Say what the letter stands for, write an expression for the situation, then test it with a number you can check. One expression then gives the answer for every case.
A party for 5 people, 50 people or 500: one expression gives the cost for all of them, if you say clearly what the letter stands for.
So far each expression was written from one sentence about a number. A real situation has parts: a fee, a price for each one, a number of people. To write one expression for all of it, first say what the letter stands for, then build the expression, then test it with a case you can check in words. After that one expression gives the answer for any number.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
What should the letter be?
If n is the number of students, what does 1 student cost? What do 10 cost?
TryTry it on the board and compare the result with your prediction.
One expression, many trips
The rule is C=12n+50. Set n to 0: what does the 50 mean? What does the 12 mean? Was your prediction right?
Try it yourselfYour edits are saved in this browser.
n is the number of students on the trip. How much does a trip of 20 students cost?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
From a situation to an expression
Here is how to build an expression for a situation.
Worked exampleBefore each step, say what you think comes next.
Hall: 60 dollars, plus 5 dollars per guestA hall costs 60 dollars to hire, and then 5 dollars for each guest.
Let g=number of guestsSay what the letter stands for. The number of guests changes, so g stands for that.
5gEach guest costs 5 dollars, so g guests cost 5×g.
5g+60Add the hire, which is paid once.
g=10:5(10)+60=110Test with a case you can check in words: 10 guests cost 50 + 60 = 110 dollars.
g=40:5(40)+60=260Now the same expression gives any case, with no counting.
Read Pia's working one line at a time. What does the letter a stand for?
Say what the letter stands for first. Then build the expression.
A phone plan costs 15 dollars a month plus 4 dollars per GB. Let g be the number of GB used. Write an expression for the monthly cost.
Pia writes the cost of a apples at 3 dollars each. Click the line where her thinking goes wrong.
ModifyChange the problem on purpose and see what happens.
New situations
The board follows the situation and shows the result. Write your expression, then check it with a number.
Try it yourselfYour edits are saved in this browser.
A taxi costs 3 dollars to get in plus 2 dollars per km. Let d be the number of km. Write an expression for the fare.
A tank holds 100 litres and loses 4 litres each minute. Let m be the minutes. Write an expression for the litres left.
Adult tickets cost 8 dollars and child tickets cost 5 dollars. Let a be the number of adults and c the number of children. Write an expression for the total cost.
Tickets cost T=12n+6 dollars for n tickets (6 dollars booking). Use the expression to find the cost of 50 tickets.
PractiseA few on your own, to check you have it.
Build and use
A plumber charges 60 dollars call-out plus 40 dollars per hour. Let h be the number of hours. Write an expression for the cost.
Using 40h+60, what does a job of 3.5 hours cost, in dollars?
Review: find 3(n+4) when n=5.
Review: write an expression for "add 5 to n, then multiply the result by 2".
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A situation of your own
Make up a situation with a fee paid once and a price for each one: a club, a game, a delivery. Write "Let ___ be the number of ___." Then write an expression for the total, and test it with one case.
Stretch: swap with a partner and see if they can write your expression from the story alone.
For teachers
Students say what a letter stands for, write an expression for a situation with a fee and a rate, test it with a case they can check, and use it for any number (AC9M7A02, node 3). About 45 minutes.
I can say what the letter in an expression stands for, a number that changes.
I can write an expression for a situation with a once-only amount and an amount for each one.
I can test my expression with a case I can check in words, then use it for a big case.
Before class
Do the Modify boards yourself first. The mystery boards follow the situation, so students check their expression against them before they answer.
The new move is saying what the letter stands for before writing anything. Two common errors: letting the letter be the thing worked out ("Let n be the cost"), and reading the letter as the object (Pia's mistake: 3a means 3 apples). Insist on the sentence "Let ___ be the number of ___".
The once-only amount is a constant added on; the amount for each one is multiplied by the letter. This is the same structure as y = mx + c in Linear relations, so say so, but do not name it yet.
Running the lesson
Predict: take a show of hands. Ask a student who chose 'the total cost' what n would be for a trip of 30 students.
Run: before moving n, ask what the cost is for 0 students. What does the 50 mean?
Investigate: model the worked example live. Write 'Let g be...' first, every time.
Modify: ask for the sentence 'Let ___ be the number of ___' before each expression.
Practise: for the plumber questions, ask which amount is paid once.
Make: have students swap situations and write each other's expressions.
When students get stuck
A student reads a letter as an object (Pia's mistake: 3a means 3 apples). Ask: If a is 5, what is 3a? Is it 3 apples or something else? Fix: a stands for a number, how many apples. 3a is 3 dollars times that number.
A student multiplies the once-only amount (60h + 40 for a 60 dollar call-out). Ask: Does the call-out change if the job is 1 hour or 10 hours? Fix: An amount paid once is added on. Only the price for each hour is multiplied by h.
Stretch and support
Stretch: A shop sells pens at 2 dollars each and rulers at 3 dollars each, with a 1 dollar bag. Let p be pens and r rulers. Write the cost, then find it for 10 pens and 4 rulers.
Support: Give a sentence frame: Let ___ be the number of ___. Paid once: ___. For each: ___. Expression: ___.
Where it leads
This finishes the Expressions unit. The Equations unit starts from expressions like these: what number of students makes the trip cost 290? In Linear relations the same structure becomes y = mx + c and its graph.
What each step is for
what should the letter be?: Committing to what the letter stands for exposes letting it stand for the thing that is being worked out, instead of the quantity that changes.
one expression, many trips: Moving n and reading off the cost shows that one expression gives every case, and the parts mean what the story says.
from a situation to an expression: Name the letter, write what changes, add what is paid once, then test with one case; the letter stands for a number, not for the objects.
new situations: One change at a time: a fee plus a rate, a quantity that goes down, two letters, and using the expression for a big case.
build and use: Naming the letter, building the expression and using it for a larger case, mixed; the review is brackets.
a situation of your own: Inventing a situation and its expression reverses the lesson: from an expression back to a story.
Now you can
Say what the letter in an expression stands for, a number that changes.
Write an expression for a situation with a once-only amount and an amount for each one.
Test my expression with a case I can check in words, then use it for a big case.