← Measure the figure in overlaps and the squares cancel out · Overlap Reduces the Total

Measure the figure in overlaps and the squares cancel out · 12 practice problems

6.G.A.16.EE.B.75.NF.B.4

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $5\frac{1}{4}\ \text{cm}^2$ . If one square's area is $3$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 5 and 1/4 square centimetres, and each square is 3 times the overlap. We want the overlap's area.

Givens

The whole figure covers 5 and 1/4 cm2.
One square is 3 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 3 units, since a square is 3 times the overlap.

\text{square} = 3 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 6 units, but the overlap sits in both of them and was counted twice, so take one back off.

3 + 3 - 1 = 5

The whole figure is 5 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

5 units cover 5 and 1/4 cm2, so one unit is that divided by 5.

5\frac{1}{4} \div 5 = 1\frac{1}{20}

The overlap is 1 and 1/20 cm2.

Answer:

1\frac{1}{20}

cm²

4 · Reviewdoes it hold up?

A square is then 3 and 3/20 cm2, two squares are 6 and 3/10 cm2, and taking the doubled overlap off leaves 5 and 1/4 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $5 \times \square = 5\frac{1}{4}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 2 easy

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $6\frac{2}{3}\ \text{cm}^2$ . If one square's area is $3$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 6 and 2/3 square centimetres, and each square is 3 times the overlap. We want the overlap's area.

Givens

The whole figure covers 6 and 2/3 cm2.
One square is 3 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 3 units, since a square is 3 times the overlap.

\text{square} = 3 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 6 units, but the overlap sits in both of them and was counted twice, so take one back off.

3 + 3 - 1 = 5

The whole figure is 5 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

5 units cover 6 and 2/3 cm2, so one unit is that divided by 5.

6\frac{2}{3} \div 5 = 1\frac{1}{3}

The overlap is 1 and 1/3 cm2.

Answer:

1\frac{1}{3}

cm²

4 · Reviewdoes it hold up?

A square is then 4 cm2, two squares are 8 cm2, and taking the doubled overlap off leaves 6 and 2/3 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $5 \times \square = 6\frac{2}{3}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 3 easy

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $7\frac{1}{5}\ \text{cm}^2$ . If one square's area is $5$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 7 and 1/5 square centimetres, and each square is 5 times the overlap. We want the overlap's area.

Givens

The whole figure covers 7 and 1/5 cm2.
One square is 5 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 5 units, since a square is 5 times the overlap.

\text{square} = 5 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 10 units, but the overlap sits in both of them and was counted twice, so take one back off.

5 + 5 - 1 = 9

The whole figure is 9 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

9 units cover 7 and 1/5 cm2, so one unit is that divided by 9.

7\frac{1}{5} \div 9 = \frac{4}{5}

The overlap is 4/5 cm2.

Answer:

\frac{4}{5}

cm²

4 · Reviewdoes it hold up?

A square is then 4 cm2, two squares are 8 cm2, and taking the doubled overlap off leaves 7 and 1/5 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $9 \times \square = 7\frac{1}{5}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 4 medium

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $4\frac{3}{8}\ \text{cm}^2$ . If one square's area is $2$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 4 and 3/8 square centimetres, and each square is 2 times the overlap. We want the overlap's area.

Givens

The whole figure covers 4 and 3/8 cm2.
One square is 2 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 2 units, since a square is 2 times the overlap.

\text{square} = 2 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 4 units, but the overlap sits in both of them and was counted twice, so take one back off.

2 + 2 - 1 = 3

The whole figure is 3 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

3 units cover 4 and 3/8 cm2, so one unit is that divided by 3.

4\frac{3}{8} \div 3 = 1\frac{11}{24}

The overlap is 1 and 11/24 cm2.

Answer:

1\frac{11}{24}

cm²

4 · Reviewdoes it hold up?

A square is then 2 and 11/12 cm2, two squares are 5 and 5/6 cm2, and taking the doubled overlap off leaves 4 and 3/8 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $3 \times \square = 4\frac{3}{8}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 5 easy

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $8\frac{1}{6}\ \text{cm}^2$ . If one square's area is $4$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 8 and 1/6 square centimetres, and each square is 4 times the overlap. We want the overlap's area.

Givens

The whole figure covers 8 and 1/6 cm2.
One square is 4 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 4 units, since a square is 4 times the overlap.

\text{square} = 4 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 8 units, but the overlap sits in both of them and was counted twice, so take one back off.

4 + 4 - 1 = 7

The whole figure is 7 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

7 units cover 8 and 1/6 cm2, so one unit is that divided by 7.

8\frac{1}{6} \div 7 = 1\frac{1}{6}

The overlap is 1 and 1/6 cm2.

Answer:

1\frac{1}{6}

cm²

4 · Reviewdoes it hold up?

A square is then 4 and 2/3 cm2, two squares are 9 and 1/3 cm2, and taking the doubled overlap off leaves 8 and 1/6 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $7 \times \square = 8\frac{1}{6}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 6 medium

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $9\frac{3}{5}\ \text{cm}^2$ . If one square's area is $6$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 9 and 3/5 square centimetres, and each square is 6 times the overlap. We want the overlap's area.

Givens

The whole figure covers 9 and 3/5 cm2.
One square is 6 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 6 units, since a square is 6 times the overlap.

\text{square} = 6 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 12 units, but the overlap sits in both of them and was counted twice, so take one back off.

6 + 6 - 1 = 11

The whole figure is 11 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

11 units cover 9 and 3/5 cm2, so one unit is that divided by 11.

9\frac{3}{5} \div 11 = \frac{48}{55}

The overlap is 48/55 cm2.

Answer:

\frac{48}{55}

cm²

4 · Reviewdoes it hold up?

A square is then 5 and 13/55 cm2, two squares are 10 and 26/55 cm2, and taking the doubled overlap off leaves 9 and 3/5 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $11 \times \square = 9\frac{3}{5}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 7 medium

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $10\frac{1}{2}\ \text{cm}^2$ . If one square's area is $4$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 10 and 1/2 square centimetres, and each square is 4 times the overlap. We want the overlap's area.

Givens

The whole figure covers 10 and 1/2 cm2.
One square is 4 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 4 units, since a square is 4 times the overlap.

\text{square} = 4 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 8 units, but the overlap sits in both of them and was counted twice, so take one back off.

4 + 4 - 1 = 7

The whole figure is 7 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

7 units cover 10 and 1/2 cm2, so one unit is that divided by 7.

10\frac{1}{2} \div 7 = 1\frac{1}{2}

The overlap is 1 and 1/2 cm2.

Answer:

1\frac{1}{2}

cm²

4 · Reviewdoes it hold up?

A square is then 6 cm2, two squares are 12 cm2, and taking the doubled overlap off leaves 10 and 1/2 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $7 \times \square = 10\frac{1}{2}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 8 medium

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $11\frac{2}{5}\ \text{cm}^2$ . If one square's area is $5$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 11 and 2/5 square centimetres, and each square is 5 times the overlap. We want the overlap's area.

Givens

The whole figure covers 11 and 2/5 cm2.
One square is 5 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 5 units, since a square is 5 times the overlap.

\text{square} = 5 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 10 units, but the overlap sits in both of them and was counted twice, so take one back off.

5 + 5 - 1 = 9

The whole figure is 9 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

9 units cover 11 and 2/5 cm2, so one unit is that divided by 9.

11\frac{2}{5} \div 9 = 1\frac{4}{15}

The overlap is 1 and 4/15 cm2.

Answer:

1\frac{4}{15}

cm²

4 · Reviewdoes it hold up?

A square is then 6 and 1/3 cm2, two squares are 12 and 2/3 cm2, and taking the doubled overlap off leaves 11 and 2/5 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $9 \times \square = 11\frac{2}{5}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 9 hard

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $12\frac{3}{4}\ \text{cm}^2$ . If one square's area is $7$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 12 and 3/4 square centimetres, and each square is 7 times the overlap. We want the overlap's area.

Givens

The whole figure covers 12 and 3/4 cm2.
One square is 7 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 7 units, since a square is 7 times the overlap.

\text{square} = 7 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 14 units, but the overlap sits in both of them and was counted twice, so take one back off.

7 + 7 - 1 = 13

The whole figure is 13 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

13 units cover 12 and 3/4 cm2, so one unit is that divided by 13.

12\frac{3}{4} \div 13 = \frac{51}{52}

The overlap is 51/52 cm2.

Answer:

\frac{51}{52}

cm²

4 · Reviewdoes it hold up?

A square is then 6 and 45/52 cm2, two squares are 13 and 19/26 cm2, and taking the doubled overlap off leaves 12 and 3/4 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $13 \times \square = 12\frac{3}{4}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 10 hard

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $13\frac{1}{2}\ \text{cm}^2$ . If one square's area is $6$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 13 and 1/2 square centimetres, and each square is 6 times the overlap. We want the overlap's area.

Givens

The whole figure covers 13 and 1/2 cm2.
One square is 6 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 6 units, since a square is 6 times the overlap.

\text{square} = 6 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 12 units, but the overlap sits in both of them and was counted twice, so take one back off.

6 + 6 - 1 = 11

The whole figure is 11 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

11 units cover 13 and 1/2 cm2, so one unit is that divided by 11.

13\frac{1}{2} \div 11 = 1\frac{5}{22}

The overlap is 1 and 5/22 cm2.

Answer:

1\frac{5}{22}

cm²

4 · Reviewdoes it hold up?

A square is then 7 and 4/11 cm2, two squares are 14 and 8/11 cm2, and taking the doubled overlap off leaves 13 and 1/2 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $11 \times \square = 13\frac{1}{2}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 11 hard

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $14\frac{2}{7}\ \text{cm}^2$ . If one square's area is $9$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 14 and 2/7 square centimetres, and each square is 9 times the overlap. We want the overlap's area.

Givens

The whole figure covers 14 and 2/7 cm2.
One square is 9 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 9 units, since a square is 9 times the overlap.

\text{square} = 9 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 18 units, but the overlap sits in both of them and was counted twice, so take one back off.

9 + 9 - 1 = 17

The whole figure is 17 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

17 units cover 14 and 2/7 cm2, so one unit is that divided by 17.

14\frac{2}{7} \div 17 = \frac{100}{119}

The overlap is 100/119 cm2.

Answer:

\frac{100}{119}

cm²

4 · Reviewdoes it hold up?

A square is then 7 and 67/119 cm2, two squares are 15 and 15/119 cm2, and taking the doubled overlap off leaves 14 and 2/7 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $17 \times \square = 14\frac{2}{7}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Variant 12 hard

The figure at the right is made by overlapping $2$ congruent squares, and the whole figure covers $15\frac{1}{3}\ \text{cm}^2$ . If one square's area is $8$ times the area of the overlap, what is the area of the overlap, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

Two identical squares overlap. The figure they make covers 15 and 1/3 square centimetres, and each square is 8 times the overlap. We want the overlap's area.

Givens

The whole figure covers 15 and 1/3 cm2.
One square is 8 times the overlap.
The two squares are congruent.

Unknowns

The area of the overlap.

Constraints

Neither square's own area is given.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram#7 Identify Subproblems

The only area named twice is the overlap, so make it the unit. Every piece of the picture is then a whole number of overlaps and the unknown squares disappear from the arithmetic.

3 · Execute3 carry out the plan

1Call the overlap one unit

#13 Convert to Algebra 6.EE.B.7

Then each square is 8 units, since a square is 8 times the overlap.

\text{square} = 8 \times \text{overlap}

Everything is now counted in overlaps.

2Count the figure in units

#1 Draw a Diagram 6.G.A.1

Two squares are 16 units, but the overlap sits in both of them and was counted twice, so take one back off.

8 + 8 - 1 = 15

The whole figure is 15 overlaps.

3Turn units into square centimetres

#7 Identify Subproblems 5.NF.B.4

15 units cover 15 and 1/3 cm2, so one unit is that divided by 15.

15\frac{1}{3} \div 15 = 1\frac{1}{45}

The overlap is 1 and 1/45 cm2.

Answer:

1\frac{1}{45}

cm²

4 · Reviewdoes it hold up?

A square is then 8 and 8/45 cm2, two squares are 16 and 16/45 cm2, and taking the doubled overlap off leaves 15 and 1/3 cm2 -- the figure we were given.

Another way: Writing the overlap as a letter and solving $15 \times \square = 15\frac{1}{3}$ is the same reasoning with the picture put into symbols.

Standardsmin grade 6

6.G.A.1 Find area of triangles, quadrilaterals, polygons by composing/decomposing — Counting the composite figure as pieces that do not overlap.
6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving one equation for the unit area.
5.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a fraction — Scaling a fractional area up and down by whole numbers.

💡Takeaway. Pick the piece that appears in both shapes as your unit, then count the picture in that unit.

Where next?

← Back to the problem: Measure the figure in overlaps and the squares cancel out