AZ Teaching Standard:
M7S5C2PO1: Analyze a problem situation to determine the question(s) to be answered.
M7S1C3PO2: Make estimates appropriate to a given situation.
M7S5C2PO4: Represent a problem situation using multiple representations, describe the process used to solve the problem, and verify the reasonableness of the solution.
Lesson Outcome: TSW change numbers to estimate by reformulation. Estimate solutions to multi-stepped word problems by rounding with decimals.
tudents will make a reasonable estimate of the range the answer must fall in for a word problem. It will typically have a lower limit that the answer must exceed, and an upper limit, which will be larger than the exact answer. Students will use counters and building mats, and move from the manipulative stage to the pictorial stage showing how estimates are arrived to formulate an interval for each word problem posed; combine two equations into one equation for each word problem posed.
Big Idea: Estimation increases the efficiency for responsibilities.
Materials: Building Mat for each pair of students, 50 small counters for each pair of students (same color or style), worksheet with base 10 drawing frames for each student, pencil, red pencil, story problems for each set
TSW use interval estimating with
80% proficiency given two whole numbers.
Illicit student responses for strategies for estimating that they are aware of.
hanging the numbers used. This is by far the most common type of strategy. It involves changing the numbers to ones that are more easily manipulated using mental strategies. Typically the original numbers are amended as tidy numbers (which are numbers that end with at least one 0). Rounding and front-end utilize tidy numbers.
Round Robin: in groups, students will write down strategies for estimating.
Student shared responses
GESD. Curriculum and Instruction Department. 03.19.10

TSW use counters to represent the numbers in a word problem
This strategy requires us to make a reasonable estimate of the range the answer must fall in. It will typically have a lower limit that the answer must exceed, and an upper limit, which will be larger than the exact answer.
One way to achieve this is to apply both rounding and frontend strategies to give an interval between which the answer lies. The front-end estimate is always too small. If all numbers are rounded up, the estimate is always too large.
Model process:
36 × 57
To get the lower limit of the interval, use front-end: 30 × 50 =
1 500.
To get the upper limit of the interval, use rounding up: 40 × 60
= 2 400.
The answer lies between 1 500 and 2 400.
Next example for Guided
Practice:
341 + 572
Students will take part in a guided practice of examples using whole numbers.
TW provide each pair of students with a building mat, counters, and two copies of the word problem sheet, which involves one-to fourdigit decimal numbers or whole numbers in two-step problems.
TW introduce objective for today.
Students place counters on the top two rows of the building mat to represent 7.8 kilometers and
6.2 kilometers. They should also place counters on the bottom row to show the two days involved.
To get the lower limit of the interval, use front-end: 300 +
500 = 800.
To get the upper limit of the interval, use rounding: 400 +
600 = 1 000.
The answer lies between 800 and 1 000.
The numbers needed for this problem should be shown on the mat appropriately.
(First Row:
One’s Column-7
GESD. Curriculum and Instruction Department. 03.19.10

TW read story problem. Kelly rode her bicycle for 7.8 kilometers on
Friday and for 6.2 kilometers on
Saturday. Estimate how many kilometers she averaged per day for the two days.
TW model thinking and process for representing numbers on the building mat.
TSW round numbers using manipulatives
TSW perform the required computations mentally or with appropriate written algorithms.
Discuss the idea that the top two numbers need to be added together, and then divided by the bottom number in order to find the average kilometers per day. Since the top two numbers have ones as their higher place value, both numbers should be rounded to the nearest one.
(The rows on the mat are only for rounding, not for computation. For convenience, if two numbers needed to be added or subtracted, they should be shown on the top two rows of the mat. If a number will serve as a multiplier or a divisor, it should be shown on the bottom row).
TW remind students that they many need to use “backward thinking” here; that is, they may have to think of the second or final step first before they can decide what the first step should be.
The rounded numbers are now ready to be computed mentally. It is possible to use facts for each step. The first step will be 8+6=14 kilometers total for two days, and the second step will be 14/2=7 kilometers estimated per day.
These two number sentences, which contain the rounded numbers and were used to solve a word problem, should be recorded
SW round 7.8 to 8 and 6.2 will round down to 6. SW show this on the building mat by removing the 8 tenths from the top row but bring in a new ones counter on that same row.
On the second row, two tenths will be removed from the mat.
SW leave the bottom row, the divisor 2 alone since it is already a 1-digit number in the ones place and does not need to be rounded.
Students will record these two equations on their word problem worksheet.
GESD. Curriculum and Instruction Department. 03.19.10 counters
Tenth’s Column –
8 counters
Second Row:
One’s Column – 6 counters
Tenth’s Column –
2 counters
Third Row:
One’s Column – 2 counters)
Counters on the mat will display the rounded decimals.
(First Row:
One’s Column – 8 counters
Second Row:
One’s Column – 6 counters
Third Row:
One’s Column – 2 counters)
8+6=14
Kilometers total for two days
14/2=7 kilometers estimated per day

below that word problem on the worksheet to show the estimated work.
TSW describe a numerical interval that contains the estimate they have found.
TSW describe a numerical interval that contains the estimate they have found.
TSW identify the numbers needed to find the estimated interval for the word problem.
SW use the building ten drawing frame to represent the decimals needed for computation in the word
An interval for the estimated solution also needs to be written and answers will vary.
My estimate is between 5 and 10 kilometers. Intervals should be reasonable. Between 0 and 50 kilometers would not be considered reasonable.
Answers may vary student by student.
Teacher will repeat process above using guided practice for problems
2 and 3.
Teacher will then have students perform problems 4 and 5 independently as pairs.
TW explain the worksheet handed out called Building Ten Drawing
Frames. The small frames look like the building mat used above.
Above each frame is a blank writing space for writing the initial numbers needed to solve the problem.
Discuss with students the following example: George has 17.9 ounces of sunflower seeds in a bag. If he triples this amount, then adds 8.5 more ounces to the bag, about how many ounces will he have in all?
TW model the process and thinking of drawing the circles in the appropriate columns to represent the three numbers found in the word problem.
Three serves as a multiplier, draw it on the bottom row of the frame.
Students will discuss by justifying their intervals among their group and record the intervals on the word problem recording sheet.
Students will use the building mats and counters to find intervals for the remaining word problems.
Students will write the numbers needed to solve the problem on the blank writing line above the building ten drawing frame for number one.
Students will draw circles in the appropriate columns to represent the three numbers.
Reasonable interval estimates
Completed recording sheet with algorithms and intervals indicated.
Numbers
17.9,3,8.5 written above the first frame
Appropriate number of circles drawn in the building ten frame representing the three numbers in the word problem.
GESD. Curriculum and Instruction Department. 03.19.10

problem
SW show my marking out and drawing new circles to demonstrate how each number of the word problem is rounded.
SW write the equations needed to solve the problem.
TW demonstrate the thinking and process of rounding these decimals. If a number must round up to the next place value, one must mark out the extra circles in the columns to the right of the new place value column, and draw a new circle in the selected place value column. If a number must round down to a certain place value, then the extra circles in the column to the right of that place value will just be marked out in red pencil.
“I should round 17.9 to the nearest ten; this will be 2 tens, or 20, since there are 7 ones.
With a red pencil, I will draw a new circle in the tens column and mark out all the circles in the ones and tenths columns.
I might round 8.5 to the nearest one or to the nearest ten. Since it will need to combine with the tens number for 17.9, it is better to round to the nearest ten, which will be 1 ten or 10, since there are 8 ones. In red pencil, I will draw a new circle in the tens column and mark out all the circles in the ones and tenths columns. Since the number 3 is already a one-digit whole number, it will not be rounded.”
Once I round my numbers I will write the equations needed to solve the problem. These equations are as follows: 3x20 = 60 and 60+10=70 ounces total of seed estimated in the bag. Next I write
Students will decide how to round each number so that mental facts might be used to compute an estimate.
Students write the equations needed to solve the problem.
GESD. Curriculum and Instruction Department. 03.19.10
First Row: 1 ten, 7 ones, 9 tenths
Second Row: 8 ones, 5 tens
Third row: 3 ones
Rounding of 17.9 to the nearest ten-20;
Rounding of 8.5 to the nearest one or to the nearest ten
Base 10 drawing frames with correct number of circles drawn, crossed out, and added
3x20=60
60+10=70 ounces estimated for seed in bag
(3x20)+10=70 ounces estimated

SW write the equations needed to solve the problem. the factors in the first equation in correct order; 3x20 means 3 of 20.
Next, TW discuss the facts used:
3x2=6 and 6+1=7.
Next, I will combine the two equations into one equation and record it as well: (3x20)+10=70 ounces estimated.
Teacher note: (this is excellent preparation for future studies in algebra).
TW guide students through similar process with number 2 on the word problem worksheet.
Last two problems can be used for independent practice.
SW combine the two equations into one equation and record it as well.
Students will display base 10 drawing frames for each of the next three problems.
Completed base ten drawing frames for the four word problems
GESD. Curriculum and Instruction Department. 03.19.10