Lesson 18
Lots of Fractions to Add
Warmup: Number Talk: A Bunch of Numbers (10 minutes)
Narrative
The purpose of this Number Talk is to encourage students to apply properties of operations (especially the commutative and associative properties of addition) to mentally find sums of three or more whole numbers. The reasoning elicited here will be helpful later in the lesson when students are to add three or more tenths and hundredths.
To mentally add several two and threedigit numbers, students need to look for and make use of structure (MP7), such as finding pairs of numbers that add up to 10 or 100, or numbers that end in 0 or 5.
Launch
 Display one expression.
 “Give me a signal when you have an answer and can explain how you got it.”
 1 minute: quiet think time
Activity
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each expression mentally.
 \(54 + 2 + 18\)
 \(61 + 104 + 39\)
 \(25 + 63 + 75 + 7\)
 \(50 + 106 + 19 + 101\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “What strategies were helpful for adding multiple numbers?” (Sample responses:
 Find two or three numbers that add up to 10 or 100.
 Add numbers that end with 5 or 0 first. Add familiar numbers first.
 Put the numbers into groups of two, and then add what’s in each group before adding the groups.)
 Consider asking:
 “Who can restate _______ 's reasoning in a different way?”
 “Did anyone have the same strategy but would explain it differently?”
 “Did anyone approach the expression in a different way?”
 “Does anyone want to add on to____’s strategy?”
Activity 1: Stack Centavos and Pesos (25 minutes)
Narrative
In this activity, students solve problems involving tenths and hundredths in a context about coins. Given information about the thickness of some Mexican coins, students compare the heights of different combinations of stacked coins. To complete the task, students need to write equivalent fractions, add tenths and hundredths, and compare fractions. Some students may choose to use multiplication to reason about the problems. Though the mathematics here is not new, the context and given information may be novel to students. Students have a wide variety of approaches available for these problems with no solution approach suggested (MP1). For example, to compare the peso coins of Diego and Lin, students could reason that they each have a 5 peso and a 20 peso coin and then compare the remaining coins, a 1 peso coin and 2 peso coin on the one hand and a 20 peso coin on the other. This method would require minimal calculations. Other students may add the thicknesses of Lin's coins and Diego's coins and compare these values.
To help students visualize stacked coins, prepare some coins of different thicknesses or include an image of stacked coins. (Access to the Mexican coins would be interesting to students but is not essential.) Some students may be curious about the equivalents of centavos or pesos in U.S. dollars. Consider checking the exchange rates before the lesson.
Advances: Speaking, Conversing
Supports accessibility for: Conceptual Processing, VisualSpatial Processing, Memory
Required Materials
Materials to Gather
Required Preparation
 Gather a few coins of different thicknesses for display.
Launch
 Display the image of Mexican coins.
 “What do you notice? What do you wonder?”
 1 minute: quiet think time
 Share responses.
 Explain that pesos and centavos are units of Mexican money, just as dollars and cents are units of American money.
 “What are the values of the American coins we use today?” (1 cent, 5 cents, 10 cents, 25 cents, half a dollar, and 1 dollar.)
 Explain that Mexico uses many more types of coins than the U.S. does.
 Display coins of different thicknesses.
 “Just like pennies, dimes, nickels, and quarters, centavo and peso coins have different weights and thicknesses.”
 Refer to the table in the task, showing the thicknesses in tenths or hundredths of a centimeter.
Activity
 “Work with your partner on the first three problems.”
 7–8 minutes: partner work time
 “Find another group of classmates and share your responses. Discuss any disagreement you might have.”
 3–4 minutes: group discussion
 “Take a few quiet minutes to answer the last question.”
 3 minutes: independent work time
 Consider asking each combined group to share their response to one assigned problem.
Student Facing
Diego and Lin each have a small collection of Mexican coins.
The table shows the thickness of different coins in centimeters (cm) and how many of each Diego and Lin have.
coin value  thickness in cm  Diego  Lin 

1 centavo  \(\frac{12}{100}\)  3  1 
10 centavos  \(\frac{22}{100}\)  0  1 
1 peso  \(\frac{16}{100}\)  0  1 
2 pesos  \(\frac{14}{100}\)  0  1 
5 pesos  \(\frac{2}{10}\)  1  1 
20 pesos  \(\frac{25}{100}\)  2  1 

If Diego and Lin each stack their centavo coins, whose stack would be taller? Show your reasoning.

If they each stack their peso coins, whose stack would be taller? Show your reasoning.

If they each stack all their coins, whose stack would be taller? Show your reasoning.

If they combine their coins to make a single stack, would it be more than 2 centimeters tall? Show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 Invite groups to share their responses and reasoning. Highlight the different approaches students took to solve the same problems.
 If no students mentioned using multiplication to find the height of Diego and Lin’s centavo stacks, ask if any of the heights could be found using multiplication.
Activity 2: More Than Two Fractions (10 minutes)
Narrative
The purpose of this activity is for students to practice finding sums of three or more fractions in tenths and hundredths and applying properties of operations to facilitate that addition. (Students are not expected to use the terms “commutative property” or “associative property,” but should recognize from the work in earlier grades that numbers can be added in different orders and in different groups.)
This activity can be done in the format of a gallery walk. Ask students to visit at least three of six posters (or as many as time permits). The last three expressions include one or more mixed numbers. In the last expression, the fractional parts add up to a sum greater than 1, which would need to be decomposed into a mixed number and a fraction before being added to the whole number. Consider assigning this as a starting expression for students who could use an extra challenge.
Required Materials
Required Preparation
 Create six posters with an addition expression from the activity on each one.
Launch
 Groups of 2
If done as a gallery walk:
 Consider assigning each group a starting poster and giving directions for rotation.
 “You’ll find six posters with addition expressions on each one. Visit at least three posters and find the value of the expressions.”
Activity
 “Work with your partner on the first two expressions and independently on at least one of them. Show your reasoning.”
 10 minutes: group work or gallery walk
Student Facing
Find the value of at least 3 of the expressions. Show your reasoning.

\(\frac{2}{100} + \frac{13}{10} + \frac{1}{10} + \frac{8}{100}\)

\(\frac{50}{10} + \frac{16}{100} + \frac{2}{10}\)

\(\frac{3}{10} + \frac{4}{100} + \frac{7}{10} + \frac{26}{100}\)

\(\frac{4}{100} + 3\frac{2}{10} + 1\frac{5}{10}\)

\(1\frac{1}{10} + 5\frac{2}{100} + \frac{78}{100}\)

\(2\frac{7}{10} + \frac{2}{100} + \frac{8}{10}\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 See lesson synthesis.
Lesson Synthesis
Lesson Synthesis
Select a group to share their response and reasoning for finding the value of each expression. Focus the discussion on whether there are other possible solution paths and on the last expression.
“Today we used what we know about equivalent fractions and addition of fractions to solve problems.”
Invite students to reflect on how their ability to find sums of fractions have improved and any areas of struggles. Consider asking:
 “In what ways has your ability to add fractions improved? What might still be challenging?”
 “Was there a kind of error you made multiple times? What was the error and why might that be?”
Cooldown: U.S. Coins (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Section Summary
Student Facing
In this section, we learned more ways to add fractions and to solve problems that involve adding, subtracting, and multiplying fractions.
We started by adding tenths and hundredths, using what we know about equivalent fractions. For example, to find the sum of \(\frac{4}{10}\) and \(\frac{30}{100}\), we can:
 Write \(\frac{4}{10}\) as \(\frac{40}{100}\), and then find \(\frac{40}{100} + \frac{30}{100}\), or
 Write \(\frac{30}{100}\) as \(\frac{3}{10}\), and then find \(\frac{4}{10} + \frac{3}{10}\).
We learned that when adding a few fractions, it may help to rearrange or group them. For instance:
 \(\frac{6}{100} + \frac{2}{10} + \frac{74}{100}\) can be rearranged as \(\frac{6}{100} + \frac {74}{100} + \frac{2}{10}\).
 Next, the hundredths can be added first, giving \(\frac{80}{100} + \frac{2}{10}\).
 Then, we can write an equivalent fraction for \(\frac{80}{100}\) and find \(\frac{8}{10} + \frac{2}{10}\), or write an equivalent fraction for \(\frac{2}{10}\) and find \(\frac{80}{100} + \frac{20}{100}\).