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Teaching video
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Grade 7 chapters and video availability01 · Read and understand
What you will learn
- Explain how to recover the constant of proportionality.
- Solve the two practice problems and explain how the assumptions affect the answers.
Before you start
Ratios and multiplication.
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
A recipe uses 6 cups of water for 4 cups of concentrate. For the same mixture, write water w as a function of concentrate c and find w when c = 10.
Why this math matters
A proportional relationship has one output-to-input ratio and passes through the origin. Use proportional rules to scale ingredient lists while checking that no fixed starting amount is present.
Set up the model
A useful answer starts with clear assumptions:
- Every batch keeps the same mixture ratio.
- Both amounts are measured with the same cup size.
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.
Recover the constant of proportionality
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A recipe uses 6 cups of water for 4 cups of concentrate. For the same mixture, write water w as a function of concentrate c and find w when c = 10.
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 quantities
k = w/c = 6/4 = 1.5
The constant is cups of water per cup of concentrate.
Apply the relationship
w = 1.5c; w(10) = 1.5 × 10
Multiplying any concentrate amount by the constant preserves the ratio.
Check and interpret
w(10) = 15 cups; 15/10 = 6/4
The original and enlarged batches have matching ratios.
The result
w(10) = 15 cups; 15/10 = 6/4
The original and enlarged batches have matching ratios.
Common mistakes to catch
- A constant difference between output and input is not a proportional constant.
- One nonzero table row alone cannot prove every possible row follows the same rule.
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
A proportional table contains (x,y) = (3,12), (5,20). Find its rule.
Show a hint
Compute y/x for both rows.
Reveal answer and explanation
y = 4x
Both ratios equal 4, so the proportional constant is 4.
Practice 2
Does y = 4x + 2 represent a proportional relationship?
Show a hint
Evaluate the output at zero input.
Reveal answer and explanation
No
At x = 0 the output is 2. A proportional relationship must give zero output at zero input.
Take the idea with you
Use proportional rules to scale ingredient lists while checking that no fixed starting amount is present.
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: Detect a fixed fee in a table
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