Demonstration 1 of 4
The denominator names the unit
Why does 2/5 + 1/5 keep the 5 on the bottom?
The top two strips are the pieces. The bottom strip shows the written result. When the bottom number stays 5, the count grows and the parts stay the same size. When it becomes 10, every part shrinks to half its size.
\[2/5 + 1/5 = 3/5\]
\[2/5 + 1/5 = 3/10\]
The bottom number names the size of each part: fifths of one pan. The top number counts how many of those parts are present.
Predict first. Switch to adding the bottoms too. Will the bottom strip reach the dashed line?
Choose an example
Constructed example: the chapter's 2/5 + 1/5 sum on its example pan, with other second pieces.
Calculated values
- Unit of each piece
- fifths
- Count of fifths
- 2 + 1 = 3
- Result written
- 3/5
- Shaded length of the result
- 3 / 5 = 0.6
- True total length
- 3 / 5 = 0.6
2/5 + 1/5: both count fifths, so add the counts: 2 + 1 = 3. The unit stays fifths: 3/5 = 0.6 of the pan, which matches the dashed line.
Use the idea
Before you add any two amounts, say the unit out loud: fifths plus fifths, dollars plus dollars. Only the counts are added.
Where the conclusion applies
Both fractions are parts of the same pan. One fifth of a small pan and one fifth of a large pan are not the same size, so matching bottoms alone would not be enough.
Check your understanding: What is 3/8 + 4/8, and why is the answer not 7/16?
Chapter 10 source: section "The denominator names the unit". Demonstration C10-D01.
Demonstration 2 of 4
Thirds and fourths become twelfths
A third and a fourth of the same pan: what common unit lets you add them?
A common unit must fit evenly inside both a third and a fourth. Twelfths do: 4 per third, 3 per fourth. Twenty-fourths also fit. Sevenths do not fit either one.
\[1/3 + 1/4 = 2/7\]
\[1/3 + 1/4 = 4/12 + 3/12 = 7/12\]
\[1/3 + 1/4 = 8/24 + 6/24 = 14/24\]
1/3 and 1/4 are parts of one pan. 4/12 and 3/12 are the same parts cut finer, so both count twelfths.
Predict first. Choose twenty-fourths. Will the total land on the dashed line too, and what will it be called?
Choose an example
Constructed example: the chapter's pan with one third and one fourth marked.
Calculated values
- 1/3 in this unit
- (1 x 4)/(3 x 4) = 4/12
- 1/4 in this unit
- (1 x 3)/(4 x 3) = 3/12
- Sum
- 4/12 + 3/12 = 7/12
- Simplest name
- 7/12
Cut each third into 4 and each fourth into 3: 1/3 = 4/12 and 1/4 = 3/12. Now both count twelfths: 4 + 3 = 7, so 1/3 + 1/4 = 7/12.
Use the idea
To add two fractions with different bottoms, list multiples of both bottoms and pick a shared one. The smallest keeps the counts short.
Where the conclusion applies
Both pieces come from the same pan. Any common multiple of 3 and 4 works. 2/7 is wrong because adding the bottoms changes the part size instead of counting parts.
Check your understanding: Add 5/6 + 3/8 using twenty-fourths?
Chapter 10 source: section "Thirds and fourths become twelfths". Demonstration C10-D02.
Demonstration 3 of 4
A sum that passes one whole
What happens when two pieces of the same pan add up to more than the pan?
The first hop is the first fraction in twelfths. The second hop starts where the first ended and moves right by the second fraction in twelfths. The dot is the sum.
\[3/4 + 2/3 = 9/12 + 8/12 = 17/12\]
3/4 and 2/3 are parts of one whole. 9/12 and 8/12 are the same amounts in twelfths. 17/12 counts seventeen twelfths, which is more than twelve.
Predict first. Change 3/4 to 1/4 and keep 2/3. Will the sum still pass 1?
Choose an example
Constructed example: the chapter's 3/4 + 2/3 sum, with other fourths and thirds.
Calculated values
- 3/4 in twelfths
- (3 x 3)/(4 x 3) = 9/12
- 2/3 in twelfths
- (2 x 4)/(3 x 4) = 8/12
- Sum
- 9/12 + 8/12 = 17/12
- Mixed number
- 1 5/12
- Past 1?
- yes
3/4 + 2/3 = 9/12 + 8/12 = 17/12, because 3 x 3 = 9 and 2 x 4 = 8. 17 twelfths = 1 x 12 + 5, so 17/12 = 12/12 + 5/12 = 1 5/12. It passes 1.
Use the idea
Before calculating, estimate: three fourths plus more than one half must pass 1. If your answer is below 1, look for the slip.
Where the conclusion applies
Both fractions are parts of equal wholes, and the line is marked in equal twelfths. A sum above 1 is a valid amount, not a mistake.
Check your understanding: Add 3/4 + 1/3 and write the result as a mixed number?
Chapter 10 source: section "A sum that passes one whole". Demonstration C10-D03.
Demonstration 4 of 4
Decimal places are common units too
Why do decimal points have to line up when you add?
Each decimal is turned into hundredths and shaded in the grid. Lining up the points makes tenths meet tenths and hundredths meet hundredths.
\[0.3 = 3/10 = 30/100\]
\[0.35 + 0.20 = 35/100 + 20/100 = 55/100\]
The first place after the point counts tenths, the second counts hundredths. One tenth is a row of ten hundredth squares.
Predict first. Change the second number to 0.65. How many of the 100 squares will be shaded?
Choose an example
Constructed example: the chapter's 0.35 + 0.20 sum, with other decimals.
Calculated values
- 0.35 by place
- 3 tenths + 5 hundredths = 35/100
- 0.20 by place
- 2 tenths + 0 hundredths = 20/100
- Sum in hundredths
- 35 + 20 = 55
- Sum as a decimal
- 0.55
0.35 + 0.20 = 35/100 + 20/100 = 55/100 = 0.55. Tenths meet tenths (3 + 2) and hundredths meet hundredths (5 + 0), which is why the columns line up.
Use the idea
When adding prices or measurements, write a zero to give both numbers the same number of places, then add column by column.
Where the conclusion applies
Decimals with at most two places, so every number is a whole count of hundredths. Lining up the right-hand digits instead of the points would add tenths to hundredths.
Check your understanding: Add 0.4 + 0.25 by renaming both in hundredths?
Chapter 10 source: section "Decimal places are common units too". Demonstration C10-D04.