent with a quadratic pattern, but finitely many points do not uniquely determine a global rule.
Common mistakes to catch
- Constant first differences test linearity only for equal input steps.
- A pattern in a finite table is not proof of a unique rule beyond the data.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Are outputs 4,9,14,19 at inputs 0,1,2,3 linear?
Show a hint
Compare successive output gaps.
Reveal answer and explanation
Yes; y=5x+4 fits
The constant increment five and initial output four determine that linear fit.
Practice 2
Can unequal input intervals be tested using raw output differences alone?
Show a hint
Rate divides by the input change.
Reveal answer and explanation
No
Use difference quotients Δy/Δx; different step widths change the expected output gap.
Take the idea with you
Use a table's rate structure to choose a candidate model and state the modeling assumption.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Index an arithmetic sequence correctly
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