Explore Big Numbers with Base-Ten Hoppers

Est. Class Sessions: 2–3

Developing the Lesson

Part 1: Represent Big Numbers

Use a Number Line. Display the first page of Explore Big Numbers with Base-Ten Hoppers in the Student Guide. Ask students to work in pairs to study the moves that the base-ten hopper can make using Questions 1 and 2. Tell students that they are Professor Peabody's assistants and that they need to prepare an oral report for the rest of the class on their observations of how the base-ten hopper moves and how Professor Peabody records its movements. A possible report follows:

“The base-ten hopper can make hops with distances of ten thousand, one hundred thousand, one million, and 10 million on the number line. These are the only distances on the two number lines. The base-ten hopper can go forward or backward. You can see that sometimes the hopper goes forward and sometimes it goes backward. Professor Peabody notes where the hopper lands after each hop by writing the number under the hop. He shows the direction of each hop by writing plus or minus above each hop with how far it goes.”

Encourage those listening to ask questions of those reporting so that everyone understands how the hopper can move and how to record the movements.

Do you think the base-ten hopper can make a 100 million hop? Why or why not? (Possible response: A hop of 100 million would be like going to the next biggest place on the place value chart, so I think it could.)
How far did the base-ten hopper move on the first number line? How do you know? (It moved 1,240,000. It started at 0 and landed on 1,240,000. It made one hop of 1,000,000, two hops of 100,000, and four hops of 10,000.)
Write a number sentence to show how the hopper moved on the first number line.
(1,000,000 + 100,000 + 100,000 + 10,000 + 10,000 + 10,000 + 10,000 = 1,240,000)
Write 1,240,000 in expanded form. (1,240,000 = 1,000,000 + 200,000 + 40,000)
How is showing the base-ten hopper's moves like using expanded form? How is it different? (Expanded form breaks numbers down into numbers that are in the ten thousands, hundred thousands, millions, tens of millions, and hundreds of millions places. The base-ten hopper can go backwards, too.)
Where did the base-ten hopper start on the second number line? Where did it land after its last hop? (It started at 83,000,000 and ended up at 100,000,000.)
How far did the base-ten hopper move on the second number line? Write a number sentence that shows how it moved. (83,000,000 + 10,000,000 + 10,000,000 − 1,000,000 − 1,000,000 − 1,000,000 = 100,000,000)
How far is it from where the base-ten hopper started to where it landed? How do you know? (Possible response: It is 17,000,000. It moved 10,000,000 forward twice or 20,000,000. Then it moved back three 1,000,000s. 20,000,000 − 3,000,000 = 17,000,000)

This lesson provides a context for students to develop skills composing and decomposing numbers. Number lines serve as tools for developing mental images of numbers decomposed into groups of hundred thousands, millions, tens of millions, hundreds of millions, and one billion. The goal is for students to develop mental math skills that will allow them to use flexible and efficient strategies to estimate, add, and subtract big numbers.

For example, to subtract 300,100,000 − 279,800,000, students can count up on a number line that starts at 279,800,000 move forward 200,000 to 280,000,000; then 20,000,000 to 300,000,000; then 100,000 more to 300,100,000. They add the moves on their hops: 200,000 + 20,000,000 + 100,000 = 20,300,000, so the difference between 279,800,000 and 30,0100,000 is 20,300,000. They can visualize:

Other students may choose to start at 300,100,000, count back by 10,000,000s until they get near the closest hundred thousands (300,100,000; 290,100,000; 280,100,000) then count back by hundred thousands to 279,800,000 (280,000,000; 279,900,000; 279,800,000). They can visualize:

Place the emphasis of this lesson on using the number line to break apart and combine numbers using hundred thousands, millions, tens of millions, hundreds of millions, and a billion.

Ask students to work on Questions 3–6 in pairs. These questions help students understand how they can show moves on the number line so that someone else can see where the hopper starts, the distance and direction of each move, and the final stopping point.

Question 5C asks students to find the distance the hopper moved. This question develops the idea of using the number line to find the difference between two numbers. That is, finding how far it is from 16,000,000 to 50,100,000 on the number line is the same as subtracting 16,000,000 from 50,100,000 and counting up from 16,000,000 to 50,100,000 to find the difference.

Ask:

What number sentence could you write to show how far it is from where the hopper started to where it landed? (Possible responses: 1,000,000 + 1,000,000 + 1,000,000 + 1,000,000 + 10,000,000 + 10,000,000 + 10,000,000 + 100,000 = 34,100,000 or 4,000,000 + 30,000,000 + 100,000 = 34,100,000)
What moves would the base-ten hopper make if it started at 50,100,000 and moved to 16,000,000? What number sentence could you write to show this? (Possible responses: 50,100,000 − 100,000 − 10,000,000 − 10,000,000 − 10,000,000 − 1,000,000 − 1,000,000 − 1,000,000 − 1,000,000 = 16,000,000 or 50,100,000 − 30,000,000 − 4,000,000 − 100,000 = 16,000,000)

In Question 6, students work with their partner to draw a number line that show how a base-ten hopper can start at 78,500,000 and land on 1,000,000,000. After students have completed this question, bring the students back together to share some of their work. See possible responses in Figure 1. As students share their solution to Question 6C, encourage them to apply what they have learned about fact families. Students should recognize that if the base-ten hopper hops 921,500,000 places to move from 78,500,000 to reach 1,000,000,000 then 78,500,000 + 921,500,000 = 1,000,000,000 and 1,000,000,000 − 921,500,000 = 78,500,000.

Equal Number Sentences. Display the Number Lines Master from the Teacher Guide as students look at the number lines in the Find Equal Number Sentences with Big Numbers section of their Student Guide. Use the vignette about Professor Peabody to discuss how the base-ten hopper moves on the number line can be represented in different ways.

Ask:

How are the two number lines alike? (They both show moves from 0 to 10,320,000. They both start with hops of 10,000,000.)
How are they different? (One shows 300,000 as one hop of 300,000 and the other shows 300,000 as 3 hops of 100,000. One shows 20,000 as one hop of 20,000 and the other shows 2 hops of 10,000.)

Display Question 7 from the Find Equal Number Sentences with Big Numbers section of the Student Guide. Use this problem to discuss how the moves of the base-ten hopper can be represented using different number sentences. Ask students to connect each hop on the number line to the symbols in the number sentences. In Question 7B, show that Professor Peabody is simply combining the hops of the same value “in his head” before writing them in the number sentence. For example, four hops of 10,000,000 is the same as 40,000,000, two hops of 1,000,000 is the same as 2,000,000, and three hops of 100,000 is the same as 300,000. Question 7C represents this thinking using multiplication within the number sentence.

Order of Operations. At first glance an expression such as 5,000,000 + 4,000,000 × 3 is open to various interpretations. One approach takes 5,000,000 + 4,000,000 × 3 to mean, “first add 5,000,000 and 4,000,000 to get 9,000,000 and then multiply by 3 to obtain 27,000,000.” This simple left-to-right interpretation is the way many four-function calculators evaluate the expressions, but it is incorrect, according to mathematical convention accepted the world over. Applying the order of operations, multiplication comes before addition.

After this discussion go over Tanya's example and definition of an equation in the Student Guide. Ask students to work in pairs to discuss whether or not the number sentences in Question 8 are true. As students work, ask them to tell you how they know their answer is correct.

Read the short vignette to introduce Question 9. For this question, students find the value of an unknown in an equation. Students should solve the problems using their knowledge of place value and methods of partitioning numbers as opposed to following a set of formal steps. Romesh applies his knowledge of expanded form to explain how he finds the unknown in a number sentence. Another method for completing the number sentences is to draw number lines to show that the values on both sides of the equation are the same. See Figure 1. Students may also apply order of operations to Questions 9B and 9D. See Figure 2 and Content Notes.

Avoiding Misconceptions. Various mnemonic devices are sometimes offered to help students remember the order of operations. While mnemonics are helpful in remembering some things, we strongly advise against their use in teaching this topic. The mnemonics that are commonly used to remember order of operations often lead to incorrect computations.

The phrase “My Dear Aunt Sally” is a mnemonic that reminds students to do the operations in this order: multiplication, division, addition, subtraction. This is an incorrect order—it suggests that multiplication is done before division and addition is done before subtraction. This would cause a student to incorrectly evaluate the expression 8 − 4 + 1 as 3. The correct answer, gotten by evaluating left to right because addition and subtraction have the same priority, is 8 − 4 + 1 = 5. Similarly, since multiplication and division have the same priority, 8 ÷ 4 × 2 = 4. A student who follows My Dear Aunt Sally's rule and does the multiplication first would incorrectly calculate 1 to be the answer.

To use Aunt Sally correctly, you need to tell students the additional condition that M and D are at the same level, and A and S are at the same level. But our experience shows that, as memory fades, many students and adults remember only the order given by the Aunt Sally mnemonic (MDAS) but not the extra condition.

Instead of offering a mnemonic, encourage students to think why the conventional order makes sense: multiplication and division are more complicated so we do them first to get them out of the way. They have the same priority level because they “go together”—one is the inverse of the other—so neither is higher than the other. Addition and subtraction “go together” too, so they have the same priority level. They are the final step—they put all the pieces together.