How to Write Expressions

How to Write Expressions exercise

• Explain how to identify the operator by keywords.

  Hints

  Keep the keywords for the operations in mind:

  • plus or more than or combine or in total ... for addition.
  • minus or less than or fewer than or take away ... for subtraction.

  If each student has $2$ books, and letting $s$ stand for the number of student there are in total, then the total number of books is given by $2s$, or $2\times s$.

  Imagine you've got $12$ candies. You want to split those candies into packages of $3$ each.

  Then you have to divide $12\div 3=4$ to see that you get $4$ packages with $3$ candies each.

  Solution

  So let's help Mateo with his business plan:

  First, he establishes the following:

  • $2.50$ dollars for each taco
  • a delivery fee of $200$ dollars

  We don't know the number of tacos, thus we assign the variable $t$ to it. The keyword for each indicates multiplication: $2.50\times t$ or $2.50t$.

  For each delivery, Mateo decides to charge a flat rate delivery fee of $200$ dollars. We know that plus stands for addition, so we have to add $200$ dollars to the term $2.50t$ to get

  $2.50+200$.

  Next, we take a look at the number of people he can feed with tacos:

  • Each person gets 3 tacos.
  • Mateo can only calculate with 10 fewer than the total number of tacos.

  Again we don't know the total number of tacos and thus assign $t$ to it.

  Fewer than indicates subtraction. So we have to subtract $10$ from $t$. Keep the order in mind. This gives us $t-10$.

  So the remaining number of tacos is given by $t-10$.

  Because each person gets 3 tacos we have to divide $t-10$ by $3$ to end up with the expression

  $\frac{t-10}3$.

• Decide the operation symbol.

  Hints

  Take care: The number of persons can't be negative.

  Keep the keywords for the operation symbols in mind:

  • plus or combine ... for addition
  • minus or fewer than ... for subtraction
  • for each or times ... for multiplication

  For example, the total of $a$ and $b$ can be written as

  $a+b$.

  Solution

  To write a word problem as an algebraic expression, you have to recognize keywords. In the following the keywords are highlighted.

  Cheyenne has one sibling and Lou has two siblings more than Cheyenne. Thus we have to add:

  $1+2$.

  So, in other words, Lou has three siblings.

  Cheyenne has ten apples fewer than Lou. Let the variable $l$ represents the number of Lou's apples. The keyword indicates subtraction. So the corresponding expression is

  $l-10$.

  Together Cheyenne and Lou would like to solve ten word problems. They want to split the problems evenly. To split evenly means division:

  $10\div 2=5$.

  So each of them has to solve five word problems.

  Well done Cheyenne and Lou: now you can order $t$ tacos, at $3$ dollars each. Together they have to pay

  $3\times t$ dollars.

• Match the keyword with the operation symbol.

  Hints

  Take any of the given keywords and think about which operation symbol you have to take, and test out what you think by making up a word problem to go along with it.

  Keep in mind:

  • Summand $+$ summand $=$ sum
  • Minuend $-$ subtrahend $=$ difference
  • Factor $\times$ factor $=$ product
  • Dividend $\div$ divisor $=$ quotient

  Solution

  Let's start with the addition sign $+$:

  The following keywords in a word problem indicate addition: combine, in total, sum, plus, more than ... Sure, there are a lot more keywords for addition as well.

  Now we take a look at subtraction, the $-$ sign:

  Keywords you usually find are difference, take away, less than, fewer than, minus ...

  Multiplication can be indicated by the following keywords:

  for each or times or product. For example, the double of can be written as $2\times ...$.

  Last, the division keywords:

  split, divide, as well as relation

• Highlight important keywords from various real world statements.

  Hints

  You only have to highlight the keyword and do not have to decide the operation.

  Be careful: there are also non keywords that you can highlight.

  If any word is a keyword, you should be able to write down a corresponding algebraic expression.

  Solution

  First, we look at the keywords and then we indicate the corresponding operation and write down the algebraic expression.

  Here only the keywords are highlighted. If any word isn't highlighted, then it isn't a keyword at all.

  $~$

  Shells

  The total number of shells is wanted. So we have to add the found shells to get $23+16=39$.

  $~$

  Classrooms

  Because $20$ students can fit in each classroom, we have to divide $320\div 20=16$. This is the number of classrooms the school needs.

  The next example is more theoretical:

  $t$ raised to the fourth power leads to $t^4$. Next it's increased: this indicates addition of the product of $2$ and $s$. So we have in total $t^4+2\times s$.

  $~$

  Books

  Sasha has nine books. Chayenne has two times more books than Sasha.

  • two times stands for multiplication
  • more than stands for addition

  Thus we can establish the following algebraic expression

  $9+2\times 9=27$.

  This is the number of books Chayenne owns.

• Summarize your knowledge about variables.

  Hints

  If you don't know the number of students in your grade, you can represent the unknown number by a variable, such as $s$.

  If you've decided one letter for an unknown value you have to use it consistently throughout the exercise.

  Let's have a look at the algebraic expression, $x+4$.

  • $x=2$ we get $2+4=6$
  • $x=10$ we get $10+4=14$
  • ...

  Solution

  If you'd like to write an algebraic expression with an unknown value, you first assign a variable to this unknown value.

  Often the variable $x$ or $y$ is used, but sure you can also use any letter. For example, $t$ for the number of tacos.

  If you've decided on using one variable you don't have to change it during an exercise.

  Let's have a look at the example of Matteo's taco company:

  $2.50t+200$

  is the expression for the money coming into Matteo's company. Now, Matteo is able to put different values in for $t$ to get:

  • $2.50\times 100+200=250+200=450$ for $t=100$
  • $2.50\times 400+200=600+200=800$ for $t=400$
  • ...

• Determine the corresponding algebraic expressions.

  Hints

  The total numbers of people is four - including Sasha.

  The total number of tacos is $12$ and the total number of drinks is $10$.

  The total bill is $61$ dollars.

  Solution

  It's also quite a good strategy to practice word problems in the real world. So, Sasha and her friends do it. Mathematics in a taco bar:

  The number of eaten tacos

  Each of them has eaten three tacos. Together we get $4\times 3=12$ tacos eaten.

  The number of drinks

  Two had two and the other two had three drinks. So we have to multiply two times and to add the products last:

  $2\times 2+2\times 3=4+6=10$, the total number of drinks.

  They ask for the bill

  $10$ tacos, $3$ dollars each, and $10$ drinks $2.50$ dollars each leads to

  $12\times 3+10\times 2.50=36+25=61$ dollars. So they have to pay $61$ dollars in total.

  They'd like to pay

  They increase the total bill by $3$ dollars tip, $61+3$, and split this sum evenly to get

  $\frac{61+3}4=\frac{64}4=16$.

  So each of them has to pay $16$ dollars.