Free Math Worksheets

- List the integers to the right of \(-6\) and to the left of \(3\). - Include zero when it lies in the interval. - Do not include the endpoints.

1. The integers greater than \(-6\) and less than \(3\) are \(-5,-4,-3,-2,-1,0,1,2\). 2. There are \(8\) integers in the list.

There are \(8\) integers: \(-5,-4,-3,-2,-1,0,1,2\).

- Look for integers to the right of \(-103\) and to the left of \(-97\). - Do not include the endpoints. - As negative integers increase, their absolute values decrease.

1. The integers must be greater than \(-103\) and less than \(-97\). 2. Listing the consecutive integers gives \(-102,-101,-100,-99,-98\).

\(-102,-101,-100,-99,-98\)

- Picture the integers on a number line. - Which integer is one unit to the left? - Which integer is one unit to the right? - Subtract or add \(1\) as appropriate.

1. The integer immediately before \(x\) is \(x-1\), and the integer immediately after \(x\) is \(x+1\). 2. For \(-15\), the integers are \(-16\) and \(-14\). 3. For \(-1\), the integers are \(-2\) and \(0\). 4. For \(0\), the integers are \(-1\) and \(1\).

For \(-15\): before \(-16\), after \(-14\) For \(-1\): before \(-2\), after \(0\) For \(0\): before \(-1\), after \(1\)

- Picture consecutive integers on a number line. - The previous integer is one unit to the left. - Complete one row at a time from the value already shown.

1. In Row 1, the integer immediately before \(-45\) is \(-46\), and the integer immediately after it is \(-44\). 2. In Row 2, the integer after \(-101\) is \(-100\), and the next integer is \(-99\). 3. In Row 3, the integer before \(1\) is \(0\), and the integer before \(0\) is \(-1\).

Row 1: \(-46,-45,-44\) Row 2: \(-101,-100,-99\) Row 3: \(-1,0,1\)

- For part a), find the total distance and move halfway from one endpoint. - What does it mean for two integers to be consecutive?

1. The distance from \(-5\) to \(1\) is \(6\) units. Half of \(6\) is \(3\), and \(-5+3=-2\). 2. The integers \(-40\) and \(-39\) are consecutive, so no integer lies strictly between them.

a) \(-2\) b) No. The numbers \(-40\) and \(-39\) are consecutive integers.

- List the integers between the endpoints of each pair. - Do not include the endpoints themselves. - Compare the two counts.

1. Between \(-10\) and \(-7\) are \(-9\) and \(-8\), so Pair A has \(2\) integers between its endpoints. 2. Between \(-2\) and \(2\) are \(-1,0,1\), so Pair B has \(3\) integers between its endpoints. 3. Since \(3>2\), Pair B has more integers between its endpoints.

Pair B has more integers between its endpoints. Pair A has \(2\), and Pair B has \(3\).

- How many degrees separate a pole from the equator? - On a signed latitude scale, what happens when you move north from \(-90^\circ\)? - Which line is halfway between the North Pole and South Pole?

1. The poles are \(90^\circ\) from the equator. The South Pole is at \(90^\circ\,\text{S}\), which can also be represented as \(-90^\circ\). 2. Moving north by \(90^\circ\) changes the signed latitude from \(-90^\circ\) to \(0^\circ\). 3. Latitude \(0^\circ\) is the equator.

a) \(90^\circ\,\text{S}\), or \(-90^\circ\) b) The equator

- Determine the value of one small interval first. - Values to the left of \(0\) are negative, and values to the right are positive. - Count from the nearest labeled tick.

1. From \(0\) to \(1\), there are \(10\) equal intervals, so each small interval represents \(1\div10=0.1\). 2. Point \(A\) is two intervals left of \(-1\), so \(A=-1-2\times0.1=-1.2\). 3. Point \(B\) is four intervals left of \(0\), so \(B=-0.4\). 4. Point \(C\) is three intervals right of \(0\), so \(C=0.3\). 5. Point \(D\) is one interval right of \(1\), so \(D=1.1\).

\(A=-1.2\), \(B=-0.4\), \(C=0.3\), \(D=1.1\)

- Use the labeled ticks to determine the value of one small interval. - A point left of zero represents a negative number, and a point right of zero represents a positive number. - Count from a nearby labeled value to each point.

1. There are \(5\) equal intervals from \(0\) to \(5\), so each tick interval represents \(1\). 2. Point \(A\) is \(7\) units left of \(0\), so \(A=-7\). 3. Point \(B\) is \(3\) units right of \(0\), so \(B=3\).

Point \(A\) represents \(-7\), and point \(B\) represents \(3\).

- Count the intervals between two consecutive integers. - Points left of \(0\) have negative values. - Count small intervals from the nearest labeled integer.

1. Each interval between consecutive integers is divided into \(10\) equal parts, so each small interval represents \(0.1\). 2. Point \(P\) is seven intervals left of \(-1\), so \(P=-1.7\). 3. Point \(Q\) is eight intervals left of \(0\), so \(Q=-0.8\). 4. Point \(R\) is three intervals left of \(0\), so \(R=-0.3\). 5. Point \(S\) is four intervals right of \(0\), so \(S=0.4\).

\(P=-1.7\), \(Q=-0.8\), \(R=-0.3\), \(S=0.4\)

- Determine the value of one small interval. - Values left of \(0\) are negative. - Count from the nearest labeled integer.

1. From \(0\) to \(1\), there are \(10\) equal intervals, so each small interval represents \(0.1\). 2. Point \(P\) is two intervals right of \(-2\), so \(P=-1.8\). 3. Point \(Q\) is three intervals right of \(-1\), so \(Q=-0.7\). 4. Point \(R\) is one interval left of \(0\), so \(R=-0.1\). 5. Point \(S\) is six intervals right of \(0\), so \(S=0.6\). 6. Point \(T\) is three intervals right of \(1\), so \(T=1.3\).

\(P=-1.8\), \(Q=-0.7\), \(R=-0.1\), \(S=0.6\), \(T=1.3\)

- Count the equal intervals from \(0\) to \(1\). - Determine the value of one small interval. - Use the point’s direction from \(0\) to determine its sign.

1. There are \(5\) equal intervals from \(0\) to \(1\), so each small interval represents \(\frac{1}{5}=0.2\). 2. Point \(A\) is two intervals left of \(-1\), so \(A=-1-2\times0.2=-1.4\). 3. Point \(B\) is three intervals left of \(0\), so \(B=-3\times0.2=-0.6\). 4. Point \(C\) is four intervals right of \(0\), so \(C=4\times0.2=0.8\).

\(A=-1.4\), \(B=-0.6\), \(C=0.8\)

- Determine the value of one small tick interval. - Points left of zero are negative. - Count from zero or from the nearest labeled tick. - Values decrease as you move left.

1. There are \(10\) equal intervals from \(0\) to \(10\), so each small interval represents \(1\). 2. Counting from nearby labeled ticks gives \(A=-14\), \(B=-2\), \(C=8\), and \(D=17\).

\(A=-14\), \(B=-2\), \(C=8\), and \(D=17\)

- Count the equal intervals between consecutive integers. - Values left of \(0\) are negative. - Count from the nearest labeled integer.

1. There are \(10\) equal intervals between consecutive integers, so each small interval represents \(0.1\). 2. Point \(A\) is six intervals left of \(-1\), so \(A=-1.6\). 3. Point \(B\) is three intervals left of \(0\), so \(B=-0.3\). 4. Point \(C\) is five intervals right of \(0\), so \(C=0.5\). 5. Point \(D\) is two intervals right of \(1\), so \(D=1.2\). 6. Point \(E\) is nine intervals right of \(1\), so \(E=1.9\).

\(A=-1.6\), \(B=-0.3\), \(C=0.5\), \(D=1.2\), \(E=1.9\)

- Determine the fraction represented by one interval. - Values left of \(0\) are negative. - Simplify each fraction.

1. The interval from \(0\) to \(1\) is divided into \(4\) equal parts, so each interval represents \(\frac{1}{4}\). 2. Point \(X\) is three intervals left of \(0\), so \(X=-\frac{3}{4}\). 3. Point \(Y\) is one interval left of \(0\), so \(Y=-\frac{1}{4}\). 4. Point \(Z\) is two intervals right of \(0\), so \(Z=\frac{2}{4}=\frac{1}{2}\).

\(X=-\frac{3}{4}\), \(Y=-\frac{1}{4}\), \(Z=\frac{1}{2}\)

- Count the equal intervals between the labeled whole numbers. - Count from \(0\) to each marker. - Use a negative sign for points to the left of \(0\), then simplify.

1. In a), the interval from \(0\) to \(1\) is divided into tenths. Thus, \(A=\frac{3}{10}\), \(B=\frac{5}{10}=\frac{1}{2}\), and \(C=\frac{8}{10}=\frac{4}{5}\). 2. In b), the interval from \(-1\) to \(0\) is divided into fourths. Thus, \(D=-\frac{3}{4}\), \(E=-\frac{2}{4}=-\frac{1}{2}\), and \(F=-\frac{1}{4}\).

a) \(A=\frac{3}{10}\), \(B=\frac{1}{2}\), \(C=\frac{4}{5}\) b) \(D=-\frac{3}{4}\), \(E=-\frac{1}{2}\), \(F=-\frac{1}{4}\)

- First list all integers between the two endpoints. - An even integer is divisible by \(2\). - Check each integer in your list.

1. The integers strictly between \(-45\) and \(-39\) are \(-44,-43,-42,-41,-40\). 2. The even integers in this list are \(-44,-42,-40\).

\(-44,-42,-40\)

- Determine the value of one tick interval. - Points to the left of zero are negative, and points to the right are positive. - Start at a nearby labeled tick and count intervals to each marker.

1. There are \(5\) equal intervals from \(0\) to \(5\), so each tick interval represents \(1\). 2. Point \(P\) is \(2\) units left of \(-10\), so \(P=-12\). 3. Point \(Q\) is \(1\) unit right of \(-5\), so \(Q=-4\). 4. Point \(R\) is \(3\) units right of \(0\), so \(R=3\). 5. Point \(S\) is \(1\) unit right of \(10\), so \(S=11\).

\(P=-12\) \(Q=-4\) \(R=3\) \(S=11\)

- Find the distance between the two labeled values. - Count the equal intervals between them to determine the scale. - Count right or left from a labeled point.

1. The distance from \(-0.8\) to \(-0.2\) is \(-0.2-(-0.8)=0.6\). 2. There are \(6\) equal intervals between those labels, so each interval represents \(0.6\div6=0.1\). 3. Point \(A\) is seven intervals right of \(-0.2\), so \(A=-0.2+7\times0.1=0.5\). 4. Point \(B\) is five intervals left of \(-0.8\), so \(B=-0.8-5\times0.1=-1.3\).

\(A=0.5\) and \(B=-1.3\)

- Use the labeled major ticks to find the difference between them. - Count the equal small intervals between two labeled ticks. - Divide the major-tick difference by the number of intervals. - Count left for lesser values and right for greater values.

1. On number line a), consecutive labeled values differ by \(10\), with \(5\) equal intervals between them. Each small interval represents \(10\div5=2\). 2. Reading the markers gives \(A=-46\), \(B=-18\), \(C=4\), and \(D=32\). 3. On number line b), consecutive labeled values differ by \(50\), with \(5\) equal intervals between them. Each small interval represents \(50\div5=10\). 4. Reading the markers gives \(E=-730\), \(F=-610\), \(G=-550\), and \(H=-480\).

a) \(A=-46\), \(B=-18\), \(C=4\), \(D=32\) b) \(E=-730\), \(F=-610\), \(G=-550\), \(H=-480\)

- Determine the value of one small interval. - Use the labeled tenths to count by hundredths. - Points left of \(0\) are negative.

1. The interval from \(0\) to \(0.1\) is divided into \(5\) equal parts, so each small interval represents \(0.1\div5=0.02\). 2. Point \(P\) is three intervals right of \(-0.5\), so \(P=-0.5+3\times0.02=-0.44\). 3. Point \(Q\) is one interval right of \(-0.2\), so \(Q=-0.2+0.02=-0.18\). 4. Point \(R\) is three intervals right of \(0\), so \(R=3\times0.02=0.06\). 5. Point \(S\) is one interval right of \(0.3\), so \(S=0.3+0.02=0.32\).

\(P=-0.44\), \(Q=-0.18\), \(R=0.06\), \(S=0.32\)

- The number of equal parts from \(0\) to \(1\) gives the denominator. - Values left of \(0\) are negative. - Simplify fractions when possible.

1. The interval from \(0\) to \(1\) is divided into \(4\) equal parts, so each small interval represents \(\frac{1}{4}\). 2. Point \(P\) is at \(-1\frac{3}{4}\). 3. Point \(Q\) is at \(-\frac{2}{4}=-\frac{1}{2}\). 4. Point \(R\) is at \(\frac{3}{4}\). 5. Point \(S\) is at \(1\frac{1}{4}\).

\(P=-1\frac{3}{4}\), \(Q=-\frac{1}{2}\), \(R=\frac{3}{4}\), \(S=1\frac{1}{4}\)

- Find the difference between the two labeled values. - Divide that difference by the number of intervals between them. - Decide whether \(0\) lies to the left or right of the negative values.

1. The distance from \(C=-0.12\) to \(D=-0.04\) is \(-0.04-(-0.12)=0.08\). 2. There are \(8\) equal intervals from \(C\) to \(D\), so each interval represents \(0.08\div8=0.01\). 3. From \(D=-0.04\) to \(0\) is a distance of \(0.04\). At \(0.01\) per interval, this is \(0.04\div0.01=4\) intervals to the right.

a) \(0.01\) b) \(0\) is \(4\) intervals to the right of \(D\).

- Determine the value of one small interval on each number line separately. - Count from a nearby labeled value. - Values to the left of \(0\) are negative.

1. On number line a), consecutive labeled thousands are divided into \(4\) equal intervals, so each small interval represents \(1000\div4=250\). 2. Therefore, \(A=-2250\), \(B=-750\), \(C=500\), and \(D=1750\). 3. On number line b), consecutive labeled tenths are divided into \(2\) equal intervals, so each small interval represents \(0.1\div2=0.05\). 4. Therefore, \(E=-0.35\), \(F=-0.15\), \(G=0.1\), and \(H=0.45\).

a) \(A=-2250\), \(B=-750\), \(C=500\), \(D=1750\) b) \(E=-0.35\), \(F=-0.15\), \(G=0.1\), \(H=0.45\)

- Find the value represented by one minor tick mark. - Read each marker carefully, including its sign. - Write each decimal as a fraction with denominator \(10\). - Simplify each fraction.

1. The minor tick marks are spaced by \(0.1\). Reading the marked locations gives \(P = -1.2\), \(Q = -0.4\), and \(R = 0.8\). 2. Convert \(P\): \(-1.2 = -\frac{12}{10} = -\frac{6}{5}\). 3. Convert \(Q\): \(-0.4 = -\frac{4}{10} = -\frac{2}{5}\). 4. Convert \(R\): \(0.8 = \frac{8}{10} = \frac{4}{5}\).

\(P = -1.2 = -\frac{6}{5}\) \(Q = -0.4 = -\frac{2}{5}\) \(R = 0.8 = \frac{4}{5}\)

- Count the equal intervals between the given labeled values. - Determine the value of one tick interval on each number line. - Values to the left of zero are negative. - The farther left a value lies, the less it is.

1. On number line a), there are \(3\) intervals from \(0\) to \(3\), so each interval represents \(1\). Therefore, \(A=-7\), \(B=-2\), \(C=5\), and \(D=9\). 2. On number line b), there are \(2\) intervals from \(0\) to \(20\), so each interval represents \(10\). Therefore, \(E=-50\), \(F=-30\), \(G=10\), and \(H=40\). 3. The least coordinate is \(-50\), so point \(E\) has the least coordinate.

a) \(A=-7\), \(B=-2\), \(C=5\), \(D=9\) b) \(E=-50\), \(F=-30\), \(G=10\), \(H=40\) c) Point \(E\), with coordinate \(-50\)

- Determine the value of one small interval on each number line. - Values to the left of \(0\) are negative. - Opposite numbers have equal distance from \(0\) on different sides.

1. On number line a), each small interval represents \(0.2\). Therefore, \(P=-1.2\), \(Q=-0.6\), and \(R=0.8\). 2. On number line b), each small interval represents \(0.25=\frac{1}{4}\). Therefore, \(S=-1.75=-1\frac{3}{4}\), \(T=-0.5=-\frac{1}{2}\), and \(U=0.5=\frac{1}{2}\). 3. The opposite of \(0.5\) is \(-0.5\), which is point \(T\).

a) \(P=-1.2\), \(Q=-0.6\), \(R=0.8\) b) \(S=-1.75=-1\frac{3}{4}\), \(T=-0.5=-\frac{1}{2}\), \(U=0.5=\frac{1}{2}\) c) Point \(T\)