Edexcel GCSE Mathematics – Nov 2017 Paper 1 Foundation

What are we being asked to find?

We are asked to convert measurements between different metric units. For part (a), we’re converting length from centimetres to metres. For part (b), we’re converting mass from kilograms to grams.

Why we do this:

We know that there are \(100 \text{ cm}\) in \(1 \text{ metre}\). Because a metre is a larger unit, we will have fewer of them, so we need to divide our centimetre value by \(100\).

What this tells us:

When dividing by 100, we move the decimal point two places to the left. The value \(365 \text{ cm}\) is exactly \(3.65 \text{ m}\).

Why we do this:

The prefix “kilo-” means one thousand. There are \(1000 \text{ grams}\) in \(1 \text{ kilogram}\). To convert from a larger unit (kg) to a smaller unit (g), we will have more of them, so we multiply by \(1000\).

What are we being asked to find?

We need to perform a calculation involving both addition and multiplication. We must remember the correct Order of Operations (often remembered by the acronym BIDMAS or BODMAS).

Why we do this:

According to BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction), multiplication must be completed *before* addition. So, we will calculate \(7 \times 10\) first, and then add \(2\) to that result.

What this tells us:

A common mistake here is to read left-to-right (\(2+7 = 9\), then \(9 \times 10 = 90\)). By carefully applying BIDMAS, we can be confident our answer of \(72\) is correct.

What are we being asked to find?

We need to solve for the unknown value, \(y\). The equation says that if we divide \(y\) by 4, the result is \(10.5\).

Why we do this:

To isolate \(y\) and find its value, we have to “undo” the division by 4. The inverse (opposite) of dividing by 4 is multiplying by 4. So, we will multiply both sides of the equation by 4.

What this tells us:

We can check our answer by putting \(42\) back into the original equation: \(\frac{42}{4}\). Half of \(42\) is \(21\), and half of \(21\) is \(10.5\). It works!

What are we being asked to find?

We are given four specific numbers: \(9\), \(-2\), \(2\), and \(-9\). We must select exactly two of these numbers that, when added together, equal \(-7\).

Why we do this:

Because the sum is negative (\(-7\)), we know we must include a negative number. Specifically, the negative number must have a larger absolute value than the positive number we pair it with.

What this tells us:

Adding a negative number is the same as subtracting a positive number. Starting at positive \(2\) and moving \(9\) steps down the number line lands us exactly at \(-7\).

What are we being asked to find?

We need to find the very next number (the 5th term) in the sequence. We have been provided with the specific rule to generate new terms, so we just need to apply it to the final given number.

Why we do this:

To find the 5th term, we apply the rule to the 4th term. The 4th term is \(23\). The rule requires us to first multiply by \(2\), and then add \(1\).

What this tells us:

We can verify we understood the rule by testing it on earlier terms: \((5 \times 2) + 1 = 11\), which matches the 3rd term. So our process is definitely correct, making \(47\) our next term.

What are we being asked to find?

We need to create an algebraic formula for the total length (\(L\)). “Total length” means we must add the lengths of all five individual rods together. The instruction to write it “as simply as possible” means we must collect the like terms to simplify the expression.

Why we do this:

We’ll write down an equation starting with \(L =\) and then list the addition of all five pieces.

Why we do this:

To simplify, we group the variable terms (the \(a\)’s) and the constant terms (the numbers) together.

What this tells us:

The final formula \(L = 5a + 3\) represents exactly the same total length as the long addition string, but it is in its simplest and most useful form. Note that you must write \(L = …\) for full marks as it asks for a “formula for L”.

Why we do this:

Coordinates are always written in the format \((x, y)\). We first move “along the corridor” (the x-axis) from the origin, then “up or down the stairs” (the y-axis).

Why we do this:

The equation \(y = 4x + 1\) is a rule. For any point to “live” on this line, substituting its \(x\) value into the rule must perfectly calculate its \(y\) value. Point B has \(x = 2\) and \(y = 9\).

Why we do this:

The equation \(x = -2\) describes a set of points where the x-coordinate is *always* \(-2\), regardless of what the y-coordinate is. For example, \((-2, 0)\), \((-2, 5)\), \((-2, -3)\). This creates a vertical line crossing the x-axis at \(-2\).

What are we being asked to find?

Before drawing, we need to mathematically find the specific length and width of the rectangle. We know two facts:
1. Area = \(32 \text{ cm}^2\)
2. Length = \(2 \times\) Width

Why we do this:

The area of a rectangle is found by multiplying Length \(\times\) Width. We can test pairs of numbers to find a combination where the length is double the width, and multiplying them together gives \(32\).

What this tells us:

Now we know exactly what to draw: a rectangle that takes up \(4\) squares on one side and \(8\) squares on the other side. Each square represents \(1 \text{ cm}^2\).

What are we being asked to find?

We need to spot the flaw in Jacqui’s proportional reasoning. She tried to use a related fact (dividing by 10) to solve a harder problem (dividing by 5).

Why we do this:

Think about dividing a pizza. If you share it among 10 people, everyone gets a small slice. If you share it among only 5 people (half as many people), each person’s slice will be twice as big. Therefore, dividing by 5 should give an answer twice as large as dividing by 10.

What this tells us:

When you divide by a smaller number, the result (quotient) gets larger. This is an inverse relationship that is very common in maths problems.

Why we do this:

In statistics, “consistency” means how spread out the numbers are. A smaller spread means the player was more consistent. We measure spread by calculating the Range (Highest value \(-\) Lowest value).

Why we do this:

The “mode” or “modal score” is indeed the score that appears most often. However, we have to read the stem and leaf diagram correctly. A leaf of “6” does not just mean “6”. It is attached to a “stem” on the left side of the line.

What this tells us:

Always use the Key when reading a Stem and Leaf diagram. The leaf alone is only part of the number!

Why we do this:

We are dividing the total number of children (\(30\)) into groups of \(8\), because each group of up to \(8\) children requires \(1\) adult. If there are any children left over, we will need an extra adult to look after them (we can’t have a fraction of an adult!).

Why we do this:

First, we find the new total number of children. Then, we check if our current number of adults (\(4\)) is enough to cover this new total.

Why we do this:

To solve a two-way table, we first insert all the exact numbers given in the bullet points. Then we can use addition and subtraction to find the missing gaps.

Why we do this:

In any row or column, the parts must add up to the total for that row or column.

Why we do this:

Probability is written as a fraction: \(\frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\). We are picking from ALL rabbits, so our denominator is the total number of rabbits. Our numerator is the specific group we want (female with short hair).

What are we being asked to find?

To find the volume of a cube, we need to know its side length. We are given the total surface area. A cube is made of exactly \(6\) identical square faces. If we divide the total area by \(6\), we will find the area of just one square face.

Why we do this:

We know the area of one square face is \(49 \text{ cm}^2\). The area of a square is calculated by \(\text{side} \times \text{side}\) (or \(\text{side}^2\)). To find the side length, we must find the square root of \(49\).

Why we do this:

The volume of a cube is found by multiplying \(\text{length} \times \text{width} \times \text{height}\). Because it’s a cube, all these dimensions are the same (\(7 \text{ cm}\)).

What this tells us:

The cube has a volume of \(343 \text{ cm}^3\). This multi-step problem tests both geometric knowledge (surface area and volume formulas) and arithmetic skills (division, square roots, and multiplication) without a calculator.

What are we being asked to find?

To see which fraction is “closer” to \(1\), we need to find the difference between each fraction and \(1\). The fraction with the smaller difference is closer.

Why we do this:

It’s hard to compare fifths and sevenths directly. By converting everything into a common denominator (a number both \(5\) and \(7\) fit into, like \(35\)), we can easily compare their sizes.

What this tells us:

Because \(\frac{10}{35}\) is a smaller gap than \(\frac{14}{35}\), we know that \(\frac{5}{7}\) is closer to \(1\).

Why we do this:

To find a percentage, it is easiest to first write the amount of orange buttons as a fraction of the total jar. We need to find the total number of “parts” in the ratio to be our denominator.

Why we do this:

“Percent” literally means “out of 100”. To convert our fraction \(\frac{9}{20}\) into a percentage, we need to find an equivalent fraction that has \(100\) at the bottom.

What this tells us:

Exactly \(45\%\) of the buttons in the jar are orange. Converting the denominator to 100 is a very quick and reliable method for percentage questions on Non-Calculator papers.

Why we do this:

When we estimate, we round numbers to make calculation easier. If we round the numbers up, our estimated answer will be bigger than the real answer (an overestimate).

Why we do this:

To find the final price, we must first find out how much the T-shirts cost at full price. We need to multiply the quantity (\(35\)) by the price per shirt (£\(5.80\)).

Why we do this:

Berenika is buying \(35\) shirts, which is more than \(30\), so she gets the \(10\%\) discount. We find \(10\%\) of our total, and subtract it from the full price to get the final actual cost.

What this tells us:

Because we’re dealing with money, we write the final answer with two decimal places as £\(182.70\), not just £\(182.7\).

What are we being asked to find?

We need to find out how many different combinations make an odd total, out of all the possible combinations. A good strategy is to list all the possible totals in a quick table.

Why we do this:

Probability is the number of successful outcomes (odd totals) divided by the total number of possible outcomes. We just need to count them up.

What this tells us:

Alternatively, we could use rules of odd/even numbers: Only an Odd + Even will make an Odd total. Box A has 2 Odds and 1 Even. Box B has 2 Odds and 1 Even. You could calculate the probability without listing all \(9\) sums, but the table guarantees we won’t miss any!

Why we do this:

The total £\(450\) is being split into “parts”. If we add up all the numbers in the ratio, we get the total number of parts. By dividing the total money by the total parts, we figure out exactly how much money is in a single part.

Why we do this:

We need to look back at the ratio to see how many parts belong to Kelan. The names (Harry, Regan, Kelan) match the numbers in the ratio (\(2 : 5 : 3\)) in order. So Kelan gets \(3\) parts.

What this tells us:

Kelan receives £\(135\). If we wanted to check, we could calculate Harry’s (\(2 \times 45 = 90\)) and Regan’s (\(5 \times 45 = 225\)). \(90 + 225 + 135 = 450\). The math is perfect.

What are we being asked to find?

We need to adjust a recipe designed for \(16\) flapjacks so that it makes \(24\) flapjacks. To do this, we need to know what our “multiplier” or scale factor is.

Why we do this:

We multiply each original quantity by \(1.5\). Multiplying a number by \(1.5\) is the same as taking the original amount and adding half of it.

What this tells us:

Proportion problems like recipes can always be solved by multiplying by a scale factor. If you ever get stuck finding the scale factor, finding the recipe for 1 flapjack (divide by 16) and then multiplying by 24 also works perfectly!

What are we being asked to find?

We are trying to check who is right. Ami’s answer is around \(27\), while Josh’s is around \(271\). They are completely different magnitudes (factor of 10). By rounding the complex numbers to simpler numbers (often to 1 significant figure), we can easily estimate the answer in our heads.

Why we do this:

We now evaluate this simplified version using standard order of operations. Calculate the denominator first (squaring the \(4\), then adding the \(5\)), and finally divide.

What this tells us:

Our rough estimate is \(30\). Looking at the two given answers, Ami’s answer of \(27.1115\) is very close to \(30\). Josh’s answer of \(271.115\) is nearly ten times larger, meaning he likely missed a decimal point or calculation step entirely.

What are we being asked to find?

We need to perform the calculation and then convert the final answer into standard form (which looks like \(A \times 10^n\), where \(A\) is between \(1\) and \(10\)). Let’s tackle the top part of the fraction first.

Why we do this:

Now we substitute our numerator back into the fraction. We must divide \(0.000018\) by \(0.01\). Dividing by \(0.01\) (which is one hundredth) is exactly the same mathematically as multiplying by \(100\).

Why we do this:

The question requires the final answer to be in standard form. This means the first number must be between \(1\) and \(10\), multiplied by a power of \(10\).

What this tells us:

Alternatively, you could have converted everything to standard form first: \(\frac{6 \times 10^{-2} \times 3 \times 10^{-4}}{1 \times 10^{-2}} = \frac{18 \times 10^{-6}}{10^{-2}} = 18 \times 10^{-4} = 1.8 \times 10^{-3}\). Both methods are perfectly valid!

Why we do this:

We cannot add fractions together unless their denominators (bottom numbers) are exactly the same. We must find a common multiple for \(5\) and \(4\). The lowest common multiple is \(20\).

What this tells us:

You could also solve this using decimals: \(\frac{2}{5} = 0.4\) and \(\frac{1}{4} = 0.25\). Adding them gives \(0.65\), which is equivalent to \(\frac{13}{20}\). The mark scheme accepts \(0.65\) too.

Why we do this:

A negative index means “one over” the positive version of that index. It creates the reciprocal. So, \(a^{-n} = \frac{1}{a^n}\).

What are we being asked to find?

We need to break the number \(36\) down into numbers that multiply together to make \(36\) (“product of factors”). We must keep breaking them down until all the numbers are prime numbers (numbers that only divide by \(1\) and themselves, like \(2, 3, 5, 7\)).

Why we do this:

The question asks for a “product”, which means things multiplied together. We list all the prime numbers we circled at the ends of our tree and put multiplication signs between them.

What this tells us:

This is the unique “prime fingerprint” of \(36\). You could also write this using indices as \(2^2 \times 3^2\), which the mark scheme accepts as equivalent (oe).

What are we being asked to find?

This is a puzzle involving unknown ages. The best way to solve it is to use a letter to represent one person’s age, and then write the other ages in terms of that letter.

Why we do this:

The problem tells us that adding all three ages together equals \(77\). We can construct an equation combining our three algebraic expressions and setting them equal to \(77\).

Why we do this:

We solve the equation to find the value of \(x\), which is Jay’s age. Once we have Jay’s age, we can easily calculate the other two.

Why we do this:

Now that we know Jay is \(14\), we calculate Kiaria and Martha’s ages, and format them as the requested ratio (Jay : Kiaria : Martha).

What this tells us:

The mark scheme accepts \(14 : 21 : 42\). If you notice they all divide by \(7\), you could also simplify this to \(2 : 3 : 6\), but writing their literal ages in ratio form earns full marks here.

What are we being asked to find?

We need to prove that angle \(ABF\) (the angle inside the bottom left triangle) is exactly \(70^{\circ}\). Because it’s a “Show that” question asking for reasons, we must write down the value of every angle we find and explicitly state the geometry rule we used to find it.

Why we do this:

To find angle \(ABF\), it helps to find the other two angles inside the triangle \(ABF\). We can find angle \(AFB\) by looking at the straight lines that cross at point \(F\).

Why we do this:

We need the third angle in the triangle to finish our calculation. Angle \(FAB\) is one of the main corners of the large parallelogram. Parallelograms have special angle properties we can use.

Why we do this:

We now know two angles inside triangle \(ABF\): \(35^{\circ}\) and \(75^{\circ}\). Since the angles in any triangle always add up to \(180^{\circ}\), we can subtract to find the last missing piece.

What this tells us:

We successfully arrived at \(70^{\circ}\) as requested, meaning our logic and geometric rules were applied perfectly. (Note: There are other valid paths, such as using alternate angles \(FBC = 35^{\circ}\) and co-interior angles, but this method is very direct!)

What are we being asked to find?

We need to determine the area of the entire logo (the largest circle) and the area of just the shaded ring. The formula for the area of a circle is \(\pi \times r^2\).

Why we do this:

The total radius of the large circle is \(4 + 3 + 3 = 10 \text{ cm}\). The shaded part is the middle circle (radius \(4 + 3 = 7 \text{ cm}\)) minus the unshaded inner circle (radius \(4 \text{ cm}\)). We leave our answers in terms of \(\pi\) to make calculation easy without a calculator.

Why we do this:

We need to test Daisy’s claim. Daisy says the fraction is exactly \(\frac{1}{3}\). We will write our accurate areas as a fraction and compare.

What this tells us:

Even though \(\frac{33}{100}\) is very close to \(\frac{1}{3}\), in mathematics “exactly” leaves no room for rounding. Proving this precisely using fractions is the safest way to get all the marks.

Why we do this:

Because the earnings are in groups (e.g., between £150 and £250), we don’t know the exact amounts. To “estimate” the mean, we pretend every person in a group earns exactly the middle amount (the midpoint). We multiply each midpoint by the frequency to find the total money earned, and then divide by the total number of people.

Why we do this:

The “mean” takes into account every single piece of data. If there are extreme values (outliers), they can pull the mean away from the typical value. The median is often better when there are outliers.

What are we being asked to find?

Because this is a rectangle, the opposite sides must be exactly the same length. We can use this geometric fact to set up an algebraic equation and find \(x\).

Why we do this:

Now that we know \(x = 5\), we can substitute it back into either side expression to find the actual vertical length of the rectangle in centimetres.

Why we do this:

We know the area is \(48 \text{ cm}^2\). Area is found by multiplying the height by the width. We know the height is \(16\), and the width is \(y\).

What this tells us:

In “show that” questions, you cannot start by assuming \(y = 3\). You must logically prove it by building up from the other evidence until \(y=3\) is the unavoidable final conclusion.

What are we being asked to find?

We need to spot a critical error in how the graph has been drawn. The equation is \(y = x^2 + 1\), which contains an \(x^2\). This makes it a quadratic function.

What are we being asked to find?

This is a “Reverse Percentages” question. We are given the price after the discount. A common mistake is to find \(30\%\) of £\(2.80\) and add it on, but percentages don’t work backwards like that. Instead, we must ask: what percentage of the original price is left over?

Why we do this:

Now that we know \(70\%\) = £\(2.80\), we can use the “unitary method”. We divide to find what \(10\%\) is worth, and then multiply by \(10\) to get up to \(100\%\) (the original price).

What this tells us:

The original price was £\(4.00\). We can check this quickly: \(10\%\) of £\(4.00\) is \(40\text{p}\). So \(30\%\) is \(3 \times 40\text{p} = \text{£}1.20\). £\(4.00 – \text{£}1.20 = \text{£}2.80\). The answer is definitely correct.