Graphical representations

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Test first differences for constancy
Explain a non-linear counterexample

Vocabulary: first difference · linear sequence · quadratic sequence · term

Retrieve previous learning

What is the difference between successive terms 4, 7, 10?

Check: Each difference is 3.

Learn the concept

For terms 4, 7, 10, 13, successive differences are 3, 3, 3. A fixed difference gives an arithmetic rule: aₙ = 4 + 3(n − 1) = 3n + 1. Plotting (n, aₙ) at positive integer positions puts the points on a straight line.

For 2, 5, 10, 17, the differences are 3, 5, 7, which are not constant. The values fit aₙ = n² + 1 for n = 1, 2, 3, 4. Their second differences are 2, 2; that supports a quadratic candidate but does not prove a unique continuation from only four terms.

Do not call every rising sequence linear. State the evidence you use: equal changes in the input position must produce equal changes in the output for a linear relationship. Sequence positions are integers; a drawn curve between them is a separate continuous extension.

Worked model

Write 2, 5, 10, 17 under positions n = 1, 2, 3, 4.
Subtract consecutive terms to get 3, 5, 7.
Subtract again to get 2, 2, suggesting a quadratic rule.
Test n² + 1 at all four displayed positions before using it as a candidate.

Guided practice

Classify 6, 10, 14, 18 and 3, 6, 11, 18 by first differences.

Hint: Subtract neighbouring terms in each list.

Check the guided answer

The first has differences 4, 4, 4 and is linear on these positions. The second has differences 3, 5, 7 and is non-linear; it fits n² + 2 for positions 1–4.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A sequence is linear whenever its points look almost straight.”

Correction: Check equal numerical differences for equal position steps; approximate appearance is insufficient.

Extend and reflect

Which formula has terms 2, 5, 10, 17, and what is its tenth value if adopted?

Guidance: aₙ = n² + 1 fits; a₁₀ = 101 under this explicitly chosen rule.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Test first differences for constancy. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Extend a sequence through zero into negative values

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Apply a term rule at positive integer positions
Plot and interpret negative sequence values

Vocabulary: negative output · zero crossing · position · domain

Retrieve previous learning

Calculate 2² − 10.

Check: 4 − 10 = −6.

Learn the concept

A sequence can have positive position numbers but negative term values. For aₙ = n² − 10, the first four terms are −9, −6, −1 and 6. The term number n stays positive; the output moves from below zero to above it between positions 3 and 4.

A table must keep the two roles separate. Points (1, −9), (2, −6), (3, −1), (4, 6) show negative vertical values, not negative term positions. The first differences 3, 5, 7 grow, so the plotted points do not follow a linear rule.

The equation n² − 10 = 0 has real solutions ±√10, but neither is a positive integer position. Thus this sequence has no zero term even though a continuous quadratic extension crosses the x-axis. This distinction prevents misreading a graph as a list of sequence terms.

Worked model

Substitute n = 1, 2, 3, 4 into n² − 10.
Record terms −9, −6, −1, 6 without changing the input positions.
Plot the four separate (n, aₙ) points with a vertical scale below zero.
Check no displayed term is zero and explain the integer-position restriction.

Guided practice

For bₙ = 2n − 7, write its first five terms and locate the first positive term.

Hint: Start at n = 1, not zero.

Check the guided answer

The terms are −5, −3, −1, 1, 3. The first positive term is at n = 4; the rule is linear despite its negative early outputs.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Negative terms require negative position numbers.”

Correction: The input position can be positive while the output term is negative.

Extend and reflect

At which positive integer position does n² − 10 first exceed 20?

Guidance: n² > 30, so n = 6 is first: 6² − 10 = 26.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Apply a term rule at positive integer positions. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Compare straight and curved graphs with a digital tool

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Predict points before graphing
Distinguish linear and quadratic shapes without trusting screen appearance alone

Vocabulary: linear · quadratic · parabola · viewing window

Retrieve previous learning

For y = x², what are the outputs at x = −2 and x = 2?

Check: Both are 4.

Learn the concept

Before graphing y = 2x + 1, calculate (−2, −3), (0, 1), (2, 5). Equal x-steps of 2 change y by 4 each time, so the graph is a straight line. For y = x², the same x-values give (−2, 4), (0, 0), (2, 4); the graph bends and is symmetric around the y-axis.

Enter both rules in a graphing tool with a window such as −3 ≤ x ≤ 3 and −4 ≤ y ≤ 10, then probe the predicted coordinates. A stretched or narrow window can disguise curvature or hide an intercept, so numerical checks matter more than apparent steepness.

A digital graph displays a continuous equation, whereas a sequence uses only selected integer x-values. Label which object you are analysing. If no device is available, a hand table and carefully scaled graph support the same comparison.

Worked model

Make a three-row table at x = −2, 0, 2 for each rule.
Predict the line's outputs −3, 1, 5 and the quadratic's outputs 4, 0, 4.
Enter rules and choose a window containing all six points.
Probe the points and explain straight versus curved structure.

Guided practice

Compare y = −x + 3 and y = x² − 1 at x = −1, 0, 1.

Hint: Substitute each x before viewing a graph.

Check the guided answer

The line outputs 4, 3, 2; the quadratic outputs 0, −1, 0. Constant line differences contrast with the curved quadratic's symmetry.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“If a graphing screen shows one intercept, the equation has only one.”

Correction: Check the window and solve algebraically; another intercept may be off-screen.

Extend and reflect

How does adding 3 to x² change the graph?

Guidance: Every y-value increases by 3, shifting the parabola upward without changing its symmetry axis.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Predict points before graphing. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Read structural features from an equation

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Identify gradient and intercept of a linear equation
Identify symmetry and intercepts of a simple quadratic

Vocabulary: gradient · y-intercept · root · axis of symmetry

Retrieve previous learning

For y = 3x − 2, what is y when x = 0?

Check: −2, so the y-intercept is (0, −2).

Learn the concept

In y = mx + c, m is the change in y for a one-unit increase in x and c is the y-intercept. For y = −2x + 3, gradient −2 means the line falls by 2 each step right; it crosses the y-axis at (0, 3). The equation can be checked at (1, 1).

The graph of y = x² − 4 is a parabola symmetric about x = 0 because x and −x give the same square. Its y-intercept is (0, −4); setting y = 0 gives x = −2 or 2. These features are derived from the equation, not guessed from a drawing.

Not every quadratic has symmetry axis x = 0. For example y = (x − 1)² has its minimum at x = 1. State the exact equation before transferring features from one graph to another.

Worked model

Read m = −2 and c = 3 from y = −2x + 3.
Check (0, 3) and (1, 1) by substitution.
For y = x² − 4, calculate the y-intercept (0, −4).
Solve x² − 4 = 0 to locate x-intercepts at −2 and 2.

Guided practice

Describe y = 4x − 5 using gradient and y-intercept.

Hint: Compare with y = mx + c.

Check the guided answer

Its gradient is 4 and y-intercept is (0, −5). A one-unit increase in x raises y by 4; (1, −1) verifies the rule.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The constant c in y = mx + c is the x-intercept.”

Correction: At x = 0, y = c, so c gives the y-intercept.

Extend and reflect

Find the symmetry axis of y = (x − 2)² + 1.

Guidance: The squared part is smallest when x − 2 = 0, so the symmetry axis is x = 2.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Identify gradient and intercept of a linear equation. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Draw a linear graph from two verified points

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Use gradient and intercept to select efficient points
Check a drawn line with an additional point

Vocabulary: straight line · gradient · intercept · scale

Retrieve previous learning

Find y at x = 0 and x = 1 for y = −2x + 3.

Check: The values are 3 and 1.

Learn the concept

Two distinct points determine a straight line. For y = −2x + 3, start at the intercept (0, 3), then move one unit right and two down to (1, 1). A second point can also be calculated rather than read from the gradient: at x = 2, y = −1.

Choose an axis scale that includes all planned points. Draw the line through the accurately marked points, extending only over the requested domain. A third point is a useful check: if (2, −1) misses the line, revisit the arithmetic, coordinates or ruler placement.

Some equations are not given in y = mx + c form. Rearrange 2y = 6 − 4x to y = 3 − 2x before plotting. Divide every term by two, not only the constant.

Worked model

Rearrange if needed into y = mx + c.
Plot (0, 3) from the intercept.
Use gradient −2 to plot (1, 1) and draw a straight line.
Check the independently calculated point (2, −1) lies on it.

Guided practice

Give two convenient points for y = 3x − 2 and a check point.

Hint: Use x = 0, 1 and 2.

Check the guided answer

Plot (0, −2) and (1, 1). The independently checked point (2, 4) must lie on the same line because 3 × 2 − 2 = 4.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“One point is enough to determine a unique straight line.”

Correction: Infinitely many lines pass through one point; use a second distinct point and verify a third.

Extend and reflect

What is the x-intercept of y = −2x + 3 exactly?

Guidance: Set y = 0: 2x = 3, so x = 3/2; the intercept is (1.5, 0).

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Use gradient and intercept to select efficient points. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Read scale, intervals and meaning from a distance-time graph

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Read coordinates with units
Distinguish rising and flat graph segments

Vocabulary: horizontal segment · rate · axis units · interpolation

Retrieve previous learning

If a journey covers 30 km in 2 hours at a constant rate, what is the average speed?

Check: 30/2 = 15 km/h.

Learn the concept

Consider a fictional cumulative-distance-travelled graph passing through (0 h, 0 km), (2 h, 30 km) and (4 h, 30 km), with straight segments between. During the first two hours 30 km is travelled, so the modelled constant speed is 15 km/h. From hours 2 to 4 no further distance is travelled, indicating a stop in this simplified model.

Read the horizontal and vertical axes with their units before interpreting a point. At t = 1 h, straight-line interpolation gives 15 km travelled; at t = 3 h, the graph gives 30 km travelled in total. A coordinate (3, 30) means time 3 h and cumulative distance 30 km, not the reverse.

Cumulative distance travelled records the length of the route completed, regardless of direction, and cannot decrease. A flat section represents zero additional travel over that interval, so the modelled traveller is stopped; the graph alone does not show the route or prove the traveller was safe.

Worked model

Label time in hours on x and cumulative distance travelled in kilometres on y.
Read points (0, 0), (2, 30), (4, 30) with units.
Calculate rise/run on the first straight segment: 30/2 = 15 km/h.
Identify zero additional distance on the second segment and interpret a stop.

Guided practice

On this graph, what cumulative distance is shown at 1 h and 3 h?

Hint: Interpolate along the first straight segment; then read the flat part.

Check the guided answer

At 1 h the straight-line model gives 15 km travelled. At 3 h the flat segment gives 30 km travelled in total since the journey began.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“The y-value of a cumulative-distance graph is the speed.”

Correction: The y-value is total distance travelled; speed over an interval comes from the segment's gradient.

Extend and reflect

How would the graph differ if the traveller went 10 km back toward the start between hours 2 and 4?

Guidance: Cumulative distance would rise from 30 km to 40 km despite the return direction. Its slope would be 10/2 = 5 km/h, not negative.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Read coordinates with units. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Model two fictional price plans with linear graphs

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Translate fixed and variable costs into equations
Interpret an intersection only within a valid domain

Vocabulary: fixed charge · rate · intersection · domain

Retrieve previous learning

What is £5 plus £2 for each of three visits?

Check: £5 + 3 × £2 = £11.

Learn the concept

A fictional plan A costs £5 plus £2 per visit: A(n) = 5 + 2n. Plan B costs £11 plus £1 per visit: B(n) = 11 + n. Graph n on the horizontal axis and cost in pounds on the vertical axis. Intercepts (0, 5) and (0, 11) are the fixed charges; gradients 2 and 1 are added cost per visit.

To compare costs, solve 5 + 2n = 11 + n, giving n = 6 and cost £17. At n = 4, A costs £13 and B costs £15, so A is cheaper; at n = 8, A costs £21 and B £19, so B is cheaper. The crossing has a practical interpretation because six visits is a nonnegative whole number.

The algebraic lines continue to negative and fractional n, but those values are not meaningful counts of visits. Show or explain a discrete nonnegative integer domain for purchases, and do not imply the prices are actual market offers.

Worked model

Define n as a nonnegative whole-number visit count.
Write A(n) = 5 + 2n and B(n) = 11 + n.
Plot two points for each plan, such as n = 0 and n = 6.
Solve equality to get (6, 17), then compare one count on each side.

Guided practice

Plan C costs £4 plus £3 per visit; D costs £12 plus £1 per visit. Find equal cost.

Hint: Set 4 + 3n = 12 + n.

Check the guided answer

Subtract n and 4 to get 2n = 8, so n = 4 visits. Both formulas then give £16.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“Every crossing of two drawn lines is a usable real-world solution.”

Correction: Check units, integer restrictions and other domain conditions before interpreting the crossing.

Extend and reflect

For the original A/B plans, what if a customer can only buy bundles of five visits?

Guidance: The algebraic crossing n = 6 is unavailable; compare allowed counts such as 5 and 10 instead.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Translate fixed and variable costs into equations. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Find the intersection of two linear relationships

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Read an intersection as a simultaneous solution
Confirm the coordinate algebraically

Vocabulary: simultaneous equations · intersection · substitution · solution

Retrieve previous learning

What is y at x = 2 for y = 2x + 1?

Check: 5.

Learn the concept

The lines y = 2x + 1 and y = −x + 7 meet where the same ordered pair satisfies both. Equate their y-values: 2x + 1 = −x + 7, hence 3x = 6 and x = 2. Either rule gives y = 5, so the intersection is (2, 5).

On a graph, trace the crossing to both axes and use the labelled scale. A sketch may give only an approximate reading, especially for fractional coordinates. Substitution into both original equations is a stronger check: 2(2) + 1 = 5 and −2 + 7 = 5.

Parallel distinct lines have no intersection; coincident lines share every point and do not have one unique intersection. For these two lines, gradients 2 and −1 differ, so one crossing is expected.

Worked model

Plot y = 2x + 1 through (0, 1) and (2, 5).
Plot y = −x + 7 through (0, 7) and (2, 5).
Read the common point (2, 5).
Equate and solve both equations to verify the exact coordinate.

Guided practice

Find where y = x + 3 and y = −x + 7 meet.

Hint: Set x + 3 equal to −x + 7.

Check the guided answer

2x = 4 gives x = 2; then y = 5 in both equations. The intersection is (2, 5).

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“An intersection solves only one of the graphed equations.”

Correction: A shared point must satisfy both equations simultaneously.

Extend and reflect

What happens if the two graph equations are both y = 2x + 1?

Guidance: The lines coincide; infinitely many points satisfy both, so there is no unique crossing.

Where this lesson could lead · Mathematics GCSE → Daskerel Mathematical Sciences

This lesson practises: Read an intersection as a simultaneous solution. Quantitative reasoning supports later mathematical, data and engineering study.

Next school subject: Mathematics GCSE. Check the subject and examination board that fit your school.
Later DASKEREL learning: Daskerel Mathematical Sciences. Explore further study and build evidence from practical work.
Later credential exploration: No subject-specific professional credential is mapped from this lesson. Explore the credential catalogue after choosing a later specialism.
Example career direction: Quantitative Analyst. Explore organisations and local support.

This is an exploratory route, not a promise of admission, a qualification, certification eligibility or employment. Years 7–9 learning is an early foundation; any later credential may require substantial study, experience or local approval.

Year 9 · Algebra · original lesson pack

DASKEREL beta · internal AI review pending

Solve and check a two-graph water model

DASKEREL wrote this teaching and practice using Oak's sequence as a source. View the source lesson at Oak National Academy. Oak has not endorsed this material. Check Oak's terms and separate rights for third-party works.

Learning outcomes

Construct equations from rates and starting values
Interpret a fractional-time crossing within model limits

Vocabulary: initial value · rate of change · intersection · model limit

Retrieve previous learning

At 3 litres per minute, how much is added in 4 minutes?

Check: 12 litres.

Learn the concept

In a fictional model, container A begins with 20 L and fills at 3 L/min, so A(t) = 20 + 3t. Container B begins with 50 L and drains at 1 L/min, so B(t) = 50 − t. Time t is measured in minutes; plot both volumes against the same horizontal time scale.

Equal volumes satisfy 20 + 3t = 50 − t. Thus 4t = 30 and t = 7.5 minutes. Both expressions then give 42.5 L: A gives 20 + 22.5, B gives 50 − 7.5. The intersection is (7.5 min, 42.5 L), and fractional time is meaningful here.

This model assumes constant rates, no overflow and measurements beginning at the same time. If the stated graph covers 0 ≤ t ≤ 20, the crossing lies in its domain and both volumes remain nonnegative. Actual tank capacity or changing flow could alter the situation.

Worked model

Define t as elapsed minutes and label volume in litres.
Write A(t) = 20 + 3t and B(t) = 50 − t.
Solve 20 + 3t = 50 − t to get t = 7.5.
Substitute into both models to check 42.5 L and test the time domain.

Guided practice

Containers C and D have C(t) = 10 + 2t and D(t) = 22 − t. When do they match?

Hint: Equate the volumes, then verify in both rules.

Check the guided answer

10 + 2t = 22 − t gives 3t = 12, so t = 4. Both volumes equal 18 L at that time.

Independent practice

Write your own response to every question before viewing the model answers. Your writing stays in this page session; this activity does not submit, store or grade it. Avoid including private information.

Find and correct a misconception

“A visually apparent graph crossing always proves equal real-world quantities.”

Correction: Verify the equations, axis units, common time and domain before interpreting it.

Extend and reflect

When would container B in the original model reach zero if the constant drain continued?

Guidance: Solve 50 − t = 0 to get t = 50 min, beyond the plotted 0–20 min interval.