After
a discussion of estimating products and number lines, the class will be
divided in half and compete to find products using mental math or
estimate products and indicate them on number lines which will be
projected on a whiteboard.
To prepare for the lesson:
- The day before the lesson, invite a few students to bring in a CD
of their favorite band or choose some songs that you think the students
will like. Use different tempos or genres of music for each round.
- Print out Number Line Dancing Cards on card stock, and cut them apart. Or, write the numbers directly on 3 × 5 cards. Sort into piles according to the shape on the cards.
Number Line Dancing Cards - Take the time to view the Number Line Dancing Presentation a few times before the lesson to get comfortable with the slides.
Number Line Dancing Presentation
Tell the story of a class who won a contest. Their prize was to go
to a concert of their favorite band. Twelve classes from schools across
the city would be going to the contest. The teacher wanted to know
about how many students would be there all together. Most classes had
about 27 students. Ask students, "How could you estimate to find the total?" [12 can be
rounded to 10, and 27 could be rounded to 30. 30 x 10 = 300 students.]
Ask if anyone can give other examples of when estimation would be
needed in real life. [Shopping, planning for parties, sports
statistics, etc.]
Explain that estimating is a great time-saver and is also a great way
to know whether an answer is reasonable. (Students using calculators
sometimes hit a button in error, but can use estimating to see whether
their answer is reasonable.)
Put the headings from the table below on
your board, and write the first problem.
Ask students:
- How do you estimate products?
[If you round both
numbers to the nearest multiple of ten, you will be able to use basic
math facts, and then put all the zeroes back in.] - Is it necessary
to round.
[No.] - What basic fact would you use to solve the problem?
[8 x
3 = 24.] - Is 240 the exact
answer or an estimate? How do you know?
[Exact. We didn’t have to round our original
numbers.]
Proceed through the other examples. Add into your list of questions (after asking for the basic math fact), "What about the zero from 30 (or 80)?" [Put it back into the product.]
Problem | Round | Basic Fact | Put zeroes back | Exact | Estimate |
8 x 30 | no | 8 x 3 = 24 | 240 | yes | |
8 x 32 | 32 to 30 | 8 x 3 = 24 | 240 | | yes |
78 x 32 | 78 to 80
32 to 30 | 8 x 3 = 24 | 2,400 | | yes |
Depending on your class’s previous experience, you may want to add
more examples to the chart. Ask students to look at the chart and
explain what the factors would look like when you need to estimate
versus being able to calculate the answer mentally. [If at least one of
the factors is 2 or more digits that don’t end in a zero, you will need
to estimate. If the factors are one digit or end in zero, you can use
mental math to calculate the answer.] As a quick formative assessment,
have students pair up. First student should give the second student a 2
digit by 2 digit problem without zero as a digit. They should use paper
or a small white board. Second student should estimate the answer and
explain each step as it is done. Students should switch roles. Once
they feel confident in the skill, they can indicate it to the teacher
with thumbs up. You can circulate and help those that still need it.
Explain that the class will be playing a game that will give them a
chance to practice quick estimation of products. Students will have to
use mental math to estimate products, and then place those products on
a number line. Use the Number Line Dancing PowerPoint presentation to
talk with students about number lines.
Begin the slide show. When you get to the second slide, ask a
student to come to the board and point to where the product of 4 × 6 is
on the first number line. The third slide shows a second number line.
Have a different student show where the same product is on the second
number line. Repeat this with the third number line, which appears on the fourth slide. As
each child shows their point, have them explain how they decided where
to place it. A good explanation might be “I knew that it had to be
between 20 and 30 and 24 is a bit less than half so I went to the
middle and put it a little before that.” Accept any answer that is
appropriate. If the class needs more practice, give another
multiplication problem to practice with all three number lines.
Explain the rules of the game:
- There will be 2 teams standing in a line on each side of the board competing for highest score. However, only one team will go at a time.
- The first person in line will dance to the board, choose a card from each pile needed for that round, and place them on the ledge
- That person will find the product, or an estimate of the factors on the cards, and draw a dot on the number line to indicate the answer.
- Each person will keep the cards with them until the end of the round, but hand the marker to the next person in line for your team.
- After drawing a dot, dance to the end of your line.
- Only the person at the board is allowed to work on the problem.
Depending on the make-up of your class, you may need to change the rules of the game. Here are some suggestions:
- Allow the team to call out answers to help a stumped student at the board.
- Send 2 students up together to solve each problem.
- Give students a lifeline (they can pick one other student to help
if they need it). You may want to limit lifelines to 3 per round.
- Allow students to have a clipboard, with paper and pencil, at the board.
Explain that there will be number lines with different intervals for
each round. You should use a different song for each round. The first
round will be practice in solving multiplication problems with mental
math, but the other rounds will ask students to estimate products.
Move to the next slide.
Demonstrate how students should show their answers on the number
line. For example, if the product is 260, put a dot on the number line
where you think 260 is, draw a line straight above or below the point,
and draw a circle with your initials in it. An example is shown on Slide 5. This will allow the class
to check the answers after the round. You may want to have a few
students practice recording an answer on this slide to make sure they
understand.
Divide the class into their teams and have them line up in the
proper place. For Round One (on Slide 6), place the square and circle piles on a desk near the
board. Remind the class that in the first section they will be solving
problems using mental math. When you are ready to begin, click on the
next slide. Start the music and invite the first student in line to
dance up to the board. As the student records his dot on the number
line, remind him her to draw a line with a circle and put his/her
initials in it. Continue until all members of the team have had a turn.
Stop the music. Have each student show their 2 cards. Ask the other
team to help determine if the answer is in the correct spot on the
number line. Award a point for each correct answer. Collect the square cards, shuffle them, and put on desk for the next team. Do the same with circle cards. Be sure to have a student erase the whiteboard. The second team
takes their turn.
For the second round (Slide 7), let the second team go first. For this part of
the game, students will round one of the factors, and then use mental
math to complete the estimation. Use the square cards and the triangle cards for this round.
The third round (Slide 8) uses the triangle cards and the star cards. Allow the team with the lower score
to go first. For this part of the game, students round both of the
factors, and then use mental math to complete the estimation.
Give the winning team the option of doing a 30 second celebration dance.
If there is time at the end of the game, have a class discussion
using the following questions. If there is no time, discuss on the next
day as a follow up.
Here are a few suggestions for discussion questions:
1. Look back at the Slide 2 with the three number lines. Can you
tell anything about what happens when you plot the same number on the
different number lines as the intervals get larger?
[It’s more difficult to find numbers that are between the larger
intervals. The number is closer to the zero mark as the intervals get
larger.]
2. Was it important to look at the intervals on the number line to
decide where to put your answer?
[I needed to look at the numbers that
were marked on the tick lines and decide between which two I needed to
put my answer. I needed to see what the middle would be and whether my
number should go close to the middle of it or nearer to one tick line
or the other.]
3. In Round One and Two, the number line goes from zero to 900. The
cards in Pile A went from 2 to 9. Pile B and C had numbers from 10 to
99. Explain why all the products had to fall somewhere on the number
line.
[The lowest possible product would be 20 (10 x 2) and the largest
possible product would be 900 (9 x 99 rounded to 100)]
4. In Round Three, the number line goes from zero to 10,000. What
did the range of numbers in Pile C and D need to be to fit on that
number line? All factors were 2 digit numbers.
[They could have been
anywhere from 10 to 99. 10 x 10 = 100 for lowest and 99 x 99 rounded to
100 x 100 = 10,000 for highest.]
Reference
Math Heads CD, Theatrix, 1996 (out of print).