--- title: "Place value and powers of 10" subtitle: "Number Foundations · Place value and estimating · Lesson 17" --- # Place value and powers of 10 **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** The 4 in 4052 is worth 4000, but the 5 is worth only 50. What does the column a digit sits in do to its value? :::teacher-note **In one sentence:** Students build numbers on place value columns to see that each column is a power of 10, write numbers in expanded notation with exponents, and convert back (AC9M7N03, nodes 1 and 2). **Students should be able to say:** I can say which power of 10 each place value column is. I can write a number in expanded notation using powers of 10. I can turn expanded notation back into a number. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict How many zeros does $10^6$ have? :::keep-together **What is $10^3$?** ☐ 100 ☐ 1000 ☐ 10 000 ☐ 30 My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 1000. 10³ = 10 × 10 × 10: three 10s multiplied, so 1 followed by three zeros. 30 is 10 × 3. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Columns and powers :::keep-together ### Read the powers The board shows the columns. Here is 4052 written out. 1. $4052 = 4000 + 0 + 50 + 2$ — Each digit is worth its digit times its column. 2. $4000 = 4 \times 1000 = 4 \times 10^3$ — The thousands column is 10³. 3. $50 = 5 \times 10 = 5 \times 10^1$ — The tens column is 10¹. 4. $2 = 2 \times 1 = 2 \times 10^0$ — The ones column is 10⁰. 5. $4052 = 4 \times 10^3 + 5 \times 10^1 + 2 \times 10^0$ — The 0 in the hundreds adds nothing, so we leave it out. Now try one: which power goes with the 5? **In 4052, the 5 is in the tens column. What is the exponent of its power of 10? (The 5 is worth $5 \times 10^{?}$.)** Answer: {{field-15}} ::: :::teacher-note **Answer:** 1 The ones column is 10⁰, the tens column 10¹, the hundreds 10². So 5 tens is 5 × 10¹ = 50. ::: :::keep-together ### Find the mistake **Sam writes 4052 in expanded notation. Click the line where the working first goes wrong.** ``` 1 Write 4052 in expanded notation. 2 4 is in the thousands: 4 × 10³ 3 0 is in the hundreds: nothing to add 4 5 is in the tens: 5 × 10² 5 2 is in the ones: 2 × 10¹ ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 4 Line 4: the tens column is 10¹, not 10². Sam counted the ones column as 10¹. The ones column is 10⁰. ::: :::keep-together ### Why 10⁰? **Why is $10^0 = 1$?** ☐ It is not allowed ☐ It is the ones column: 1 one, so the power is 1 ☐ Any number to the power 0 is 0 ☐ 10⁰ means 10 × 0 ::: :::teacher-note **Answer:** It is the ones column: 1 one, so the power is 1 Each step down divides by 10: 1000, 100, 10, then 1. So 10⁰ = 1, the value of the ones column. ::: :::typical ## Build, read, convert **Far apart.** **Back to a number.** **Which exponent?.** **Shift the digits.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Expanded notation :::keep-together ### The value of a digit **What is the value of the 2 in 52 300?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 2000 The 2 is in the thousands column: 2 × 10³ = 2000. ::: :::keep-together ### Back to a number **Write $6 \times 10^5 + 4 \times 10^2$ as a number.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 600400 600 000 + 400 = 600 400. ::: :::keep-together ### Times 100 **Why is $6300 \times 100$ the same as $6300 \times 10^2$, and why is it easy?** ☐ × 10² means × 10 × 2 = × 20 ☐ It is not the same ☐ 10² = 100, and × 10² moves every digit two columns left (add two zeros) ☐ 10² = 20, so it doubles ::: :::teacher-note **Answer:** 10² = 100, and × 10² moves every digit two columns left (add two zeros) 10² = 100. Multiplying by 100 moves each digit two columns left: 630 000. ::: :::keep-together ### Review: powers **Review: what is $2^5$?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 32 2 × 2 × 2 × 2 × 2 = 32. ::: :::keep-together ### Millions **Write 3 000 000 as $3 \times 10^{?}$. What is the exponent?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 The exponent counts the zeros: 3 000 000 = 3 × 10⁶, and 10⁶ is the millions column. ::: ## Make: three terms Then read your number out in expanded notation. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Write 7 304 in expanded notation using powers of 10.** {{lines-3}} ::: :::teacher-note **Answer:** 7 × 10³ + 3 × 10² + 4 × 10⁰. (The 0 in the tens column is left out.) :::