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Section 1.1 Integers
An integer is defined as a whole number that does not a fraction or decimal part. Example of integers are
\(1\,,\,2\,,\,500 \) e.t.c
Subsection 1.1.1 Basic operation of integers
Basic operation of integers is either addition, subtraction, addition, division and multiplication.
Subsubsection 1.1.1.1 Addition of integers
Addition is losely defined as putting together. In addition, adding a positive and positive, gives a positive.
An example is
\(2+4=6\text{.}\)
While adding a positive to a negative, the result and the sign may varry depending on the presidence of a negative or positive sign.
Activity 1.1.1 .
\(\textbf{Work in groups}\)
Paper made number line with values from
\(-10\) to
\(10\)
Sheets of papers containing questions.
A central book for recording marks scored by each student.
The teacher to give question papers containing randomized questions.
Each student to attempt the given questions.
The teacher to call out each student for provide his or her answer.
Teacher to validate the answers given.
The student with the highest number of correct answers emerges the winner!
Example 1.1.1 .
Jane had
\(3000\) bags of rice in January. In February she bought an additional
\(430\) bags. What is the current number of bags she has?
Solution .
\begin{align*}
\text{January} =\amp 3000\, \text{bags} \\
\text{February}=\amp 430\, \text{bags} \\
3000 + 430=\amp 3430\, \text{bags}
\end{align*}
Example 1.1.2 .
Solution .
Simply add
\(3+5+8=\amp 16\)
Example 1.1.3 .
During a hiking trip, a group of students recorded their changes in elevation throughout the day. In the morning, they ascended
\(120\) meters up a hill.Then, they descended
\(85\) meters into a small valley. After lunch, they climbed up again by
\(40\) meters, but later slipped and went down by
\(65\) meters due to a steep slope. What was the group’s final elevation change relative to their starting point? Explain your answer using integer operations.
Solution .
\begin{align*}
120-85=\amp 35\, \text{meters} \\
35+40=\amp 75\, \text{meters} \\
75-65=\amp 10\, \text{meters}
\end{align*}
Subsubsection 1.1.1.2 Subtraction of integers
Subtraction is deducting one number from the other. For instance, if Olivia had 10 sweets then she ate 3, she remains with 7.
In subtraction a negative and a negative, gives a negative result. While a positive and a negative gives a negative result.
For example
\(-2-10=-12\) and
\(3-9=-6\)
Activity 1.1.2 .
\(\textbf{Work in groups}\)
Sheets of paper for working out
The teacher to introduce subtraction through a real-life budgeting scenario with 10,000 shillings.
In groups, students choose expenses and subtract them from their budget.
Students to continue subtracting each item step-by-step and calculate the remaining balance.
Groups share what they spent on, how much is left, and why.
The teacher to introduce surprise expenses to challenge their subtraction and decision-making.
Example 1.1.4 .
Crown had
\(12\) pieces of apples. He finds out that
\(4\text{.}\) are spoilt. How many apples are edible?
Solution .
\begin{align*}
12\, \text{apples} - 4\, \text{apples}=\amp 8\, \text{apples}
\end{align*}
Example 1.1.5 .
Solve for :
\(-60-40-20\)
Solution .
\begin{align*}
-60-40-20=\amp -120
\end{align*}
Example 1.1.6 .
Solve for :
\(-40--20-10\)
Solution .
HINT : (- -) = +
\begin{align*}
-40+20-10=\amp -30
\end{align*}
Subsubsection 1.1.1.3 Multiplication of integers
Multiplication is defined as repeated addition of a number to itself for the given number of times.
Activity 1.1.3 .
\(\textbf{Work in groups}\)
Manually prepared Bingo cards
The teacher to create bingo cards with a grid.
Teacher to fill in each square with answers to the questions being tested.
The teacher to call out multiplication problems.
Students to look for product of called out problem in the square grid and mark it off.
The first student to get the answer calls out Bingo! And wins
Example 1.1.7 .
Brian earns 450 shillings per day working at a hardware shop. If he works for 22 days in a month, how much money does he earn in that month?
Solution .
\begin{align*}
450\, \text{shillings} \times 22 =\amp 9900\, \text{shilligs}
\end{align*}
Example 1.1.8 .
Evaluate:
\(20 \times 4 \times2\)
Solution .
\begin{align*}
20 \times 4 \times2 =\amp 160
\end{align*}
Example 1.1.9 .
Solve:
\(-10 \times -2 \times 4\)
Solution .
\begin{align*}
-10 \times -2=\amp 20 \\
20 \times 4=\amp 80
\end{align*}
Subsubsection 1.1.1.4 Division of integers
Division means sharing among a given quantity. A basic example is how a mother decides to sha]re oranges she buys among her children, so that each one gets the same number of oranges.
Activity 1.1.4 .
\(\textbf{Work in groups}\)
Under the teachers guidance, the learners to divide themselves in groups of
\(4\text{.}\)
Let each group select a group leader.
Let each group leader from all the groups pick marbles from the teacher, until the marbles are finished.
Let all members of each group confirm the number of marbles the leader has.
The results to be recorded by each group on the different sheets of paper.
The groups to compare their results to see whether they are the same.
Example 1.1.10 .
Wangechi has 12 pencils. She decides to arrange them in groups of 3. How many pencils were in each group?
Solution .
\begin{align*}
1\, \text{group} =\amp 3\, \text{pencils} \\
12\, \text{pencils}=\amp ? \\
12 \div 3=\amp 4
\end{align*}
Example 1.1.11 .
Evaluate :
\(-20 \div -5\)
Solution .
\begin{align*}
-20 \div -5=\amp 4
\end{align*}
Example 1.1.12 .
Solution .
\begin{align*}
-60 \div 2=\amp -30
\end{align*}
Technology 1.1.13 .
\(\textbf{Technology Integration:Basic operations of integers}\)
\(\textbf{YouTube: Math and Eng TV }\)
This channel guides through the steps of working out basic integer operations.
Subsection 1.1.2 Combined operation of Integers
Combined operation of integers refers to representation and calculation of basic mathematics operations in a single question.
Subsubsection 1.1.2.1 Mixed operation of integers
In mixed operation of integers, the four basic operations of addition,subtraction,multiplication and division are carried out in the same question. Below is an example
Example 1.1.14 .
Victoria had 12 boxes each containing 10 apples. She gave 6 apples to her friends.She decided to pack the remaining apples in 2 boxes, how many apples will be contained in each box?
Solution .
\begin{align*}
1\, \text{box} =\amp 10\, \text{apples} \\
12\, \text{boxes}=\amp ? \\
12 \times 10=\amp 120
\end{align*}
From the 120 apples, she gave 6 to her friends,
\(120-6=114 \text{apples}\)
When division is carried out,
\(114 \div 2=57 \text{apples} \)
Example 1.1.15 .
Evaluate the expression :
\((-3)\times(-4)+2-4\div2 \)
Solution .
Observe the
\(BODMAS\) rule:
\begin{align*}
(-3)\times(-4)=\amp 12 \\
12 \div 2=\amp 6\\
6+2-4=\amp 4
\end{align*}
Subsection 1.1.3 Real life application of integers
Integers are a part of our daily lives, indeed more so outside school. Integers help us express and deal with real-life situations from positive and negative numbers.
Integers find use in tracking of scores, conducting census, counting items that can’t be counted in halves or decimals and even in computer programming and coding.
