Free math worksheets

- Count the equal-area parts in the whole badge. - A unit fraction has \(1\) as its numerator. - Use the number of equal parts as the denominator.

1. The badge is divided into \(2\) equal-area parts. 2. One of \(2\) equal parts is \(\frac{1}{2}\) of the whole.

Each part is \(\frac{1}{2}\) of the badge.

- Count the unit squares inside the shaded region. - Count how many equal-size regions make the whole. - Use the number of equal-size regions as the denominator.

1. The shaded strip contains \(2\) small squares. 2. The rectangle is divided into \(4\) equal vertical strips. 3. One of \(4\) equal-area strips is \(\frac{1}{4}\) of the rectangle.

The shaded strip has \(2\) small squares and is \(\frac{1}{4}\) of the rectangle.

- Compare the two regions on the grid. - Look for a way one region could fit exactly onto the other. - Use the number of equal-area regions as the fraction denominator.

1. A diagonal of a square divides it into \(2\) matching triangles, so the triangles have equal area. 2. Share the total area equally: \(16 \div 2 = 8\). 3. Each triangle has area \(8\) square units. 4. One of \(2\) equal-area parts is \(\frac{1}{2}\) of the square.

Each triangle has area \(8\) square units and is \(\frac{1}{2}\) of the square.

- Compare the sizes of the regions in the diagram. - Count how many equal regions make the whole. - Use the number of equal regions as the denominator of a unit fraction.

1. The three regions are the same size, so they have equal area. 2. The whole is divided into \(3\) equal-area regions. 3. One of \(3\) equal parts is \(\frac{1}{3}\) of the whole.

a) Yes. The three regions are the same size. b) Each region is \(\frac{1}{3}\) of the whole.

- Compare every listed region area. - An equal-area partition needs one common area value. - Change only the value that does not match the others.

1. Five regions have area \(8\) square units, but one has area \(9\) square units. 2. Because the areas are not all the same, the partition is not equal in area. 3. Changing the \(9\)-square-unit region to \(8\) square units makes all six areas equal.

a) No. b) Change \(9\) square units to \(8\) square units.

- Count or group the unit squares in each region. - Look for a larger familiar shape that can help you compare two regions. - Equal-area regions do not have to have the same outline.

1. The left rectangle is \(2\) squares wide and \(2\) squares tall, so its area is \(2 \times 2 = 4\) square units. 2. Each triangle on the right is half of a \(4 \times 2\) rectangle. That rectangle has area \(8\) square units, so each triangle has area \(8 \div 2 = 4\) square units. 3. All three parts have area \(4\) square units, so they are equal in area even though they are not the same shape. 4. The whole is divided into \(3\) equal-area parts, so each part is \(\frac{1}{3}\) of the whole.

Yes. Each part has an area of \(4\) square units, so each part is \(\frac{1}{3}\) of the whole.

- Count or group the unit squares in A. - Count or group the unit squares in B. - When two regions have equal area and make the whole, each is one half.

1. Part A is \(3\) squares wide and \(2\) squares tall, so its area is \(3 \times 2 = 6\) square units. 2. Part B contains the other \(6\) unit squares. 3. The parts have equal area even though their shapes are different. 4. Each of the \(2\) equal-area parts is \(\frac{1}{2}\) of the whole.

a) A has area \(6\) square units, and B has area \(6\) square units. b) Yes. A and B have equal area. c) Each part is \(\frac{1}{2}\) of the whole.

- Check whether the three numbers in each row are equal. - Do not confuse having the correct total with having equal parts. - Use the number of equal regions to name one region as a unit fraction.

1. Each plan totals \(24\) square units, but an equal-area partition needs all three region areas to match. 2. Only Plan A has three equal areas: \(8, 8, 8\). 3. One of \(3\) equal regions is \(\frac{1}{3}\) of the board.

Plan A. Each region is \(\frac{1}{3}\) of the board.

- Use the grid to find the area of each region. - Compare the area values before naming the fraction. - Equal area describes the amount of space, not the outline shape.

1. The left and right regions are each \(2\) squares wide and \(4\) squares tall, so each has area \(2 \times 4 = 8\) square units. 2. The two middle regions together make a \(4 \times 4\) rectangle with area \(16\) square units. The diagonal divides that rectangle into two equal-area triangles, so each has area \(16 \div 2 = 8\) square units. 3. All four regions have area \(8\) square units, so they are equal in area. 4. One of \(4\) equal-area regions is \(\frac{1}{4}\) of the whole. 5. The regions include rectangles and triangles, so equal-area regions do not have to have the same shape.

a) Yes. Each region has area \(8\) square units. b) Each region is \(\frac{1}{4}\) of the whole. c) No. Equal-area regions can have different shapes.

- Count the final regions shown by the boundary lines. - Compare how many unit squares are in each region. - For part c), compare one smallest region with all the unit squares in the whole rectangle.

1. Replacing one region with two regions changes the count from \(4\) to \(4 - 1 + 2 = 5\) regions. 2. Three regions contain \(2\) unit squares each, while the two smallest regions contain \(1\) unit square each, so the regions are not all equal in area. 3. The whole rectangle contains \(8\) equal unit squares. Each smallest region contains \(1\) of them, so each is \(\frac{1}{8}\) of the original rectangle.

a) \(5\) regions b) No. c) Each smallest region is \(\frac{1}{8}\) of the original rectangle.

- Use the grid to find the dimensions of the large region. - Find the area of one narrow region from its width and height. - Compare one narrow region with the area of the whole grid.

1. The large region is \(3\) squares wide and \(4\) squares tall, so its area is \(3 \times 4 = 12\) square units. 2. The three smaller regions together also cover a \(3 \times 4\) area, so their total area is \(12\) square units. 3. Each smaller region is \(1\) square wide and \(4\) squares tall, so each has area \(1 \times 4 = 4\) square units. 4. The whole has area \(6 \times 4 = 24\) square units. Since \(24 \div 4 = 6\), each smaller region is \(\frac{1}{6}\) of the whole.

a) The large region has area \(12\) square units, and the three smaller regions have total area \(12\) square units. b) Each smaller region has area \(4\) square units. c) Each smaller region is \(\frac{1}{6}\) of the whole.

- Focus on the quantity named in “equal-area partition.” - Compare the four area values directly. - Decide whether perimeter has to match when area matches.

1. Equal-area parts must have the same area. 2. All four regions have area \(6\) square units. 3. Different perimeters do not change the regions' areas. 4. Therefore, the partition is an equal-area partition.

Yes. All four regions have area \(6\) square units, so their areas are equal even though their perimeters differ.

- Use the grid to find the area of each region. - Compare the wider regions with the narrower regions. - Equal-area regions must all have the same area value.

1. Each of the two wider regions is \(2\) squares wide and \(2\) squares tall, so each has area \(2 \times 2 = 4\) square units. 2. Each of the four narrower regions is \(1\) square wide and \(2\) squares tall, so each has area \(1 \times 2 = 2\) square units. 3. The regions do not all have the same area because \(4 \ne 2\). 4. The whole has area \(8 \times 2 = 16\) square units. The wider regions are each \(\frac{4}{16}=\frac{1}{4}\) of the whole, while the narrower regions are each \(\frac{2}{16}=\frac{1}{8}\) of the whole.

No. Two regions have area \(4\) square units and are each \(\frac{1}{4}\) of the whole; four regions have area \(2\) square units and are each \(\frac{1}{8}\) of the whole.

- Find one piece's area before thinking about the rearrangement. - Ask what measurements stay unchanged when pieces are moved. - Count the same pieces in the new whole.

1. Each original piece has area \(24 \div 6 = 4\) square units. 2. Rearranging the pieces does not change their areas or the total area. 3. The new whole still contains the same \(6\) equal-area pieces. 4. Each piece is \(\frac{1}{6}\) of the new whole.

a) \(4\) square units b) \(\frac{1}{6}\)

- Compare the region labels before adding them. - Check whether their total matches the whole area. - Divide the whole area by the required number of equal regions.

1. The labels use two different region areas, \(6\) and \(5\), so they do not describe equal areas. 2. Their total is \(5 \times 6 + 3 \times 5 = 30 + 15 = 45\) square units, not \(48\). 3. Eight equal regions in an area of \(48\) must each have area \(48 \div 8 = 6\) square units. 4. Every region should be labeled \(6\) square units.

a) No. b) No; the labels total \(45\) square units. c) \(6\) square units per region.

- Equal parts would need the same whole-number count of unit squares. - Test nearby equal group sizes. - Compare their totals with \(21\).

1. If each part had \(5\) unit squares, the total would be \(4 \times 5 = 20\). 2. If each part had \(6\) unit squares, the total would be \(4 \times 6 = 24\). 3. There is no whole-number square count between \(5\) and \(6\) for each part. 4. Therefore, \(21\) unit squares cannot be split into \(4\) equal-area parts without cutting a square.

No. Four equal groups of whole unit squares cannot total \(21\).

- Count the equal strips separately from the unit squares. - Find the area of the shaded strip from its width and height. - A fraction denominator names the number of equal parts in the partition.

1. The whole grid has area \(8 \times 8 = 64\) square units. 2. The partition has \(8\) equal strips, not \(64\) equal strips. 3. One strip has area \(64 \div 8 = 8\) square units. 4. One of \(8\) equal strips is \(\frac{1}{8}\) of the grid. Sam used the number of unit squares instead of the number of equal parts as the denominator.

One strip has area \(8\) square units and is \(\frac{1}{8}\) of the grid. Sam incorrectly used the total number of unit squares as the fraction denominator.