Connections Between Adding and Subtracting, Multiplying and Dividing

Connections Between Adding and Subtracting, Multiplying and Dividing exercise

• Explain how the Inverse Properties of Addition and Multiplication work.

  Hints

  To split $10$ pieces of cake between $5$ people, you have to divide $10$ by $5$:

  $10\div 5=2$.

  So each person gets $2$ pieces of cake.

  Instead of adding a number, you can also use multiplication:

  $4+4+4+4+4+4=6\times 4=4\times 6$.

  If you have $5$ pieces of cake and you give away exactly this number of pieces, then you have no pieces of cake left over at all: $5-5=0$ or $5+-5=0$.

  Solution

  This $\mathbf{+}$ is the sign for addition.

  Let's have a look at the following example Sunflower seeds:

  Jerome had $200$ seeds in his knapsack, but he lost $40$ seeds. So he has to collect $40$ seeds to get back to $200$ seeds.

  This lead to

  • $200-40=160$
  • $200-40+40=200$

  To figure this out, we used the inverse property of addition:

  $a+-a=0$ or $-a+a=0$.

  This $\mathbf{\times}$ is the sign for multiplication.

  Once again let's have a look at Jerome and what he needs for his potion. He still needs forget-me-not petals.

  Each flower Jerome shakes has $5$ petals. Together he gets

  $5+5+5=5\times 3=15$ petals.

  He splits the whole number on $3$ jars. So he has to put

  $15\div3=5$

  petals in each jar.

  To figure this out, we used the inverse property of multiplication:

  $a\times b=ab$ and $\frac{ab}b=a$.

• Summarize your knowledge about the Definition of Multiplication.

  Hints

  Compare the expressions:

  • $3+3+3+3+3=?$
  • $3\times 5=?$
  • $3+5=?$
  • $3^5=?$

  One beagle plus one beagle, plus one beagle plus one beagle, is the same as four beagles.

  $1+1+1+1=1\times 4=4$

  Solution

  Jerome needs $12$ ladybugs.

  He can catch $4$ ladybugs at a time. How many times does Jerome have to catch ladybugs to get $12$ ladybugs in total:

  • $4+4=8$
  • $4+4+4=12$

  Here we see that $4$ is added three times. We can rewrite this as a product:

  $4+4+4=4\times 3$.

  In general, you can write the sum of $n$ times the summand $a$ as follows:

  $a+a+\cdots+a=n\times a$.

• Place the equations with the correct properties and definition.

  Hints

  The inverse property of addition is given by $a+-a=0$ or $-a+a=0$.

  The inverse property of multiplication is given by $a\times b=ab$ and $\frac{ab}b=a$.

  Multiplication is a shorter way to write a sum of repeating summands

  $a+a+\cdots+a=n\times a$.

  For example, $2+2+2+2+2+2=6\times 2=2\times 6=12$.

  Solution

  Let's start with the inverse property of addition:

  $a+-a=0$ or $-a+a=0$.

  • $4+-4=0$
  • $27+3-3=27$
  • $-b+b=0$

  Multiplication is a shorter way to write sums with repeating summands $a+a+\cdots+a=n\times a$:

  • $2\times 7=7+7$
  • $k+k+k+k=4\times k$
  • $3+3+3+3=4\times 3$

  We finish with the inverse property of multiplication:

  $a\times b=ab$ and $\frac{ab}b=a$.

  • $5\times 3\div 3=5$
  • $m\times 2\div 2=m$
  • $(bc)\div c=b$

• Complete the equations to make true statements. (+ multiplication)

  Hints

  Here is an example for the inverse property of addition:

  $27+5-5=27$.

  Here is an example for the inverse property of multiplication:

  $3\div 4\times 4=3$.

  Instead of adding $5+5+5+5+5$ you can also write a product:

  $5\times 5=25$.

  Solution

  Here we use...

  • ...the inverse property of addition: $a+-a=0$ or $-a+a=0$.
  • ...the inverse property of multiplication: $a\times b=ab$ and $\frac{ab}b=b$ or $a\times b\div b=a$.
  • ...the definition of multiplication: $a+a+a+a+a=5\times a$ - just count the number of summands.

  1. $5\times 3\div 3 = 5$ - this is the inverse property of multiplication.
  2. Once again the inverse property of multiplication: $p\div p\times r=r$.
  3. The next three points are examples for the inverse property of addition $5+-5=0$.
  4. $-7+7=0$.
  5. $33+5-5=33$.
  6. The definition of multiplication: $6+6+6+6=4\times 6=24$.

• Determine how many ladybugs Jerome found.

  Hints

  Jerome collects $4$ ladybugs each time:

  • $4$ ladybugs one time
  • $4+4$ ladybugs two times
  • ...

  You can also multiply $4$ by $3$.

  Just add $4$ three times.

  Solution

  Jerome collects $4$ ladybugs at one time:

  So he collects

  • $4+4=8$ ladybugs two times
  • $4+4+4=12$ ladybugs three times

  You can shorten the calculation by multiplying $4$ by $3$:

  $4+4+4=4\times 3=12$.

  Keep in mind that multiplication is a shorter way to write a sum with repeating summand $a$:

  $a+a+\cdots+a=n\times a$.

• Figure out which equations represent how many of each item Jerome has lost.

  Hints

  The inverse property of addition: $5+6-6=5$.

  The definition of multiplication:

  $6+6+6+6+6=5\times 6=6\times 5=30$.

  The inverse property of multiplication: $7\times 3\div 3=7$.

  Solution

  • Blueberries: Loosing $5$ blueberries can be written as $-5$. So, using the inverse property of addition, we conclude that $20-5+5=20$.

  • Twigs: Once again we use the inverse property of addition: $25+-13+13=25$.

  • Coffee beans: $3$ times $5$ can be written as $5+5+5=3\times 5=15$, using the definition of multiplication.

  • Fish scales: Lastly, we use the inverse property of multiplication to examine the number of fish scales: $6\times 4\div 4=6$.