# Lesson 12

Represent Problems on the Coordinate Grid

## Warm-up: True or False: Addition and Multiplication (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate understandings they have of the properties of operations. These understandings will be helpful later when students will need to be able to use addition and multiplication to solve problems involving money. Each expression here is chosen to represent the total value of a set of coins (nickels, dimes, and quarters).

### 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.

• $$(2 \times 10) + (3 \times 5) = (3 \times 10) + (1 \times 5)$$
• $$(3 \times 25) + (5 \times 5) = 8 \times 25$$
• $$(4 \times 25) + (10 \times 5) = (2 \times 25) + (10 \times 10)$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Which statement was your favorite to think about and why?” (I liked the first one because I could calculate all the values mentally.)

## Activity 1: Heads or Tails (15 minutes)

### Narrative

The purpose of this activity is for students to plot and interpret points that represent the result of flipping a coin 10 times (MP2). Students also interpret points that are already on the graph, representing the number of heads and tails two other students got when they flipped a coin.

Students may wonder what to do if they get the same result twice or the same result as their partner since that point is already plotted on the graph. They may:

• put a letter for their name next to the point
• put a number to indicate that they got that result on their first and second coin tosses

Students may notice that the points all lie on a line (MP7). It is not necessary for students to understand why the points form a line. Focus students’ attention on the meaning of each point.

Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Attention, Organization

### Required Materials

Materials to Gather

### Required Preparation

• Gather pennies, nickels, dimes, and quarters to show students during the launch.

### Launch

• Groups of 2

• Display only the problem stem, without revealing the grid or question(s):
• “We are going to read this problem 3 times.”
• 1st Read: “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails.”
• “What is this situation about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• 2nd Read: “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails.”
• “Name the quantities. What can we count or measure in this situation?”
• 30 seconds: quiet think time
• 2 minutes: partner discussion
• Share and record all quantities.
• Reveal the question(s).
• “What are some strategies we can use to solve this problem?”

### Activity

• 2 minutes: independent think time
• 8 minutes: partner work time

### Student Facing

Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails. Their results are plotted on the graph.

1. How many heads did Jada get? How many tails did Jada get? Explain or show how you know.
2. How many heads did Han get? How many tails did Han get? Explain or show how you know.
3. Flip the coin 10 times and record how many heads and tails you get. Plot the point on the coordinate grid that represents your coin flips.
4. Show your partner the point you plotted on the coordinate grid. Look at your partner's coordinate grid. How many heads did your partner flip? How many tails did your partner flip? Explain or show your reasoning.
5. Do any of the points you plotted lie on the horizontal axis? What would a point on the horizontal axis mean in this situation?
6. If time allows, toss the coin 10 more times and record your results and your partner’s results on the coordinate grid.

### Student Response

For access, consult one of our IM Certified Partners.

If students say they aren’t sure where to plot a point to represent their coin tosses, refer to the point that represents Jada’s data and ask, “What does this point represent?”

### Activity Synthesis

• Display the coordinate grid from the activity.
• As each student shares, ask them to explain where to plot their point on the displayed graph.
• “What are the coordinates of Jada’s point?” ($$(6,3)$$)
• “How many times did Jada toss the coin? How do you know?” (9 times because she got 6 heads and 3 tails.)
• Highlight the point $$(10,0)$$ on the grid. “What does this point mean?” (10 heads and no tails)
• “Did anyone get this result?” (most likely no)
• “Do you think all heads happens very often?” (This probably does not happen very often because that means that you can get only heads on every toss.)

## Activity 2: Coin Values (15 minutes)

### Narrative

The purpose of this activity is for students to plot and interpret points on the coordinate grid. The context remains coins but there is a variety of coins and the vertical coordinate is determined by the value of the coins. Students plot points corresponding to different combinations of coins. They identify the coordinates of plotted points and interpret them in terms of the context of coins and their value (MP2). During the activity synthesis, students discuss how they decided where to plot points and how they interpreted points on the graph.

### Launch

• Groups of 2
• “What do you know about coins?” (They're round. I can buy things with them. There are different kinds and they have different values.)
• Record responses for all to see.
• Display a penny, dime, nickel, and quarter.
• If no student mentions it, say and record the value of each coin.

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time

### Student Facing

The graph shows the number and value of coins some students had with them.

1. Tyler has 1 dime, 3 nickels, and 2 pennies. Which point represents Tyler's coins? Label the point.
2. Lin has 3 quarters, 1 dime, and 1 penny. Which point represents Lin's coins? Label the point.
3. Diego has 1 quarter and 1 dime. Write the coordinates of the point that represents Diego's coins. Explain or show your reasoning.
4. Clare has 5 coins and does not have a quarter. Write the coordinates of the point that represents Clare's coins.
5. Which coins might Clare have? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

If students are not familiar with American coins or need support determining the total value of the coins, display only the problem stem, without revealing the question and ask, “What do you know about each student?”

### Activity Synthesis

• “How did you know which point represents Tyler’s coins?” (Tyler has 6 coins so I looked for the 6 below the horizontal line at the bottom of the graph. The value of the coins is 27 cents so I looked for a point between 20 and 30 in the vertical direction, closer to 30 than 20.)
• “How did you know which point represents Clare's coins?” (There are three points that represent 5 coins. Two of them have a vertical coordinate of more than 70. That can’t be Clare because she has no quarters. So Clare is the other point representing 5 coins.)

## Lesson Synthesis

### Lesson Synthesis

“Today we represented real world and mathematical problems by graphing points in the first quadrant of the coordinate grid and interpreting the points.”

Display the image from the second activity.

“Which point on the graph represents the smallest number of coins? How do you know?” (The point at the bottom right since it’ ;s just 1 coin. All the others represent more than one coin.)

“Which coin does it represent? How do you know?” (It’ s a nickel because it’ s less than 10 cents but more than 1 cent.)

“Which point represents the most money? How do you know?” (The one to the top right because it’ s almost 100 cents. Everything else is below 90.)

“How many coins does that point represent? How do you know?” (9, because the horizontal coordinate is 9.)

## Cool-down: Half Dollar (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.