# Lesson 12

Equations with Unknowns

## Warm-up: True or False: Making Tens (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for making it easier to find the value of expressions by making a ten. These understandings help students deepen their understanding of the properties of operations and will be helpful later when students will need to fluently add within 100.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$40 = 10 + 27 + 3$$

• $$47 = 20 + 7 + 3 + 10$$

• $$60 = 3 + 47 + 10$$

### Activity Synthesis

• “How could we change the second equation to make it true?” (We could change the 10 to 17 or the 20 to 27 because we need 7 more.)

## Activity 1: Number Line Riddles (20 minutes)

### Narrative

The purpose of this activity is for students to solve addition and subtraction problems within 100 with the unknown in all positions. Students write equations with a ? for the unknown and find the number that makes the equations true. The mathematical context of each problem encourages students to use the number line to reason about what is unknown and how they may represent the problem with an equation (MP2).

Representation: Access for Perception. Use a small toy animal or cut out animal (bird, frog, rabbit, or another animal) to demonstrate the jumping on the number line.
Supports accessibility for: Attention, Organization

### Required Materials

Materials to Copy

• Number Line to 100

### Launch

• Groups of 2
• Give each student a copy of the blackline master.
• Display an image of a blank number line  or draw a number line.
• “Today you will be solving riddles to find a mystery number.”
• “For each riddle, you will write an equation that represents the riddle, and write a ? for the unknown.”
• “Then you will represent the equation on the number line.”
• “Let’s try one together.”
• Demonstrate on a number line with input from the students.
• “I started on a number, jumped 10 to the left. My jump ended at 42. What equation could I write with a ? for the unknown?” (? - 10 = 42)
• “How could I find the value of the mystery number?” (Do the opposite. Start at 42 and move 10 to the right.)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share responses and record on the number line.

### Activity

• “Now you will have a chance to solve riddles to find a missing number, and then represent your thinking on a number line. You and your partner can take turns reading the riddle, while the other person follows along on the number line.”
• 12 minutes: partner work time

### Student Facing

Solve riddles to find the mystery number.

For each riddle:

• Write an equation that represents the riddle and write a ? for the unknown.

• Write the mystery number. Represent the equation on the number line.

1. I started at 15 and jumped 17 to the right. Where did I end?

Equation: _______________________________

Mystery number: _______________________

2. I started at a number and jumped 20 to the left. I ended at 33. Where did I start?

Equation: _______________________________

Mystery number: _______________________

3. I started on 42 and ended at 80. How far did I jump?

Equation: _______________________________

Mystery number: _______________________

4. I started at 76 and jumped 27 to the left. Where did I end?

Equation: _______________________________

Mystery number: _______________________

5. I started at a number and jumped 19 to the right. I ended at 67. Where did I start?

Equation: _______________________________

Mystery number: _______________________

6. I started at 92 and ended at 33. How far did I jump?

Equation: _______________________________

Mystery number: _______________________

### Student Response

If students write equations other than those represented by the riddle or number line representation, consider asking:
• “How do you decide where to put your starting or ending points?”
• “What part of the riddle could help you choose the direction for your arrows and jumps?”

### Activity Synthesis

• Invite students to share the answer to each riddle and display their number line.
• “Which riddles did you and your partner find to be most challenging? Explain.”

## Activity 2: Make the Equations True (15 minutes)

### Narrative

The purpose of this activity is for students to find the value of an unknown in addition and subtraction equations. Students can choose to find the unknown number using either operation and represent their thinking on a number line. Listen for the ways students use the number line to make sense of the relationship between the numbers in each equation and use methods that show they are thinking about ways to use the structure of the number line and their understanding of place value (MP7).

MLR8 Discussion Supports.Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.

### Required Materials

Materials to Copy

• Number Line to 100

### Launch

• Groups of 2
• Give each student a copy of the blackline master.

### Activity

• “Find the number that makes each equation true in a way that makes sense to you.”
• “Represent your thinking on the number line.”
• Monitor for a student who finds the value for $${?} + 57 = 72$$ by:
• starting at 57, drawing a jump to 72, and counting each length unit in between
• staring at 57, drawing a jump of 3 and a jump of 12 to 72
• starting at 72 and jumping back 57 in one jump or multiple jumps

### Student Facing

Find the number that makes each equation true.

Show your thinking on the number line.

1. $${?} - 48 = 19$$

2. $$86 - {?} = 39$$

3. $${?} + 57 = 72$$

4. $$73 + {?} = 91$$

### Activity Synthesis

• Invite previously selected students to share how they found the number that makes $$? + 57 =72$$ true.
• “How does ______'s number line show the numbers that we knew? How does it show the unknown number?”
• “How are these methods the same? How are they different?”
• As time permits, continue with other equations.

## Lesson Synthesis

### Lesson Synthesis

“Today you solved all different types of problems on the number line with the unknown in all different positions by using addition and subtraction. You used equations with a symbol for the unknown and found the number that made them true.”

Display $${?} + 14 = 24$$ and $${?} - 14 = 24$$

“How could I find the number that makes each of these equations true?” (For the addition equation, you could start at 24 and go to the left 14 on the number line, but for the subtraction equation you could start at 24 and go to the right 14.)

“How did the number line help with these types of equations?” (I could start with the result and jump in the opposite direction and land on the answer.)