We included HMH Into Math Grade 4 Answer Key PDF Module 15 Lesson 5 Apply Properties of Addition to Add Fractions and Mixed Numbers to make students experts in learning maths.

I Can add fractions and mixed numbers using the properties of addition.

Step It Out

Question 1.
The Watson family cross-country skis on all three trails in this park. How many miles do they ski?

A. Write an addition equation for the problem. Use d for the total distance.
d = 10.5 miles
4$$\frac{8}{10}$$ + 3$$\frac{5}{10}$$ + 2$$\frac{2}{10}$$
Explanation:
Watson family cross wood trail 4$$\frac{8}{10}$$
They cross mountain trail 3$$\frac{5}{10}$$
They cross wolf trail 2$$\frac{2}{10}$$
Total miles they ski = 4$$\frac{8}{10}$$ + 3$$\frac{5}{10}$$ + 2$$\frac{2}{10}$$
= $$\frac{48}{10}$$ + $$\frac{35}{10}$$ + $$\frac{22}{10}$$
= $$\frac{48 + 35 + 22}{10}$$
= $$\frac{105}{10}$$

B. Use the Commutative Property to order the mixed numbers so the fractions that add to 1 are next to each other.
3$$\frac{5}{10}$$ + 4$$\frac{8}{10}$$ + 2$$\frac{2}{10}$$
= $$\frac{35}{10}$$ + $$\frac{48}{10}$$ + $$\frac{22}{10}$$
= $$\frac{35 + 48 + 22}{10}$$
= $$\frac{105}{10}$$
Explanation:
The commutative property of addition states that when finding a sum,
changing the order of the addends will not change their sum.
For example; a + b = b + a

C. Use the Associative Property to group the mixed numbers you can add mentally.
(4$$\frac{8}{10}$$ + 3$$\frac{5}{10}$$) + 2$$\frac{2}{10}$$
= $$\frac{48}{10}$$ + $$\frac{35}{10}$$ + $$\frac{22}{10}$$
= $$\frac{48 + 35 + 22}{10}$$
= $$\frac{105}{10}$$

Explanation:
In associative property of addition when finding a sum, changing the way addends are grouped will not change their sum.
In symbols, the associative property of addition says that for numbers a, b and c:
For example; (a + b)+c = a + (b + c)

D. Find the sum. Show your work.

10.5 miles
Explanation:
According to associative property

$(a + b) + c = a + (b + c)$

Let, a = 4$$\frac{8}{10}$$
b = 3$$\frac{5}{10}$$
c = 2$$\frac{2}{10}$$
(4$$\frac{8}{10}$$ + 3$$\frac{5}{10}$$) + 2$$\frac{2}{10}$$
= $$\frac{48}{10}$$ + $$\frac{35}{10}$$ + $$\frac{22}{10}$$
= $$\frac{48 + 35 + 22}{10}$$
= $$\frac{105}{10}$$
= 10.5 miles

E. How far does the Watson family ski?
10.5 miles
Explanation:
(4$$\frac{8}{10}$$ + 3$$\frac{5}{10}$$) + 2$$\frac{2}{10}$$
= $$\frac{48}{10}$$ + $$\frac{35}{10}$$ + $$\frac{22}{10}$$
= $$\frac{48 + 35 + 22}{10}$$
= $$\frac{105}{10}$$
= 10.5

There are many mathematical rules and properties that are necessary or helpful to know when trying to solve math problems.
Learning and understanding these rules helps students form a foundation and they can use to solve problems and tackle more advanced mathematical concepts.

Question 2.
Allie practices ice skating for 1$$\frac{1}{4}$$ hours on Monday, $$\frac{2}{4}$$ hour on Tuesday, and $$\frac{3}{4}$$ hour on Wednesday. How many hours does she spend practicing skating?

A. Write an equation you could use to solve the problem.
1$$\frac{1}{4}$$ + $$\frac{2}{4}$$ + $$\frac{3}{4}$$
=1$$\frac{1}{2}$$ hour
Explanation:
1$$\frac{1}{4}$$ + $$\frac{2}{4}$$ + $$\frac{3}{4}$$
= $$\frac{1 + 2 + 3}{4}$$
= $$\frac{6}{4}$$
=1$$\frac{2}{4}$$
=1$$\frac{1}{2}$$

B. What property can you use to change the order of the addends? Rewrite your equation so the fractions and fractional parts that add to 1 are next to each other or are added first.
Explanation:
Changing the order of addends does not change the sum.
According to the commutative property of addition, changing the order of the numbers we are adding, does not change the sum

C. What property can you use to group the addends that are easy to add mentally? Use parentheses to group the addends.
Associative property
Explanation:
In associative property of addition when finding a sum, changing the way addends are grouped will not change their sum.
In symbols, the associative property of addition says that for numbers a, b and c:
For example; (a + b)+c = a + (b + c)

=1$$\frac{1}{2}$$ hour
Explanation:
(1$$\frac{1}{4}$$ + $$\frac{2}{4}$$) + $$\frac{3}{4}$$
= $$\frac{3}{4}$$ + $$\frac{3}{4}$$
= $$\frac{6}{4}$$
= 1$$\frac{2}{4}$$
= 1$$\frac{1}{2}$$

Check Understanding

Question 1.
Ross has 4$$\frac{5}{8}$$ pounds of tomatoes, 2$$\frac{1}{8}$$ pounds of green peppers, and 1$$\frac{3}{8}$$ pounds of onions that he will use to make salsa. How many pounds of ingredients does he have? Show your work.
9$$\frac{1}{8}$$ pounds
Explanation:
Ross has 4$$\frac{5}{8}$$ pounds of tomatoes,
2$$\frac{1}{8}$$ pounds of green peppers,
1$$\frac{3}{8}$$ pounds of onions
Total pounds of ingredients he have in all =
4$$\frac{5}{8}$$ + 2$$\frac{1}{8}$$ + 1$$\frac{3}{8}$$
= $$\frac{45}{8}$$ + $$\frac{17}{8}$$ + $$\frac{11}{8}$$
= $$\frac{45+17+11}{8}$$ = $$\frac{73}{8}$$
= 9$$\frac{1}{8}$$

Use the properties and mental math to find the sum.

Question 2.
1$$\frac{3}{4}$$ + 1$$\frac{3}{4}$$ + 1$$\frac{1}{4}$$ = ___________
4$$\frac{3}{4}$$
Explanation:
1$$\frac{3}{4}$$ + 1$$\frac{3}{4}$$ + 1$$\frac{1}{4}$$
= $$\frac{7}{4}$$ + $$\frac{7}{4}$$ + $$\frac{5}{4}$$
= $$\frac{7+7+5}{4}$$ = $$\frac{19}{4}$$ = 4$$\frac{3}{4}$$

Question 3.
4$$\frac{1}{6}$$ + 3$$\frac{2}{6}$$ + 1$$\frac{5}{6}$$ = _____________
9$$\frac{2}{6}$$
Explanation:
4$$\frac{1}{6}$$ + 3$$\frac{2}{6}$$ + 1$$\frac{5}{6}$$
= $$\frac{25}{6}$$ + $$\frac{20}{6}$$ + $$\frac{11}{6}$$
= $$\frac{25+20+11}{6}$$ = $$\frac{56}{6}$$ = 9$$\frac{2}{6}$$

Use the properties and mental math to solve the problem.

Question 4.
Use Structure Yuri makes a shopping list. How many pounds of vegetables does Yuri want to buy?

6$$\frac{2}{4}$$ pounds
Explanation:
3$$\frac{1}{4}$$ potatoes + 1$$\frac{2}{4}$$ turnips + 1$$\frac{3}{4}$$ carrots
= $$\frac{13}{4}$$ + $$\frac{6}{4}$$ + $$\frac{7}{4}$$
= $$\frac{13+6+7}{4}$$ = $$\frac{26}{4}$$ = 6$$\frac{2}{4}$$

Question 5.
Cayla walks 1$$\frac{1}{10}$$ miles on Monday, 1$$\frac{3}{10}$$ miles on Wednesday, and mile on Friday. Cayla wants to know how many miles she walks on these three days. Complete Cayla’s work and name the properties she uses.
1$$\frac{1}{10}$$ + 1$$\frac{3}{10}$$ + $$\frac{9}{10}$$ = 1$$\frac{3}{10}$$ + 1 $$\frac{1}{10}$$ + $$\frac{9}{10}$$ _______________
= 1$$\frac{3}{10}$$ + (1$$\frac{1}{10}$$ + $$\frac{9}{10}$$) ____________
= 1$$\frac{3}{10}$$ + ___________
= _____________
Cayla uses associative property and commutative property 1$$\frac{5}{10}$$
Explanation:
Commutative property – Changing the order of addends does not change the sum.
1$$\frac{1}{10}$$ + 1$$\frac{3}{10}$$ + $$\frac{9}{10}$$
= 1$$\frac{3}{10}$$ + 1 $$\frac{1}{10}$$ + $$\frac{9}{10}$$
= $$\frac{13}{10}$$ +$$\frac{11}{10}$$ + $$\frac{9}{10}$$
= $$\frac{13+11+9}{10}$$ = $$\frac{33}{10}$$
= 3$$\frac{3}{10}$$
Associative property – regrouping the addends doesn’t change the sum.
= 1$$\frac{3}{10}$$ + (1$$\frac{1}{10}$$ + $$\frac{9}{10}$$)
= $$\frac{13}{10}$$ + $$\frac{20}{10}$$
= $$\frac{13+20}{10}$$ = $$\frac{33}{10}$$ = 3$$\frac{3}{10}$$

Question 6.
When preparing for a recital, Xiao practices playing the violin for 1$$\frac{1}{6}$$ hours on Friday, 1$$\frac{2}{6}$$ hours on Saturday, and 1$$\frac{5}{6}$$ hours on Sunday. How many hours does Xiao practice? Explain how you can use the Associative and Commutative Properties of Addition to solve mentally.
4$$\frac{2}{6}$$ hours
Explanation:
Xiao practices playing the violin for 1$$\frac{1}{6}$$ hours on Friday,
1$$\frac{2}{6}$$ hours on Saturday and
1$$\frac{5}{6}$$ hours on Sunday.
Associative property:
1$$\frac{1}{6}$$ +(1$$\frac{2}{6}$$ + 1$$\frac{5}{6}$$)
= $$\frac{7}{6}$$ + ($$\frac{8}{6}$$ + $$\frac{11}{6}$$)
= $$\frac{7}{6}$$ + $$\frac{19}{6}$$
= $$\frac{26}{6}$$ = 4$$\frac{2}{6}$$
Commutative property:
Given, 1$$\frac{1}{6}$$ +1$$\frac{2}{6}$$ + 1$$\frac{5}{6}$$
1$$\frac{2}{6}$$ +1$$\frac{5}{6}$$ + 1$$\frac{1}{6}$$
= $$\frac{8}{6}$$ + $$\frac{11}{6}$$ + $$\frac{7}{6}$$
= $$\frac{8+11+7}{6}$$ = $$\frac{26}{6}$$
= 4$$\frac{2}{6}$$

Question 7.
Reason Mr. Cahill drives 1$$\frac{3}{12}$$ hours before stopping for gas. He drives another 1$$\frac{6}{12}$$ hours before stopping for lunch. Then he drives the remaining 1$$\frac{3}{12}$$ hours to his destination. How many hours does he drive?
4 hours
Explanation:
Mr. Cahill drives 1$$\frac{3}{12}$$ hours before stopping for gas.
He drives another 1$$\frac{6}{12}$$ hours before stopping for lunch.
Then he drives the remaining 1$$\frac{3}{12}$$ hours to his destination.
Total hours he drive = 1$$\frac{3}{12}$$ + 1$$\frac{6}{12}$$ + 1$$\frac{3}{12}$$
= $$\frac{15}{12}$$ + $$\frac{18}{12}$$ + $$\frac{15}{12}$$
= $$\frac{15+18+15}{12}$$ = $$\frac{48}{12}$$ = 4

Question 8.
Attend to Precision Dylan wants to solve $$\frac{3}{8}$$ + 1$$\frac{7}{8}$$ + 2$$\frac{5}{8}$$. Explain how he can use the Commutative and Associative Properties to add fractions and mixed numbers mentally.
The associative property of addition means we can group the addends in different ways without changing the order to get the sum.
The commutative property of addition means we can reorder the addends without changing the outcome to get the sum.
Explanation:
Given, $$\frac{3}{8}$$ + 1$$\frac{7}{8}$$ + 2$$\frac{5}{8}$$
By using associative property first group the 2 addends and add the other one to get the sum.
$$\frac{3}{8}$$ + 1$$\frac{7}{8}$$ + 2$$\frac{5}{8}$$.
= 1$$\frac{7}{8}$$ + $$\frac{3}{8}$$ + 2$$\frac{5}{8}$$.
= $$\frac{14}{8}$$ + 1$$\frac{3}{8}$$ + 2$$\frac{5}{8}$$.
= $$\frac{14+3}{8}$$+ 2$$\frac{5}{8}$$.
= $$\frac{17}{8}$$ + $$\frac{21}{8}$$.
= $$\frac{17+21}{8}$$ = $$\frac{38}{8}$$ = 4$$\frac{6}{8}$$.
By using Commutative property change the order of addends to get the sum.
Given, $$\frac{3}{8}$$ + 1$$\frac{7}{8}$$ + 2$$\frac{5}{8}$$.
1$$\frac{7}{8}$$ + 2$$\frac{5}{8}$$ + $$\frac{3}{8}$$.
= $$\frac{14}{8}$$ + $$\frac{21}{8}$$ + $$\frac{3}{8}$$.
= $$\frac{14+21+3}{8}$$ = $$\frac{38}{8}$$ = 4$$\frac{6}{8}$$.

Use the properties and mental math to find the sum. If possible, write the answer as a mixed number with a fractional part less than 1.

Question 9.
Paz has 1$$\frac{7}{10}$$ yards of red fabric, 2$$\frac{2}{10}$$ yards of blue fabric, and 8$$\frac{3}{10}$$ yards of tan fabric to make a wreath. How many yards of fabric does he have?

12$$\frac{2}{10}$$ yards
Explanation:
Red fabric 1$$\frac{7}{10}$$ yards
Blue fabric 2$$\frac{2}{10}$$ yards
Tan fabric 8$$\frac{3}{10}$$ yards
Total fabric need to make wreath,
= 1$$\frac{7}{10}$$ + 2$$\frac{2}{10}$$ + 8$$\frac{3}{10}$$
= $$\frac{17}{10}$$ + $$\frac{22}{10}$$ + $$\frac{83}{10}$$
= $$\frac{17+22+83}{10}$$ = $$\frac{122}{10}$$
= 12$$\frac{2}{10}$$ yards
Question 10.
1$$\frac{6}{10}$$ + 2$$\frac{5}{10}$$ + 4$$\frac{4}{10}$$ = _____________
8$$\frac{5}{10}$$
Explanation:
1$$\frac{6}{10}$$ + 2$$\frac{5}{10}$$ + 4$$\frac{4}{10}$$
= $$\frac{16}{10}$$ + $$\frac{25}{10}$$ + $$\frac{44}{10}$$
= $$\frac{16+25+44}{10}$$ = $$\frac{85}{10}$$ = 8$$\frac{5}{10}$$

Question 11.
$$\frac{9}{12}$$ + 3$$\frac{2}{12}$$ + 4$$\frac{3}{12}$$ = _____________
8$$\frac{2}{12}$$
Explanation:
$$\frac{9}{12}$$ + 3$$\frac{2}{12}$$ + 4$$\frac{3}{12}$$
= $$\frac{9}{12}$$ + $$\frac{38}{12}$$ + $$\frac{51}{12}$$
= $$\frac{9+38+51}{10}$$ = $$\frac{98}{12}$$ = 8$$\frac{2}{12}$$

Question 12.
1$$\frac{3}{4}$$ + 3$$\frac{2}{4}$$ + 1$$\frac{1}{4}$$ = ____________
6$$\frac{2}{4}$$
Explanation:
1$$\frac{3}{4}$$ + 3$$\frac{2}{4}$$ + 1$$\frac{1}{4}$$
= $$\frac{7}{4}$$ + $$\frac{14}{4}$$ + $$\frac{5}{4}$$
= $$\frac{7+14+5}{4}$$ = $$\frac{26}{4}$$ + 6$$\frac{2}{4}$$

Question 13.
$$\frac{7}{8}$$ + 1$$\frac{5}{8}$$ + $$\frac{3}{8}$$ = _____________
2$$\frac{7}{8}$$
Explanation:
$$\frac{7}{8}$$ + 1$$\frac{5}{8}$$ + $$\frac{3}{8}$$
= $$\frac{7}{8}$$ + $$\frac{13}{8}$$ + $$\frac{3}{8}$$
= $$\frac{7+13+3}{8}$$ = $$\frac{23}{8}$$ = 2$$\frac{7}{8}$$

Question 14.
4$$\frac{3}{12}$$ + 1$$\frac{7}{12}$$ + $$\frac{9}{12}$$ = ____________
6$$\frac{7}{12}$$
Explanation:
4$$\frac{3}{12}$$ + 1$$\frac{7}{12}$$ + $$\frac{9}{12}$$
= $$\frac{51}{12}$$ + $$\frac{19}{12}$$ + $$\frac{9}{12}$$
= $$\frac{51+19+9}{12}$$ = $$\frac{79}{12}$$ = 6$$\frac{7}{12}$$

Question 15.
9 + $$\frac{5}{3}$$ + 6$$\frac{1}{3}$$ = ___________
10$$\frac{3}{3}$$ = 11
Explanation:
9 + $$\frac{5}{3}$$ + 6$$\frac{1}{3}$$
= 9 + $$\frac{5}{3}$$ + $$\frac{19}{3}$$
= $$\frac{9+5+19}{3}$$ = $$\frac{33}{3}$$
= 10$$\frac{3}{3}$$

Question 16.
6$$\frac{4}{6}$$ + 2$$\frac{1}{6}$$ + 3$$\frac{2}{6}$$ = ______________
12$$\frac{1}{6}$$
Explanation:
6$$\frac{4}{6}$$ + 2$$\frac{1}{6}$$ + 3$$\frac{2}{6}$$
= $$\frac{40}{6}$$ + $$\frac{13}{6}$$ + $$\frac{20}{6}$$
= $$\frac{40+13+20}{6}$$ = $$\frac{73}{6}$$ = 12$$\frac{1}{6}$$

Question 17.
$$\frac{5}{4}$$ + $$\frac{3}{4}$$ + 7$$\frac{1}{4}$$ = ____________
9$$\frac{1}{4}$$
Explanation:
$$\frac{5}{4}$$ + $$\frac{3}{4}$$ + 7$$\frac{1}{4}$$
= $$\frac{5}{4}$$ + $$\frac{3}{4}$$ + $$\frac{29}{4}$$
= $$\frac{5+3+29}{4}$$ = $$\frac{37}{4}$$ = 9$$\frac{1}{4}$$

Question 18.
2$$\frac{1}{10}$$ + $$\frac{7}{10}$$ + $$\frac{9}{10}$$ = ___________
3$$\frac{7}{10}$$
Explanation:
2$$\frac{1}{10}$$ + $$\frac{7}{10}$$ + $$\frac{9}{10}$$
= $$\frac{21}{10}$$ + $$\frac{7}{10}$$ + $$\frac{9}{10}$$
= $$\frac{21+7+9}{10}$$ = $$\frac{37}{10}$$ = 3$$\frac{7}{10}$$

Question 19.
5$$\frac{1}{8}$$ + $$\frac{6}{8}$$ + 3$$\frac{7}{8}$$ = ____________
9$$\frac{6}{8}$$
5$$\frac{1}{8}$$ + $$\frac{6}{8}$$ + 3$$\frac{7}{8}$$
= $$\frac{41}{8}$$ + $$\frac{6}{8}$$ + $$\frac{31}{8}$$
= $$\frac{41+6+31}{8}$$ = $$\frac{78}{8}$$ = 9$$\frac{6}{8}$$