# Lesson 18

## Warm-up: 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 three-digit 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

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$54 + 2 + 18$$
• $$61 + 104 + 39$$
• $$25 + 63 + 75 + 7$$
• $$50 + 106 + 19 + 101$$

### Activity Synthesis

• 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.)
• “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.

MLR8 Discussion Supports. Synthesis: Display sentence frames to agree or disagree. “I agree because . . .” “I disagree because . . . .”
Representation: Access for Perception. Invite students to model the situation using sticky notes or scrap paper to represent the coins. Students can label each sticky note with the thickness, value, and owner of the coins, then move the sticky notes around as they model and solve each problem. Encourage students to use the sticky notes to solve strategically. For example, they might group like denominators before adding or layer repeated fractions to represent multiplication.
Supports accessibility for: Conceptual Processing, Visual-Spatial 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
1. If Diego and Lin each stack their centavo coins, whose stack would be taller? Show your reasoning.

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

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

4. If they combine their coins to make a single stack, would it be more than 2 centimeters tall? Show your reasoning.

### 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

Materials to Gather

Materials to Copy

• More Than Two Fractions

### 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.

1. $$\frac{2}{100} + \frac{13}{10} + \frac{1}{10} + \frac{8}{100}$$

2. $$\frac{50}{10} + \frac{16}{100} + \frac{2}{10}$$

3. $$\frac{3}{10} + \frac{4}{100} + \frac{7}{10} + \frac{26}{100}$$

4. $$\frac{4}{100} + 3\frac{2}{10} + 1\frac{5}{10}$$

5. $$1\frac{1}{10} + 5\frac{2}{100} + \frac{78}{100}$$

6. $$2\frac{7}{10} + \frac{2}{100} + \frac{8}{10}$$

### 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?”

## 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 $$\frac{80}{100} + \frac{20}{100}$$.