June 2025 Paper 1 Q5
5 Scientists are comparing \(h\), the average height of a child in cm, with \(t\), the age of the child in years.
They suggest the model \(h = at + b\) for \(t \geqslant 2\), where \(a\) and \(b\) are constants.
They find that the average height of a 2 year old child is 87 cm and the average height of a 5 year old child is 108 cm.
Use the model to predict Sam’s height. [2]
| Scheme | Marks | AO |
|---|---|---|
| \(2a + b = 87\), \(5a + b = 108\) | B1 | 3.3 |
| \(3a = 21\) | M1 | 3.1a |
| \(a = 7\) | A1 | 1.1 |
| \(14 + b = 87\) \(b = 73\) | A1 | 1.1 |
| [4] |
Notes
B1: State two correct equations soi
M1: Attempt to solve their two linear simultaneous equations. As far as eliminating one variable. M1 implied by correct final answer, if no method shown
A1: Obtain correct value for \(a\)
A1: Obtain correct value for \(b\)
Alternative method (graphical approach)
| Scheme | Marks | AO |
|---|---|---|
| gradient \(= \frac{108-87}{5-2}\) oe | M1 | |
| gradient \(= a = 7\) | A1 | |
| \(87 = 14 + b\) or \(108 = 35 + b\) | M1 | |
| \(b = 73\) | A1 |
M1: Attempt gradient of line. M1 implied by correct \(a\)
A1: Obtain \(a = 7\)
M1: Attempt to find value for \(b\). Intercept of line or algebraic method. M1 implied by correct \(b\)
A1: Obtain \(b = 73\)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(h = 7 \times 4 + 73\) | M1 | 3.4 |
| \(= 101\) (cm) | A1FT | 1.1 |
| [2] | ||
| (ii) Model only for ‘average’ height so will be variations Sam could be exactly 4 years old or 4 years and 364 days – their height won’t stay the same throughout | B1 | 3.5a |
| [1] |
Notes
(b)(i)
M1: Substitute \(t = 4\) into their model
A1FT: Condone no units, but A0 if incorrect eg 101m. FT their model. Or 1.01 (m)
(b)(ii)
B1: Any sensible reason as to why the model for ‘average height’ may, or may not, be accurate in predicting Sam’s height
eg height varies a lot between children of the same age
eg could depend on different diets and lifestyle
eg might be accurate as long as Sam is ‘average’
B0 for interpolation, including comments about Sam’s height being between 87cm and 108cm
Exemplar responses for Q5(b)(ii)
| Response | Mark | Comment |
|---|---|---|
| Doesn’t take into account factors that affect height, such as genetics | B1 | Assume relevant to Sam |
| Not accurate as not all people will be the same height at the same age. | B1 | Implies that Sam may not be the average height. |
| Not likely to be accurate as equation only predicts the average height. People have different heights so unlikely Sam has exactly the same height as the average. | B1 | Comment that model is about average height and Sam may not be average |
| Sam may have a condition that stunts his growth. Therefore, he may not have the average height of a 4 year old. | B1 | Comment that model is about average height and Sam may not be average |
| Not accurate as a lot of children will be below or above the average height. | B1 | Comment that model is about average height and Sam may not be average |
| It is an average height prediction of 4 yr olds, so won’t be exactly accurate | B1 | Implies Sam may differ from the model |
| Model doesn’t consider if Sam has just turned 4 or is nearly 5. | B1 | Comment dealing with Sam's age not being a discrete value |
| Inaccurate as children are unlikely to grow at a constant rate throughout their life. | B0 | Not relevant to predicting Sam's height at 4 |
| Accurate since it is within the range of data. | B0 | Interpolation |
| Not quite accurate as things like gender could affect height. | B0 | Commenting on model not prediction |
| Inaccurate as children are unlikely to grow at a constant rate throughout their life. | B0 | Not relevant to predicting Sam's height at 4 |
| Scheme | Marks | AO |
|---|---|---|
| Increase in growth unlikely to continue as a linear model | B1 | 3.5b |
| [1] |
Notes
B1: Any sensible comment
eg growth spurt at 12 due to puberty
eg people grow quicker when younger
Extrapolation not reliable
B0 for comments about individual variations as question refers to ‘average height’ model
Exemplar responses for Q5(c)
| Response | Mark | Comment |
|---|---|---|
| 157cm is a very large average for a 12 year old as growth slows and comes to a stop at an age which this model doesn’t show. | B1 | For second comment acknowledging a change in the growth rate. |
| At 12 children start having growth spurts so don't grow at the same rate as those aged 2 - 5 | B1 | References change in growth rate |
| The child will not grow at the same rate throughout childhood | B1 | References change in growth rate for children |
| Does not take into account the gender of the child as boys and girls can have growth spurts at different ages due to puberty | B1 | References growth rate changing in all children due to puberty (only gender is not enough) |
| 12 year olds will grow at a slower rate than toddlers | B1 | References change in growth rate |
| Child may stop growing at age 12 | B1 | References change in growth rate |
| Children do not grow at a constant speed | B0 | Does not reference a change in the model and seems to imply original model is wrong. "...continue to grow at constant speed.." would be B1 |
| 12 years of age is when children tend to hit puberty therefore the average height tends to vary very much | B0 | Does not reference change in growth rate |
| 7(12) + 73 = 157cm which is too tall for a 12 year old | B0 | Does not refer to a limitation of the model |
| Around 12 years old the different sexes start to display large differences in height so this would not be an accurate average | B0 | Does not reference growth rate |
| Some children may grow slower than others due to different lifestyle | B0 | References individual children rather than the average |
| At puberty boys tend to grow faster than girls | B0 | Gender difference is not relevant as model is about average |
| Does not take into account the gender of the child | B0 | Gender difference is not relevant as model is about average |