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so it describes both how steep the line is and its direction."},{"text":"The length of the line (how far it travels between its two endpoints).","correct":false,"explanation":"Length measures the distance between two points, not the rate at which the line rises or falls. Gradient represents steepness: the vertical change per 1 unit across."},{"text":"The intercept of the line (where it crosses the horizontal or vertical axis).","correct":false,"explanation":"An intercept gives a point where the line crosses an axis, rather than its steepness. Gradient measures how much the line rises or falls per unit across."},{"text":"The position of the line (how far it is from the coordinate axes).","correct":false,"explanation":"A line’s position does not determine its gradient because parallel lines can have different positions but equal gradients. Gradient represents vertical change for every 1 unit across."}]},{"id":"2b23e261-2ffc-1fac-5285-4735c56c1d58","cardType":"cloze","points":null,"front":"A line with a positive gradient slopes [ ] from left to right, while a negative gradient slopes downwards.","back":"A line with a positive gradient slopes upwards from left to right, while a negative gradient slopes downwards.","mcqOptions":[{"text":"upwards","correct":true,"explanation":"A positive gradient means $y$ increases as $x$ increases, so the line slopes upwards when read from left to right."},{"text":"downwards","correct":false,"explanation":"This reverses the signs of gradients. A line sloping downwards from left to right has a negative gradient; a positive gradient slopes upwards."},{"text":"horizontally","correct":false,"explanation":"A horizontal line has no vertical change, so its gradient is zero rather than positive. A positive-gradient line slopes upwards from left to right."},{"text":"vertically","correct":false,"explanation":"A vertical line has zero horizontal change, making its gradient undefined rather than positive. A line with a positive gradient slopes upwards from left to right."}]},{"id":"17801eb1-c839-e5d2-76cf-b9ed66aa36f8","cardType":"cloze","points":null,"front":"A line with a positive gradient slopes upwards from left to right, while a negative gradient slopes [ ].","back":"A line with a positive gradient slopes upwards from left to right, while a negative gradient slopes downwards.","mcqOptions":[{"text":"downwards","correct":true,"explanation":"A negative gradient means $y$ decreases as $x$ increases, so the line slopes downwards when read from left to right."},{"text":"upwards","correct":false,"explanation":"This confuses negative and positive gradients. An upward slope from left to right has a positive gradient, while a negative gradient slopes downwards."},{"text":"horizontally","correct":false,"explanation":"A horizontal line has zero vertical change and therefore a gradient of zero, not a negative gradient. A negative-gradient line slopes downwards."},{"text":"vertically","correct":false,"explanation":"A vertical line has an undefined gradient because its horizontal change is zero. A line with a negative gradient slopes downwards from left to right."}]},{"id":"25ff1e11-599a-83b3-710e-a300b982529c","cardType":"basic","points":1,"front":"What is the rise over run formula for gradient?","back":"$$$\\text{gradient} = \\frac{\\text{change in } y}{\\text{change in } x}$$","mcqOptions":[{"text":"$$$\\text{gradient} = \\frac{\\text{change in } y}{\\text{change in } x}$$","correct":true,"explanation":"Gradient is rise over run: divide the vertical change, represented by the change in $y$, by the horizontal change, represented by the change in $x$."},{"text":"$$$\\text{gradient} = \\frac{\\text{change in } x}{\\text{change in } y}$$","correct":false,"explanation":"This reverses rise over run and produces the reciprocal of the gradient. The correct formula divides the change in $y$ by the change in $x$."},{"text":"$$$\\text{gradient} = \\frac{\\text{change in } y}{\\text{change in } x + \\text{change in } y}$$","correct":false,"explanation":"Adding the two changes in the denominator does not calculate steepness. Gradient is rise over run, so it is the change in $y$ divided by the change in $x$."},{"text":"$$$\\text{gradient} = \\frac{\\text{change in } x + \\text{change in } y}{\\text{change in } x}$$","correct":false,"explanation":"This incorrectly includes the horizontal change in the numerator. Gradient compares vertical change alone with horizontal change, using change in $y$ divided by change in $x$."}]},{"id":"d746e1a4-fafc-f4d3-39ca-8f6aa11502b7","cardType":"basic","points":1,"front":"In the coordinate pair $(x, y)$, which value determines the vertical position?","back":"The y-coordinate (the second number).","mcqOptions":[{"text":"The y-coordinate (the second number).","correct":true,"explanation":"In an ordered pair $(x, y)$, the second value is the $y$-coordinate. It gives the point’s vertical position, measured up or down."},{"text":"The x-coordinate (the first number).","correct":false,"explanation":"The first number is the $x$-coordinate, which determines horizontal position rather than vertical position. The second number, the $y$-coordinate, determines vertical position."},{"text":"The origin (the point $(0, 0)$).","correct":false,"explanation":"The origin is the fixed point $(0, 0)$ and does not determine another point’s vertical position. That position is given by the second number, the $y$-coordinate."},{"text":"The gradient (the rate of change).","correct":false,"explanation":"Gradient describes a line’s steepness, not the vertical position of an individual point. In $(x, y)$, vertical position is determined by the second value, the $y$-coordinate."}]},{"id":"46a2046c-721b-f4b5-05f0-b41b3156099c","cardType":"cloze","points":null,"front":"When using a grid to find gradient, the run refers to the [ ] change and the rise refers to the vertical change.","back":"When using a grid to find gradient, the run refers to the horizontal change and the rise refers to the vertical change.","mcqOptions":[{"text":"horizontal","correct":true,"explanation":"The run is the horizontal change between two points, corresponding to the change in $x$. It forms the denominator in the rise-over-run gradient calculation."},{"text":"vertical","correct":false,"explanation":"This swaps rise and run. Vertical change is the rise and corresponds to the change in $y$; the run is the horizontal change in $x$."},{"text":"diagonal","correct":false,"explanation":"The diagonal distance between two points is not used as the run. Run specifically means horizontal change, while rise means vertical change."},{"text":"total","correct":false,"explanation":"Run does not mean the total distance travelled between two points. It is specifically the horizontal change, or change in $x$, used in the gradient formula."}]},{"id":"771b8348-72a3-62be-63c9-d11425efe4d8","cardType":"cloze","points":null,"front":"When using a grid to find gradient, the run refers to the horizontal change and the rise refers to the [ ] change.","back":"When using a grid to find gradient, the run refers to the horizontal change and the rise refers to the vertical change.","mcqOptions":[{"text":"vertical","correct":true,"explanation":"The rise is the vertical change between two points, corresponding to the change in $y$. It forms the numerator in the rise-over-run gradient calculation."},{"text":"horizontal","correct":false,"explanation":"This swaps rise and run. Horizontal change is the run and corresponds to the change in $x$; the rise is the vertical change in $y$."},{"text":"diagonal","correct":false,"explanation":"Rise is not the diagonal distance between the points. It specifically means the vertical change, or change in $y$, while run is the horizontal change."},{"text":"sideways","correct":false,"explanation":"A sideways change is horizontal and describes the run, not the rise. The rise is the vertical change between the two selected points."}]},{"id":"f6f5be18-1d2b-611b-b433-0a92565fe8f2","cardType":"basic","points":2,"front":"A line passes through $(0, 0)$ and $(2, 6)$. Calculate its gradient.","back":"3 (calculated as $\\frac{6}{2}$)","mcqOptions":[{"text":"3 (calculated as $\\frac{6}{2}$)","correct":true,"explanation":"The vertical change is $6-0=6$ and the horizontal change is $2-0=2$. Dividing rise by run gives the gradient $\\frac{6}{2}=3$."},{"text":"$\\frac{1}{3}$ (calculated as $\\frac{2}{6}$)","correct":false,"explanation":"This divides run by rise, giving the reciprocal of the gradient. Gradient is rise over run, so the correct calculation is $\\frac{6}{2}=3$."},{"text":"4 (calculated as $6-2$)","correct":false,"explanation":"Subtracting the coordinates from each other does not compare vertical and horizontal changes. Gradient is rise divided by run, so $\\frac{6}{2}=3$."},{"text":"8 (calculated as $6+2$)","correct":false,"explanation":"Adding the coordinate values does not calculate a gradient. The vertical change must be divided by the horizontal change, giving $\\frac{6}{2}=3$."}]},{"id":"fd71553d-8511-649b-5207-a60c65c9e2a4","cardType":"basic","points":2,"front":"A line passes through $(0, 4)$ and $(2, -2)$. Why is the gradient negative?","back":"Because the line moves downwards (the change in $y$ is $-6$) as it moves to the right.","mcqOptions":[{"text":"Because the line moves downwards (the change in $y$ is $-6$) as it moves to the right.","correct":true,"explanation":"As $x$ increases from 0 to 2, $y$ decreases from 4 to $-2$, so the vertical change is $-6$. A downward movement to the right gives a negative gradient."},{"text":"Because the line moves upwards (the change in $y$ is $6$) as it moves to the right.","correct":false,"explanation":"This reverses the direction of the vertical change. From 4 to $-2$, $y$ goes down by 6, so the change in $y$ is $-6$ and the gradient is negative."},{"text":"Because the line moves downwards (the change in $x$ is $-6$) as it moves to the right.","correct":false,"explanation":"This confuses the vertical change with the horizontal change. The $-6$ is the change in $y$, while the change in $x$ is $2$, so the line slopes downwards to the right."},{"text":"Because the line moves downwards (the change in $y$ is $-2$) as it moves to the right.","correct":false,"explanation":"This uses the final $y$-coordinate instead of the change in $y$. The change is $-2 - 4 = -6$, so the downward movement to the right gives a negative gradient."}]},{"id":"6246192c-6c11-9bb6-1bef-710ff77f24e5","cardType":"basic","points":1,"front":"What is the formal coordinate formula for gradient $m$ using points $(x_1, y_1)$ and $(x_2, y_2)$?","back":"$$$m = \\frac{y_2 - y_1}{x_2 - x_1}$$","mcqOptions":[{"text":"$$$m = \\frac{y_2 - y_1}{x_2 - x_1}$$","correct":true,"explanation":"Gradient is vertical change divided by horizontal change. Using the labelled coordinates, that is the difference in $y$-values over the matching difference in $x$-values."},{"text":"$$$m = \\frac{x_2 - x_1}{y_2 - y_1}$$","correct":false,"explanation":"This swaps the numerator and denominator, giving the reciprocal of the gradient. The correct formula puts the change in $y$ on top and the change in $x$ underneath."},{"text":"$$$m = \\frac{y_2 + y_1}{x_2 + x_1}$$","correct":false,"explanation":"This adds coordinates instead of finding their differences. Gradient measures change, so the correct formula is $m = \\frac{y_2 - y_1}{x_2 - x_1}$."},{"text":"$$$m = \\frac{y_1 - y_2}{x_2 - x_1}$$","correct":false,"explanation":"This reverses only the $y$-subtraction, which changes the sign of the gradient. If you reverse the order on top, you must also reverse it on the bottom."}]},{"id":"68cc360b-a48b-b319-54c5-98053d757bfe","cardType":"cloze","points":null,"front":"When using the gradient formula, the [ ] must always be on the top of the fraction and the x-values on the bottom.","back":"When using the gradient formula, the y-values must always be on the top of the fraction and the x-values on the bottom.","mcqOptions":[{"text":"y-values","correct":true,"explanation":"The numerator of the gradient formula is the vertical change, so it uses the $y$-values. The denominator uses the $x$-values for the horizontal change."},{"text":"x-values","correct":false,"explanation":"Putting the $x$-values on top swaps horizontal and vertical change, giving the reciprocal gradient. The correct numerator is the change in the y-values."},{"text":"$$x$-coordinates","correct":false,"explanation":"This is the same swap as putting horizontal change in the numerator. In the gradient formula, the top must use the y-values, not the x-coordinates."},{"text":"points","correct":false,"explanation":"The formula does not put whole points on top of the fraction. It uses the difference between the y-values on top and the difference between the x-values underneath."}]},{"id":"e4c75f53-c521-9274-8d6c-5b4299850ade","cardType":"cloze","points":null,"front":"When using the gradient formula, the y-values must always be on the top of the fraction and the [ ] on the bottom.","back":"When using the gradient formula, the y-values must always be on the top of the fraction and the x-values on the bottom.","mcqOptions":[{"text":"x-values","correct":true,"explanation":"The denominator of the gradient formula is the horizontal change, so it uses the $x$-values. The $y$-values stay on top as the vertical change."},{"text":"y-values","correct":false,"explanation":"Putting the y-values on the bottom repeats the vertical change in the wrong place. The denominator must be the change in x-values, because gradient is rise over run."},{"text":"$$y$-coordinates","correct":false,"explanation":"This confuses the denominator with the numerator. The y-coordinates give the vertical change on top, while the x-values give the horizontal change on the bottom."},{"text":"gradients","correct":false,"explanation":"The bottom of the formula is not another gradient. It is the change in the x-values, which measures the horizontal distance between the two points."}]},{"id":"735ca7b7-2b87-d608-5f7d-4f5744a37839","cardType":"basic","points":2,"front":"Calculate the gradient of the line passing through $(3, 2)$ and $(7, 10)$.","back":"$$$m = \\frac{10 - 2}{7 - 3} = \\frac{8}{4} = 2$$","mcqOptions":[{"text":"$$$m = \\frac{10 - 2}{7 - 3} = \\frac{8}{4} = 2$$","correct":true,"explanation":"The change in $y$ is $10 - 2 = 8$ and the change in $x$ is $7 - 3 = 4$. Dividing gives $m = 2$."},{"text":"$$$m = \\frac{7 - 3}{10 - 2} = \\frac{4}{8} = \\frac{1}{2}$$","correct":false,"explanation":"This swaps the change in $x$ and the change in $y$, giving the reciprocal. Gradient is vertical change over horizontal change, so the answer is $2$."},{"text":"$$$m = \\frac{10 - 2}{7 + 3} = \\frac{8}{10} = \\frac{4}{5}$$","correct":false,"explanation":"This adds the x-coordinates instead of subtracting them. The horizontal change is $7 - 3 = 4$, so the correct gradient is $\\frac{8}{4} = 2$."},{"text":"$$$m = \\frac{2 - 10}{7 - 3} = \\frac{-8}{4} = -2$$","correct":false,"explanation":"This reverses the y-values but not the x-values, causing a sign error. The order must match, so $\\frac{10 - 2}{7 - 3} = 2$."}]},{"id":"af03f52d-fc5d-8efc-b555-434e5ed8a215","cardType":"cloze","points":null,"front":"The gradient of a vertical line is [ ] because the change in $x$ is 0.","back":"The gradient of a vertical line is undefined because the change in $x$ is 0.","mcqOptions":[{"text":"undefined","correct":true,"explanation":"A vertical line has no horizontal change, so the denominator in the gradient formula is 0. Division by 0 is undefined, so the gradient is undefined."},{"text":"zero","correct":false,"explanation":"Zero gradient belongs to a horizontal line, where the change in $y$ is 0. A vertical line has change in $x$ equal to 0, making the gradient undefined."},{"text":"positive","correct":false,"explanation":"A positive gradient means the line rises as it moves to the right. A vertical line has no run at all, so its gradient is undefined rather than positive."},{"text":"negative","correct":false,"explanation":"A negative gradient means the line falls as it moves to the right. A vertical line has zero horizontal change, so division by 0 makes its gradient undefined."}]},{"id":"5c2fad0c-935f-56fc-b4e6-d4ee8b13c8ad","cardType":"cloze","points":null,"front":"The gradient of a vertical line is undefined because the change in $x$ is [ ].","back":"The gradient of a vertical line is undefined because the change in $x$ is 0.","mcqOptions":[{"text":"0","correct":true,"explanation":"On a vertical line, all points have the same $x$-coordinate. That means the change in $x$ is 0, so the gradient formula would involve division by 0."},{"text":"1","correct":false,"explanation":"This assumes there is one unit of horizontal movement, but a vertical line has no horizontal movement. Its change in $x$ is 0, which makes the gradient undefined."},{"text":"-1","correct":false,"explanation":"A negative change in $x$ would mean moving left between the points. On a vertical line the $x$-coordinate does not change, so the change in $x$ is 0."},{"text":"2","correct":false,"explanation":"This treats the vertical line as if it has horizontal run. For any two points on a vertical line, the $x$-coordinates are equal, so the change in $x$ is 0."}]},{"id":"8b3b234e-dbe6-ccfb-4c69-76efc58f7ab5","cardType":"basic","points":1,"front":"What is a common mistake when substituting negative coordinates into the gradient formula?","back":"Forgetting that subtracting a negative becomes addition (e.g., $2 - (-4) = 6$).","mcqOptions":[{"text":"Forgetting that subtracting a negative becomes addition (e.g., $2 - (-4) = 6$).","correct":true,"explanation":"When a negative coordinate is subtracted, the two minus signs combine to make addition. This is why $2 - (-4)$ equals $6$, not $-2$."},{"text":"Forgetting that adding a negative becomes subtraction (e.g., $2 + (-4) = -2$).","correct":false,"explanation":"This statement is a correct rule for adding a negative, not the key substitution mistake here. The common gradient slip is missing that subtracting a negative becomes addition."},{"text":"Forgetting that subtracting a positive becomes addition (e.g., $2 - 4 = 6$).","correct":false,"explanation":"Subtracting a positive does not become addition; $2 - 4 = -2$. The relevant mistake is with negative coordinates, where $2 - (-4)$ should become $2 + 4$."},{"text":"Forgetting that multiplying a negative becomes addition (e.g., $2 \\times (-4) = 6$).","correct":false,"explanation":"This mixes up multiplication with subtraction in the gradient formula. The common issue is subtracting a negative coordinate, where the operation becomes addition, such as $2 - (-4) = 6$."}]},{"id":"70872edc-78df-d075-b65b-004ec4bdb055","cardType":"basic","points":1,"front":"When using the formula $m = \\frac{y_2 - y_1}{x_2 - x_1}$, does it matter which point is $(x_1, y_1)$?","back":"No, as long as the subtraction order is consistent for both $x$ and $y$.","mcqOptions":[{"text":"No, as long as the subtraction order is consistent for both $x$ and $y$.","correct":true,"explanation":"Changing which point is labelled first reverses the signs of both the numerator and denominator. The two negatives cancel, so the gradient remains unchanged."},{"text":"Yes, the point with the larger coordinates must be labelled first for both $x$ and $y$.","correct":false,"explanation":"A common misconception is that the larger coordinates must come first. Either point can be labelled first, provided the subtraction order is consistent for both $x$ and $y$."},{"text":"Yes, the left-hand point must always be labelled first for both $x$ and $y$.","correct":false,"explanation":"Labelling the left-hand point first is convenient but not compulsory. Either point can be $(x_1,y_1)$ because reversing both subtractions leaves the gradient unchanged."},{"text":"No, as long as the subtraction order is opposite for $x$ and $y$.","correct":false,"explanation":"Using opposite subtraction orders changes the sign of only one difference, producing the negative of the correct gradient. The order must be consistent for $x$ and $y$."}]},{"id":"50ef8fb7-85f1-941c-c369-a3783703d824","cardType":"cloze","points":null,"front":"To find the gradient between $(-4, 1)$ and $(2, 4)$, the calculation is $$\\text{[...]}$$ resulting in a gradient of $$\\frac{1}{2}$$.","back":"To find the gradient between $(-4, 1)$ and $(2, 4)$, the calculation is $$\\boxed{m = \\frac{4 - 1}{2 - (-4)}}$$ resulting in a gradient of $$\\frac{1}{2}$$.","mcqOptions":[{"text":"$$$m = \\frac{4 - 1}{2 - (-4)}$$","correct":true,"explanation":"Gradient is the change in $y$ divided by the change in $x$. Here, $4-1=3$ and $2-(-4)=6$, giving $m=\\frac{3}{6}=\\frac{1}{2}$."},{"text":"$$$m = \\frac{2 - (-4)}{4 - 1}$$","correct":false,"explanation":"This swaps the changes in $x$ and $y$, producing the reciprocal gradient. The correct calculation is $m=\\frac{4-1}{2-(-4)}=\\frac{1}{2}$."},{"text":"$$$m = \\frac{4 - 1}{2 - 4}$$","correct":false,"explanation":"This incorrectly uses $4$ instead of the first point’s $x$-coordinate, $-4$. The denominator should be $2-(-4)=6$, so the gradient is $\\frac{1}{2}$."},{"text":"$$$m = \\frac{4 - (-1)}{2 - (-4)}$$","correct":false,"explanation":"This changes the first point’s $y$-coordinate from $1$ to $-1$. The numerator should be $4-1=3$, giving the correct gradient $\\frac{3}{6}=\\frac{1}{2}$."}]},{"id":"caca94d3-21b8-a4bd-e2f5-c6166386280c","cardType":"cloze","points":null,"front":"To find the gradient between $(-4, 1)$ and $(2, 4)$, the calculation is $$m = \\frac{4 - 1}{2 - (-4)}$$ resulting in a gradient of $$\\text{[...]}$$.","back":"To find the gradient between $(-4, 1)$ and $(2, 4)$, the calculation is $$m = \\frac{4 - 1}{2 - (-4)}$$ resulting in a gradient of $$\\boxed{\\frac{1}{2}}$$.","mcqOptions":[{"text":"$$$\\frac{1}{2}$$","correct":true,"explanation":"The change in $y$ is $4-1=3$ and the change in $x$ is $2-(-4)=6$. Therefore, the gradient is $\\frac{3}{6}=\\frac{1}{2}$."},{"text":"$$$2$$","correct":false,"explanation":"This is the reciprocal of the gradient, caused by dividing the change in $x$ by the change in $y$. The correct calculation gives $\\frac{3}{6}=\\frac{1}{2}$."},{"text":"$$$-\\frac{1}{2}$$","correct":false,"explanation":"This sign error usually comes from treating $2-(-4)$ as negative. Subtracting a negative gives $2+4=6$, so the gradient is positive: $\\frac{1}{2}$."},{"text":"$$$\\frac{1}{6}$$","correct":false,"explanation":"This result overlooks that the numerator is $4-1=3$, not $1$. The fraction is $\\frac{3}{6}$, which simplifies to $\\frac{1}{2}$."}]},{"id":"dda0766d-50ae-a815-9b80-898f50d8e309","cardType":"basic","points":2,"front":"A line with gradient 3 passes through $(1, -2)$ and $(5, k)$. Outline the first step to find $k$.","back":"Set up the equation using the formula: $$3 = \\frac{k - (-2)}{5 - 1}$$","mcqOptions":[{"text":"Set up the equation using the formula: $$3 = \\frac{k - (-2)}{5 - 1}$$","correct":true,"explanation":"The gradient equals change in $y$ divided by change in $x$. Using the points in the same order gives $3=\\frac{k-(-2)}{5-1}$."},{"text":"Set up the equation using the formula: $$3 = \\frac{5 - 1}{k - (-2)}$$","correct":false,"explanation":"This reverses the gradient formula by placing the change in $x$ above the change in $y$. The correct setup is $3=\\frac{k-(-2)}{5-1}$."},{"text":"Set up the equation using the formula: $$3 = \\frac{k - 2}{5 - 1}$$","correct":false,"explanation":"The first $y$-coordinate is $-2$, so subtracting it gives $k-(-2)$, not $k-2$. The correct equation is $3=\\frac{k-(-2)}{5-1}$."},{"text":"Set up the equation using the formula: $$3 = \\frac{k - (-2)}{5 + 1}$$","correct":false,"explanation":"The change in $x$ is found by subtraction, so it is $5-1=4$, not $5+1$. The correct setup is $3=\\frac{k-(-2)}{5-1}$."}]},{"id":"c3c80d39-27c9-bdde-c114-e4cfeccbee63","cardType":"basic","points":3,"front":"A line has a gradient of 4 and passes through $(-2, 3)$ and $(k, 15)$. Solve for $k$.","back":"

Set up: $4 = \\frac{15 - 3}{k + 2}$
Simplify: $4 = \\frac{12}{k + 2}$
Solve: $k = 1$

","mcqOptions":[{"text":"

Set up: $4 = \\frac{15 - 3}{k + 2}$
Simplify: $4 = \\frac{12}{k + 2}$
Solve: $k = 1$

","correct":true,"explanation":"The coordinate differences are $15-3=12$ and $k-(-2)=k+2$. Solving $4=\\frac{12}{k+2}$ gives $4(k+2)=12$, so $k=1$."},{"text":"

Set up: $4 = \\frac{15 - 3}{k - 2}$
Simplify: $4 = \\frac{12}{k - 2}$
Solve: $k = 5$

","correct":false,"explanation":"This treats subtracting the negative coordinate $-2$ as subtracting $2$. The denominator should be $k-(-2)=k+2$, leading to $k=1$."},{"text":"

Set up: $4 = \\frac{k + 2}{15 - 3}$
Simplify: $4 = \\frac{k + 2}{12}$
Solve: $k = 46$

","correct":false,"explanation":"This swaps the changes in $x$ and $y$, using the reciprocal of the gradient formula. The correct equation is $4=\\frac{12}{k+2}$, which gives $k=1$."},{"text":"

Set up: $4 = \\frac{15 + 3}{k + 2}$
Simplify: $4 = \\frac{18}{k + 2}$
Solve: $k = \\frac{5}{2}$

","correct":false,"explanation":"The change in $y$ requires subtraction, $15-3=12$, rather than addition. Using $4=\\frac{12}{k+2}$ and solving gives the correct value $k=1$."}]},{"id":"845348dc-a011-db21-74df-e1aa737c87f9","cardType":"cloze","points":null,"front":"On a coordinate grid, you should choose points where the line passes exactly through [ ] to avoid estimating values.","back":"On a coordinate grid, you should choose points where the line passes exactly through grid intersections to avoid estimating values.","mcqOptions":[{"text":"grid intersections","correct":true,"explanation":"Grid intersections provide points whose $x$- and $y$-coordinates can both be read exactly. Using these points makes the gradient calculation more accurate and reliable."},{"text":"axis crossings","correct":false,"explanation":"A line may have only two axis crossings, and they are not always clearly readable. Grid intersections provide exact coordinate pairs and are the best points to choose."},{"text":"coordinate labels","correct":false,"explanation":"Coordinate labels identify values on the axes rather than points through which the line passes. The useful points are grid intersections, where both coordinates can be read exactly."},{"text":"square centres","correct":false,"explanation":"The centres of grid squares are difficult to identify precisely and may require estimating half-units. Grid intersections give exact, clearly readable coordinate pairs instead."}]},{"id":"a776d32c-0f9a-87b7-0840-0b387346f4ad","cardType":"cloze","points":null,"front":"On a coordinate grid, you should choose points where the line passes exactly through grid intersections to avoid [ ] values.","back":"On a coordinate grid, you should choose points where the line passes exactly through grid intersections to avoid estimating values.","mcqOptions":[{"text":"estimating","correct":true,"explanation":"At a grid intersection, both coordinates can be read directly from the grid. This avoids estimating coordinates between marked values and reduces errors in the calculated gradient."},{"text":"plotting","correct":false,"explanation":"Choosing grid intersections does not avoid plotting values; plotting is the process of placing points. It avoids estimating coordinates because the values can be read exactly."},{"text":"labelling","correct":false,"explanation":"Grid intersections do not remove the need to label coordinates or axes. Their advantage is that they avoid estimating values between grid markings."},{"text":"calculating","correct":false,"explanation":"A gradient calculation is still required after choosing the points. Grid intersections specifically avoid estimating coordinate values, making that calculation more accurate."}]},{"id":"e798ffbf-461b-e6ab-2738-a5770ebaeb05","cardType":"basic","points":1,"front":"What does it mean if you find the reciprocal of the gradient by mistake?","back":"You accidentally swapped the x and y values in the formula (put $x$ on top).","mcqOptions":[{"text":"You accidentally swapped the x and y values in the formula (put $x$ on top).","correct":true,"explanation":"Gradient is change in $y$ divided by change in $x$. Putting the $x$-difference on top instead gives the reciprocal of the correct gradient."},{"text":"You accidentally reversed both subtraction orders in the formula (used the second point first).","correct":false,"explanation":"Reversing both subtraction orders multiplies the numerator and denominator by $-1$, so the gradient is unchanged. A reciprocal comes from swapping the $x$- and $y$-differences."},{"text":"You accidentally changed the sign of the gradient in the formula (made $m$ negative).","correct":false,"explanation":"Changing the sign produces the negative of the gradient, not its reciprocal. A reciprocal results from putting the change in $x$ above the change in $y$."},{"text":"You accidentally added the coordinate values in the formula (used sums instead).","correct":false,"explanation":"Adding coordinates gives unrelated differences and does not systematically produce a reciprocal. The reciprocal occurs when the $x$-difference and $y$-difference are swapped in the fraction."}]},{"id":"da6c951a-5c12-b455-a869-63b170e9ea3d","cardType":"basic","points":2,"front":"Describe the movement between $(1, 2)$ and $(4, 8)$ in terms of $x$ and $y$ changes.","back":"

$x$ change: 3 units right
$y$ change: 6 units up

","mcqOptions":[{"text":"

$x$ change: 3 units right
$y$ change: 6 units up

","correct":true,"explanation":"Subtract the starting coordinates from the ending coordinates: $4-1=3$ and $8-2=6$. Both differences are positive, giving 3 units right and 6 units up."},{"text":"

$x$ change: 4 units right
$y$ change: 8 units up

","correct":false,"explanation":"This uses the ending coordinates as the changes instead of subtracting the starting coordinates. The actual changes are $4-1=3$ units right and $8-2=6$ units up."},{"text":"

$x$ change: 3 units right
$y$ change: 6 units down

","correct":false,"explanation":"The $y$-coordinate increases from 2 to 8, so the movement is upwards, not downwards. The correct movement is 3 units right and 6 units up."},{"text":"

$x$ change: 6 units right
$y$ change: 3 units up

","correct":false,"explanation":"This swaps the horizontal and vertical changes. The $x$-difference is $4-1=3$, while the $y$-difference is $8-2=6$, so the movement is 3 units right and 6 units up."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Gradient of a Line","title":"Gradient of a Line Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"e3056fba-8790-472d-b2f8-43deab97cac5","walled":"$L37"}],"$L38","$L39"]