Minds On
Number pairs
Examine the following number pairs:
- 5 and 6
- 6 and 12
- all the following numbers: 1, 2, 3, 4, 5, and 8
- 6 and 6
Record answers to the following questions using a method of your choice. Share your ideas with a partner, if possible.
- What do you notice?
- How are they connected?
- How can you compare them?
Action
Connecting numbers
As you may have noticed, the numbers in the Minds On exercise can be connected in many ways. One way that we could describe them is to use terms such as equal, greater than and less than.
For example: 5 is greater than 2. Examine the following number line. 5 is further away from 0 than 2, therefore it is a larger number.
5 + 3 is greater than 4 because 5 + 3 = 8. 8 is greater than 4.
We can also use symbols <, >, and =. For example, represents greater than and represents less than.
5 + 3 > 4 means the same as 8 is greater than 4.
However, sometimes an inequality can represent many numbers. For example,
- X > 3 + 2 or X > 5
X is any number greater than 5 (6, 7, 8, 9, 10, ...21...). When showing less than or greater than on a number line we use an open dot to start the line with an arrow to indicate the possible numbers.
Inequalities on a number line
To show that a number on a number line is greater than or less than, an open circle is used. Let’s examine the following examples and image to help us gain an understanding of how this works.
X > 5, where X is greater than 5 and can be any number after 5 on a number line.
X < 4, where X is less than 4 and can be any number before 4 on a number line.
A chart showing how the greater than and less than symbols are used to indicate inequalities and how they are represented on a number line. Title: Inequalities on a number line. First Symbol: greater than symbol is 2 connected line segments forming the shape of an arrow pointing to the right. Number line example: x greater than 5 is demonstrated by an unfilled circle above the line at 5 and a stretched arrow to the right of the number line. Second Symbol: less than symbol is 2 connected line segments forming the shape of an arrow pointing to the left. Number line example: x less than negative 4 is demonstrated with an unfilled circle above the line at negative 4 and a stretched arrow to the left of the number line.
Creating number lines
Complete the following printable document Number Lines. You can also complete this activity in your notebook or using the method of your choice. You can include pictures, images, or words to help explain your thinking.
Show the following on the number lines provided:
- x = 3
- x > 3
- x < 4 + 10
Press the ‘Activity’ button to access the Number Line Sheet.
Greater than, less than, equal to
At times we also have equations that may be greater than or equal to, or less than or equal to. We can identify those with a greater than or equal to sign or with a less than or equal to sign.
For example, X ≥ 5. This means X is greater than or equal to 5. So, X is 5 and any number larger than 5.
Let’s also examine X ≤ 4, which means X is less than or equal to 4. So, X is 4 and any number less than 4.
When showing that a number on a number line is greater than or equal to, or less than or equal to a number a closed (or coloured in) circle is used.
Let’s examine the following examples and image to help us gain an understanding of how this works.
X ≥ 3, where x is greater than or equal to 3.
X ≤ 5, where x is less than or equal to 5.
Notice how the different circles are used on the number lines below to indicate the value of X.
A chart showing how different symbols are used to indicate inequalities and how they are represented on a number line. Title: Inequalities on a number line. First symbol: greater than or equal to symbol is 2 connected line segments forming the shape of an arrow pointing to the right and a horizontal line right under it. Number line example: X greater than or equal to 3 demonstrated with a filled circle above the line at 3 and an arrow that stretches to the right of the number line. Second symbol: less than or equal to symbol Is 2 connected line segments forming the shape of an arrow pointing to the left and a horizontal line right under it. Number line example: x less than or equal to 5 is demonstrated with a filled circle above the line at 5 and a stretched arrow to the left of the number line.
Solving inequalities
Sometimes expressions that looks like this: 4 + x > 12. 4 plus a number must be less than 12. Can you think of a value that when added to 4 will give less than 12?
We can also solve these expressions by using the opposite operation.
- 4 + x < 12
- 4 – 4 + x < 12 – 4
- x < 8
Now we can check to see if our answer makes sense by plugging some numbers less than 8 into the inequality.
- 4 + 5 < 12, 9 < 12
- 4 + 7 < 12, 11 < 12
- 4 + 1 < 12, 5 < 12
Your turn
Solve the following inequalities. Try using the opposite operation. Be sure to test your answer to make sure it’s true! Record your thoughts using a method of your choice.
- x ≤ 12 – 3
- 20 ≥ x – 5
- x + 7 ≤ 10
After you have solved the inequalities on your own, press ‘Answer’ to access solutions to the inequalities to check your answer.
Question 1: x ≤ 9
Question 2: 25 ≥ x
Question 3: x ≤ 3
Consolidation
Solve the following inequalities
Complete the following questions using a method of your choice.
Solve the inequalities. You may wish to use a number line or opposite operations. Be sure to verify your solution.
- X > 4
- X + 5 > 10 + 5
- X − 2 ≥ 7
- X + 3 ≤ 16
Reflection
As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.
I feel...
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.