Free math worksheets

- Write the value represented by each \(4\). - Compare \(4\) with \(0.4\). - Think about what happens between adjacent place values.

1. The \(4\) in the ones place has value \(4\). 2. The \(4\) in the tenths place has value \(0.4\). 3. Since \(4 = 10 \times 0.4\), the ones-place \(4\) is \(10\) times the value of the tenths-place \(4\).

The \(4\) in the ones place is \(10\) times the value of the \(4\) in the tenths place.

- Identify the place-value column containing the chips in each panel. - Write the total value represented in each panel. - Compare adjacent place values.

1. Panel a) shows \(3\) tenths, which has value \(0.3\). 2. Panel b) shows \(3\) hundredths, which has value \(0.03\). 3. Since \(0.3 = 10 \times 0.03\), panel a) has \(10\) times the value of panel b).

Panel a) has \(10\) times the value of panel b).

- Compare the number of zeros in the divisor with the change in place value. - Division by a number greater than \(1\) makes this positive number smaller.

1. \(15.4\div10=1.54\). Dividing by \(10\) makes the value represented by every digit one tenth as large. 2. \(15.4\div1000=0.0154\). Dividing by \(1000\) makes the value represented by every digit one thousandth as large. 3. In general, dividing by \(10^n\) makes each digit's value \(\frac{1}{10^n}\) as large. Zeros are used as placeholders when a place has no nonzero digit.

\(15.4\div10=1.54\) and \(15.4\div1000=0.0154\). Dividing by \(10^n\) makes each digit's value \(\frac{1}{10^n}\) as large.

- How many powers of \(10\) are needed so every nonzero decimal-place value becomes a whole-number place value? - What happens to each digit's value when you multiply by \(10\), \(100\), or \(1000\)? - How is the exponent in a power of ten related to repeated tenfold changes in place value?

1. For \(0.8\), multiplying by \(10^1\) makes each digit's value \(10\) times as large, so \(0.8 \times 10 = 8\). 2. For \(0.045\), multiplying by \(10^3\) makes each digit's value \(1000\) times as large, so \(0.045 \times 1000 = 45\). 3. For \(1.203\), multiplying by \(10^3\) makes each digit's value \(1000\) times as large, so \(1.203 \times 1000 = 1203\).

a) \(10^1\); product: \(8\) b) \(10^3\); product: \(45\) c) \(10^3\); product: \(1203\)

- Does a zero at the far right of a decimal change its value? - Rewrite the decimal without any unnecessary trailing zero. - Test powers of ten in order, starting with \(10^1\).

1. The decimals \(0.750\) and \(0.75\) have the same value because a trailing zero does not change a decimal's value. 2. Multiplying by \(10^1\) gives \(0.75 \times 10 = 7.5\), which is not a whole number. 3. Multiplying by \(10^2\) gives \(0.75 \times 100 = 75\), which is a whole number. 4. Therefore, \(10^2\), not \(10^3\), is the smallest power of ten that works.

The claim is incorrect. Since \(0.750 = 0.75\), the smallest power of ten is \(10^2\), because \(0.75 \times 100 = 75\).

- Compare how the value represented by each nonzero digit changes. - Multiplication by \(10^n\) makes every digit's value \(10^n\) times as large. - Division by \(10^n\) makes every digit's value \(\frac{1}{10^n}\) as large. - Use the required value factor to determine the missing power of ten.

1. For a), \(0.082\times1000=82\). 2. For b), \(740\div1000=0.74\). 3. For c), \(0.005\times100{,}000=500\). 4. For d), \(1.23\div100=0.0123\).

a) \(1000\) b) \(1000\) c) \(100{,}000\) d) \(0.0123\)

- Identify the place of the \(7\) in each value. - The two places are adjacent. - Compare one tenth with one hundredth.

1. \(0.07\) is seven hundredths. 2. \(0.7\) is seven tenths. 3. A tenth is \(10\) times a hundredth, so \(0.7\) is \(10\) times \(0.07\).

\(0.7\) is \(10\) times \(0.07\).

- Identify the column that currently contains the chips. - Write the current total as a decimal. - Compare hundredths with tenths.

1. The panel shows \(6\) hundredths, or \(0.06\). 2. Six tenths is \(0.6\). 3. Since \(0.6 = 10 \times 0.06\), the new value would be \(10\) times as large.

The new value would be \(0.6\), which is \(10\) times the shown value \(0.06\).

- Compare the value of the nonzero digits before and after division. - Ask what factor makes each digit's value one tenth, one hundredth, or one thousandth as large. - Zeros may be needed to hold places that contain no nonzero digit.

1. In a), the value \(68\) must become one hundredth as large: \(68\div100=0.68\). The missing divisor is \(100\). 2. In b), \(0.009\) is one hundredth of \(0.9\), so \(0.9\div100=0.009\). The missing divisor is \(100\). 3. The zeros are placeholders for the tenths and hundredths places, showing that the \(9\) has thousandths value.

a) \(100\) b) \(100\). The zeros are place-value placeholders.

- Express each operation as a multiplicative change in value. - Combine the factors \(\frac{1}{1000}\) and \(100\). - Identify the single power-of-ten operation with the same combined factor.

1. Dividing by \(1000\) makes every digit's value \(\frac{1}{1000}\) as large. 2. Multiplying that result by \(100\) makes every digit's value \(100\) times as large as it was after the division. 3. The combined factor is \(\frac{100}{1000}=\frac{1}{10}\), so the final number is one tenth as large as the original. 4. A single division by \(10\) has the same effect.

The overall effect is division by \(10\): every digit's value is one tenth as large as in the original number.

- Find the smallest power of ten for each decimal separately. - Begin with \(10^1\) and test larger powers only as needed. - Compare the exponents after you have found all four powers.

1. For \(A = 0.002\), the smallest power is \(10^3\), because \(0.002 \times 1000 = 2\). 2. For \(B = 0.05\), the smallest power is \(10^2\), because \(0.05 \times 100 = 5\). 3. For \(C = 0.8\), the smallest power is \(10^1\), because \(0.8 \times 10 = 8\). 4. For \(D = 1.23\), the smallest power is \(10^2\), because \(1.23 \times 100 = 123\). 5. Comparing the exponents gives \(C\), then \(B\) and \(D\), then \(A\).

\(C\) requires \(10^1\); \(B\) and \(D\) each require \(10^2\); \(A\) requires \(10^3\). Therefore, the order is \(C\), then \(B\) and \(D\), then \(A\).

- Track how the place value of each digit changes. - Think about what happens when a positive number is divided by a number less than \(1\). - Rewrite a decimal divisor using an equivalent power-of-ten relationship when useful.

1. For a), \(3.4\div100=0.034\). Each digit's value becomes one hundredth as large. 2. For b), \(0.005\) is one hundredth of \(0.5\), so the divisor is \(100\). 3. For c), \(20\) is \(10{,}000\) times \(0.002\), so the multiplier is \(10{,}000\). 4. For d), \(0.8\div0.01=80\) and \(0.8\times100=80\). Dividing by \(0.01\) is equivalent to multiplying by \(100\).

a) \(0.034\); each digit's value is one hundredth as large. b) \(100\) c) \(10{,}000\) d) Both equal \(80\). Dividing by \(0.01\) is equivalent to multiplying by \(100\).

- Start with the divisions by \(10\) and \(1\). - Look for a pattern in the quotients. - Track how the divisor changes from one line to the next. - Compare the place value of the digits in consecutive quotients.

1. The quotients are \(0.045\), \(0.45\), \(4.5\), \(45\), and \(450\). 2. Each time the divisor becomes one tenth as large, the quotient becomes ten times as large.

a) \(0.045\) b) \(0.45\) c) \(4.5\) d) \(45\) e) \(450\) The quotient becomes \(10\) times as large each time the divisor becomes one tenth as large.

- Write the value contributed by each \(3\). - Compare neighboring places, not just the digit symbols. - What happens to a place value when you move one place to the right?

1. A digit's value depends on its place, not only on the digit itself. 2. The four values are \(3\), \(0.3\), \(0.03\), and \(0.003\). 3. Moving one place to the right makes the value one tenth as large, so each value is \(10\) times the value immediately to its right.

Riley is incorrect because equal digits can have different values in different places. The values are \(3\), \(0.3\), \(0.03\), and \(0.003\), and each is \(10\) times the value immediately to its right.