Twój koszyk jest pusty
Open sentences (supplementary material)
Example 1
You have some money in both of your pockets. If you move \latex{ 20 }¢ from your left to your right pocket, you will have the same amount in both pockets.
- Write an open formula to describe the relationship between the pieces of information.
- Find amounts of money that make the open formula true.
- How much money did you have in your pockets if you had \latex{ 500 }¢ in total?
Solution
- Represent the money in one of your pockets with a \latex{ \triangle } , while that in the other with a \latex{ \square }. Make a table.
left pocket
right pocket
before
after
\latex{ -20 }
\latex{ +20 }
You removed \latex{ 20 }¢ from your left pocket.
You put \latex{ 20 }¢ in your right pocket.
After rearranging the money in your pockets, you had the same amount in both of them:
\latex{ \triangle -20=\square +20. }
- If you had \latex{ 50 }¢ in your left pocket at the beginning, then \latex{ \triangle =50}. After the rearrangement, you would have \latex{ 50 – 20 = 30 }¢ left in it.
After putting \latex{ 20 }¢ into your right pocket, you had \latex{ 30 }¢ in it; therefore, \latex{ \square +20=30 }. This means you originally had \latex{ 30 – 20 = 10 }¢ in it \latex{\square =10. }
These numbers make the open formula true, as \latex{ 50 – 20 = 10 + 20 }.
These numbers make the open formula true, as \latex{ 50 – 20 = 10 + 20 }.
You can also try with other numbers, organising them into a table.
\latex{ 50 }
\latex{ 80 }
\latex{ 90 }
\latex{ 100 }
\latex{ 120 }
\latex{ 160 }
\latex{ 200 }
\latex{ 300 }
\latex{ 260 }
\latex{ 160 }
\latex{ 120 }
\latex{ 80 }
\latex{ 60 }
\latex{ 50 }
\latex{ 40 }
\latex{ 10 }
Note that you initially needed \latex{ 40 }¢ more in your left pocket to equalise the amounts in both after moving \latex{ 20 }¢ to your right pocket.
Expressing it with an open formula: \latex{ \triangle =\square +40. }
Expressing it with an open formula: \latex{ \triangle =\square +40. }
- If you had \latex{ 500 }¢ in total, you would have \latex{ 250 }¢ in both pockets after moving \latex{ 20 }¢ from your left pocket to the right.
Before moving \latex{ 20 }¢ to your right pocket, you had \latex{ 250 + 20 = 270 }¢ in your left pocket.
Before moving \latex{ 20 }¢ to your right pocket, you had \latex{ 250 – 20 = 230 }¢ in it.
Check: \latex{ 270 + 230 = 500 } and \latex{ 270 -20 = 230 + 20 }.
Before moving \latex{ 20 }¢ to your right pocket, you had \latex{ 250 – 20 = 230 }¢ in it.
Check: \latex{ 270 + 230 = 500 } and \latex{ 270 -20 = 230 + 20 }.
So, initially, you had \latex{ 270 }¢ in your left pocket and \latex{ 230 }¢ in your right pocket.
Example 2
Match the open formulas to the corresponding descriptions. Show the numbers that make the open sentences true on a number line.
- Rational numbers greater than \latex{ 2 }.
- Rational numbers not smaller than \latex{ -1 }.
- Rational numbers that are at least \latex{ -1 } but not greater than \latex{ 2 }.
- Rational numbers that are greater than \latex{ -1 } and not greater than \latex{ 2 }.
\latex{ \gt 2 }
\latex{ -1\lt }
\latex{ 2\leq }
\latex{ -1\lt }
\latex{ \leq 2}
\latex{ \geq -1}
\latex{ -1\leq }
\latex{ \leq 2 }
\latex{ -1\leq }
\latex{ \lt2 }
Solution
- \latex{ \square \gt 2; }
- \latex{ \square \geq -1; }
- \latex{ -1\leq \square \leq 2; }
- \latex{ -1\lt \square \leq 2. }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ -3 }
\latex{ -2 }
\latex{ -1 }
\latex{ 0 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
Exercises
{{exercise_number}}. A postcard with a stamp costs \latex{ \triangle }¢. The stamp is \latex{ \bigcirc }¢ more expensive than the postcard. Make a diagram and choose the open formulas that represent the relationship correctly if the price of the postcard is indicated by \latex{ \square }.
\latex{ 2\times }
\latex{ \div2 }
{{exercise_number}}. Which of the following open formulas correspond to the figure? Write a word problem for the figure.
- \latex{ \triangle +\triangle +\triangle +2=86 }
- \latex{ 4\times \triangle +2=86 }
- \latex{ \triangle +(3\times \triangle +2)=86 }
- \latex{ (86-2)\div 4=\triangle }
\latex{ 2 }
\latex{ 86 }
{{exercise_number}}. Three children took \latex{ \square } sandwiches on a trip. Dora took half as many as Zack and \latex{ 2 } fewer than Zoe. Make a diagram and write an open formula.
{{exercise_number}}. How many natural numbers make the following open formulas true? Show them on a number line.
a) \latex{ 0\lt \triangle -2}
- \latex{\bigcirc\div 2\leq 6}
c) \latex{ -8\leq \square +5\lt 8 }
Quiz
What number could \latex{ \square } be if \latex{ \square \times \square \lt \square } ?