Introduction
The first chapter of Math Mammoth Grade 4-A Complete Worktext covers addition and subtraction topics, word problems, graphs, and money problems.
At first, we review the “technical aspects” of adding and subtracting: mental math techniques plus adding and subtracting in columns. If these are fairly easy for your student(s), you can choose to skip some problems.
Going beyond those, the chapter includes lessons in addition and subtraction terminology. These lessons are already preparing your child for algebraic thinking.
In the next lessons, the student reviews the addition/subtraction connection, and solves word problems with the help of bar models. Next, we solve simple missing addend equations using subtraction, such as x + 20 = 60. We use bar models to illustrate these and connect them with fact families.
The lesson on the order of operations contains some review but it goes beyond that. In many of the problems, the student builds the mathematical expression (calculation) needed for a certain real-life situation.
Going towards applications of math, the chapter contains lessons on bar graphs, line graphs, rounding, estimating, and money problems.
The Lessons in Chapter 1
page span
Addition Review ... 12 3 pages Adding in Columns ... 15 1 pages Subtraction Review ... 16 3 pages Subtract in Columns ... 19 3 pages Mental Math Workout and
Pascal's Triangle ... 22 3 pages Subtraction Terms ... 25 2 pages Word Problems and Bar Models ... 27 3 pages Missing Addend Solved With Subtraction ... 30 4 pages Order of Operations ... 33 2 pages Bar Graphs ... 35 3 pages Line Graphs ... 38 3 pages Rounding ... 41 4 pages Estimating ... 45 2 pages Reviewing Money ... 47 3 pages Review ... 50 1 page
Helpful Resources on the Internet Calculator Chaos
Most of the keys have fallen off the calculator but you have to make certain numbers using the keys that are left.
http://www.mathplayground.com/calculator_chaos.html ArithmeTiles
Use the four operations and numbers on neighboring tiles to make target numbers.
http://www.primarygames.com/math/arithmetiles/index.htm Choose Math Operation
Choose the mathematical operation(s) so that the number sentence is true. Practice the role of zero and one in basic operations or operations with negative numbers. Helps develop number sense and logical thinking.
http://www.homeschoolmath.net/operation-game.php MathCar Racing
Keep ahead of the computer car by thinking logically, and practice any of the four operations at the same time.
http://www.funbrain.com/osa/index.html Fill and Pour
Fill and pour liquid with two containers until you get the target amount. A logical thinking puzzle.
http://nlvm.usu.edu/en/nav/frames_asid_273_g_2_t_4.html Estimate Addition Quiz
Scroll down the page to find this quiz plus some others. Fast loading.
http://www.quiz-tree.com/Math_Practice_main.html Mental Addition and Subtraction
A factsheet, quiz, game, and worksheet about basic mental addition and subtraction.
http://www.bbc.co.uk/skillswise/numbers/wholenumbers/addsubtract/mental/
Shop 'Til You Drop
Get as many items as you can and be left with the least amount of change, and practices your addition skills. The prices are in English pounds and pennies.
http://www.channel4.com/learning/microsites/P/puzzlemaths/shop.shtml Change Maker
Determine how many of each denomination you need to make the exact change. Good and clear pictures! Playable in US, Canadian, Mexican, UK, or Australian money.
http://www.funbrain.com/cashreg/index.html Cash Out
Give correct change by clicking on the bills and coins.
http://www.mrnussbaum.com/cashd.htm
Piggy bank
When coins fall from the top of the screen, choose those that add up to the given amount, and the piggy bank fills.
http://fen.com/studentactivities/Piggybank/piggybank.html Bar Chart Virtual Manipulative
Build your bar chart online using this interactive tool.
http://nlvm.usu.edu/en/nav/frames_asid_190_g_1_t_1.html?from=category_g_1_t_1.html An Interactive Bar Grapher
Graph data sets in bar graphs. The color, thickness and scale of the graph are adjustable. You can put in your own data, or you can use or alter pre-made data sets.
http://illuminations.nctm.org/ActivityDetail.aspx?ID=63
Addition Review
1. Add mentally. You can add in parts (tens and ones separately).
2. Write the numbers as a sum of whole thousands, whole hundreds, whole tens, and ones.
3. Solve the problems.
a. Two of the addends are 56 and 90. The sum is 190.
What is the third addend?
b. Four of the addends equal 70 and five other addends equal 80.
What is the sum?
Remember addition?
You can write any number
as a SUM of the different units such as whole thousands, whole hundreds, whole tens, and ones.
5,248 = 5,000 + 200 + 40 + 8
thousands hundreds tens ones
You can add in parts:
56 + 124
= 100 + 50 + 20 + 6 + 4
= 100 + 70 + 10 = 180
Add in any order:
7 + 90 + 91 + 3
= 7 + 3 + 90 + 91
= 10 + 90 + 91 = 191
Trick: add first a bigger number, then subtract to correct the error:
76 + 89
= 76 + 90 − 1
= 166 − 1 = 165
a.
70 + 80 = ___
77 + 80 = ___
77 + 82 = ___
b.
140 + 50 = ___
141 + 50 = ___
144 + 55 = ___
c.
50 + 60 = ___
54 + 65 = ___
58 + 62 = ___
d.
80 + 90 = ___
82 + 93 = ___
88 + 91 = ___
a.
487 =
c.8,045 =
b.
2,103 =
d.650 =
4. Add and compare the results. The addition problems are “related”!
5. Write here four different addition problems that are “related” to the problem 5 + 8 = 13.
8. Add in parts, or use other “tricks”.
9. Continue the patterns.
a.
7 + 8 = ___
10. Double and halve the numbers.
11. a. There are five people in the Brill family and they went to a concert. Children's tickets were $20 each and the two parents' tickets were $28 a piece.
What was the total cost of the tickets for the family?
b. In another concert, adult ticket cost $30 and children's tickets were half that price.
What was the total cost for the Brill family?
12. Fill in the table - add 999 each time.
Half the number 10
Number
20 90 110 120 480 500 900 1,600 4,010
Its double 40
n
56
69 125 156 287 569 788 950 999n + 999
John is writing very simple “missing addend” problems for first graders.
For example, he wrote the problem 2 + ___ = 8. The first addend is given, and the second addend is missing.
John uses whole numbers from 0 on up to the number that is the sum.
a. How many such problems can he write when the sum is 8?
b. How many such problems can he write when the sum is 10?
c. How many such problems can he write when the sum is 20?
d. You should see a pattern in the above answers. Now use the pattern to solve this:
How many such problems could he write when the sum is 100 (for second-graders)?
Adding in Columns
1. Add in columns. Check by adding the numbers in each column in different order (for example from down up).
2. Write the numbers under each other carefully, and add in columns.
a.
5,609 + 1,388 + 89 + 402
b.
$8.05 + $0.29 + $38.40 + $293 + $203.20 + $46.49 + $94
3. The map shows some Kentucky cities and distances between them. For example, from Louisville to Frankfort is 54 miles.
The one distance not marked is written below the map: from Frankfort to Lexington is 28 miles.
Calculate the total driving distance, if a family goes on a field trip like this:
a. Covington - Lexington - Paducah - Lexington - Covington
b. A round trip from Lexington via Covington, Louisville, and Frankfort, and back to Lexington.
a.
3 8 4 2 9 1 2 2 0 0 8 2 0 9 + 2 6
b.
$1.8 2 4 0.5 9 9.9 7 1 0.2 9 1.0 9 + 0.4 3
c.
2 4 5
1 3 9 3 0 2 9 3 1 5 9 4 9 5 9 3 + 5 2 6
d.
1 7 3 8
2 3 9 0
1 0 7 8
3 6 4
2 8 0 3
2 1 1
+ 9 9
Subtraction Review
1. Subtract from whole hundreds. You can subtract in parts.
2. Subtract. Use the helping problem.
3. Subtract and compare the results. The problems are “related” – can you see how?
4. Write here four different subtraction problems that are “related” to the problem 14 – 8 = 6.
See the examples above!
Compare the methods.
Marie: “I subtract in parts: first to the previous whole ten, then the rest.”
35 − 7
5. Fill in the table - subtract 99 each time.
6. Subtract in parts, use a helping problem, add up to find the difference, or use other “tricks”.
7. Fill in the table - subtract 27 each time.
Trick: subtract first a bigger number, then add back some to correct the error:
705 − 99
Strategy: Add up to find the difference of two numbers.
To solve 93 – 28, start at 28 and add until you reach 93. However much you added is the
8. Subtract the same number repeatedly. Multiplication tables can help!
Repeated Subtraction Game!
Jane and Jim are playing a repeated subtraction game. Each player has various number cards. A player pairs his cards together, two by two. With each two cards, the player subtracts the smaller number as many times as possible from the bigger number.
For example, Jane pairs together cards 20 and 4. Jane subtracts 20 – 4 – 4 – 4 – 4 – 4 = 0.
Jim pairs the cards 45 and 11, and subtracts 45 – 11 – 11 – 11 – 11 = 1. He can't subtract any more.
Each player gets as many “points” as is the “remainder” number (the final difference).
Above, Jane got 0 points and Jim got 1. The player who first accumulates 25 points loses the game.
Write the subtractions that Jane does with these cards:
With four cards, you need to choose which two will make a pair. Pair the cards for subtractions so that you will get the least possible points. Then write the subtractions.
e. Play the game yourself! Try number cards from 2-30 for an easier game. Try numbers from 2 to 60 for a challenge. Give each player 4-8 cards, depending on the difficulty level you wish to have.
a.
240
Subtract in Columns
1. This is review. Subtract in columns. Check by adding!
It is time to review borrowing over zeros!
2. Subtract in columns. Check by adding!
a.
You can't subtract 3 from 0.
You can't borrow a ten - there are none!
First borrow one hundred.
You get 10 tens in the tens column.
Then borrow 1 ten into the ones column.
Now you can subtract. You can't borrow from the tens
nor from the hundreds. So borrow 1 thousand.
Next, borrow one hundred into the tens column.
Then borrow one ten into the ones column. You're
3. Subtract in columns. Check by adding!
5. Write the numbers under each other carefully, and subtract in columns.
a. 4,400 − 2,745 − 493 b. 5,604 − 592 − 87 c. $45.60 − $12.36 − $1.69
6. You can solve the problem 5,200 − 592 − 87 − 345 − 99 by subtracting the numbers one at a time. That means four separate subtractions. Can you find a quicker way?
8120 – 2653 – 754 = ?
When subtracting two numbers, you can continue the subtraction under your first answer.
Check by adding the answer and all the numbers you subtracted.
7 10 11 10
8 1 2 0
–2 6 5 3 5 4 6 7
–
7 5 4
4 7 1 3
Check:
4 7 1 3 7 5 4
+2 6 5 3 8 1 2 0
Little Hannah has almost learned to read the (analog) clock, but she can't remember which hand is the hour hand and which is the minute hand. So when the time is 1:15, she might say, “It is 3:05”, mixing the hours and the minutes.
One day mom was lying in bed, sick, and she asked Hannah what time it was. Hannah said, “It is 2:20.” Just a few minutes later mom asked again for the time. Hannah claimed it was now 4:25.
Remembering that each time Hannah either tells the time right, or mixes the hour and minute hands, mom was able to figure out what time it was in reality. Can you?
Mental Math Workout and Pascal's Triangle
1. Fill in the table - add 29 each time.
2. Fill in the table - subtract 39 each time.
3. Subtract - and be careful!
4. Figure out the patterns and continue them.
n
9 18 27 36 45 54
n + 29
n
660 600 540 480 420
n – 39
a.
500 – 3 = 500 – 30 = 500 – 300 = 500 – 33 = 500 – 303 =
b.
600 – 2 = 600 – 20 = 600 – 200 = 600 – 22 = 600 – 202 =
c.
300 – 3 = 400 – 40 = 500 – 5 = 600 – 60 = 700 – 7 =
d.
1,000 – 7 = 1,000 – 70 = 1,000 – 700 = 1,000 – 77 = 1,000 – 707 =
+ + + + + + + +
5 28 51 74 ____ ____ __ ____ ____
– – – – – – – –
1000 900 810 730 660 ____ ____ ____ ____
5. Continue the patterns.
6. This will be a Pascal's triangle but you need to fill it in. On the left and right sides are ones.
Any other number is gotten by adding the two numbers right above it (slightly to the right and to the left). For example, the colored number 3 comes from adding the 1 and 2 above it.
+ 300 + 300 + 300 + + + + +
3,000
___ ___ ____ ____ ____ ____ ____ ___– 400 – 400 – 400 – – – – –
10,000
___ ___ ____ ____ ____ ____ ____ ___7. a. After filling the triangle, add the numbers in each row and make a list. For example, the first row just has 1. In the second row, add 1 + 1 = 2. In the third row, add 1 + 2 + 1 = 4.
The row sums are: 1, 2, 4, ____, ____, ____, ____, ____, ____, ____, ____, ____.
What do you notice about these numbers?
b. Can you find a diagonal with the numbers 1, 2, 3, 4, 5, 6, 7?
c. Can you find a diagonal with triangular numbers?
(Triangular numbers start like this:)
Read more about Pascal's triangle and its patterns at http://ptri1.tripod.com/
Below you will find an empty Pascal's triangle to explore with. You can fill it with some other number on all the sides, such as 2, 3, or 20.
Subtraction Terms
1. The minuend is missing! Find a general idea that always works to solve these kind of problems.
2. The subtrahend is missing! Find a general idea that always works to solve these kind of problems.
3. a. Write three subtraction problems where the difference is 10.
b. The subtrahend is 12 and the difference is 58. What is the minuend?
c. The minuend is 55 and the difference is 17. What is the subtrahend?
4. Explain an easy to way to subtract 999 from any number mentally.
For example, explain how to do easily 1,446 – 999.
5. The difference of two numbers is 20, and one of the numbers is 25.
What can the other number be?
Remember subtraction terms?
Just like “m” comes before “s” in the alphabet, the minuend comes before the subtrahend.
a.
____ − 8 = 7 ____ − 4 = 20
b.
____ − 15 = 17 ____ − 24 = 48
c.
____ − 22 − 7 = 70 ____ − 300 − 50 = 125
a.
20 − ____ = 12 6 − ____ = 5
b.
55 −
____ = 34 100 − ____ = 72
c.
234 − ____ = 100
899 − ____ = 342
6. Solve the problems. You will need addition AND subtraction.
a. A package of cheese costs $6 and a package of ham costs $2 less.
How much do the two cost together?
b. One alarm clock costs $11 and another costs $8 more.
How much would the two cost together?
c. Of the 45 students, 18 are girls. How many are boys?
How many more boys are there than girls?
d. Jack gave the clerk $50 for his purchases, and got $13 as his change.
How much did his purchases cost?
e. It rained five days in June and six days in July.
How many non-rainy days did those two months have?
f. Amy is 134 cm tall and her mom is 162 cm tall.
What is the difference in their heights?
g. Jack bicycled his favorite 28 km route on Tuesday and on Wednesday.
On Thursday and Saturday he bicycled along a route that was 6 km shorter.
How many kilometers did he bicycle all totalled?
Subtraction is used:
z To find the difference
z In “less than” or “more than” situations
z In “take away” situations
z To find one part when you have a “whole” and several “parts”.
Find the missing numbers.
a.
200 − 45 − ____ − 70 = 25
b._____ − 5 − 55 − 120 = 40
Word Problems and Bar Models
Mark the numbers given in the problem in the diagram. Mark what is asked with “?”.
Then solve the problem.
Bar models help you see how the numbers in a problem relate to each other. Whenever you get stumped by a word problem, try drawing a bar model.
On Monday, Dad drove 277 miles, and on Tuesday he drove 25 miles more than he did on Monday. How many miles did he drive in the two days?
On Tuesday he drove 277 + 25 = 302 miles.
Altogether he drove 277 + 302 = 579 miles.
The bracket “}” means addition or the total of the two bars. We do not know the total or the sum of the two days' journey, so it is marked with a question mark.
Monday Tuesday
After driving 20 miles, Dad says, “I still have 15 more miles to go to the half-way point.” How long is the trip?
20 mi + 15 mi = 35 miles, and that is the first half of the trip. So, the total trip is 2 × 35 = 70 miles.
We do not know the total length, so it is marked with “?”.
1. Jake worked for 56 days on a farm, and Ed worked for 14 days less.
How many days did Ed work?
2. Of his paycheck, Dad paid $250 on taxes, and spent $660 on other bills and purchases. Then, half of his paycheck was gone. How much was his paycheck?
3. Dad bought two hammers. One cost $18 and the other cost $28 more.
What was his total bill?
Mark the numbers given in the problem in the diagram. Mark what is asked with “?”.
Then solve the problem.
6. Eric and Angela did yard work together. They earned $80 and split it so that Eric got $12 more
than Angela. How much did each one get?
Draw a bar diagram.
To solve it, you can think this way. If you took away (subtracted) the “additional” $10, then the total would be $90, and we would only have the two equal parts (the two green parts). So, $90 ÷ 2 = $45 gives us the amount Rebecca got, and then Angi got $45 + $10 = $55.
Angi and Rebecca split a $100 paycheck so that Angi got $10 more than Rebecca.
How much did each one get?
The bar diagram shows the situation. Angi got $10 more than Rebecca, and together they earned $100.
We can then see Angi got $50 + $5 = $55 and Rebecca got $50 − $5 = $45.
Here's another way of looking at the same situation.
We draw just one bar for the paycheck, and divide it into two halves in the middle (the dashed line). Then we draw half of the $10, or $5, on either side of that middle line.
4. Mary and Luisa bought a $46 gift together.
Mary spent $6 more on it than Luisa.
How many dollars did each spend?
5. Henry bought two circular saws. One saw was $100 cheaper than the other.
His total bill was $590. What did each saw cost?
You can solve the rest of the problems any way you like best.
7. Mark bought four towels for $7 each, and a blanket for $17.
He paid, and the clerk handed him back $5.
What denomination was the bill Mark used to pay?
8. One plain yogurt costs $2.40, strawberry yogurt costs $0.15 less than plain yogurt, and plum yogurt costs $0.30 more than plain yogurt.
What is your total bill if you buy all three?
9. Erica was 132 cm tall when she was 9 years old.
In the next year, she grew 6 cm, and the next year 2 cm less than the previous year. How tall was she at the age of 11?
10. John's monthly phone service bill is $48. John said that with the money he earned on his summer job, he could pay his phone service for two months, spend $120 for a bike, and still have half his money left. How much did he earn?
11. Melissa found a nice shirt for $11.50, another for $2.55 less, and yet another for $2 less.
If she buys all three, what will her total bill be?
Missing Addend Solved with Subtraction
1. The missing addend is solved with subtraction. Solve.
2. Write a missing addend sentence using x, and a subtraction sentence to solve it.
From this simple diagram, we can write two addition and two subtraction sentences.
Those four form a fact family.
x stands for a number, too. We just don't know it yet.
Which fact in the family makes it easy to find the value of x?
z x + 15 = 56
z 15 + x = 56
z 56 – x = 15
z 56 – 15 = x
Here is missing addend problem:
769 + x = 1,510.
You can solve it by subtracting
the one part (769) from the total (1,510):
x = 1,510 – 769 = 741
a.
78 + x = 145
x = 145 – 78 = _____
b.
128 + x = 400
x = ____ – ____ = _____
c.
x + 385 = 999
x = ____ – ____ = _____
a. A car costs $1,200 and dad has $890.
How much more does he need?
b. The school has 547 students, of which
265 are girls. How many are boys?
3. a. Write a fact family using these
4. Write a missing addend sentence with x. Solve.
5. Which number sentence fits the problem? Find x.
a. A school's teachers and students filled a 450-seat auditorium. If the school had 43 teachers, how many students did it have?
x =
students + teachers = total ______ + ______ = ______
b. Mom went shopping with $250 and came back home with $78.
How much did she spend?
x =
spent + left = had originally ______ + ______ = ______
c. Janet had $200. She bought an item for $54 and another for $78.
How much is left?
x =
item 1 + item 2 + left = total ______ + ______ + ______ = ______
d. Jean bought one item for $23 and another for $29, and she had $125 left. How much did she have
initially? x =
______ + ______ + ______ = ______
a. Jane had $15. Dad gave Jane her allow-
6. Pick a number sentence that you can use to find x. Then solve for x.
7. Solve for x.
8. Write the numbers and x to the picture. Write a missing addend sentence. Solve.
a. Problem: 253 + x = 2056
2056 – 253 = x OR x – 253 = 2056
b. Problem: x + 148 = 397
148 – 397 = x OR 397 – 148 = x
c. Problem: x – 23 = 45
45 – 23 = x OR 45 + 23 = x
d. Problem: 120 – x = 55
120 – 55 = x OR 120 + 55 = x
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a.
x 1,750 b. 23 + 56 + x = 110
a. The Jones' family had traveled 420 miles of their 1,200-mile journey. How many miles were left to travel?
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b. The store is expecting a shipment of 4,000 blank CDs. Two boxes of 500 arrived.
How many are still to come?
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c. A 250 cm board is divided into three parts:
two 20 cm parts at the ends and a part in the middle. How long is the middle part?
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d. After traveling 56 miles, Dad said, “We have 118 miles left.”
How long is the journey?
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Order of Operations
1. Do the calculations in the right order.
1. Do the calculations in the right order.