A Peek Beyond the Point · Lesson 2 of 11
Adding and Subtracting Tenths
“Use equal-sized parts to combine lengths, find differences, and extend patterns.”
• Add measurements expressed in whole units and tenths. • Explain exchanging ten tenths for one unit. • Subtract tenths by regrouping a whole unit when needed. • Use addition and subtraction to explain simple sequences.
Combining Whole Units and Tenths
Suppose a lower arm measures 2 units and 7 tenths, while the upper arm measures 3 units and 6 tenths. To find the combined length, we must add like-sized parts. Whole units combine with whole units, and tenths combine with tenths. This is the same idea used when adding tens to tens and ones to ones.
The whole-unit total is 2 + 3 = 5 units. The fractional total is 7 + 6 = 13 tenths. Thirteen tenths is larger than one unit, so the answer can be regrouped: ten tenths form a unit and three tenths remain. Five units and thirteen tenths therefore become six units and three tenths.
Problem
Add 2 + 7/10 units and 3 + 6/10 units.
- 1.Add whole units: 2 + 3 = 5.
- 2.Add tenths: 7 + 6 = 13. Split these into 10 tenths and 3 tenths.
- 3.Exchange 10 tenths for 1 unit. The sum is 6 + 3/10 units.
- 4.Check by counting entirely in tenths: 27 + 36 = 63 tenths, which is 6 units and 3 tenths.
The exchange does not add extra length. It only changes the way an existing quantity is counted. A useful check is to express both starting measurements and the result entirely in tenths. If the counts agree, the regrouping preserved the total.
Problem
The head, thorax, and abdomen measure 2 + 3/10, 5 + 4/10, and 7 + 5/10 units. Find their total length.
- 1.Add whole units: 2 + 5 + 7 = 14.
- 2.Add tenths: 3 + 4 + 5 = 12 tenths.
- 3.Twelve tenths equal 1 unit and 2 tenths. The total is 15 + 2/10 units.
- 4.Alternatively, 23 + 54 + 75 = 152 tenths. These group into 15 units and 2 tenths.
Finding a Difference
If a hand measures 12 units and 4 tenths from wrist to fingertip and its palm measures 6 units and 7 tenths, subtracting the palm length gives the remaining finger length. The difficulty is that the larger total has fewer tenths: four tenths cannot directly supply seven tenths. We can exchange one of its whole units for ten tenths.
After the exchange, 12 units and 4 tenths becomes 11 units and 14 tenths. It is still the same length. Now subtract 7 tenths from 14 tenths, then 6 units from 11 units. The remainder is 5 units and 7 tenths.
Problem
Find (12 + 4/10) − (6 + 7/10) units.
- 1.Rewrite the first quantity as 11 units and 14 tenths by exchanging one whole unit.
- 2.Subtract tenths: 14 − 7 = 7.
- 3.Subtract units: 11 − 6 = 5. Difference = 5 + 7/10 units.
- 4.Check in tenths: 124 − 67 = 57 tenths. Adding the difference to the palm gives 57 + 67 = 124 tenths, the hand length.
Another method keeps the whole and fractional differences separate. Here, 12 − 6 is 6 units, but 4 tenths − 7 tenths leaves a shortfall of 3 tenths. Removing that shortfall from 6 units gives 5 units and 7 tenths. This reasoning explains the result, although exchanging first is often easier to organise.
Problem
One fish is 2 + 4/10 cm long and another is 9/10 cm long. Find the difference.
- 1.Express the first length as 24 tenths of a centimetre.
- 2.Subtract the second count: 24 − 9 = 15 tenths.
- 3.Group 15 tenths into 1 cm and 5 tenths of a centimetre. Difference = 1 + 5/10 cm.
- 4.A ruler comparison would show the longer fish exceeds the shorter one by this distance.
During subtraction, do not reverse the tenths merely because their digits are easier to subtract that way. For 12 + 4/10 minus 6 + 7/10, using 7 − 4 changes the problem. Regroup the first quantity instead.
Patterns Built from Tenths
A sequence is an ordered list of quantities following a stated or inferred rule. To investigate a simple sequence, compare neighbouring terms. If the same amount is added or subtracted each time, that repeated change allows us to predict the next terms. Check more than one pair before deciding on a rule.
Problem
Extend 4, 4 + 3/10, 4 + 6/10 by three terms using a constant change.
- 1.The change is +3 tenths from the first term to the second and again to the third.
- 2.Add 3 tenths to 4 + 6/10 to obtain 4 + 9/10.
- 3.Add again: 4 units and 12 tenths regroup to 5 + 2/10.
- 4.One more addition gives 5 + 5/10. The next terms are 4 + 9/10, 5 + 2/10, and 5 + 5/10.
Start at 5 units and 7 tenths. Subtract four tenths repeatedly. Write five terms and explain the exchange when a step passes below a whole unit. Then invent a different constant-change sequence for someone else to continue.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
What is (2 + 7/10) + (3 + 6/10)?
Which quantity equals 12 units and 4 tenths?
What is (12 + 4/10) − (6 + 7/10)?
What is the next term after 8 + 2/10, 8 + 7/10, 9 + 2/10, using a constant change?
What does an exchange of ten tenths for one unit do?
The total is 5 units and 13 tenths, which regroups to 6 units and 3 tenths.
Practice Problems
- Add 4 + 8/10 and 2 + 5/10 units.
- Find (10 + 2/10) − (3 + 8/10) units.
- Extend 7 + 6/10, 8 + 7/10 by three terms using a constant change.
- Extend 13 + 5/10, 13, 12 + 5/10 by three terms.
- Extend 11 + 5/10, 10 + 4/10, 9 + 3/10 by three terms.
- A model has parts of length 1 + 9/10, 2 + 8/10, and 3 + 6/10 units. Find its total length and explain the exchange.
- Explain why 8 units and 2 tenths minus 3 units and 7 tenths is not 5 units and 5 tenths.
Whole units give 6; tenths give 13. Exchange 10 tenths for a unit to get 7 + 3/10 units.
Key Takeaways
• Combine corresponding whole and fractional units. • Ten tenths can be exchanged for one whole unit. • Subtraction may require exchanging one unit for ten tenths. • An exchange preserves the total quantity. • Counting entirely in tenths offers another calculation method. • Check a constant-change pattern using consecutive differences.