Monday
Monday: From a Loft Problem to an Equation
Comet tips a bag of screws onto the workbench. "Thirty-eight. Every support block under the plank takes four, and the end stop takes six."
"So how many blocks can we build?" Wren asks. "What do you notice we know?"
"Blocks times four, plus six, is thirty-eight," Comet says. "Call the blocks n. Then 4n + 6 = 38."
Nova projects the equation over the bench and lets the 6 blink. "Would you like a hint? What undoes adding six?"
"Subtracting six, from both sides," Wren says. "4n = 32. Then divide both sides by four. n = 8."
"Eight blocks," Comet says. "Check it: four times eight is thirty-two, plus six is thirty-eight."
"The equation was the problem in symbols," Wren says. "Solving it was just keeping both sides equal."
Name the unknown, say what is true
A Loft problem becomes an equation in three moves. Name the unknown with a letter. Write what the story says must be true. Solve.
The screws: n blocks use 4n screws, the end stop uses 6, and the bag holds 38. So 4n + 6 = 38.
An equation is a statement that two expressions are equal. A solution is a value of n that makes the statement true.
A solved problem to study
Here is how Wren solved the screw equation. Each line says what he did and why it is allowed.
| Step | Equation | Why it is allowed |
|---|---|---|
| Start | 4n + 6 = 38 | The story says this is true. |
| Subtract 6 from both sides | 4n = 32 | Equal amounts minus the same amount stay equal. |
| Divide both sides by 4 | n = 8 | Equal amounts divided by the same number stay equal. |
| Check | 4 × 8 + 6 = 38 | Substitution shows the solution works. |
The coat-hanger picture
A level coat hanger is an equation. Each envelope hides x beans, and loose beans sit beside it.
The balance below shows x + 3 = 8. Take 3 beans off both cups and it stays level: x = 5.
- Read the question.
- Tap your answer.
The level hanger shows x + 3 = 8. How many beans are in the envelope?
A level hanger shows 2x + 1 = 9. How many beans are in each envelope?- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| An equation says two expressions are equal. | ? |
| n = 8 is a solution of 4n + 6 = 38. | ? |
| n = 7 is a solution of 4n + 6 = 38. | ? |
| 4 × 8 + 6 = 38 | ? |
| Subtracting 6 from only one side keeps an equation true. | ? |
Write an equation with a letter for the group count, solve it and check by counting again.
Strong start. Tomorrow you justify every step and compare two ways to solve the same equation.