1.1: Notice and Wonder: A Number Line (5 minutes)
The purpose of this warm-up is to introduce students to a number line diagram they will be using to represent addition and subtraction of integers in future lessons. To introduce this idea, students write a story and equation that a given number line diagram could represent.
Tell students they are going to see a number line diagram and that their job is to think of at least one thing they notice and at least one thing they wonder about the picture. Display the problem for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have noticed or wondered about something.
What do you notice? What do you wonder?
After students’ notice and wonder ideas are displayed, tell them that the diagram is about money, and invite students to share a possible story and equation that the diagram represents. For example, someone owes \$4. Then they earn \$7 doing chores and another \$9 helping out the neighbor with their yard. The equation would be \(\text-4+7+9 = 12\).
1.2: Telling Temperatures (15 minutes)
The purpose of this activity is for students to solve number puzzles using any representation they choose. Students then make sense of other representations for the same problems, starting with those of a partner. The whole-class discussion should focus on the strengths and weaknesses of different representations (MP5). For example, tape diagrams only work for problems with all positive values, so you could use one for the distance puzzle, but a tape diagram would not work for the temperature puzzle.
Identify students using different strategies, such as number line diagrams, tape diagrams, written out reasoning, and equations, to share during the whole-class discussion.
Arrange students in groups of 2. Give 5–6 minutes of quiet work time followed by partner discussion. During their discussion, partners explain their representations of the problems to one another, including any representations they started out with that didn’t work. If partners used the same representation, ask them to find another representation they could use to solve the puzzle. Follow with a whole-class discussion.
Supports accessibility for: Visual-spatial processing
Design Principle(s): Support sense-making; Maximize meta-awareness
Solve each puzzle. Show your thinking. Organize it so it can be followed by others.
The temperature was very cold. Then the temperature doubled.
Then the temperature dropped by 10 degrees. Then the temperature increased by 40 degrees. The temperature is now 16 degrees. What was the starting temperature?
- Lin ran twice as far as Diego. Diego ran 300 m farther than Jada. Jada ran \(\frac13\) the distance that Noah ran. Noah ran 1200 m. How far did Lin run?
Students who write the equation \(2x-10+40=16\) for the temperature puzzle may incorrectly follow that up with \(2x-50=16\). By comparing their process with a different representation, help them notice that a decrease of 10 and an increase of 40 is, overall, an increase of 30.
The goal of this discussion is for students to make connections between different representations of problems and, more centrally to this unit as a whole, to see equations as an efficient way to represent problems.
Select previously identified students to share their representations. Record and display a visual of the representations for each problem. If not shared by students, make sure at least one equation representation for each problem is included. Once multiple representations from the class are displayed, ask 2–3 students to explain which one(s) they prefer and why. If not brought up in discussion, note that some representations, such as tape diagrams, do not work all the time. For example, a tape diagram is not possible for the temperature puzzle due to the negative values in the problem. Other representations, such as equations, can work for almost any type of problem.
1.3: Making a Puzzle (15 minutes)
In this task, students create their own number puzzle to trade with a partner to solve. The purpose of this task is for students to practice writing and solving multi-step number puzzles and compare their representations with the representations of others to decide which are more efficient. While these problems are phrased using the words “number puzzle,” it is important to note the mathematical work students are doing here thinking about, creating, and solving situations that are, essentially, linear equations in one variable, even if not all students are using equations to represent them.
While students are sharing representations, identify partners who solved at least one of their puzzles using different representations to share during the whole-class discussion. If possible, select groups who used an equation.
Keep students in the same groups. Give 5 minutes for students to write their own puzzle and make a representation of their solution before trading their puzzle with a partner to solve. Make sure students write their puzzle and solution in such a way that when they trade, their partner cannot see the solution.
If both partners created the same (or very similar) representation for their solutions, ask them to work together to create a different representation. If they created different representations, ask partners to discuss which one they prefer and to be ready to explain why during the whole-class discussion.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Write another number puzzle with at least three steps. On a different piece of paper, write a solution to your puzzle.
Trade puzzles with your partner and solve theirs. Make sure to show your thinking.
With your partner, compare your solutions to each puzzle. Did they solve them the same way you did? Be prepared to share with the class which solution strategy you like best.
Are you ready for more?
Here is a number puzzle that uses math. Some might call it a magic trick!
- Think of a number.
- Double the number.
- Add 9.
- Subtract 3.
- Divide by 2.
- Subtract the number you started with.
- The answer should be 3.
Why does this always work? Can you think of a different number
puzzle that uses math (like this one) that will always result in 5?
Select partners previously identified to share a puzzle with the class and the two representations they created. Ask which representation they prefer and why. If students do not bring it up in their explanations, ask which of their representations was the most efficient one for solving the puzzle.
During this discussion, students may ask you to state which representation is best and, if so, it is important to note that there is no one correct answer for the “best representation.” The “best representation” is the one that makes sense to the student and helps them solve the problem. However, as problems grow more and more complex, students are likely to find that certain representations are more useful for solving problems than others.
Ask students to think about what their number line diagrams, tape diagrams, and equations represented in each of the activities. Guide them in seeing that stories with an unknown quantity usually involve actions, like the temperature rising, earning money by doing chores, or relationships, like Diego’s distance being half of Lin’s distance and 300 m more than Jada’s. Ask them to think about the puzzle they wrote in the last activity and whether they described actions or relationships. Invite their opinions about which representations best represent actions and which best represent relationships.
Tell students to think about the expression \(x+5\). Ask: “What could this mean in terms of two numbers being related to each other? What could this represent as an action?” (One number is 5 more than the other. A sample action: the temperature increases by 5 degrees.)
If time allows, ask students to make up another puzzle that describes actions if they previously chose relationships, and vice versa, and to represent their puzzles with a diagram and an equation. Display diagrams and equations for all to see.
1.4: Cool-down - Seeing the Puzzle (5 minutes)
Cool-downs for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
Here is an example of a puzzle problem:
Twice a number plus 4 is 18. What is the number?
There are many different ways to represent and solve puzzle problems.
- We can reason through it.
Twice a number plus 4 is 18.
Then twice the number is \(18 - 4 =14\).
That means the number is 7.
- We can draw a diagram.
- We can write and solve an equation. \(\displaystyle 2x +4 = 18\) \(\displaystyle 2x = 14\) \(\displaystyle x = 7\)
Reasoning and diagrams help us see what is going on and why the answer is what it is. But as number puzzles and story problems get more complex, those methods get harder, and equations get more and more helpful. We will use different kinds of diagrams to help us understand problems and strategies in future lessons, but we will also see the power of writing and solving equations to answer increasingly more complex mathematical problems.