Addition: regrouping tens and reordering
🎯 In this topic you will
- Estimate and add two 2-digit numbers, regrouping the tens or the ones.
- Estimate and add numbers up to 3 digits, regrouping the tens or the ones.
- Reorder numbers when adding to support efficient calculation.
🧠 Key Words
- associative
- column addition
- commutative
Show Definitions
- associative: A property of addition that means numbers can be grouped in different ways without changing the total (for example, $(2 + 3) + 5 = 2 + (3 + 5)$).
- column addition: A written method for adding numbers by lining up digits in columns and adding ones, tens, and hundreds separately, regrouping when needed.
- commutative: A property of addition that shows numbers can be added in any order and the answer stays the same (for example, $4 + 7 = 7 + 4$).
🧮 Building on What You Already Know
Y ou can already add some 3-digit numbers, regrouping the ones when you have too many for the ones place. In this unit, you will learn how to add numbers with up to three digits, regrouping the tens when you have too many tens for the tens place.
❓ EXERCISES
1. Complete these $2$-digit additions. Show your method.
a. estimate: $43 + 56 =$
b. estimate: $67 + 29 =$
👀 Show answer
a. $43 \approx 40$, $56 \approx 60$, so estimate $40 + 60 = 100$. Exact total: $43 + 56 = 99$.
b. $67 \approx 70$, $29 \approx 30$, so estimate $70 + 30 = 100$. Exact total: $67 + 29 = 96$.
2. Estimate the total. Find and represent the total in the place value grid. Complete the number sentence.
👀 Show answer
Top grid shows $6$ tens and $4$ ones ($64$). Bottom grid shows $5$ tens and $3$ ones ($53$).
Estimate: $60 + 50 = 110$.
Exact total: $64 + 53 = 117$.
Regrouping gives $1$ hundred, $1$ ten, and $7$ ones.
3. Estimate and then solve these $2$-digit additions. Show your method.
a. estimate: $84 + 62 =$
b. estimate: $71 + 87 =$
c. estimate: $64 + 72 =$
d. estimate: $75 + 34 =$
👀 Show answer
a. Estimate: $80 + 60 = 140$. Exact: $84 + 62 = 146$.
b. Estimate: $70 + 90 = 160$. Exact: $71 + 87 = 158$.
c. Estimate: $60 + 70 = 130$. Exact: $64 + 72 = 136$.
d. Estimate: $80 + 30 = 110$. Exact: $75 + 34 = 109$.
🧠 Think like a Mathematician
Challenge: I added two $2$-digit numbers. Their total is one hundred and thirty-seven. The ones digits were $6$ and $1$. What could my numbers have been?
👀 show answer
- The ones digits are $6$ and $1$, so together they make $7$.
- Since the total is $137$, the tens parts must add to $130$.
- One possible pair is $56$ and $81$, because $56 + 81 = 137$.
- Another possible pair is $66$ and $71$, because $66 + 71 = 137$.
- There are several correct answers — any two $2$-digit numbers ending in $6$ and $1$ whose tens add to $130$ will work.
❓ EXERCISES
4. Estimate and then solve these additions. Show your method.
a. estimate: $233 + 50 =$
b. estimate: $178 + 60 =$
c. estimate: $154 + 65 =$
d. estimate: $191 + 56 =$
e. estimate: $286 + 352 =$
f. estimate: $473 + 166 =$
Share and discuss your methods with your partner.
👀 Show answer
a. Estimate: $230 + 50 = 280$. Exact: $233 + 50 = 283$.
b. Estimate: $180 + 60 = 240$. Exact: $178 + 60 = 238$.
c. Estimate: $150 + 70 = 220$. Exact: $154 + 65 = 219$.
d. Estimate: $190 + 60 = 250$. Exact: $191 + 56 = 247$.
e. Estimate: $290 + 350 = 640$. Exact: $286 + 352 = 638$.
f. Estimate: $470 + 170 = 640$. Exact: $473 + 166 = 639$.
5.
You will need a paper clip and a pencil to use the spinner.
Spin the spinner three times to find three single digit numbers. Add them. Show your method.
Now spin the spinner four times to find four single digit numbers to add. Show your method.
Reorder your numbers to help you add. Remember, addition is commutative and associative!
👀 Show answer
Answers will vary.
Example (three spins): $6 + 2 + 4 = 12$.
Example (four spins): $9 + 1 + 5 + 3 = 18$.
Numbers can be reordered to make tens first, such as $(9 + 1) + (5 + 3) = 10 + 8 = 18$.
6. Choose three different $3$-digit numbers to add by reading across or reading down the grid. Estimate and then solve your addition. Show your method.
👀 Show answer
Example: across the first row gives $132$, second row $243$, third row $341$.
Estimate: $130 + 240 + 340 \approx 710$.
Exact total: $132 + 243 + 341 = 716$.
Numbers were grouped by hundreds first to make adding easier.
7. Record in words any two numbers from question $6$. How is the place value of a number shown in words and in numbers?
👀 Show answer
Example: $132$ is written as “one hundred and thirty-two”.
$243$ is written as “two hundred and forty-three”.
In numbers, place value is shown by the position of digits (hundreds, tens, ones). In words, it is shown using “hundred”, “ten”, and the remaining ones.